Abstract. This paper addresses the challenging problem of modeling and computing multivalued mappings with varying cardinality, where a single input can correspond to multiple valid outputs, and the number of possible outputs may vary for different inputs. Such scenarios are prevalent in various imaging applications where multiple plausible solutions exist for a given input. We introduce a deep neural network framework to model multivalued mappings with varying cardinality. The framework integrates a discrete codebook with a generative network to produce valid outputs for each input. The discrete codebook variables are combined with the input to guide the generator in producing different valid solutions. The discrete nature of the codebook enables the framework to efficiently estimate the conditional probability distribution of possible outputs through a fixed equiangular tight frame classifier. By jointly optimizing the discrete codebook and its uncertainty estimation during training using a specially designed covariance loss function, an accurate computation of multiple solution candidates with reliable confidence measures can be achieved. We demonstrate the effectiveness of the proposed framework on various imaging applications, using both synthetic and real datasets. Experimental results show the efficacy of our proposed model to generate multiple high-quality outputs while providing meaningful uncertainty estimates for each solution.

Abstract. This paper presents a novel shape prior segmentation model guided by the harmonic Beltrami signature (HBS) that integrates shape information to enhance the segmentation accuracy. The HBS is a robust shape representation that fully captures a 2D simply connected shape. It exhibits resilience to perturbations and is invariant under translation, rotation, and scaling. These properties make it a suitable candidate for incorporating general shape information into the segmentation model, instead of relying on primitive properties such as convexity or topology. Our shape prior segmentation model embeds the HBS within a quasi-conformal, topology-preserving segmentation framework. With the shape prior knowledge, our proposed model significantly enhances the segmentation performance, especially for low-quality or occluded images. The robustness of HBS to shape perturbations allows for the use of a simple [Formula: see text] distance metric to define shape dissimilarity, which simplifies the optimization process. Besides, the invariance of HBS to translation, rotation, and scaling allows the integration of shape prior information without requiring the knowledge of orientation, size and position of the shape in the image. This enhances its practicality for real-world applications. Extensive experiments have been carried out on both synthetic and real images. Results demonstrate that our proposed model significantly improves segmentation accuracy, particularly for degraded or corrupted images.

Abstract. Deep learning on non-Euclidean domains is important for analyzing complex geometric data that lacks common coordinate systems and familiar Euclidean properties. A central challenge in this field is to define convolution on domains, which inherently possess irregular and non-Euclidean structures. In this work, we introduce quasi-conformal convolution (QCC), a novel framework for defining convolution on simply-connected open surfaces using quasi-conformal theories. Each QCC operator is linked to a specific quasi-conformal mapping, enabling the adjustment of the convolution operation through manipulation of this mapping. By utilizing trainable estimator modules that produce quasi-conformal mappings, QCC facilitates adaptive and learnable convolution operators that can be dynamically adjusted according to the underlying data structured on the surfaces. QCC unifies a broad range of spatially defined convolutions, facilitating the learning of tailored convolution operators on each underlying surface optimized for specific tasks. Building on this foundation, we develop the quasi-conformal convolutional neural network (QCCNN) to address a variety of tasks related to geometric data. We validate the efficacy of QCCNN through the classification of images defined on curvilinear simply-connected open Riemann surfaces, demonstrating superior performance in this context. Additionally, we explore its potential in medical applications, including craniofacial analysis using 3D facial data and lesion segmentation on 3D human faces, achieving enhanced accuracy and reliability.

Abstract. Interactive segmentation allows users to provide meaningful input to guide the segmentation process. However, an important problem in interactive segmentation lies in determining how to incorporate minimal yet meaningful user guidance into the segmentation model. In this paper, we propose the quasi-conformal interactive segmentation (QIS) model, which incorporates user input in the form of positive and negative clicks. Users mark a few pixels belonging to the object region as positive clicks, indicating that the segmentation model should include a region around these clicks. Conversely, negative clicks are provided on pixels belonging to the background, instructing the model to exclude the region near these clicks from the segmentation mask. By solving our proposed theoretical supported model, the segmentation mask is obtained by deforming a template mask with the same topology as the object of interest using a quasi-conformal mapping. This approach makes each user input effectively used and helps to avoid topological errors in the segmentation results. We provide a thorough theoretical analysis of the proposed model for its ability to include or exclude regions of interest or disinterest based on the user’s indication. To evaluate the performance of QIS, we conduct experiments on synthesized images, medical images, and natural images. The results demonstrate the efficacy of our proposed method.

