We present a topology-informed inverse rendering approach for reconstructing high-genus surface meshes from multi-view images. Compared to 3D representations like voxels and point clouds, mesh-based representations are preferred as they enable the application of differential geometry theory and are optimized for modern graphics pipelines. However, existing inverse rendering methods often fail catastrophically on high-genus surfaces, leading to the loss of key topological features, and tend to oversmooth low-genus surfaces, resulting in the loss of surface details. This failure stems from their overreliance on Adambased optimizers, which can lead to vanishing and exploding gradients. To overcome these challenges, we introduce an adaptive V-cycle remeshing scheme in conjunction with a re-parametrized Adam optimizer to enhance topological and geometric awareness. By periodically coarsening and refining the deforming mesh, our method informs mesh vertices of their current topology and geometry before optimization, mitigating gradient issues while preserving essential topological features. Additionally, we enforce topological consistency by constructing topological primitives with genus numbers that match those of ground truth using Gauss-Bonnet theorem. Experimental results demonstrate that our inverse rendering approach outperforms the current state-of-the-art method, achieving significant improvements in Chamfer Distance and Volume IoU, particularly for high-genus surfaces, while also enhancing surface details for low-genus surfaces.

Neural representations for 3D meshes are emerging as an effective solution for compact storage and efficient processing. Existing methods often rely on neural overfitting, where a coarse mesh is stored and progressively refined through multiple decoder networks. While this can restore high-quality surfaces, it is computationally expensive due to successive decoding passes and the irregular structure of mesh data. In contrast, images have a regular structure that enables powerful super-resolution and restoration frameworks, but applying these advantages to meshes is difficult because their irregular connectivity demands complex encoder–decoder architectures. Our key insight is that a geometry image–based representation transforms irregular meshes into a regular image grid, making efficient image-based neural processing directly applicable. Building on this idea, we introduce our neural geometry image–based representation, which is decoder-free, storage-efficient, and naturally suited for neural processing. It stores a low-resolution geometry-image mipmap of the surface, from which high-quality meshes are restored in a single forward pass. To construct geometry images, we leverage Optimal Transport (OT), which resolves oversampling in flat regions and undersampling in feature-rich regions, and enables continuous levels of detail (LoD) through geometry-image mipmapping. Experimental results demonstrate state-of-the-art storage efficiency and restoration accuracy, measured by compression ratio (CR), Chamfer distance (CD), and Hausdorff distance (HD).

Animating 3D head meshes using audio inputs has significant applications in AR/VR, gaming, and entertainment through 3D avatars. However, bridging the modality gap between speech signals and facial dynamics remains a challenge, often resulting in incorrect lip syncing and unnatural facial movements. To address this, we propose OT-Talk, the first approach to leverage optimal transportation to optimize the learning model in talking head animation. Building on existing learning frameworks, we utilize a pre-trained Hubert model to extract audio features and a transformer model to process temporal sequences. Unlike previous methods that focus solely on vertex coordinates or displacements, we introduce Chebyshev Graph Convolution to extract geometric features from triangulated meshes. To measure mesh dissimilarities, we go beyond traditional mesh reconstruction errors and velocity differences between adjacent frames. Instead, we represent meshes as probability measures and approximate their surfaces. This allows us to leverage the sliced Wasserstein distance for modeling mesh variations. This approach facilitates the learning of smooth and accurate facial motions, resulting in coherent and natural facial animations. Our experiments on two public audio-mesh datasets demonstrate that our method outperforms state-of-the-art techniques both quantitatively and qualitatively in terms of mesh reconstruction accuracy and temporal alignment. In addition, we conducted a user perception study with 20 volunteers to further assess the effectiveness of our approach.

We present the design and construction of the Silo, a fully immersive stereoscopic cylindrical tiled-display visualization facility. Comprising 168 high-density LCD displays, the facility provides an ultra-high-resolution image of 619 million pixels, and close to 360 horizontal field-of-regards (FoR), aiming to maximize visual acuity and completely engage the human visual sensorium and its periphery. In this article, we outline the motivations, design principles, hardware selection and software systems, and interaction modalities used in constructing the Silo. To address missing visual information due to the absence of a ceiling and floor, we have designed a method that utilizes conformal mapping and optimal mass transport to reproject the entire 360 volumetric FoR of the virtual scene to the available display real estate. We showcase several applications demonstrating the utility of the Silo and report the findings of our user studies that highlight the effectiveness of the Silo layout compared to curved mono and flat powerwall display facilities. Our user evaluations and studies have shown that the Silo supports natural exploration and enhanced visualization due to its capability to render surround ultra-high-resolution stereoscopic views.

Recent CLIP-guided 3D generation methods have achieved promising results but struggle with generating faithful 3D shapes that conform with input text due to the gap between text and image embeddings. To this end, this paper proposes HOTS3D which makes the first attempt to effectively bridge this gap by aligning text features to the image features with spherical optimal transport (SOT). However, in high-dimensional situations, solving the SOT remains a challenge. To obtain the SOT map for high-dimensional features obtained from CLIP encoding of two modalities, we mathematically formulate and derive the solution based on Villani's theorem, which can directly align two hyper-sphere distributions without manifold exponential maps. Furthermore, we implement it by leveraging input convex neural networks (ICNNs) for the optimal Kantorovich potential. With the optimally mapped features, a diffusion-based generator is utilized to decode them into 3D shapes. Extensive quantitative and qualitative comparisons with state-of-the-art methods demonstrate the superiority of HOTS3D for text-to-3D generation, especially in the consistency with text semantics.

Intelligent Mesh Generation (IMG) represents a novel and promising field of research, utilizing machine learning techniques to generate meshes. Despite its relative infancy, IMG has significantly broadened the adaptability and practicality of mesh generation techniques, delivering numerous breakthroughs and unveiling potential future pathways. However, a noticeable void exists in the contemporary literature concerning comprehensive surveys of IMG methods. This paper endeavors to fill this gap by providing a systematic and thorough survey of the current IMG landscape. With a focus on 113 preliminary IMG methods, we undertake a meticulous analysis from various angles, encompassing core algorithm techniques and their application scope, agent learning objectives, data types, targeted challenges, as well as advantages and limitations. We have curated and categorized the literature, proposing three unique taxonomies based on key techniques, output mesh unit elements, and relevant input data types. This paper also underscores several promising future research directions and challenges in IMG.

Recently, there has been a significant surge of interest in medical image generation. In this study, we developed a model known as AE-COT-GAN (autoencoder-based conditional optimal transport generative adversarial network) to generate medical images that belong to specific categories. The primary objective of our research is to address the prevalent challenges often encountered during the training of generative adversarial networks (GANs), including issues such as mode collapse and mode mixing. The training process of our model encompasses three fundamental components. First, we employ an autoencoder model to obtain a low-dimensional manifold representation of real images. Second, we apply extended semi-discrete optimal transport to map Gaussian noise distribution to the latent space distribution and obtain corresponding labels effectively. This procedure leads to the generation of new latent codes with known labels. Finally, we integrate a GAN to train the decoder further to generate medical images. To evaluate the performance of the AE-COT-GAN model, we conducted experiments on two medical image datasets, namely DermaMNIST and BloodMNIST. The model’s performance was compared with state-of-the-art generative models. Results show that the AE-COT-GAN model had excellent performance in generating medical images. Moreover, it effectively addressed the common issues associated with traditional GANs.

Word-gesture production models that can synthesize word-gestures are critical to the training and evaluation of word-gesture keyboard decoders. We propose WordGesture-GAN, a conditional generative adversarial network that takes arbitrary text as input to generate realistic word-gesture movements in both spatial (i.e., (x, y) coordinates of touch points) and temporal (i.e., timestamps of touch points) dimensions. WordGesture-GAN introduces a Variational Auto-Encoder to extract and embed variations of user-drawn gestures into a Gaussian distribution which can be sampled to control variation in generated gestures. Our experiments on a dataset with 38k gesture samples show that WordGesture-GAN outperforms existing gesture production models including the minimum jerk model [37] and the style-transfer GAN [31, 32] in generating realistic gestures. Overall, our research demonstrates that the proposed GAN structure can learn variations in user-drawn gestures, and the resulting WordGesture-GAN can generate word-gesture movement and predict the distribution of gestures. WordGesture-GAN can serve as a valuable tool for designing and evaluating gestural input systems.

Sampling from diffusion probabilistic models (DPMs) can be viewed as a piecewise distribution transformation, which generally requires hundreds or thousands of steps of the inverse diffusion trajectory to get a high-quality image. Recent progress in designing fast samplers for DPMs achieves a trade-off between sampling speed and sample quality by knowledge distillation or adjusting the variance schedule or the denoising equation. However, it can’t be optimal in both aspects and often suffer from mode mixture in short steps. To tackle this problem, we innovatively regard inverse diffusion as an optimal transport (OT) problem between latents at different stages and propose the DPM-OT, a unified learning framework for fast DPMs with a direct expressway represented by OT map, which can generate high-quality samples within around 10 function evaluations. By calculating the semi-discrete optimal transport map between the data latents and the white noise, we obtain an expressway from the prior distribution to the data distribution, while significantly alleviating the problem of mode mixture. In addition, we give the error bound of the proposed method, which theoretically guarantees the stability of the algorithm. Extensive experiments validate the effectiveness and advantages of DPM-OT in terms of speed and quality (FID and mode mixture), thus representing an efficient solution for generative modeling. Source codes are available at https://github.com/cognaclee/DPM-OT.

Models that can generate touch typing tasks are important to the development of touch typing keyboards. We propose TouchType-GAN, a Conditional Generative Adversarial Network that can simulate locations and time stamps of touch points in touch typing. TouchType-GAN takes arbitrary text as input to generate realistic touch typing both spatially (i.e., (x, y) coordinates of touch points) and temporally (i.e., timestamps of touch points). TouchType-GAN introduces a variational generator that estimates Gaussian Distributions for every target letter to prevent mode collapse. Our experiments on a dataset with 3k typed sentences show that TouchType-GAN outperforms existing touch typing models, including the Rotational Dual Gaussian model [36] for simulating the distribution of touch points, and the Finger-Fitts Euclidean Model [30] for simulating typing time. Overall, our research demonstrates that the proposed GAN structure can learn the distribution of user typed touch points, and the resulting TouchType-GAN can also estimate typing movements. TouchType-GAN can serve as a valuable tool for designing and evaluating touch typing input systems.

Virtual simulated scenes are becoming a critical part of autonomous driving. In the context of knowledge automation and machine learning, simulated images are widely used for visual environmental perception. However, even the most inspirational applications have not fully exploited the potential of simulated images in solving real-world problems. In this article, we propose a novel framework “ParallelEye Pipeline,” which uses image-to-image translation and simulated images to automatically generate realistic synthetic images with multiple ground-truth annotations. Specifically, this method has three steps: first, we use Unity3D software to simulate driving scenarios and generate simulated image pairs (including raw images and six ground-truth labels) from the simulated scenes; second, advanced image-to-image translation algorithms can generate realistic and high-resolution synthetic images from simulated image pairs; third, we exploit publicly datasets, simulated images, and synthetic images to conduct experiments for visual perception. The experimental results suggest: 1) synthetic images and simulated images can improve the performance of detectors in real autonomous driving scenarios and 2) image-to-image translation algorithms can be affected by occlusion condition.

Optimal transport (OT) plays an essential role in various areas like machine learning and deep learning. However, computing discrete optimal transport plan for large scale problems with adequate accuracy and efficiency is still highly challenging. Recently, methods based on the Sinkhorn algorithm add an entropy regularizer to the prime problem and get a trade off between efficiency and accuracy. In this paper, we propose a novel algorithm to further improve the efficiency and accuracy based on Nesterov's smoothing technique. Basically, the non-smooth c-transform of the Kantorovich potential is approximated by the smooth Log-Sum-Exp function, which finally smooths the original non-smooth Kantorovich dual functional. The smooth Kantorovich functional can be optimized by the fast proximal gradient algorithm (FISTA) efficiently. Theoretically, the computational complexity of the proposed method is lower than current estimation of the Sinkhorn algorithm in terms of the precision. Empirically, compared with the Sinkhorn algorithm, our experimental results demonstrate that the proposed method achieves faster convergence and better accuracy with the same parameter.

Demand for efficient image transmission and storage is increasing rapidly because of the continuing growth of multimedia technology and VR and AR applications. In this paper, we proposed an image compression method based on the recognition of importance of regions in images. As not all the information in an image is equally useful, we can identify important regions in an image for high fidelity compression and accept a comparatively more lossy compression about less important regions of the image. First, we segment images to two parts, namely, foreground and background, where the foreground represents the more important component and the background is of less importance. Second, we apply optimal mass transportation mapping in a GAN (generative adversarial network) framework to both the foreground and background to magnify the foreground and shrink the background while keeping the shape and total image area unchanged. As a result, in the processed image, the ratio of foreground to background is larger than the corrresponding ratio in the original image. This ratio is controllable in our process, giving users the ability to control the degree of compression. The GAN-processed image is then used for compression. To restore the image, we apply a GAN model to the compressed image and recover the ratio of foreground and background using an optimal mass transportation map. Test results show that our method is highly effective in reconstructing detail of important components in compressed images while achieving a high compression ratio.

Craniofacial reconstruction is fundamental in resolving forensic cases. It is rather challenging due to the complex topology of the craniofacial model and the ambiguous relationship between a skull and the corresponding face. In this paper, we propose a novel approach for 3D craniofacial reconstruction by utilizing Conditional Generative Adversarial Networks (CGAN) based on craniofacial depth map. More specifically, we treat craniofacial reconstruction as a mapping problem from skull to face. We represent 3D cran- iofacial shapes with depth maps, which include most craniofacial features for identification purposes and are easy to generate and apply to neural networks. We designed an end-to-end neural networks model based on CGAN then trained the model with paired craniofacial data to automatically learn the complex nonlinear relationship between skull and face. By introducing body mass index classes(BMIC) into CGAN, we can realize objective reconstruction of 3D facial geometry according to its skull, which is a complicated 3D shape generation task with different topologies. Through comparative experiments, our method shows accuracy and verisimilitude in craniofacial reconstruction results.

An optimal transportation map finds the most economical way to transport one probability measure to the other. It has been applied in a broad range of applications in vision, deep learning and medical images. By Brenier theory, computing the optimal transport map is equivalent to solving a Monge-Ampère equation. Due to the highly non-linear nature, the computation of optimal transportation maps in large scale is very challenging.This work proposes a simple but powerful method, the FFT-OT algorithm, to tackle this difficulty based on three key ideas. First, solving Monge-Ampère equation is converted to a fixed point problem; Second, the obliqueness property of optimal transportation maps are reformulated as Neumann boundary conditions on rectangular domains; Third, FFT is applied in each iteration to solve a Poisson equation in order to improve the efficiency.Experiments on surfaces captured from 3D scanning and reconstructed from medical imaging are conducted, and compared with other existing methods. Our experimental results show that the proposed FFT-OT algorithm is simple, general and scalable with high efficiency and accuracy.

Shape analysis has been playing an important role in early diagnosis and prognosis of neurodegenerative diseases such as Alzheimer's diseases (AD). However, obtaining effective shape representations remains challenging. This paper proposes to use the Alexandrov polyhedra as surface-based shape signatures for cortical morphometry analysis. Given a closed genus-0 surface, its Alexandrov polyhedron is a convex representation that encodes its intrinsic geometry information. We propose to compute the polyhedra via a novel spherical optimal transport (OT) computation. In our experiments, we observe that the Alexandrov polyhedra of cortical surfaces between pathology-confirmed AD and cognitively unimpaired individuals are significantly different. Moreover, we propose a visualization method by comparing local geometry differences across cortical surfaces. We show that the proposed method is effective in pinpointing regional cortical structural changes impacted by AD.

Optimal transportation plays a fundamental role in many fields in engineering and medicine, including surface parameterization in graphics, registration in computer vision, and generative models in deep learning. For quadratic distance cost, optimal transportation map is the gradient of the Brenier potential, which can be obtained by solving the Monge-Ampère equation. Furthermore, it is induced to a geometric convex optimization problem. The Monge-Ampère equation is highly non-linear, and during the solving process, the intermediate solutions have to be strictly convex. Specifically, the accuracy of the discrete solution heavily depends on the sampling pattern of the target measure. In this work, we propose a self-adaptive sampling algorithm which greatly reduces the sampling bias and improves the accuracy and robustness of the discrete solutions. Experimental results demonstrate the efficiency and efficacy of our method.

Soft active materials can generate flexible locomotion and change configurations through large deformations when subjected to an external environmental stimulus. They can be engineered to design 'soft machines' such as soft robots, compliant actuators, flexible electronics, or bionic medical devices. By embedding ferromagnetic particles into soft elastomer matrix, the ferromagnetic soft matter can generate flexible movement and shift morphology in response to the external magnetic field. By taking advantage of this physical property, soft active structures undergoing desired motions can be generated by tailoring the layouts of the ferromagnetic soft elastomers. Structural topology optimization has emerged as an attractive tool to achieve innovative structures by optimizing the material layout within a design domain, and it can be utilized to architect ferromagnetic soft active structures. In this paper, the level-set-based topology optimization method is employed to design ferromagnetic soft robots (FerroSoRo). The objective function comprises a sub-objective function for the kinematics requirement and a sub-objective function for minimum compliance. Shape sensitivity analysis is derived using the material time derivative and adjoint variable method. Three examples, including a gripper, an actuator, and a flytrap structure, are studied to demonstrate the effectiveness of the proposed framework.

Abstract Based on mesh deformation, we present a unified mesh parametrization algorithm for both planar and spherical domains. Our approach can produce intermediate frames from the original meshes to the targets. We derive and define a novel geometric flow: ‘unit normal flow (UNF)’ and prove that if UNF converges, it will deform a surface to a constant mean curvature (CMC) surface, such as planes and spheres. Our method works by deforming meshes of disk topology to planes, and spherical meshes to spheres. Our algorithm is robust, efficient, simple to implement. To demonstrate the robustness and effectiveness of our method, we apply it to hundreds of models of varying complexities. Our experiments show that our algorithm can be a competing alternative approach to other state‐of‐the‐art mesh parametrization methods. The unit normal flow also suggests a potential direction for creating CMC surfaces.

Conformal surface parameterization is useful in graphics, imaging, and visualization, with applications to texture mapping, atlas construction, registration, remeshing, and so on. With the increasing capability in scanning and storing data, dense 3D surface meshes are common nowadays. While meshes with higher resolution better resemble smooth surfaces, they pose computational difficulties for the existing parameterization algorithms. In this work, we propose a novel parallelizable algorithm for computing the global conformal parameterization of simply-connected surfaces via partial welding maps. A given simply-connected surface is first partitioned into smaller subdomains. The local conformal parameterizations of all subdomains are then computed in parallel. The boundaries of the parameterized subdomains are subsequently integrated consistently using a novel technique called partial welding, which is developed based on conformal welding theory. Finally, by solving the Laplace equation for each subdomain using the updated boundary conditions, we obtain a global conformal parameterization of the given surface, with bijectivity guaranteed by quasi-conformal theory. By including additional shape constraints, our method can be easily extended to achieve disk conformal parameterization for simply-connected open surfaces and spherical conformal parameterization for genus-0 closed surfaces. Experimental results are presented to demonstrate the effectiveness of our proposed algorithm. When compared to the state-of-the-art conformal parameterization methods, our method achieves a significant improvement in both computational time and accuracy.

Wasserstein GANs are increasingly used in Computer Vision applications as they are easier to train. Previous WGAN variants mainly use the lι transport cost to compute the Wasserstein distance between the real and synthetic data distributions. The lι transport cost restricts the discriminator to be 1-Lipschitz. However, WGANs with lι transport cost were recently shown to not always converge. In this paper, we propose WGAN-QC, a WGAN with quadratic transport cost. Based on the quadratic transport cost, we propose an Optimal Transport Regularizer (OTR) to stabilize the training process of WGAN-QC. We prove that the objective of the discriminator during each generator update computes the exact quadratic Wasserstein distance between real and synthetic data distributions. We also prove that WGAN-QC converges to a local equilibrium point with finite discriminator updates per generator update. We show experimentally on a Dirac distribution that WGAN-QC converges, when many of the lι cost WGANs fail to [22]. Qualitative and quantitative results on the CelebA, CelebA-HQ, LSUN and the ImageNet dog datasets show that WGAN-QC is better than state-of-art GAN methods. WGAN-QC has much faster runtime than other WGAN variants.

Skull registration plays a fundamental role in forensic science and is crucial for craniofacial reconstruction. The complicated topology, lack of anatomical features, and low quality reconstructed mesh make skull registration challenging. In this work, we propose an automatic skull registration method based on the discrete uniformization theory, which can handle complicated topologies and is robust to low quality meshes. We apply dynamic Yamabe flow to realize discrete uniformization, which modifies the mesh combinatorial structure during the flow and conformally maps the multiply connected skull surface onto a planar disk with circular holes. The 3D surfaces can be registered by matching their planar images using harmonic maps. This method is rigorous with theoretic guarantee, automatic without user intervention, and robust to low mesh quality. Our experimental results demonstrate the efficiency and efficacy of the method.

Abstract Conformal parameterization for surfaces into various parameter domains is a fundamental task in computer graphics. Prior research on discrete Ricci flow provided us with promising inspirations from methods derived via Riemannian geometry, which is rigorous in theory and effective inpractice. In this paper, we propose a unified conformal parameterization approachfor turning triangle meshes into planar and spherical domains using discrete Calabi flow onpiecewise linear metric. We incorporate edge‐flipping surgery to guarantee convergence as well as other significant improvements including approximate Newton's method, optimal step‐lengths, priority embedding and boundary customizing, which achieve better performance and functionality with robustness and accuracy.

Abstract There are many methods proposed for generating polycube polyhedrons, but it lacks the study about the possibility of generating polycube polyhedrons. In this paper, we prove a theorem for characterizing the necessary condition for the skeleton graph of a polycube polyhedron, by which Steinitz's theorem for convex polyhedra and Eppstein's theorem for simple orthogonal polyhedra are generalized to polycube polyhedra of any genus and with non‐simply connected faces. Based on our theorem, we present a faster linear algorithm to determine the dimensions of the polycube shape space for a valid graph, for all its possible polycube polyhedrons. We also propose a quadratic optimization method to generate embedding polycube polyhedrons with interactive assistance. Finally, we provide a graph‐based framework for polycube mesh generation, quadrangulation, and all‐hex meshing to demonstrate the utility and applicability of our approach.

In this paper, we propose a two-step method to compute the Wasserstein distance in Wasserstein Generative Adversarial Networks (WGANs): 1) The convex part of our objective can be solved by linear programming; 2) The non-convex residual can be approximated by a deep neural network. We theoretically prove that the proposed formulation is equivalent to the discrete Monge-Kantorovich dual formulation. Furthermore, we give the approximation error bound of the Wasserstein distance and the error bound of generalizing the Wasserstein distance from discrete to continuous distributions. Our approach optimizes the exact Wasserstein distance, obviating the need for weight clipping previously used in WGANs. Results on synthetic data show that the our method computes the Wasserstein distance more accurately. Qualitative and quantitative results on MNIST, LSUN and CIFAR-10 datasets show that the proposed method is more efficient than state-of-the-art WGAN methods, and still produces images of comparable quality.

Polycube construction and deformation are essential problems in computer graphics. In this paper, we present a robust, simple, efficient, and automatic algorithm to deform the meshes of arbitrary shapes into polycube form. We derive a clear relationship between a mesh and its corresponding polycube shape. Our algorithm is edge-preserving, and works on surface meshes with or without boundaries. Our algorithm outperforms previous ones with respect to speed, robustness, and efficiency. Our method is simple to implement. To demonstrate the robustness and effectivity of our method, we have applied it to hundreds of models of varying complexity and topology. We demonstrate that our method compares favorably to other state-of-the-art polycube deformation methods.

Visualization of medical organs and biological structures is a challenging task because of their complex geometry and the resultant occlusions. Global spherical and planar mapping techniques simplify the complex geometry and resolve the occlusions to aid in visualization. However, while resolving the occlusions these techniques do not preserve the geometric context, making them less suitable for mission-critical biomedical visualization tasks. In this paper, we present a shape-preserving local mapping technique for resolving occlusions locally while preserving the overall geometric context. More specifically, we present a novel visualization algorithm, LMap, for conformally parameterizing and deforming a selected local region-of-interest (ROI) on an arbitrary surface. The resultant shape-preserving local mappings help to visualize complex surfaces while preserving the overall geometric context. The algorithm is based on the robust and efficient extrinsic Ricci flow technique, and uses the dynamic Ricci flow algorithm to guarantee the existence of a local map for a selected ROI on an arbitrary surface. We show the effectiveness and efficacy of our method in three challenging use cases: (1) multimodal brain visualization, (2) optimal coverage of virtual colonoscopy centerline flythrough, and (3) molecular surface visualization.

The alterations of brain structures and functions have been considered closely correlated to the change of cognitive performance due to neurodegenerative diseases such as Alzheimer's disease. In this paper, we introduce a variational framework to compute the optimal transformation (OT) in 3D space and propose a univariate neuroimaging index based on OT to measure such alterations. We compute the OT from each image to a template and measure the Wasserstein distance between them. By comparing the distances from all the images to the common template, we obtain a concise and informative index for each image. Our framework makes use of the Newton's method, which reduces the computational cost and enables itself to be applicable to large-scale datasets. The proposed work is a generic approach and thus may be applicable to various volumetric brain images, including structural magnetic resonance (sMR) and fluorodeoxyglucose positron emission tomography (FDG-PET) images. In the classification between Alzheimer's disease patients and healthy controls, our method achieves an accuracy of 82:30% on the Alzheimers Disease Neuroimaging Initiative (ADNI) baseline sMRI dataset and outperforms several other indices. On FDG-PET dataset, we boost the accuracy to 88:37% by leveraging pairwise Wasserstein distances. In a longitudinal study, we obtain a 5% significance with p-value = 1:13 ×105 in a t-test on FDG-PET. The results demonstrate a great potential of the proposed index for neuroimage analysis and the precision medicine research.

3D dynamic surface tracking is an important research problem and plays a vital role in many computer vision and medical imaging applications. However, it is still challenging to efficiently register surface sequences which has large deformations and strong noise. In this paper, we propose a novel automatic method for non-rigid 3D dynamic surface tracking with surface Ricci flow and Teichmüller map methods. According to quasi-conformal Teichmüller theory, the Techmüller map minimizes the maximal dilation so that our method is able to automatically register surfaces with large deformations. Besides, the adoption of Delaunay triangulation and quadrilateral meshes makes our method applicable to low quality meshes. In our work, the 3D dynamic surfaces are acquired by a high speed 3D scanner. We first identified sparse surface features using machine learning methods in the texture space. Then we assign landmark features with different curvature settings and the Riemannian metric of the surface is computed by the dynamic Ricci flow method, such that all the curvatures are concentrated on the feature points and the surface is flat everywhere else. The registration among frames is computed by the Teichmüller mappings, which aligns the feature points with least angle distortions. We apply our new method to multiple sequences of 3D facial surfaces with large expression deformations and compare them with two other state-of-the-art tracking methods. The effectiveness of our method is demonstrated by the clearly improved accuracy and efficiency.

This work introduces a novel surface registration method based on foliation. A foliation decomposes the surface into a family of closed loops, such that the decomposition has local tensor product structure. By projecting each loop to a point, the surface is collapsed into a graph. Two homeomorphic surfaces with consistent foliations can be registered by first matching their foliation graphs, then matching the corresponding leaves. This foliation based method is capable of handling surfaces with complicated topologies and large non-isometric deformations, rigorous with solid theoretic foundation, easy to implement, robust to compute. The result mapping is diffeomorphic. Our experimental results show the efficiency and efficacy of the proposed method.

Robot Coverage Path planning (i.e., the process of providing full coverage of a given domain by one or multiple robots) is a classical problem in the field of robotics and motion planning. The goal of such planning is to provide nearly full coverage while also minimize duplicately visited area. In this paper, we focus on the scenario of path planning on general surface, including planar domains with complex topology, complex terrain, and general surface in 3D space. Our approach described in this paper adopts a natural, intrinsic and global parametrization of the surface for robot path planning, namely the holomorphic quadratic differentials. We give each point on the surface a uv-coordinates naturally represented by a complex number, except for a small number of zero points (singularities). We show that natural, efficient robot paths can be obtained by using such coordinate systems. The method is based on intrinsic geometry and thus can be adapted to general surface exploration in 3D.

Automatic computation of surface correspondence via harmonic map is an active research field in computer vision, computer graphics and computational geometry. It may help document and understand physical and biological phenomena and also has broad applications in biometrics, medical imaging and motion capture industries. Although numerous studies have been devoted to harmonic map research, limited progress has been made to compute a diffeomorphic harmonic map on general topology surfaces with landmark constraints. This work conquers this problem by changing the Riemannian metric on the target surface to a hyperbolic metric so that the harmonic mapping is guaranteed to be a diffeomorphism under landmark constraints. The computational algorithms are based on Ricci flow and nonlinear heat diffusion methods. The approach is general and robust. We employ our algorithm to study the constrained surface registration problem which applies to both computer vision and medical imaging applications. Experimental results demonstrate that, by changing the Riemannian metric, the registrations are always diffeomorphic and achieve relatively high performance when evaluated with some popular surface registration evaluation standards.