We propose the Topology-Preserving Segmentation Network, a deformation-based model that can extract objects in an image while maintaining their topological properties. This network generates segmentation masks that have the same topology as the template mask, even when trained with limited data. The network consists of two components: the Deformation Estimation Network, which produces a deformation map that warps the template mask to enclose the region of interest, and the Beltrami Adjustment Module, which ensures the bijectivity of the deformation map by truncating the associated Beltrami coefficient based on Quasiconformal theories. The proposed network can also be trained in an unsupervised manner, eliminating the need for labeled training data. This is achieved by incorporating an unsupervised segmentation loss. Our experimental results on various image datasets show that TPSN achieves better segmentation accuracy than state-of-the-art models with correct topology. Furthermore, we demonstrate TPSN’s ability to handle multiple object segmentation.

Abstract. Many imaging problems can be formulated as mapping problems. A general mapping problem aims to obtain an optimal mapping that minimizes an energy functional subject to the given constraints. Existing methods to solve the mapping problems are often inefficient and can sometimes get trapped in local minima. An extra challenge arises when the optimal mapping is required to be diffeomorphic. In this work, we address the problem by proposing a deep-learning framework based on the Quasiconformal (QC) Teichmüller theories. The main strategy is to learn the Beltrami coefficient (BC) that represents a mapping as the latent feature vector in the deep neural network. The BC measures the local geometric distortion under the mapping, with which the interpretability of the deep neural network can be enhanced. Under this framework, the diffeomorphic property of the mapping can be controlled via a simple activation function within the network. The optimal mapping can also be easily regularized by integrating the BC into the loss function. A crucial advantage of the proposed framework is that once the network is successfully trained, the optimized mapping corresponding to each input data information can be obtained in real time. To examine the efficacy of the proposed framework, we apply the method to the diffeomorphic image registration problem. Experimental results outperform other state-of-the-art registration algorithms in both efficiency and accuracy, which demonstrate the effectiveness of our proposed framework to solve the mapping problem.

Abstract. This paper proposes a novel method for computing bijective density-equalizing quasiconformal flattening maps for multiply connected open surfaces. In conventional density-equalizing maps, shape deformations are solely driven by prescribed constraints on the density distribution, defined as the population per unit area, while the bijectivity and local geometric distortions of the mappings are uncontrolled. Also, prior methods have primarily focused on simply connected open surfaces but not surfaces with more complicated topologies. Our proposed method overcomes these issues by formulating the density diffusion process as a quasiconformal flow, which allows us to effectively control the local geometric distortion and guarantee the bijectivity of the mapping by solving an energy minimization problem involving the Beltrami coefficient of the mapping. To achieve an optimal parameterization of multiply connected surfaces, we develop an iterative scheme that optimizes both the shape of the target planar circular domain and the density-equalizing quasiconformal map onto it. In addition, landmark constraints can be incorporated into our proposed method for consistent feature alignment. The method can also be naturally applied to simply connected open surfaces. By changing the prescribed population, a large variety of surface flattening maps with different desired properties can be achieved. The method is tested on both synthetic and real examples, demonstrating its efficacy in various applications in computer graphics and medical imaging.

Abstract. Density-equalizing maps are a class of mapping methods in which the shape deformation is driven by prescribed density information. In recent years, they have been widely used for data visualization on planar domains and planar parameterization of open surfaces. However, the theory and computation of density-equalizing maps for closed surfaces are much less explored. In this work, we develop a novel method for computing spherical density-equalizing maps for genus-0 closed surfaces. Specifically, we first compute a conformal parameterization of the given genus-0 closed surface onto the unit sphere. Then we perform density equalization on the spherical domain based on the given density information to achieve a spherical density-equalizing map. The bijectivity of the mapping is guaranteed by introducing an overlap correction scheme based on quasi-conformal theory throughout the density-equalizing iterative process. We further propose a method for incorporating the harmonic energy and landmark constraints into our formulation to achieve landmark-aligned spherical density-equalizing maps balancing different distortion measures. Using the proposed methods, a large variety of spherical parameterizations can be achieved. Applications to surface registration, remeshing, and data visualization are presented to demonstrate the effectiveness of our methods.