We present a method for registration and visualization of corresponding supine and prone virtual colonoscopy scans based on eigenfunction analysis and fold modeling. In virtual colonoscopy, CT scans are acquired with the patient in two positions, and their registration is desirable so that physicians can corroborate findings between scans. Our algorithm performs this registration efficiently through the use of Fiedler vector representation (the second eigenfunction of the Laplace-Beltrami operator). This representation is employed to first perform global registration of the two colon positions. The registration is then locally refined using the haustral folds, which are automatically segmented using the 3D level sets of the Fiedler vector. The use of Fiedler vectors and the segmented folds presents a precise way of visualizing corresponding regions across datasets and visual modalities. We present multiple methods of visualizing the results, including 2D flattened rendering and the corresponding 3D endoluminal views. The precise fold modeling is used to automatically find a suitable cut for the 2D flattening, which provides a less distorted visualization. Our approach is robust, and we demonstrate its efficiency and efficacy by showing matched views on both the 2D flattened colons and in the 3D endoluminal view. We analytically evaluate the results by measuring the distance between features on the registered colons, and we also assess our fold segmentation against 20 manually labeled datasets. We have compared our results analytically to previous methods, and have found our method to achieve superior results. We also prove the hot spots conjecture for modeling cylindrical topology using Fiedler vector representation, which allows our approach to be used for general cylindrical geometry modeling and feature extraction.

This work presents a novel framework for spherical mesh parameterization. An efficient angle-preserving spherical parameterization algorithm is introduced, which is based on dynamic Yamabe flow and the conformal welding method with solid theoretic foundation. An area-preserving spherical parameterization is also discussed, which is based on discrete optimal mass transport theory. Furthermore, a spherical parameterization algorithm, which is based on the polar decomposition method, balancing angle distortion and area distortion is presented. The algorithms are tested on 3D geometric data and the experiments demonstrate the efficiency and efficacy of the proposed methods.

We consider the problem of capacitated kinetic clustering in which n mobile terminals and k base stations with respective operating capacities are given. The task is to assign the mobile terminals to the base stations such that the total squared distance from each terminal to its assigned base station is minimized and the capacity constraints are satisfied. This paper focuses on the development of distributed and computationally efficient algorithms that adapt to the motion of both terminals and base stations. Suggested by the optimal transportation theory, we exploit the structural property of the optimal solution, which can be represented by a power diagram on the base stations such that the total usage of nodes within each power cell equals the capacity of the corresponding base station. We show by using the kinetic data structure framework the first analytical upper bound on the number of changes in the optimal solution, i.e., its stability. On the algorithm side, using the power diagram formulation we show that the solution can be represented in size proportional to the number of base stations and can be solved by an iterative, local algorithm. In particular, this algorithm can naturally exploit the continuity of motion and has orders of magnitude faster than existing solutions using min-cost matching and linear programming, and thus is able to handle large scale data under mobility.

This paper casts surface registration as the problem of finding a set of discrete correspondences through the minimization of an energy function, which is composed of geometric and appearance matching costs, as well as higher-order deformation priors. Two higher-order graph-based formulations are proposed under different deformation assumptions. The first formulation encodes isometric deformations using conformal geometry in a higher-order graph matching problem, which is solved through dual-decomposition and is able to handle partial matching. Despite the isometry assumption, this approach is able to robustly match sparse feature point sets on surfaces undergoing highly anisometric deformations. Nevertheless, its performance degrades significantly when addressing anisometric registration for a set of densely sampled points. This issue is rigorously addressed subsequently through a novel deformation model that is able to handle arbitrary diffeomorphisms between two surfaces. Such a deformation model is introduced into a higher-order Markov Random Field for dense surface registration, and is inferred using a new parallel and memory efficient algorithm. To deal with the prohibitive search space, we also design an efficient way to select a number of matching candidates for each point of the source surface based on the matching results of a sparse set of points. A series of experiments demonstrate the accuracy and the efficiency of the proposed framework, notably in challenging cases of large and/or anisometric deformations, or surfaces that are partially occluded.

Motivated by mobile sensor networks as in participatory sensing applications, we are interested in developing a practical, lightweight solution for routing in a mobile network. While greedy routing is robust to mobility, it may get stuck in a local minimum, which then requires non-trivial recovery methods. We find an embedding of the network such that greedy routing using the virtual coordinates guarantees delivery, thus eliminating the necessity of any recovery methods. Our contribution is to replace the in-network computation of the embedding by a preprocessing of the domain before network deployment and encode the map of network domain to virtual coordinate space by using a small number of parameters which can be preloaded to all sensor nodes. As a result, the map is only dependent on the network domain and is independent of the network connectivity. Each node can directly compute or update its virtual coordinates by applying the locally stored map on its geographical coordinates. This represents the first practical solution for using virtual coordinates for greedy routing in a sensor network and could be easily extended to the case of a mobile network. The paper describes algorithmic innovations as well as implementations on a real testbed.

By the Riemann mapping theorem, one can bijectively map the interior of an n-gon P to that of another n-gon Q conformally (i.e., in an angle preserving manner). However, when this map is extended to the boundary it need not necessarily map the vertices of P to those of Q. For many applications it is important to find the "best" vertex-preserving mapping between two polygons, i.e., one that minimizes the maximum angle distortion (the so-called dilatation). Such maps exist, are unique, and are known as extremal quasiconformal maps or Teichmüller maps. There are many efficient ways to approximate conformal maps, and the recent breakthrough result by Bishop computes a (1+epsilon)-approximation of the Riemann map in linear time. However, only heuristics have been studied in the case of Teichmüller maps. We present two results in this paper. One studies the problem in the continuous setting and another in the discrete setting. In the continuous setting, we solve the problem of finding a finite time procedure for approximating Teichmüller maps. Our construction is via an iterative procedure that is proven to converge in O(poly(1/epsilon)) iterations to a (1+epsilon)-approximation of the Teichmuller map. Our method uses a reduction of the polygon mapping problem to the marked sphere problem, thus solving a more general problem. In the discrete setting, we reduce the problem of finding an approximation algorithm for computing Teichmüller maps to two basic subroutines, namely, computing discrete 1) compositions and 2) inverses of discretely represented quasiconformal maps. Assuming finite-time solvers for these subroutines we provide a (1+epsilon)-approximation algorithm.

With the recent development of localization and tracking systems for both indoor and outdoor settings, we consider the problem of analyzing and representing the huge amount of natural trajectories from human movements that we expect to gather in the near future. In this paper we argue the topological representation, which records how a target moves around the natural obstacles in the underlying environment, can be sufficiently descriptive for many applications and efficient enough for both storing, comparing and classifying these natural human trajectories. Technically, the representation uses the homotopy type of the trajectory. By using harmonic one-forms and Hodge decomposition, we pre-process the sensor network with a purely decentralized algorithm such that the homology class of a trajectory can be obtained by a simple integration along the trajectory. This supports real-time classification of trajectories up to the homology accuracy with minimum communication cost. We test the effectiveness of our approach by showing how to classify randomly generated trajectories in a multi-level arts museum layout as well as how to distinguish real world taxi trajectories in a large city.

Analysis of Internet topologies has shown that the Internet topology has negative curvature, measured by Gromov's “thin triangle condition”, which is tightly related to core congestion and route reliability. In this work we analyze the discrete Ricci curvature of the Internet, defined by Ollivier [1], Lin et al. [2], etc. Ricci curvature measures whether local distances diverge or converge. It is a more local measure which allows us to understand the distribution of curvatures in the network. We show by various Internet data sets that the distribution of Ricci cuvature is spread out, suggesting the network topology to be non-homogenous. We also show that the Ricci curvature has interesting connections to both local measures such as node degree and clustering coefficient, global measures such as betweenness centrality and network connectivity, as well as auxilary attributes such as geographical distances. These observations add to the richness of geometric structures in complex network theory.

Surface based 3D shape analysis plays a fundamental role in computer vision and medical imaging. This work proposes to use optimal mass transport map for shape matching and comparison, focusing on two important applications including surface registration and shape space. The computation of the optimal mass transport map is based on Monge-Brenier theory, in comparison to the conventional method based on Monge-Kantorovich theory, this method significantly improves the efficiency by reducing computational complexity from O(n(2)) to O(n) . For surface registration problem, one commonly used approach is to use conformal map to convert the shapes into some canonical space. Although conformal mappings have small angle distortions, they may introduce large area distortions which are likely to cause numerical instability thus resulting failures of shape analysis. This work proposes to compose the conformal map with the optimal mass transport map to get the unique area-preserving map, which is intrinsic to the Riemannian metric, unique, and diffeomorphic. For shape space study, this work introduces a novel Riemannian framework, Conformal Wasserstein Shape Space, by combing conformal geometry and optimal mass transport theory. In our work, all metric surfaces with the disk topology are mapped to the unit planar disk by a conformal mapping, which pushes the area element on the surface to a probability measure on the disk. The optimal mass transport provides a map from the shape space of all topological disks with metrics to the Wasserstein space of the disk and the pullback Wasserstein metric equips the shape space with a Riemannian metric. We validate our work by numerous experiments and comparisons with prior approaches and the experimental results demonstrate the efficiency and efficacy of our proposed approach.

In this paper, we introduce the definition of discrete conformality for triangulated surfaces with flat cone metrics and describe an algorithm for solving the problem of prescribing curvature, which is to deform the metric discrete conformally so that the curvature of the resulting metric coincides with the prescribed curvature. We explicitly construct a discrete conformal map between the input triangulated surface and the deformed triangulated surface. Our algorithm can handle a surface with any topology, with or without boundary, and can find a deformed metric for any prescribed curvature satisfying the Gauss--Bonnet formula. In addition, we present the numerical examples to show the convergence of our discrete conformality and to demonstrate the efficiency and the robustness of our algorithm.

This work proposes a novel framework for optimization in the constrained diffeomorphism space for deformable surface registration. First the diffeomorphism space is modeled as a special complex functional space on the source surface, the Beltrami coefficient space. The physically plausible constraints, in terms of feature landmarks and deformation types, define subspaces in the Beltrami coefficient space. Then the harmonic energy of the registration is minimized in the constrained subspaces. The minimization is achieved by alternating two steps: 1) optimization - diffuse the Beltrami coefficient, and 2) projection - first deform the conformal structure by the current Beltrami coefficient and then compose with a harmonic map from the deformed conformal structure to the target. The registration result is diffeomorphic, satisfies the physical landmark and deformation constraints, and minimizes the conformality distortion. Experiments on human facial surfaces demonstrate the efficiency and efficacy of the proposed registration framework.

Load balanced routing in a network, i.e., minimizing the maximum traffic load any node carries for unsplittable flows, is a well known NP-hard problem. Finding practical algorithms remains a long standing challenge. In this paper we propose greedy routing using virtual coordinates that achieves both small path stretch ratio (compared to shortest path) and small load balancing ratio (compared to optimal load balanced routing), in a large scale wireless sensor network deployed densely inside a geometric domain with complex shape. We first provide a greedy routing scheme on a disk with a stretch ratio of at most 2, and under which the maximum load is a factor 4√2 smaller than the maximum load under shortest path routing. This is the first simple routing scheme with a small stretch that has been proven to outperform shortest path routing in terms of load balancing. Then we transform a network of arbitrary shape to a disk by an area preserving map φ. We show that both the path length and the maximum traffic load in the original network only increases by an additional factor of d2, where d is the maximum length stretch of φ. Combined with the result on a disk we again achieve both bounded stretch and bounded load balancing ratio. Our simulation results evaluated the practical performance on both quality measures.

Shape analysis is a central problem in the field of computer vision. In 2D shape analysis, classification and recognition of objects from their observed silhouettes are extremely crucial but difficult. It usually involves an efficient representation of 2D shape space with a metric, so that its mathematical structure can be used for further analysis. Although the study of 2D simply-connected shapes has been subject to a corpus of literatures, the analysis of multiply-connected shapes is comparatively less studied. In this work, we propose a representation for general 2D multiply-connected domains with arbitrary topologies using conformal welding. A metric can be defined on the proposed representation space, which gives a metric to measure dissimilarities between objects. The main idea is to map the exterior and interior of the domain conformally to unit disks and circle domains (unit disk with several inner disks removed), using holomorphic 1-forms. A set of diffeomorphisms of the unit circle S(1) can be obtained, which together with the conformal modules are used to define the shape signature. A shape distance between shape signatures can be defined to measure dissimilarities between shapes. We prove theoretically that the proposed shape signature uniquely determines the multiply-connected objects under suitable normalization. We also introduce a reconstruction algorithm to obtain shapes from their signatures. This completes our framework and allows us to move back and forth between shapes and signatures. With that, a morphing algorithm between shapes can be developed through the interpolation of the Beltrami coefficients associated with the signatures. Experiments have been carried out on shapes extracted from real images. Results demonstrate the efficacy of our proposed algorithm as a stable shape representation scheme.

Registration, which aims to find an optimal 1-1 correspondence between shapes, is an important process in different research areas. Landmark-based surface registration has been widely studied to obtain a mapping between shapes that matches important features. Obtaining a unique and bijective surface registration that matches features consistently is generally challenging, especially when a large number of landmark constraints are enforced. This motivates us to search for a unique landmark matching surface diffeomorphism, which minimizes the local geometric distortion. For this purpose, we propose a special class of diffeomorphisms called the Teichmüller mappings (T-Maps). Under suitable conditions on the landmark constraints, a unique T-Map between two surfaces can be obtained, which minimizes the maximal conformality distortion. The conformality distortion measures how far the mapping deviates from a conformal mapping, and hence it measures the local geometric distortion. In this paper, we propose an efficient iterative algorithm, called the quasi-conformal (QC) iteration, to compute the T-Map. The basic idea is to represent the set of diffeomorphisms using Beltrami coefficients (BCs) and look for an optimal BC associated to the desired T-Map. The associated diffeomorphism can be efficiently reconstructed from the optimal BC using the linear Beltrami solver (LBS). Using BCs to represent diffeomorphisms guarantees the diffeomorphic property of the registration, even with very large deformation. Using our proposed method, the T-Map can be accurately and efficiently computed. The obtained registration is guaranteed to be bijective. The proposed algorithm can also be extended to compute T-Map with soft landmark constraints. We applied the proposed algorithm to real applications, such as brain landmark matching registration, constrained texture mapping, and human face registration. Experimental results shows that our method is both effective and efficient in computing a nonoverlap landmark matching registration with the least amount of conformality distortion.

Surface meshing plays a fundamental role in graphics and visualization. Many geometric processing tasks involve solving geometric PDEs on meshes. The numerical stability, convergence rates and approximation errors are largely determined by the mesh qualities. In practice, Delaunay refinement algorithms offer satisfactory solutions to high quality mesh generations. The theoretical proofs for volume based and surface based Delaunay refinement algorithms have been established, but those for conformal parameterization based ones remain wide open. This work focuses on the curvature measure convergence for the conformal parameterization based Delaunay refinement algorithms. Given a metric surface, the proposed approach triangulates its conformal uniformization domain by the planar Delaunay refinement algorithms, and produces a high quality mesh. We give explicit estimates for the Hausdorff distance, the normal deviation, and the differences in curvature measures between the surface and the mesh. In contrast to the conventional results based on volumetric Delaunay refinement, our stronger estimates are independent of the mesh structure and directly guarantee the convergence of curvature measures. Meanwhile, our result on Gaussian curvature measure is intrinsic to the Riemannian metric and independent of the embedding. In practice, our meshing algorithm is much easier to implement and much more efficient. The experimental results verified our theoretical results and demonstrated the efficiency of the meshing algorithm.