Diffusion-weighted imaging (DWI) has been extensively explored in guiding the clinic management of patients with breast cancer. However, due to the limited resolution, accurately characterizing tumors using DWI and the corresponding apparent diffusion coefficient (ADC) is still a challenging problem. In this paper, we aim to address the issue of super-resolution (SR) of ADC images and evaluate the clinical utility of SR-ADC images through radiomics analysis. To this end, we propose a novel double transformer-based network (DTformer) to enhance the resolution of ADC images. More specifically, we propose a symmetric U-shaped encoder-decoder network with two different types of transformer blocks, named as UTNet, to extract deep features for super-resolution. The basic backbone of UTNet is composed of a locally-enhanced Swin transformer block (LeSwin-T) and a convolutional transformer block (Conv-T), which are responsible for capturing long-range dependencies and local spatial information, respectively. Additionally, we introduce a residual upsampling network (RUpNet) to expand image resolution by leveraging initial residual information from the original low-resolution (LR) images. Extensive experiments show that DTformer achieves superior SR performance. Moreover, radiomics analysis reveals that improving the resolution of ADC images is beneficial for tumor characteristic prediction, such as histological grade and human epidermal growth factor receptor 2 (HER2) status.

Deformation analysis is crucial in many applications, especially in medical image analysis. Analyzing the deformation pattern of anatomical structures provides important information for disease analysis. With degraded images or uncertainties, getting a deterministic solution of the deformation is challenging. In some cases, there may also be multiple solutions with different probabilities. As such, it is important to analyze the probability distribution of deformations, given data information with uncertainty. In this work, we propose to use computational Quasiconformal (QC) Teichmuller theories to parameterize the space of deformations. A distribution over the space of special features, called the QC features, can be computed and applied for deformation analysis. Extensive experiments are carried out on both synthetic data and real medical images, which demonstrate the efficacy of the proposed framework.

There has been a growing interest in shape analysis in recent years. We present a novel shape signature for 2D Jordan domains. The proposed signature is based on Sharon's conformal welding signature [E. Sharon and D. Mumford, Internat. J. Comput. Vis., 70 (2006), pp. 55--75], which is one of the main building blocks of our proposed shape signature. The conformal welding signature is a well-known shape signature used to represent 2D shapes. Nevertheless, it is not invariant under rotation. It is also sensitive to the choice of particular feature points and shape perturbations. Motivated by this, we propose in this paper an invariant shape signature under rigid transformations and scaling. The proposed signature does not require the delineation of feature points and is robust under shape perturbations. More specifically, the proposed signature is a Beltrami coefficient of the harmonic extension of the conformal welding. We show that there is a one-to-one correspondence between a quotient space of Beltrami coefficients and the space of 2D Jordan domains up to a translation, rotation, and scaling. With a suitable normalization, each equivalence class in the quotient space is associated with a unique representative named the Harmonic Beltrami Signature (HBS). As such, each shape is associated with a unique HBS. Conversely, the associated shape of an HBS can be reconstructed based on quasiconformal Teichmüller theories, which are uniquely determined up to a translation, rotation, and scaling. The HBS is thus an effective fingerprint to represent a 2D shape. The robustness of the HBS is studied both theoretically and experimentally. With the HBS, simple metrics, such as $L^2$, can measure geometric dissimilarity between shapes. Experiments have been carried out to classify shapes into different classes using HBS. Results show good classification performance, which demonstrates the efficacy of our proposed shape signature.

With the advancement of computer technology, there is a surge of interest in effective mapping methods for objects in higher-dimensional spaces. To establish a one-to-one correspondence between objects, higher-dimensional quasi-conformal theory can be utilized for ensuring the bijectivity of the mappings. In addition, it is often desirable for the mappings to satisfy certain prescribed geometric constraints and possess low distortion in conformality or volume. In this work, we develop a unifying framework for computing $n$-dimensional quasi-conformal mappings. More specifically, we propose a variational model that integrates quasi-conformal distortion, volumetric distortion, landmark correspondence, intensity mismatch, and volume prior information to handle a large variety of deformation problems. We further prove the existence of a minimizer for the proposed model and devise efficient numerical methods to solve the optimization problem. We demonstrate the effectiveness of the proposed framework using various experiments in two and three dimensions, with applications to medical image registration, adaptive remeshing, and shape modeling.

Conformal and quasi-conformal mappings have widespread applications in imaging science, computer vision, and computer graphics and can be used in surface registration, segmentation, remeshing, and texture map compression. While various conformal and quasi-conformal parameterization methods for simply connected surfaces have been proposed, efficient parameterization algorithms for multiply connected surfaces have been less explored. In this paper, we propose a novel parallelizable algorithm for computing the global conformal and quasi-conformal parameterizations of multiply connected surfaces onto a 2D circular domain using variants of the partial welding method and the Koebe's iteration. The main idea is to first partition a multiply connected surface into several subdomains and compute the free-boundary conformal and quasi-conformal parameterizations of them, respectively, and then apply a variant of the partial welding algorithm to reconstruct the global mapping. We apply the Koebe's iteration, together with the geodesic algorithm, to the boundary points and welding paths before and after the global welding to transform all the boundaries into circles conformally. After getting all the updated boundary conditions, we obtain the global parameterization of the multiply connected surface by solving the Laplace equation for each subdomain. Using this divide-and-conquer approach, the global conformal and quasi-conformal parameterizations of surfaces can be efficiently computed. Experimental results are presented to demonstrate the effectiveness of our proposed algorithm. More broadly, the proposed shift in perspective from solving a global quasi-conformal mapping problem to solving multiple local mapping problems paves a new way for computational quasi-conformal geometry.