Automatic computation of surface correspondence via harmonic map is an active research field in computer vision, computer graphics and computational geometry. It may help document and understand physical and biological phenomena and also has broad applications in biometrics, medical imaging and motion capture. Although numerous studies have been devoted to harmonic map research, limited progress has been made to compute a diffeomorphic harmonic map on general topology surfaces with landmark constraints. This work conquer this problem by changing the Riemannian metric on the target surface to a hyperbolic metric, so that the harmonic mapping is guaranteed to be a diffeomorphism under landmark constraints. The computational algorithms are based on the Ricci flow method and the method is general and robust. We apply our algorithm to study constrained human brain surface registration problem. Experimental results demonstrate that, by changing the Riemannian metric, the registrations are always diffeomorphic, and achieve relative high performance when evaluated with some popular cortical surface registration evaluation standards.

Brain mapping transforms the brain cortical surface to canonical planar domains, which plays a fundamental role in morphological study. Most existing brain mapping methods are based on angle preserving maps, which may introduce large area distortions. This work proposes an area preserving brain mapping method based on Monge-Brenier theory. The brain mapping is intrinsic to the Riemannian metric, unique, and diffeomorphic. The computation is equivalent to convex energy minimization and power Voronoi diagram construction. Comparing to the existing approaches based on Monge-Kantorovich theory, the proposed one greatly reduces the complexity (from n2unknowns to n ), and improves the simplicity and efficiency. Experimental results on caudate nucleus surface mapping and cortical surface mapping demonstrate the efficacy and efficiency of the proposed method. Conventional methods for caudate nucleus surface mapping may suffer from numerical instability, in contrast, current method produces diffeomorpic mappings stably. In the study of cortical surface classification for recognition of Alzheimer's Disease, the proposed method outperforms some other morphometry features.

Surface registration plays a fundamental role in many applications in computer vision and aims at finding a one-to-one correspondence between surfaces. Conformal mapping based surface registration methods conformally map 2D/3D surfaces onto 2D canonical domains and perform the matching on the 2D plane. This registration framework reduces dimensionality, and the result is intrinsic to Riemannian metric and invariant under isometric deformation. However, conformal mapping will be affected by inconsistent boundaries and non-isometric deformations of surfaces. In this work, we quantify the effects of boundary variation and non-isometric deformation to conformal mappings, and give the theoretical upper bounds for the distortions of conformal mappings under these two factors. Besides giving the thorough theoretical proofs of the theorems, we verified them by concrete experiments using 3D human facial scans with dynamic expressions and varying boundaries. Furthermore, we used the distortion estimates for reducing search range in feature matching of surface registration applications. The experimental results are consistent with the theoretical predictions and also demonstrate the performance improvements in feature tracking.

We propose a novel approach for dense non-rigid 3D surface registration, which brings together Riemannian geometry and graphical models. To this end, we first introduce a generic deformation model, called Canonical Distortion Coefficients (CDCs), by characterizing the deformation of every point on a surface using the distortions along its two principle directions. This model subsumes the deformation groups commonly used in surface registration such as isometry and conformality, and is able to handle more complex deformations. We also derive its discrete counterpart which can be computed very efficiently in a closed form. Based on these, we introduce a higher-order Markov Random Field (MRF) model which seamlessly integrates our deformation model and a geometry/texture similarity metric. Then we jointly establish the optimal correspondences for all the points via maximum a posteriori (MAP) inference. Moreover, we develop a parallel optimization algorithm to efficiently perform the inference for the proposed higher-order MRF model. The resulting registration algorithm outperforms state-of-the-art methods in both dense non-rigid 3D surface registration and tracking.

In this paper we propose an algorithm to construct a “space filling” curve for a sensor network with holes. Mathematically, for a given multi-hole domain R, we generate a path P that is provably aperiodic (i.e., any point is covered at most a constant number of times) and dense (i.e., any point of R is arbitrarily close to P). In a discrete setting as in a sensor network, the path visits the nodes with progressive density, which can adapt to the budget of the path length. Given a higher budget, the path covers the network with higher density. With a lower budget the path becomes proportional sparser. We show how this density-adaptive space filling curve can be useful for applications such as serial data fusion, motion planning for data mules, sensor node indexing, and double ruling type in-network data storage and retrieval. We show by simulation results the superior performance of using our algorithm vs standard space filling curves and random walks.

Motivated by mobile sensor networks as in participatory sensing applications, we are interested in developing a practical, lightweight solution for routing in a mobile network. While greedy routing is robust to mobility, location errors and link dynamics, it may get stuck in a local minimum, which then requires non-trivial recovery methods. We follow the approach taken by Sarkar et. al. [24] to find an embedding of the network such that greedy routing using the virtual coordinates guarantees delivery, thus eliminating the necessity of any recovery methods. Our new contribution is to replace the in-network computation of the embedding by a preprocessing of the domain before network deployment and encode the map of network domain to virtual coordinate space by using a small number of parameters which can be pre-loaded to all sensor nodes. As a result, the map is only dependent on the network domain and is independent of the network connectivity. Each node can directly compute or update its virtual coordinates by applying the locally stored map on its geographical coordinates. This represents the first practical solution for using virtual coordinates for greedy routing in a sensor network and could be easily extended to the case of a mobile network. Being extremely light-weight, greedy routing on the virtual coordinates is shown to be very robust to mobility, link dynamics and non-unit disk graph connectivity models.

Random walk on a graph is a Markov chain and thus is `memoryless' as the next node to visit depends only on the current node and not on the sequence of events that preceded it. With these properties, random walk and its many variations have been used in network routing to `randomize' the traffic pattern and hide the location of the data sources. In this paper we examine a myth in common understanding of the memoryless property of a random walk applied for protecting source location privacy in a wireless sensor network. In particular, if one monitors only the network boundary and records the first boundary node hit by a random walk, this distribution can be related to the location of the source node. For the scenario of a single data source, a very simple algorithm by integrating along the network boundary would reveal the location of the source. We also develop a generic algorithm to reconstruct the source locations for various sources that have simple descriptions (e.g., k source locations, sources on a line segment, sources in a disk). This represents a new type of traffic analysis attack for invading sensor data location privacy and essentially re-opens the problem for further examination.

Abstract The emergence of laser/LiDAR sensors, reliable multi‐view stereo techniques and more recently consumer depth cameras have brought point clouds to the forefront as a data format useful for a number of applications. Unfortunately, the point data from those channels often incur imperfection, frequently contaminated with severe outliers and noise. This paper presents a robust consolidation algorithm for low‐quality point data from outdoor scenes, which essentially consists of two steps: 1) outliers filtering and 2) noise smoothing. We first design a connectivity‐based scheme to evaluate outlierness and thereby detect sparse outliers. Meanwhile, a clustering method is used to further remove small dense outliers. Both outlier removal methods are insensitive to the choice of the neighborhood size and the levels of outliers. Subsequently, we propose a novel approach to estimate normals for noisy points based on robust partial rankings, which is the basis of noise smoothing. Accordingly, a fast approach is exploited to smooth noise, while preserving sharp features. We evaluate the effectiveness of the proposed method on the point clouds from a variety of outdoor scenes.

Surface parameterizations and registrations are important in computer graphics and imaging, where 1-1 correspondences between meshes are computed. In practice, surface maps are usually represented and stored as three-dimensional coordinates each vertex is mapped to, which often requires lots of memory. This causes inconvenience in data transmission and data storage. To tackle this problem, we propose an effective algorithm for compressing surface homeomorphisms using Fourier approximation of the Beltrami representation. The Beltrami representation is a complex-valued function defined on triangular faces of the surface mesh with supreme norm strictly less than 1. Under suitable normalization, there is a 1-1 correspondence between the set of surface homeomorphisms and the set of Beltrami representations. Hence, every bijective surface map is associated with a unique Beltrami representation. Conversely, given a Beltrami representation, the corresponding bijective surface map can be exactly reconstructed using the linear Beltrami solver introduced in this paper. Using the Beltrami representation, the surface homeomorphism can be easily compressed by Fourier approximation, without distorting the bijectivity of the map. The storage requirement can be effectively reduced, which is useful for many practical problems in computer graphics and imaging. In this paper, we propose applying the algorithm to texture map compression and video compression. With our proposed algorithm, the storage requirement for the texture properties of a textured surface can be significantly reduced. Our algorithm can further be applied to compressing motion vector fields for video compression, which effectively improves the compression ratio.

We propose a new colon flattening algorithm that is efficient, shape-preserving, and robust to topological noise. Unlike previous approaches, which require a mandatory topological denoising to remove fake handles, our algorithm directly flattens the colon surface without any denoising. In our method, we replace the original Euclidean metric of the colon surface with a heat diffusion metric that is insensitive to topological noise. Using this heat diffusion metric, we then solve a Laplacian equation followed by an integration step to compute the final flattening. We demonstrate that our method is shape-preserving and the shape of the polyps are well preserved. The flattened colon also provides an efficient way to enhance the navigation and inspection in virtual colonoscopy. We further show how the existing colon registration pipeline is made more robust by using our colon flattening. We have tested our method on several colon wall surfaces and the experimental results demonstrate the robustness and the efficiency of our method.

We present a novel area-preservation mapping/flattening method using the optimal mass transport technique, based on the Monge-Brenier theory. Our optimal transport map approach is rigorous and solid in theory, efficient and parallel in computation, yet general for various applications. By comparison with the conventional Monge-Kantorovich approach, our method reduces the number of variables from O(n2) to O(n), and converts the optimal mass transport problem to a convex optimization problem, which can now be efficiently carried out by Newton's method. Furthermore, our framework includes the area weighting strategy that enables users to completely control and adjust the size of areas everywhere in an accurate and quantitative way. Our method significantly reduces the complexity of the problem, and improves the efficiency, flexibility and scalability during visualization. Our framework, by combining conformal mapping and optimal mass transport mapping, serves as a powerful tool for a broad range of applications in visualization and graphics, especially for medical imaging. We provide a variety of experimental results to demonstrate the efficiency, robustness and efficacy of our novel framework.

We study scalable routing for a sensor network deployed in complicated 3D settings such as underground tunnels in gas system or water system. The nodes are in general 3D space but they are very sparsely located and the network has complex topology. We propose a routing scheme by first embdding the network on a surface with possibly non-zero genus. Then we compute a canonical hyperbolic metric of the embedded surface, and use geodesics to decompose the network into canonical components called pairs of `pants' whose topology is simpler (with genus zero). The adjacency of the pants components is extracted as a high level routing map and stored at every node. With the hyperbolic metric one can use greedy routing to navigate within and across pants. Altogether this leads to a two-level routing scheme by first finding a sequence of pants and then realizing the route with greedy steps. We show by simulation that the number of pants is closely related to the true `genus' of the network and that the routing scheme is efficient and scalable.

In brain mapping research, parameterized 3-D surface models are of great interest for statistical comparisons of anatomy, surface-based registration, and signal processing. Here, we introduce the theories of continuous and discrete surface Ricci flow, which can create Riemannian metrics on surfaces with arbitrary topologies with user-defined Gaussian curvatures. The resulting conformal parameterizations have no singularities and they are intrinsic and stable. First, we convert a cortical surface model into a multiple boundary surface by cutting along selected anatomical landmark curves. Secondly, we conformally parameterize each cortical surface to a parameter domain with a user-designed Gaussian curvature arrangement. In the parameter domain, a shape index based on conformal invariants is computed, and inter-subject cortical surface matching is performed by solving a constrained harmonic map. We illustrate various target curvature arrangements and demonstrate the stability of the method using longitudinal data. To map statistical differences in cortical morphometry, we studied brain asymmetry in 14 healthy control subjects. We used a manifold version of Hotelling's T(2) test, applied to the Jacobian matrices of the surface parameterizations. A permutation test, along with the cumulative distribution of p-values, were used to estimate the overall statistical significance of differences. The results show our algorithm's power to detect subtle group differences in cortical surfaces.

In Virtual Reality, immersive systems such as the CAVE provide an important tool for the collaborative exploration of large 3D data. Unlike head-mounted displays, these systems are often only partially immersive due to space, access, or cost constraints. The resulting loss of visual information becomes a major obstacle for critical tasks that need to utilize the users' entire field of vision. We have developed a conformal visualization technique that establishes a conformal mapping between the full 360° field of view and the display geometry of a given visualization system. The mapping is provably angle-preserving and has the desirable property of preserving shapes locally, which is important for identifying shape-based features in the visual data. We apply the conformal visualization to both forward and backward rendering pipelines in a variety of retargeting scenarios, including CAVEs and angled arrangements of flat panel displays. In contrast to image-based retargeting approaches, our technique constructs accurate stereoscopic images that are free of resampling artifacts. Our user study shows that on the visual polyp detection task in Immersive Virtual Colonoscopy, conformal visualization leads to improved sensitivity at comparable examination times against the traditional rendering approach. We also develop a novel user interface based on the interactive recreation of the conformal mapping and the real-time regeneration of the view direction correspondence.

We present the conformal magnifier, a novel interactive focus+context visualization technique that magnifies a region of interest (ROI) using conformal mapping. Our framework supports the arbitrary shape design of magnifiers for the user to enlarge the ROI while globally deforming the context region without any cropping. By using the mathematically well-defined conformal mapping theory and algorithm, the ROI is magnified with local shape preservation (angle distortion minimization), while the transition area between the focus and context regions is deformed smoothly and continuously. After the selection of a specified magnifier shape, our system can automatically magnify the ROI in real time with full resolution even for large volumetric data sets. These properties are important for many visualization applications, especially for the computer aided detection and diagnosis (CAD). Our framework is suitable for diverse applications, including the map visualization, and volumetric visualization. Experimental results demonstrate the effectiveness, robustness, and efficiency of our framework.

A novel method for registering 3D surfaces with large deformations is presented, which is based on quasi-conformal geometry. A general diffeomorphism distorts the conformal structure of the surface, which is represented as the Beltrami coefficient. Inversely, the diffeomorphism can be determined by the Beltrami coefficient in an essentially unique way. Our registration method first extracts the features on the surfaces, then estimates the Beltrami coefficient, and finally uniquely determines the registration mapping by solving Beltrami equations using curvature flow. The method is 1) general, it can search the desired registration in the whole space of diffeomorphisms, which includes the conventional searching spaces, such as rigid motions, isometric transformations or conformal mappings; 2) global optimal, the global optimum is determined by the method unique up to a 3 dimensional transformation group; 3) robust, it handles large surfaces with complicated topologies; 4) rigorous, it has solid theoretic foundation. Experiments on the real surfaces with large deformations and complicated topologies demonstrate the efficiency, robustness of the proposed method.

This paper presents a novel intrinsic 3D surface distance and its use in a complete probabilistic tracking framework for dynamic 3D data. Registering two frames of a deforming 3D shape relies on accurate correspondences between all points across the two frames. In the general case such correspondence search is computationally intractable. Common prior assumptions on the nature of the deformation such as near-rigidity, isometry or learning from a training set, reduce the search space but often at the price of loss of accuracy when it comes to deformations not in the prior assumptions. If we consider the set of all possible 3D surface matchings defined by specifying triplets of correspondences in the uniformization domain, then we introduce a new matching cost between two 3D surfaces. The lowest feature differences across this set of matchings that cause two points to correspond, become the matching cost of that particular correspondence. We show that for surface tracking applications, the matching cost can be efficiently computed in the uniformization domain. This matching cost is then combined with regularization terms that enforce spatial and temporal motion consistencies, into a maximum a posteriori (MAP) problem which we approximate using a Markov Random Field (MRF). Compared to previous 3D surface tracking approaches that either assume isometric deformations or consistent features, our method achieves dense, accurate tracking results, which we demonstrate through a series of dense, anisometric 3D surface tracking experiments.

This paper presents a multiscale, curvature-based shape representation technique for general genus zero closed surfaces. The method is invariant under rotation, translation, scaling, or general isometric deformations; it is robust to noise and preserves intrinsic symmetry. The method is a direct generalization of the Curvature Scale Space (CSS) shape descriptor for planar curves. In our method, the Riemannian metric of the surface is deformed under Ricci flow, such that the Gaussian curvature evolves according to a heat diffusion process. Eventually the surface becomes a sphere with constant positive curvature everywhere. The evolution of zero curvature curves on the surface is utilized as the shape descriptor. Our experimental results on a 3D geometric database with about 80 shapes demonstrate the efficiency and efficacy of the method.

In this paper, we propose novel algorithms for inpainting and refinement of diffeomorphisms. We first represent a diffeomorphism by its Beltrami coefficient. Then it is possible to refine and inpaint the diffeomorphism by processing this Beltrami coefficient. With the inpainted/refined Beltrami coefficient, we construct a new diffeomorphism using the exact Beltrami holomorphic flow algorithm proposed in this paper. We apply our algorithms on several practical applications, which include the inpainting of a highly distorted diffeomorphism, the inpainting of image sequences of deforming shapes, the super-resolution of diffeomorphisms and the global parameterization of cortical surfaces by combining local parameterizations. Experiments show that our algorithm can solve these problems with natural and smooth results. We demonstrate how our proposed method can be widely applied in areas from texture mapping to video processing, and from computer graphics to medical imaging.

This work proposes a novel connectivity-based localization algorithm, well suitable for large-scale sensor networks with complex shapes and non-uniform nodal distribution. In contrast to current state-of-art connectivity-based localization methods, the proposed algorithm is fully distributed, where each node only needs the information of its neighbors, without cumbersome partitioning and merging process. The algorithm is highly scalable, with limited error propagation and linear computation and communication cost with respect to the size of the network. Moreover, the algorithm is theoretically guaranteed and numerically stable. Extensive simulations and comparison with other methods under various representative network settings are carried out, showing superior performance of the proposed algorithm.

In this paper we address the problem of scalable and load balanced routing for wireless sensor networks. Motivated by the analog of the continuous setting that geodesic routing on a sphere gives perfect load balancing, we embed sensor nodes on a convex polyhedron in 3D and use greedy routing to deliver messages between any pair of nodes with guaranteed success. This embedding is known to exist by the Koebe-Andreev-Thurston Theorem for any 3-connected planar graphs. In our paper we use discrete Ricci flow to develop a distributed algorithm to compute this embedding. Further, such an embedding is not unique and differs from one another by a Möbius transformation. We employ an optimization routine to look for the Möbius transformation such that the nodes are spread on the polyhedron as uniformly as possible. We evaluated the load balancing property of this greedy routing scheme and showed favorable comparison with previous schemes.

With both computational complexity and storage space bounded by a small constant, greedy routing is recognized as an appealing approach to support scalable routing in wireless sensor networks. However, significant challenges have been encountered in extending greedy routing from 2D to 3D space. In this research we develop decentralized solutions to achieve greedy routing in 3D sensor networks. Our proposed approach is based on a unit tetrahedron cell (UTC) mesh structure. We propose a distributed algorithm to realize volumetric harmonic mapping of the UTC mesh under spherical boundary condition. It is a one-to-one map that yields virtual coordinates for each node in the network. Since a boundary has been mapped to a sphere, node-based greedy routing is always successful thereon. At the same time, we exploit the UTC mesh to develop a face-based greedy routing algorithm, and prove its success at internal nodes. To deliver a data packet to its destination, face-based and node-based greedy routing algorithms are employed alternately at internal and boundary UTCs, respectively. As far as we know, this is the first work that realizes truly deterministic greedy routing with constant-bounded storage and computation in 3D wireless sensor networks.

Current immersive VR systems such as the CAVE provide an effective platform for the immersive exploration of large 3D data. A major limitation is that in most cases at least one display surface is missing due to space, access or cost constraints. This partially-immersive visualization results in a substantial loss of visual information that may be acceptable for some applications, however it becomes a major obstacle for critical tasks, such as the analysis of medical data. We propose a conformal deformation rendering pipeline for the visualization of datasets on partially-immersive platforms. The angle-preserving conformal mapping approach is used to map the 360°3D view volume to arbitrary display configurations. It has the desirable property of preserving shapes under distortion, which is important for identifying features, especially in medical data. The conformal mapping is used for rasterization, realtime raytracing and volume rendering of the datasets. Since the technique is applied during the rendering, we can construct stereoscopic images from the data, which is usually not true for image-based distortion approaches. We demonstrate the stereo conformal mapping rendering pipeline in the partially-immersive 5-wall Immersive Cabin (IC) for virtual colonoscopy and architectural review.

When visualizing tubular 3D structures, external representations are often used for guidance and display, and such views in 2D can often contain occlusions. Virtual dissection methods have been proposed where the entire 3D structure can be mapped to the 2D plane, though these will lose context by straightening curved sections. We present a new method of creating maps of 3D tubular structures that yield a succinct view while preserving the overall geometric structure. Given a dominant view plane for the structure, its curve skeleton is first projected to a 2D skeleton. This 2D skeleton is adjusted to account for distortions in length, modified to remove intersections, and optimized to preserve the shape of the original 3D skeleton. Based on this shaped 2D skeleton, a boundary for the map of the object is obtained based on a slicing path through the structure and the radius around the skeleton. The sliced structure is conformally mapped to a rectangle and then deformed via harmonic mapping to match the boundary placement. This flattened map preserves the general geometric context of a 3D object in a 2D display, and rendering of this flattened map can be accomplished using volumetric ray casting. We have evaluated our method on real datasets of human colon models.

Centroidal Voronoi tessellations (CVT) are widely used in computational science and engineering. The most commonly used method is Lloyd's method, and recently the L-BFGS method is shown to be faster than Lloyd's method for computing the CVT. However, these methods run on the CPU and are still too slow for many practical applications. We present techniques to implement these methods on the GPU for computing the CVT on 2D planes and on surfaces, and demonstrate significant speedup of these GPU-based methods over their CPU counterparts. For CVT computation on a surface, we use a geometry image stored in the GPU to represent the surface for computing the Voronoi diagram on it. In our implementation a new technique is proposed for parallel regional reduction on the GPU for evaluating integrals over Voronoi cells.

Parameterization of complex surfaces constitutes a major means of visualizing highly convoluted geometric structures as well as other properties associated with the surface. It also enables users with the ability to navigate, orient, and focus on regions of interest within a global view and overcome the occlusions to inner concavities. In this paper, we propose a novel area-preserving surface parameterization method which is rigorous in theory, moderate in computation, yet easily extendable to surfaces of non-disc and closed-boundary topologies. Starting from the distortion induced by an initial parameterization, an area restoring diffeomorphic flow is constructed as a Lie advection of differential 2-forms along the manifold, which yields equality of the area elements between the domain and the original surface at its final state. Existence and uniqueness of result are assured through an analytical derivation. Based upon a triangulated surface representation, we also present an efficient algorithm in line with discrete differential modeling. As an exemplar application, the utilization of this method for the effective visualization of brain cortical imaging modalities is presented. Compared with conformal methods, our method can reveal more subtle surface patterns in a quantitative manner. It, therefore, provides a competitive alternative to the existing parameterization techniques for better surface-based analysis in various scenarios.

Surface registration is widely used in machine vision and medical imaging, where 1-1 correspondences between surfaces are computed to study their variations. Surface maps are usually stored as the 3D coordinates each vertex is mapped to, which often requires lots of storage memory. This causes inconvenience in data transmission and data storage, especially when a large set of surfaces are analyzed. To tackle this problem, we propose a novel representation of surface diffeomorphisms using Beltrami coefficients, which are complex-valued functions defined on surfaces with supreme norm less than 1. Fixing any 3 points on a pair of surfaces, there is a 1-1 correspondence between the set of surface diffeomorphisms between them and the set of Beltrami coefficients on the source domain. Hence, every bijective surface map can be represented by a unique Bel-trami coefficient. Conversely, given a Beltrami coefficient, we can reconstruct the unique surface map associated to it using the Beltrami Holomorphic flow (BHF) method introduced in this paper. Using this representation, 1/3 of the storage space is saved. We can further reduce the storage requirement by 90% by compressing the Beltrami coefficients using Fourier approximations. We test our algorithm on synthetic data, real human brain and hippocampal surfaces. Our results show high accuracy in the reconstructed data, while the amount of storage is greatly reduced. Our approach is compared with the Fourier compression of the coordinate functions using the same amount of data. The latter approach often shows jaggy results and cannot guarantee to preserve diffeomorphisms.

In this paper, we propose a high-order graph matching formulation to address non-rigid surface matching. The singleton terms capture the geometric and appearance similarities (e.g., curvature and texture) while the high-order terms model the intrinsic embedding energy. The novelty of this paper includes: 1. casting 3D surface registration into a graph matching problem that combines both geometric and appearance similarities and intrinsic embedding information, 2. the first implementation of high-order graph matching algorithm that solves a non-convex optimization problem, and 3. an efficient two-stage optimization approach to constrain the search space for dense surface registration. Our method is validated through a series of experiments demonstrating its accuracy and efficiency, notably in challenging cases of large and/or non-isometric deformations, or meshes that are partially occluded.

In this paper, we introduce a new approach for partial 3D face recognition, which makes use of shape decomposition over the rigid part of a face. To explore the descriptiveness of shape dissimilarity over an isometric part of a face, which has lower probability to be influenced by expression, we transform a 3D shape to a 2D domain using conformal mapping and use shape decomposition as a similarity measurement. In our work we investigate several classifiers as well as several shape descriptors for recognition purposes. Recognition tests on a subset of the FRGC data set show approximately 80% rank-one recognition rate using only the eyes and nose part of the face.

We study how to characterize the families of paths between any two nodes s, t in a sensor network with holes. Two paths that can be deformed to one another through local changes are called homotopy equivalent. Two paths that pass around holes in different ways have different homotopy types. With a distributed algorithm we compute an embedding of the network in hyperbolic space by using Ricci flow such that paths of different homotopy types are mapped naturally to paths connecting s with different images of t. Greedy routing to a particular image is guaranteed with success to find a path with a given homotopy type. This leads to simple greedy routing algorithms that are resilient to both local link dynamics and large scale jamming attacks and improve load balancing over previous greedy routing algorithms.

For in-network storage schemes, one maps data, indexed in a logical space, to the distributed sensor locations. When the physical sensor network has an irregular shape and possibly holes, the mapping of data to sensors often creates unbalanced storage load with high data concentration on nodes near network boundaries. In this paper we propose to map data to a covering space, which is a tiling of the plane with copies of the sensor network, such that the sensors receive uniform storage load and traffic. We propose distributed algorithms to construct the covering space with Ricci flow and Möbius transforms. The use of the covering space improves the performance of many in-network storage and retrieval schemes such as geographical hash tables (GHTs) or the double rulings (quorum based schemes), and provides better load balanced routing.

Volumetric parameterization plays an important role for geometric modeling. Due to the complicated topological nature of volumes, it is much more challenging than the surface case. This work focuses on the parameterization of volumes with a boundary surface embedded in 3D space. The intuition is to decompose the volume as the direct product of a two dimensional surface and a one dimensional curve. We first partition the boundary surface into ceiling, floor and walls. Then we compute the harmonic field in the volume with a Dirichlet boundary condition. By tracing the integral curve along the gradient of the harmonic function, we can parameterize the volume to the parametric domain. The method is guaranteed to produce bijection for handle bodies with complex topology, including topological balls as a degenerate case. Furthermore, the parameterization is regular everywhere. We apply the proposed parameterization method to construct hexahedral mesh.

Ricci flow is a powerful curvature flow method, which is invariant to rigid motion, scaling, isometric, and conformal deformations. We present the first application of surface Ricci flow in computer vision. Previous methods based on conformal geometry, which only handle 3D shapes with simple topology, are subsumed by the Ricci flow-based method, which handles surfaces with arbitrary topology. We present a general framework for the computation of Ricci flow, which can design any Riemannian metric by user-defined curvature. The solution to Ricci flow is unique and robust to noise. We provide implementation details for Ricci flow on discrete surfaces of either euclidean or hyperbolic background geometry. Our Ricci flow-based method can convert all 3D problems into 2D domains and offers a general framework for 3D shape analysis. We demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems, such as 3D shape matching and registration, and shape indexing. Surfaces with large nonrigid anisotropic deformations can be registered using Ricci flow with constraints of feature points and curves. We show how conformal equivalence can be used to index shapes in a 3D surface shape space with the use of Teichmüller space coordinates. Experimental results are shown on 3D face data sets with large expression deformations and on dynamic heart data.

Designing rotational symmetry fields on surfaces is an important task for a wide range of graphics applications. This work introduces a rigorous and practical approach for automatic N-RoSy field design on arbitrary surfaces with user-defined field topologies. The user has full control of the number, positions, and indexes of the singularities (as long as they are compatible with necessary global constraints), the turning numbers of the loops, and is able to edit the field interactively. We formulate N-RoSy field construction as designing a Riemannian metric such that the holonomy along any loop is compatible with the local symmetry of N-RoSy fields. We prove the compatibility condition using discrete parallel transport. The complexity of N-RoSy field design is caused by curvatures. In our work, we propose to simplify the Riemannian metric to make it flat almost everywhere. This approach greatly simplifies the process and improves the flexibility such that it can design N-RoSy fields with single singularity and mixed-RoSy fields. This approach can also be generalized to construct regular remeshing on surfaces. To demonstrate the effectiveness of our approach, we apply our design system to pen-and-ink sketching and geometry remeshing. Furthermore, based on our remeshing results with high global symmetry, we generate Celtic knots on surfaces directly.

In virtual colonoscopy, CT scans are typically acquired with the patient in both supine (facing up) and prone (facing down) positions. The registration of these two scans is desirable so that the user can clarify situations or confirm polyp findings at a location in one scan with the same location in the other, thereby improving polyp detection rates and reducing false positives. However, this supine-prone registration is challenging because of the substantial distortions in the colon shape due to the patient's change in position. We present an efficient algorithm and framework for performing this registration through the use of conformal geometry to guarantee that the registration is a diffeomorphism (a one-to-one and onto mapping). The taeniae coli and colon flexures are automatically extracted for each supine and prone surface, employing the colon geometry. The two colon surfaces are then divided into several segments using the flexures, and each segment is cut along a taenia coli and conformally flattened to the rectangular domain using holomorphic differentials. The mean curvature is color encoded as texture images, from which feature points are automatically detected using graph cut segmentation, mathematic morphological operations, and principal component analysis. Corresponding feature points are found between supine and prone and are used to adjust the conformal flattening to be quasi-conformal, such that the features become aligned. We present multiple methods of visualizing our results, including 2D flattened rendering, corresponding 3D endoluminal views, and rendering of distortion measurements. We demonstrate the efficiency and efficacy of our registration method by illustrating matched views on both the 2D flattened colon images and in the 3D volume rendered colon endoluminal view. We analytically evaluate the correctness of the results by measuring the distance between features on the registered colons.

Recently, various conformal geometric methods have been presented for non-rigid surface matching and registration. This work proposes to improve the robustness of conformal geometric methods to the boundaries by incorporating the symmetric information of the input surface. We presented two symmetric conformal mapping methods, which are based on solving Riemann-Cauchy equation and curvature flow respectively. Experimental results on geometric data acquired from real life demonstrate that the symmetric conformal mapping is insensitive to the boundary occlusions. The method outperforms all the others in terms of robustness. The method has the potential to be generalized to high genus surfaces using hyperbolic curvature flow.

All surfaces can be classified by the conformal equivalence relation. Conformal invariants, which are shape indices that can be defined intrinsically on a surface, may be used to identify which surfaces are conformally equivalent, and they can also be used to measure surface deformation. Here we propose to compute a conformal invariant, or shape index, that is associated with the perimeter of the inner concentric circle in the hyperbolic parameter plane. With the surface Ricci flow method, we can conformally map a multiply connected domain to a multi-hole disk and this conformal map can preserve the values of the conformal invariant. Our algorithm provides a stable method to map the values of this shape index in the 2D (hyperbolic space) parameter domain. We also applied this new shape index for analyzing abnormalities in brain morphology in Alzheimer's disease (AD) and Williams syndrome (WS). After cutting along various landmark curves on surface models of the cerebral cortex or hippocampus, we obtained multiple connected domains. We conformally projected the surfaces to hyperbolic plane with surface Ricci flow method, accurately computed the proposed conformal invariant for each selected landmark curve, and assembled these into a feature vector.We also detected group differences in brain structure based on multivariate analysis of the surface deformation tensors induced by these Ricci flow mappings. Experimental results with 3D MRI data from 80 subjects demonstrate that our method powerfully detects brain surface abnormalities when combined with a constrained harmonic map based surface registration method.

Two surfaces are conformally equivalent if there exists a bijective angle-preserving map between them. The Teichmüller space for surfaces with the same topology is a finite-dimensional manifold, where each point represents a conformal equivalence class, and the conformal map is homotopic to the identity map. In this paper, we propose a novel method to apply conformal equivalence based shape index to study brain morphometry. The shape index is defined based on Teichmüller space coordinates. It is intrinsic, and invariant under conformal transformations, rigid motions and scaling. It is also simple to compute; no registration of surfaces is needed. Using the Yamabe flow method, we can conformally map a genus-zero open boundary surface to the Poincaré disk. The shape indices that we compute are the lengths of a special set of geodesics under hyperbolic metric. By computing and studying this shape index and its statistical behavior, we can analyze differences in anatomical morphometry due to disease or development. Study on twin lateral ventricular surface data shows it may help detect generic influence on lateral ventricular shapes. In leave-one-out validation tests, we achieved 100% accurate classification (versus only 68% accuracy for volume measures) in distinguishing 11 HIV/AIDS individuals from 8 healthy control subjects, based on Teichmüller coordinates for lateral ventricular surfaces extracted from their 3D MRI scans.Our conformal invariants, the Teichmüller coordinates, successfully classified all lateral ventricular surfaces, showing their promise for analyzing anatomical surface morphometry.

Constructing smooth freeform surfaces of arbitrary topology with higher order continuity is one of the most fundamental problems in shape and solid modeling. This paper articulates a novel method to construct C∞ smooth surfaces with negative Euler numbers based on hyperbolic geometry and discrete curvature flow. According to Riemann uniformization theorem, every surface with negative Euler number has a unique conformal Riemannian metric, which induces Gaussian curvature of --1 everywhere. Hence, the surface admits hyperbolic geometry. Such uniformization metric can be computed using the discrete curvature flow method: hyperbolic Ricci flow. Consequently, the basis function for each control point can be naturally defined over a hyperbolic disk, and through the use of partition-of-unity, we build a freeform surface directly over hyperbolic domains while having C∞ property. The use of radial, exponential basis functions gives rise to a true meshless method for modeling freeform surfaces with greatest flexibilities, without worrying about control point connectivity. Our algorithm is general for arbitrary surfaces with negative Euler characteristic. Furthermore, it is C∞ continuous everywhere across the entire hyperbolic domain without singularities. Our experimental results demonstrate the efficiency and efficacy of the proposed new approach for shape and solid modeling.

Surface parameterization refers to the process of mapping the surface to canonical planar domains, which plays crucial roles in texture mapping and shape analysis purposes. Most existing techniques focus on simply connected surfaces. It is a challenging problem for multiply connected genus zero surfaces. This work generalizes conventional Koebe's method for multiply connected planar domains. According to Koebe's uniformization theory, all genus zero multiply connected surfaces can be mapped to a planar disk with multiply circular holes. Furthermore, this kind of mappings are angle preserving and differ by Möbius transformations. We introduce a practical algorithm to explicitly construct such a circular conformal mapping. Our algorithm pipeline is as follows: suppose the input surface has n boundaries, first we choose 2 boundaries, and fill the other n -- 2 boundaries to get a topological annulus; then we apply discrete Yamabe flow method to conformally map the topological annulus to a planar annulus; then we remove the filled patches to get a planar multiply connected domain. We repeat this step for the planar domain iteratively. The two chosen boundaries differ from step to step. The iterative construction leads to the desired conformal mapping, such that all the boundaries are mapped to circles. In theory, this method converges quadratically faster than conventional Koebe's method. We give theoretic proof and estimation for the converging rate. In practice, it is much more robust and efficient than conventional non-linear methods based on curvature flow. Experimental results demonstrate the robustness and efficiency of the method.

Teichmuller shape space is a finite dimensional Riemannian manifold, where each point represents a class of surfaces, which are conformally equivalent, and a path represents a deformation process from one shape to the other. Two surfaces in the real world correspond to the same point in the Teichmuller space, only if they can be conformally mapped to each other. Teichmuller shape space can be used for surface classification purpose in shape modeling. This work focuses on the computation of the coordinates of high genus surfaces in the Teichmuller space. The coordinates are called as Fenchel-Nielsen coordinates. The main idea is to decompose the surface to pairs of hyperbolic pants. Each pair of pants is a genus zero surface with three boundaries, equipped with hyperbolic metric. Furthermore, all the boundaries are geodesics. Each pair of hyperbolic pants can be uniquely described by the lengths of its boundaries. The way of gluing different pairs of pants can be represented by the twisting angles between two adjacent pairs of pants which share a common boundary. The algorithms are based on Teichmuller space theory in conformal geometry, and they utilize the discrete surface Ricci flow. Most computations are carried out using hyperbolic geometry. The method is automatic, rigorous and efficient. The Teichmuller shape space coordinates can be used for surface classification and indexing. Experimental results on surfaces acquired from real world showed the potential value of the method for geometric database indexing, shape comparison and classification.

Homotopy group plays a role in computational topology with a fundamental importance. Each homotopy equivalence class contains an infinite number of loops. Finding a canonical representative within a homotopy class will simplify many computational tasks in computational topology, such as loop homotopy detection, pants decomposition. Furthermore, the canonical representative can be used as the shape descriptor. This work introduces a rigorous and practical method to compute a unique representative for each homotopy class. The main strategy is to use hyperbolic structure, such that each homotopy class has a unique closed geodesic, which is the representative. The following is the algorithm pipeline: for a given surface with negative Euler number, we apply hyperbolic Yamabe curvature flow to compute the unique Riemannian metric, which has constant negative one curvature everywhere and is conformal to the original metric. Then we compute the Fuchsian group generators of the surface on the hyperbolic space. For a given loop on the surface, we lift it to the universal covering space, to obtain the Fuchsian transformation corresponding to the homotopy class of the loop. The unique closed geodesic inside the homotopy class is the axis of the Fuchsian transformation, which is the canonical representative. Theories and algorithms are explained thoroughly in details. Experimental results are reported to show the efficiency and efficacy of the algorithm. The unique homotopy class representative can be applied for homotopy detection and shape comparison.

Abstract Surface Ricci flow is a powerful tool to design Riemannian metrics by user defined curvatures. Discrete surface Ricci flow has been broadly applied for surface parameterization, shape analysis, and computational topology. Conventional discrete Ricci flow has limitations. For meshes with low quality triangulations, if high conformality is required, the flow may get stuck at the local optimum of the Ricci energy. If convergence to the global optimum is enforced, the conformality may be sacrificed. This work introduces a novel method to generalize the traditional discrete Ricci flow. The generalized Ricci flow is more flexible, more robust and conformal for meshes with low quality triangulations. Conventional method is based on circle packing, which requires two circles on an edge intersect each other at an acute angle. Generalized method allows the two circles either intersect or separate from each other. This greatly improves the flexibility and robustness of the method. Furthermore, the generalized Ricci flow preserves the convexity of the Ricci energy, this ensures the uniqueness of the global optimum. Therefore the algorithm won't get stuck at the local optimum. Generalized discrete Ricci flow algorithms are explained in details for triangle meshes with both Euclidean and hyperbolic background geometries. Its advantages are demonstrated by theoretic proofs and practical applications in graphics, especially surface parameterization.

Harmonic volumetric mapping aims to establish a smooth bijective correspondence between two solid shapes with the same topology. In this paper, we develop an automatic meshless method for creating such a mapping between two given objects. With the shell surface mapping as the boundary condition, we first solve a linear system constructed by a boundary method called themethodoffundamentalsolution, and then represent the mapping using a set of points with different weights in the vicinity of the shell of the given model. Our algorithm is a true meshless method (without the need of any specific meshing structure within the solid interior) and the behavior of the interior region is directly determined by the boundary, which can improve the computational efficiency and robustness significantly. Therefore, our algorithm can be applied to massive volume data sets with various geometric primitives and topological types. We demonstrate the utility and efficacy of our algorithm in information transfer, shape registration, deformation sequence analysis, tetrahedral remeshing, and solid texture synthesis.

Shape indexing, classification, and retrieval are fundamental problems in computer graphics. This work introduces a novel method for surface indexing and classification based on Teichmuller theory. The Teichmuller space for surfaces with the same topology is a finite dimensional manifold, where each point represents a conformal equivalence class, a curve represents a deformation process from one class to the other. We apply Teichmuller space coordinates as shape descriptors, which are succinct, discriminating and intrinsic; invariant under the rigid motions and scalings, insensitive to resolutions. Furthermore, the method has solid theoretic foundation, and the computation of Teichmuller coordinates is practical, stable and efficient. This work focuses on the surfaces with negative Euler numbers, which have a unique conformal Riemannian metric with -1 Gaussian curvature. The coordinates which we will compute are the lengths of a special set of geodesics under this special metric. The metric can be obtained by the curvature flow algorithm, the geodesics can be calculated using algebraic topological method. We tested our method extensively for indexing and comparison of about one hundred of surfaces with various topologies, geometries and resolutions. The experimental results show the efficacy and efficiency of the length coordinate of the Teichmuller space.

Surface mapping is fundamental to shape computing and various downstream applications. This paper develops a pants decomposition framework for computing maps between surfaces with arbitrary topologies. The framework first conducts pants decomposition on both surfaces to segment them into consistent sets of pants patches (a pants patch is intuitively defined as a genus-0 surface with three boundaries), then composes global mapping between two surfaces by using harmonic maps of corresponding patches. This framework has several key advantages over existing techniques. First, it is automatic. It can automatically construct mappings for surfaces with complicated topology, guaranteeing the one-to-one continuity. Second, it is general and powerful. It flexibly handles mapping computation between surfaces with different topologies. Third, it is flexible. Despite topology and geometry, it can also integrate semantics requirements from users. Through a simple and intuitive human-computer interaction mechanism, the user can flexibly control the mapping behavior by enforcing point/curve constraints. Compared with traditional user-guided, piecewise surface mapping techniques, our new method is less labor intensive, more intuitive, and requires no user's expertise in computing complicated surface maps between arbitrary shapes. We conduct various experiments to demonstrate its modeling potential and effectiveness.

This paper formalizes a novel, intrinsic geometric scale space (IGSS) of 3D surface shapes. The intrinsic geometry of a surface is diffused by means of the Ricci flow for the generation of a geometric scale space. We rigorously prove that this multiscale shape representation satisfies the axiomatic causality property. Within the theoretical framework, we further present a feature-based shape representation derived from IGSS processing, which is shown to be theoretically plausible and practically effective. By integrating the concept of scale-dependent saliency into the shape description, this representation is not only highly descriptive of the local structures, but also exhibits several desired characteristics of global shape representations, such as being compact, robust to noise and computationally efficient. We demonstrate the capabilities of our approach through salient geometric feature detection and highly discriminative matching of 3D scans.

Automatic non-rigid registration of 3D time-varying data is fundamental in many vision and graphics applications such as facial expression analysis, synthesis, and recognition. Despite many research advances in recent years, it still remains to be technically challenging, especially for 3D dynamic, densely-sampled facial data with a large number of degrees of freedom (necessarily used to represent rich and subtle facial expressions). In this paper, we present a new method for automatic non-rigid registration of 3D dynamic facial data using least-squares conformal maps, and based on this registration method, we also develop a new framework of facial expression synthesis and transfer. Nowadays more and more 3D dynamic, densely-sampled data become prevalent with the advancement of novel 3D scanning techniques. To analyze and utilize such huge 3D data, an efficient non-rigid registration algorithm is needed to establish one-to-one inter frame correspondences. Towards this goal, a non-rigid registration algorithm of 3D dynamic facial data is developed by using least-squares conformal maps with additional feature correspondences detected by employing active appearance models (AAM). The proposed method with additional, interior feature constraints guarantees that the non-rigid data will be accurately registered. The least-squares conformal maps between two 3D surfaces are globally optimized with the least angle distortion and the resulting 2D maps are stable and one-to-one. Furthermore, by using this non-rigid registration method, we develop a new system of facial expression synthesis and transfer. Finally, we perform a series of experiments to evaluate our non-rigid registration method and demonstrate its efficacy and efficiency in the applications of facial expression synthesis and transfer.

Surface matching is fundamental to shape computing and various downstream applications. This paper develops a powerful pants decomposition framework for computing maps between surfaces with arbitrary topologies. We first conduct pants decomposition on both surfaces to segment them into consistent sets of pants patches (here a pants patch is intuitively defined as a genus-zero surface with three boundaries). Then we compose global mapping between two surfaces by harmonic maps of corresponding patches. This framework has several key advantages over other state-of-the-art techniques. First, the surface decomposition is automatic and general. It can automatically construct mappings for surfaces with same but complicated topology, and the result is guaranteed to be one-to-one continuous. Second, the mapping framework is very flexible and powerful. Not only topology and geometry, but also the semantics can be easily integrated into this framework with a little user involvement. Specifically, it provides an easy and intuitive human-computer interaction mechanism so that mapping between surfaces with different topologies, or with additional point/curve constraints, can be properly obtained within our framework. Compared with previous user-guided, piecewise surface mapping techniques, our new method is more intuitive, less labor-intensive, and requires no user's expertise in computing complicated surface map between arbitrary shapes. We conduct various experiments to demonstrate its modeling potential and effectiveness. © 2008 ACM.

Polycube T-spline has been formulated elegantly that can unify T-splines and manifold splines to define a new class of shape representations for surfaces of arbitrary topology by using polycube map as its parametric domain. In essense, The data fitting quality using polycube T-splines hinges upon the construction of underlying polycube maps. Yet, existing methods for polycube map construction exhibit some disadvantages. For example, existing approaches for polycube map construction either require projection of points from a 3D surface to its polycube approximation, which is therefore very difficult to handle the cases when two shapes differ significantly; or compute the map by conformally deforming the surfaces and polycubes to the common canonical domain and then construct the map using function composition, which is challenging to control the location of singularities and makes it hard for the data-fitting and hole-filling processes later on.

This paper presents a novel and efficient surface matching and visualization framework through the geodesic distance-weighted shape vector image diffusion. Based on conformal geometry, our approach can uniquely map a 3D surface to a canonical rectangular domain and encode the shape characteristics (e.g., mean curvatures and conformal factors) of the surface in the 2D domain to construct a geodesic distance-weighted shape vector image, where the distances between sampling pixels are not uniform but the actual geodesic distances on the manifold. Through the novel geodesic distance-weighted shape vector image diffusion presented in this paper, we can create a multiscale diffusion space, in which the cross-scale extrema can be detected as the robust geometric features for the matching and registration of surfaces. Therefore, statistical analysis and visualization of surface properties across subjects become readily available. The experiments on scanned surface models show that our method is very robust for feature extraction and surface matching even under noise and resolution change. We have also applied the framework on the real 3D human neocortical surfaces, and demonstrated the excellent performance of our approach in statistical analysis and integrated visualization of the multimodality volumetric data over the shape vector image.

This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by user-defined Gaussian curvatures. Furthermore, the target metrics are conformal (angle-preserving) to the original metrics. A Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature becomes the user defined curvature. Discrete Ricci flow algorithms are based on a variational framework. Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope. The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy. Furthermore, the Ricci energy can be optimized using Newton's method more efficiently. Discrete Ricci flow algorithms are rigorous and efficient. Our experimental results demonstrate the efficiency, accuracy and flexibility of the algorithms. They have the potential for a wide range of applications in graphics, geometric modeling, and medical imaging. We demonstrate their practical values by global surface parameterizations.

Computing smooth and optimal one-to-one maps between surfaces of same topology is a fundamental problem in computer graphics and such a method provides us a ubiquitous tool for geometric modeling and data visualization. Its vast variety of applications includes shape registration/matching, shape blending, material/data transfer, data fusion, information reuse, etc. The mapping quality is typically measured in terms of angular distortions among different shapes. This paper proposes and develops a novel quasi-conformal surface mapping framework to globally minimize the stretching energy inevitably introduced between two different shapes. The existing state-of-the-art inter-surface mapping techniques only afford local optimization either on surface patches via boundary cutting or on the simplified base domain, lacking rigorous mathematical foundation and analysis. We design and articulate an automatic variational algorithm that can reach the global distortion minimum for surface mapping between shapes of arbitrary topology, and our algorithm is sorely founded upon the intrinsic geometry structure of surfaces. To our best knowledge, this is the first attempt towards numerically computing globally optimal maps. Consequently, our mapping framework offers a powerful computational tool for graphics and visualization tasks such as data and texture transfer, shape morphing, and shape matching.

Mesh parameterization is a fundamental technique in computer graphics. Our paper focuses on solving the problem of finding the best discrete conformal mapping that also minimizes area distortion. Firstly, we deduce an exact analytical differential formula to represent area distortion by curvature change in the discrete conformal mapping, giving a dynamic Poisson equation. Our result shows the curvature map is invertible. Furthermore, we give the explicit Jacobi matrix of the inverse curvature map. Secondly, we formulate the task of computing conformal parameterizations with least area distortions as a constrained nonlinear optimization problem in curvature space. We deduce explicit conditions for the optima. Thirdly, we give an energy form to measure the area distortions, and show it has a unique global minimum. We use this to design an efficient algorithm, called free boundary curvature diffusion, which is guaranteed to converge to the global minimum. This result proves the common belief that optimal parameterization with least area distortion has a unique solution and can be achieved by free boundary conformal mapping. Major theoretical results and practical algorithms are presented for optimal parameterization based on the inverse curvature map. Comparisons are conducted with existing methods and using different energies. Novel parameterization applications are also introduced.

Summary form only given. In this paper we generalize the shortest path algorithm to the shortest cycles in each homotopy class on a surface with arbitrary topology, utilizing the universal covering space (UCS) in algebraic topology. In order to store and handle the UCS, we propose a two-level data structure which is efficient for storage and easy to process. We also pointed several practical applications for our shortest cycle algorithms and the UCS data structure.

Ricci flow is a powerful curvature flow method in geometric analysis. This work is the first application of surface Ricci flow in computer vision. We show that previous methods based on conformal geometries, such as harmonic maps and least-square conformal maps, which can only handle 3D shapes with simple topology are subsumed by our Ricci flow based method which can handle surfaces with arbitrary topology. Because the Ricci flow method is intrinsic and depends on the surface metric only, it is invariant to rigid motion, scaling, and isometric and conformal deformations. The solution to Ricci flow is unique and its computation is robust to noise. Our Ricci flow based method can convert all 3D problems into 2D domains and offers a general framework for 3D surface analysis. Large non-rigid deformations can be registered with feature constraints, hence we introduce a method that constrains Ricci flow computation using feature points and feature curves. Finally, we demonstrate the applicability of this intrinsic shape representation through standard shape analysis problems, such as 3D shape matching and registration.

This paper develops a novel computational technique to define and construct powerful manifold splines with only one singular point by employing the rigorous mathematical theory of Ricci flow. The central idea and new computational paradigm of manifold splines are to systematically extend the algorithmic pipeline of spline surface construction from any planar domain to arbitrary topology. As a result, manifold splines can unify planar spline representations as their special cases. Despite their earlier success, the existing manifold spline framework is plagued by the topology-dependent, large number of singular points (i.e., |2g -- 2| for any genus-g surface), where the analysis of surface behaviors such as continuity remains extremely difficult. The unique theoretical contribution of this paper is that we devise new mathematical tools so that manifold splines can now be constructed with only one singular point, reaching their theoretic lower bound of singularity for real-world applications. Our new algorithm is founded upon the concept of discrete Ricci flow and associated techniques. First, Ricci flow is employed to compute a special metric of any manifold domain (serving as a parametric domain for manifold splines), such that the metric becomes flat everywhere except at one point. Then, the metric naturally induces an affine atlas covering the entire manifold except this singular point. Finally, manifold splines are defined over this affine atlas. The Ricci flow method is theoretically sound, and practically simple and efficient. We conduct various shape experiments and our new theoretical and algorithmic results alleviate the modeling difficulty of manifold splines, and hence, promising to promote the widespread use of manifold splines in surface and solid modeling, geometric design, and reverse engineering.

Surface classification is one of the most fundamental problems in geometric modeling. Surfaces can be classified according to their conformal structures. In general, each topological equivalent class has infinite conformally equivalent classes.

Harmonic volumetric mapping for two solid objects establishes a one-to-one smooth correspondence between them. It finds its applications in shape registration and analysis, shape retrieval, information reuse, and material/texture transplant. In sharp contrast to harmonic surface mapping techniques, little research has been conducted for designing volumetric mapping algorithms due to its technical challenges. In this paper, we develop an automatic and effective algorithm for computing harmonic volumetric mapping between two models of the same topology. Given a boundary mapping between two models, the volumetric (interior) mapping is derived by solving a linear system constructed from a boundary method called the fundamental solution method. The mapping is represented as a set of points with different weights in the vicinity of the solid boundary. In a nutshell, our algorithm is a true meshless method (with no need of specific connectivity) and the behavior of the interior region is directly determined by the boundary. These two properties help improve the computational efficiency and robustness. Therefore, our algorithm can be applied to massive volume data sets with various geometric primitives and topological types. We demonstrate the utility and efficacy of our algorithm in shape registration, information reuse, deformation sequence analysis, tetrahedral remeshing and solid texture synthesis.

This paper proposes a new concept of polycube splines and develops novel modeling techniques for using the polycube splines in solid modeling and shape computing. Polycube splines are essentially a novel variant of manifold splines which are built upon the polycube map, serving as its parametric domain. Our rationale for defining spline surfaces over polycubes is that polycubes have rectangular structures everywhere over their domains except a very small number of corner points. The boundary of polycubes can be naturally decomposed into a set of regular structures, which facilitate tensor-product surface definition, GPU-centric geometric computing, and image-based geometric processing. We develop algorithms to construct polycube maps, and show that the introduced polycube map naturally induces the affine structure with a finite number of extraordinary points. Besides its intrinsic rectangular structure, the polycube map may approximate any original scanned data-set with a very low geometric distortion, so our method for building polycube splines is both natural and necessary, as its parametric domain can mimic the geometry of modeled objects in a topologically correct and geometrically meaningful manner. We design a new data structure that facilitates the intuitive and rapid construction of polycube splines in this paper. We demonstrate the polycube splines with applications in surface reconstruction and shape computing.

Three-dimensional shape matching is a fundamental issue in computer vision with many applications such as shape registration, 3D object recognition, and classification. However, shape matching with noise, occlusion, and clutter is a challenging problem. In this paper, we analyze a family of quasi-conformal maps including harmonic maps, conformal maps, and least-squares conformal maps with regards to 3D shape matching. As a result, we propose a novel and computationally efficient shape matching framework by using least-squares conformal maps. According to conformal geometry theory, each 3D surface with disk topology can be mapped to a 2D domain through a global optimization and the resulting map is a diffeomorphism, i.e., one-to-one and onto. This allows us to simplify the 3D shape-matching problem to a 2D image-matching problem, by comparing the resulting 2D parametric maps, which are stable, insensitive to resolution changes and robust to occlusion, and noise. Therefore, highly accurate and efficient 3D shape matching algorithms can be achieved by using the above three parametric maps. Finally, the robustness of least-squares conformal maps is evaluated and analyzed comprehensively in 3D shape matching with occlusion, noise, and resolution variation. In order to further demonstrate the performance of our proposed method, we also conduct a series of experiments on two computer vision applications, i.e., 3D face recognition and 3D nonrigid surface alignment and stitching.

In medical imaging, parameterized 3-D surface models are useful for anatomical modeling and visualization, statistical comparisons of anatomy, and surface-based registration and signal processing. Here we introduce a parameterization method based on Riemann surface structure, which uses a special curvilinear net structure (conformal net) to partition the surface into a set of patches that can each be conformally mapped to a parallelogram. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable (their solutions tend to be smooth functions and the boundary conditions of the Dirichlet problem can be enforced). Conformal parameterization also helps transform partial differential equations (PDEs) that may be defined on 3-D brain surface manifolds to modified PDEs on a two-dimensional parameter domain. Since the Jacobian matrix of a conformal parameterization is diagonal, the modified PDE on the parameter domain is readily solved. To illustrate our techniques, we computed parameterizations for several types of anatomical surfaces in 3-D magnetic resonance imaging scans of the brain, including the cerebral cortex, hippocampi, and lateral ventricles. For surfaces that are topologically homeomorphic to each other and have similar geometrical structures, we show that the parameterization results are consistent and the subdivided surfaces can be matched to each other. Finally, we present an automatic sulcal landmark location algorithm by solving PDEs on cortical surfaces. The landmark detection results are used as constraints for building conformal maps between surfaces that also match explicitly defined landmarks.

Conveying shape using feature lines is an important visualization tool in visual computing. The existing feature lines (e.g., ridges, valleys, silhouettes, suggestive contours, etc.) are solely determined by local geometry properties (e.g., normals and curvatures) as well as the view position. This paper is strongly inspired by the observation in human vision and perception that a sudden change in the luminance plays a critical role to faithfully represent and recover the 3D information. In particular, we adopt the edge detection techniques in image processing for 3D shape visualization and present Photic Extremum Lines (PELs) which emphasize significant variations of illumination over 3D surfaces. Comparing with the existing feature lines, PELs are more flexible and offer users more freedom to achieve desirable visualization effects. In addition, the user can easily control the shape visualization by changing the light position, the number of light sources, and choosing various light models. We compare PELs with the existing approaches and demonstrate that PEL is a flexible and effective tool to illustrate 3D surface and volume for visual computing.

3D surface matching is a fundamental issue in computer vision with many applications such as shape registration, 3D object recognition and classification. However, surface matching with noise, occlusion and clutter is a challenging problem. In this paper, we analyze a family of conformal geometric maps including harmonic maps, conformal maps and least squares conformal maps with regards to 3D surface matching. As a result, we propose a novel and computationally efficient surface matching framework that uses least squares conformal maps. According to conformal geometry theory, each 3D surface with disk topology can be mapped to a 2D domain through a global optimization and the resulting map is a diffeomorphism, i.e., one-to-one and onto. This allows us to simplify the 3D surface-matching problem to a 2D image-matching problem, by comparing the resulting 2D conformal geometric maps, which are stable, insensitive to resolution changes and robust to occlusion and noise. Therefore, highly accurate and efficient 3D surface matching algorithms can be achieved by using conformal geometric maps. Finally, the performance of conformal geometric maps is evaluated and analyzed comprehensively in 3D surface matching with occlusion, noise and resolution variation. We also provide a series of experiments on real 3D face data that achieve high recognition rates.

Recent advances in imaging technologies, such as magnetic resonance imaging (MRI), positron emission tomography (PET) and diffusion tensor imaging (DTI) have accelerated brain research in many aspects. In order to better understand the synergy of the many processes involved in normal brain function, integrated modeling and analysis of MRI, PET, and DTI across subjects is highly desirable. The current state-of-art computational tools fall short in offering an analytic approach for intersubject brain registration and analysis. In this paper we present an approach which is based on landmark constrained conformal parameterization of a brain surface from high-resolution structural MRI data to a canonical spherical domain. This model allows natural integration of information from co-registered PET as well as DTI data and lays a foundation for the quantitative analysis of the relationship among diverse datasets across subjects. Consequently, the approach can be extended to provide a software environment able to facilitate detection of abnormal functional brain patterns in patients with neurological disorder.

We present an efficient colon flattening algorithm using a conformal structure, which is angle-preserving and minimizes the global distortion. Moreover, our algorithm is general as it can handle high genus surfaces. First, the colon wall is segmented and extracted from the CT data set of the abdomen. The topology noise (i.e., minute handle) is located and removed automatically. The holomorphic 1-form, a pair of orthogonal vector fields, is then computed on the 3D colon surface mesh using the conjugate gradient method. The colon surface is cut along a vertical trajectory traced using the holomorphic 1-form. Consequently, the 3D colon surface is conformally mapped to a 2D rectangle. The flattened 2D mesh is then rendered using a direct volume rendering method accelerated with the GPU. Our algorithm is tested with a number of CT data sets of real pathological cases, and gives consistent results. We demonstrate that the shape of the polyps is well preserved on the flattened colon images, which provides an efficient way to enhance the navigation of a virtual colonoscopy system.

Geometric structures are natural structures of surfaces, which enable different geometries to be defined on the surfaces. Algorithms designed for planar domains based on a specific geometry can be systematically generalized to surface domains via the corresponding geometric structure. For example, polar form splines with planar domains are based on affine invariants. Polar form splines can be generalized to manifold splines on the surfaces which admit affine structures and are equipped with affine geometries.Surfaces with negative Euler characteristic numbers admit hyperbolic structures and allow hyperbolic geometry. All surfaces admit real projective structures and are equipped with real projective geometry. Because of their general existence, both hyperbolic structures and real projective structures have the potential to replace the role of affine structures in defining manifold splines.This paper introduces theoretically rigorous and practically simple algorithms to compute hyperbolic structures and real projective structures for general surfaces. The method is based on a novel geometric tool - discrete variational Ricci flow. Any metric surface admits a special uniformization metric, which is conformal to its original metric and induces constant curvature. Ricci flow is an efficient method to calculate the uniformization metric, which determines the hyperbolic structure and real projective structure.The algorithms have been verified on real surfaces scanned from sculptures. The method is efficient and robust in practice. To the best of our knowledge, this is the first work of introducing algorithms based on Ricci flow to compute hyperbolic structure and real projective structure.More importantly, this work introduces the framework of general geometric structures, which enable different geometries to be defined on manifolds and lay down the theoretical foundation for many important applications in geometric modeling.

We develop a new surface matching framework to handle surface comparisons based on the mathematical analysis of curves on surfaces, and propose a unique signature for any closed curve on a surface. The signature describes not only the shape of the curve, but also the intrinsic relationship between the curve and its embedding surface; and furthermore, the signature metric is stable across surfaces sharing similar Riemannian geometry metrics. Based on this theoretical advance, we analyze and align features defined as closed curves on surfaces using their signatures. These curves segment a surface into different regions which are mapped onto canonical domains for the matching purpose. The experimental results are very promising, demonstrating that the curve signatures and the comparison framework are robust and discriminative for the effective shape comparison. Besides its utility in our current framework, we believe the curve signature will also serve as a powerful shape segmentation/mapping tool and can be used to aid in many existing techniques towards effective shape analysis

This paper presents a new approach to the physically-based thin-shell simulation of point-sampled geometry via explicit, global conformal point-surface parameterization and meshless dynamics. The point-based global parameterization is founded upon the rigorous mathematics of Riemann surface theory and Hodge theory. The parameterization is globally conformal everywhere except for a minimum number of zero points. Within our parameterization framework, any well-sampled point surface is functionally equivalent to a manifold, enabling popular and powerful surface-based modeling and physically-based simulation tools to be readily adapted for point geometry processing and animation. In addition, we propose a meshless surface computational paradigm in which the partial differential equations (for dynamic physical simulation) can be applied and solved directly over point samples via Moving Least Squares (MLS) shape functions defined on the global parametric domain without explicit connectivity information. The global conformal parameterization provides a common domain to facilitate accurate meshless simulation and efficient discontinuity modeling for complex branching cracks. Through our experiments on thin-shell elastic deformation and fracture simulation, we demonstrate that our integrative method is very natural, and that it has great potential to further broaden the application scope of point-sampled geometry in graphics and relevant fields.

We propose a general method that parameterizes general surfaces with complex (possible branching) topology using Riemann surface structure. Rather than evolve the surface geometry to a plane or sphere, we instead use the fact that all orientable surfaces are Riemann surfaces and admit conformal structures, which induce special curvilinear coordinate systems on the surfaces. We can then automatically partition the surface using a critical graph that connects zero points in the global conformal structure on the surface. The trajectories of iso-parametric curves canonically partition a surface into patches. Each of these patches is either a topological disk or a cylinder and can be conformally mapped to a parallelogram by integrating a holomorphic I-form defined on the surface. The resulting surface subdivision and the parameterizations of the components are intrinsic and stable. For surfaces with similar topology and geometry, we show that the parameterization results are consistent and the subdivided surfaces can be matched to each other using constrained harmonic maps. The surface similarity can be measured by direct computation of distance between each pair of corresponding points on two surfaces. To illustrate the technique, we computed conformal structures for anatomical surfaces in MRI scans of the brain and human face surfaces. We found that the resulting parameterizations were consistent across subjects, even for branching structures such as the ventricles, which are otherwise difficult to parameterize. Our method provides a surface-based framework for statistical comparison of surfaces and for generating grids on surfaces for PDE-based signal processing.

We present a novel fully automatic method for high resolution, nonrigid dense 3D point tracking. High quality dense point clouds of nonrigid geometry moving at video speeds are acquired using a phase-shifting structured light ranging technique. To use such data for the temporal study of subtle motions such as those seen in facial expressions, an efficient nonrigid 3D motion tracking algorithm is needed to establish inter-frame correspondences. The novelty of this paper is the development of an algorithmic framework for 3D tracking that unifies tracking of intensity and geometric features, using harmonic maps with added feature correspondence constraints. While the previous uses of harmonic maps provided only global alignment, the proposed introduction of interior feature constraints guarantees that nonrigid deformations are accurately tracked as well. The harmonic map between two topological disks is a diffeomorphism with minimal stretching energy and bounded angle distortion. The map is stable, insensitive to resolution changes and is robust to noise. Due to the strong implicit and explicit smoothness constraints imposed by the algorithm and the high-resolution data, the resulting registration/deformation field is smooth, continuous and gives dense one-to-one inter-frame correspondences. Our method is validated through a series of experiments demonstrating its accuracy and efficiency.

Constructing splines whose parametric domain is an arbitrary manifold and effectively computing such splines in real-world applications are of fundamental importance in solid and shape modeling, geometric design, graphics, etc. This paper presents a general theoretical and computational framework, in which spline surfaces defined over planar domains can be systematically extended to manifold domains with arbitrary topology with or without boundaries. We study the affine structure of domain manifolds in depth and prove that the existence of manifold splines is equivalent to the existence of a manifold's affine atlas. Based on our theoretical breakthrough, we also develop a set of practical algorithms to generalize triangular B-spline surfaces from planar domains to manifold domains. We choose triangular B-splines mainly because of its generality and many of its attractive properties. As a result, our new spline surface defined over any manifold is a piecewise polynomial surface with high parametric continuity without the need for any patching and/or trimming operations. Through our experiments, we hope to demonstrate that our novel manifold splines are both powerful and efficient in modeling arbitrarily complicated geometry and representing continuously-varying physical quantities defined over shapes of arbitrary topology.

Traditional approaches for modeling a closed manifold surface with either regular tensor-product or triangular splines (defined over an open planar domain) require decomposing the acquired geometric data into a group of charts, mapping each chart to a planar parametric domain, fitting an open surface patch of certain degree to each chart, and finally, trimming the patches (if necessary) and stitching all of them together to form a closed manifold. In this paper, we develop a novel modeling method which does not need any cutting or patching operations for genus zero surfaces. Our new approach is founded upon the concept of spherical splines proposed by Pfeifle and Seidel. Our work is strongly inspired by the fact that, for genus zero surfaces, it is both intuitive and necessary to employ spheres as their natural domains. Using this framework, we can convert genus zero mesh to a single rational spherical spline whose maximal error deviated from the original data is less than a user-specified tolerance. With the rational spherical splines, we can model sharp features and edit both the global shape and the local details with ease. Furthermore, we can accurately compute the differential quantities without resorting to any numerical approximations. We conduct several experiments in order to demonstrate the efficacy of our approach for reverse engineering, shape modeling, and interactive graphics.

Topological concepts and techniques have been broadly applied in computer graphics and geometric modeling. However, the homotopy type of a mapping between two surfaces has not been addressed before. In this paper, we present a novel solution to the problem of computing continuous maps with different homotopy types between two arbitrary triangle meshes with the same topology. Inspired by the rich theory of topology as well as the existing body of work on surface mapping, our newly-developed mapping techniques are both fundamental and unique, offering many attractive advantages. First, our method allows the user to change the homotopy type or global structure of the mapping with minimal intervention. Moreover, to locally affect shape correspondence, we articulate a new technique that robustly satisfies hard feature constraints, without the use of heuristics to ensure validity. In addition to acting as a useful tool for computer graphics applications, our method can be used as a rigorous and practical mechanism for the visualization of abstract topological concepts such as homotopy type of surface mappings, homology basis, fundamental domain, and universal covering space. At the core of our algorithm is a procedure for computing the canonical homology basis and using it as a common cut graph for any surface with the same topology. We demonstrate our results by applying our algorithm to shape morphing in this paper.

All orientable metric surfaces are Riemann surfaces and admit global conformal parameterizations. Riemann surface structure is a fundamental structure and governs many natural physical phenomena, such as heat diffusion and electro-magnetic fields on the surface. A good parameterization is crucial for simulation and visualization. This paper provides an explicit method for finding optimal global conformal parameterizations of arbitrary surfaces. It relies on certain holomorphic differential forms and conformal mappings from differential geometry and Riemann surface theories. Algorithms are developed to modify topology, locate zero points, and determine cohomology types of differential forms. The implementation is based on a finite dimensional optimization method. The optimal parameterization is intrinsic to the geometry, preserves angular structure, and can play an important role in various applications including texture mapping, remeshing, morphing and simulation. The method is demonstrated by visualizing the Riemann surface structure of real surfaces represented as triangle meshes.

We developed a general method for global conformal parameterizations based on the structure of the cohomology group of holomorphic one-forms for surfaces with or without boundaries (Gu and Yau, 2002), (Gu and Yau, 2003). For genus zero surfaces, our algorithm can find a unique mapping between any two genus zero manifolds by minimizing the harmonic energy of the map. In this paper, we apply the algorithm to the cortical surface matching problem. We use a mesh structure to represent the brain surface. Further constraints are added to ensure that the conformal map is unique. Empirical tests on magnetic resonance imaging (MRI) data show that the mappings preserve angular relationships, are stable in MRIs acquired at different times, and are robust to differences in data triangulation, and resolution. Compared with other brain surface conformal mapping algorithms, our algorithm is more stable and has good extensibility.

3D surface classification is a fundamental problem in computer vision and computational geometry. Surfaces can be classified by different transformation groups. Traditional classification methods mainly use topological transformation groups and Euclidean transformation groups. We introduce a novel method to classify surfaces by conformal transformation groups. Conformal equivalent class is refiner than topological equivalent class and coarser than isometric equivalent class, making it suitable for practical classification purposes. For general surfaces, the gradient fields of conformal maps form a vector space, which has a natural structure invariant under conformal transformations. We present an algorithm to compute this conformal structure, which can be represented as matrices, and use it to classify surfaces. The result is intrinsic to the geometry, invariant to triangulation and insensitive to resolution. To the best of our knowledge, this is the first paper to classify surfaces with arbitrary topologies by global conformal invariants. The method introduced here can also be used for surface matching problems.

Parameterization of 3D mesh data is important for many graphics applications, in particular for texture mapping, remeshing and morphing. Closed manifold genus-0 meshes are topologically equivalent to a sphere, hence this is the natural parameter domain for them. Parameterizing a triangle mesh onto the sphere means assigning a 3D position on the unit sphere to each of the mesh vertices, such that the spherical triangles induced by the mesh connectivity are not too distorted and do not overlap. Satisfying the non-overlapping requirement is the most difficult and critical component of this process. We describe a generalization of the method of barycentric coordinates for planar parameterization which solves the spherical parameterization problem, prove its correctness by establishing a connection to spectral graph theory and show how to compute these parameterizations.

Surface geometry is often modeled with irregular triangle meshes. The process of remeshing refers to approximating such geometry using a mesh with (semi)-regular connectivity, which has advantages for many graphics applications. However, current techniques for remeshing arbitrary surfaces create only semi-regular meshes. The original mesh is typically decomposed into a set of disk-like charts, onto which the geometry is parametrized and sampled. In this paper, we propose to remesh an arbitrary surface onto a completely regular structure we call a geometry image. It captures geometry as a simple 2D array of quantized points. Surface signals like normals and colors are stored in similar 2D arrays using the same implicit surface parametrization --- texture coordinates are absent. To create a geometry image, we cut an arbitrary mesh along a network of edge paths, and parametrize the resulting single chart onto a square. Geometry images can be encoded using traditional image compression algorithms, such as wavelet-based coders.

Approximating detailed with coarse, texture-mapped meshes results in polygonal silhouettes. To eliminate this artifact, we introduce silhouette clipping, a framework for efficiently clipping the rendering of coarse geometry to the exact silhouette of the original model. The coarse mesh is obtained using progressive hulls, a novel representation with the nesting property required for proper clipping. We describe an improved technique for constructing texture and normal maps over this coarse mesh. Given a perspective view, silhouettes are efficiently extracted from the original mesh using a precomputed search tree. Within the tree, hierarchical culling is achieved using pairs of anchored cones. The extracted silhouette edges are used to set the hardware stencil buffer and alpha buffer, which in turn clip and antialias the rendered coarse geometry. Results demonstrate that silhouette clipping can produce renderings of similar quality to high-resolution meshes in less rendering time.