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.

Image segmentation aims to partition an image into meaningful regions and extract important objects therein. In real applications, the given images may contain multiple overlapping objects with noisy background, creating great challenges to the segmentation task. In these cases, prior information of the target object is essential for an accurate and meaningful segmentation result. In this paper, we present a new convexity shape prior segmentation framework to guarantee the segmented region to be fully or partially convex according to the user's preference. The basic idea is to incorporate a registration-based segmentation model with a specially designed convexity constraint. The convexity constraint is based on the discrete conformality structures of the image mesh. To solve the segmentation model, we propose an iterative scheme, which smoothly deforms a template object to trace the boundary of the target object. A projection is carried out to enforce the convexity constraint. The target object is then captured by a (fully or partially) convex region. Convexity is the only prior information needed for a (fully) convex shape, whereas the location of partial convexity is needed for a partially convex shape. Experiments have been carried out on both synthetic and real images and the results demonstrate the effectiveness of our proposed framework.

Computing surface folding maps has numerous applications ranging from computer graphics to material design. In this work we propose a novel way of computing surface folding maps via solving a linear PDE. This framework is a generalization of the existing computational quasi-conformal geometry and allows precise control of the geometry of folding. This property comes from a crucial quantity that occurs as the coefficient of the equation, namely, the alternating Beltrami coefficient. This approach also enables us to solve an inverse problem of parametrizing the folded surface given only partial data with known folding topology. Various interesting applications such as fold sculpting on 3 dimensional models, study of Miura-ori patterns, and self-occlusion reasoning are demonstrated to show the effectiveness of our method.

Image retargeting aims to resize an image to one with a prescribed aspect ratio. Simple scaling inevitably introduces unnatural geometric distortions on the important content of the image. In this paper, we propose a simple and yet effective method to resize an image, which preserves the geometry of the important content, using the Beltrami representation. Our algorithm allows users to interactively label content regions as well as line structures. Image resizing can then be achieved by warping the image by an orientation-preserving bijective warping map with controlled distortion. The warping map is represented by its Beltrami representation, which captures the local geometric distortion of the map. By carefully prescribing the values of the Beltrami representation, images with different complexity can be effectively resized. Our method does not require solving any optimization problems and tuning parameters throughout the process. This results in a simple and efficient algorithm to solve the image retargeting problem. Extensive experiments have been carried out, which demonstrate the efficacy of our proposed method.

The point cloud is the most fundamental representation of three-dimensional geometric objects. Analyzing and processing point cloud surfaces is important in computer graphics and computer vision. However, most of the existing algorithms for surface analysis require connectivity information. Therefore, it is desirable to develop a mesh structure on point clouds. This task can be simplified with the aid of a parameterization. In particular, conformal parameterizations are advantageous in preserving the geometric information of the point cloud data. In this paper, we extend a state-of-the-art spherical conformal parameterization algorithm for genus-0 closed meshes to the case of point clouds, using an improved approximation of the Laplace--Beltrami operator on data points. Then, we propose an iterative scheme called the north-south reiteration for achieving a spherical conformal parameterization. A balancing scheme is introduced to enhance the distribution of the spherical parameterization. High-quality triangulations and quadrangulations can then be built on the point clouds with the aid of the parameterizations. Also, the meshes generated are guaranteed to be genus-0 closed meshes. Moreover, using our proposed spherical conformal parameterization, multilevel representations of point clouds can be easily constructed. Experimental results demonstrate the effectiveness of our proposed framework.

In recent decades, the use of three-dimensional point clouds has been widespread in the computer industry. The development of techniques for analyzing point clouds is increasingly important. In particular, mapping of point clouds has been a challenging problem. In this paper, we develop a discrete analogue of the Teichmüller extremal mappings, which guarantees uniform conformality distortions on point cloud surfaces. Based on the discrete analogue, we propose a novel method called TEMPO for computing Teichmüller extremal mappings between feature-endowed point clouds. Using our proposed method, the Teichmüller metric is introduced for evaluating the dissimilarity of point clouds. Consequently, our algorithm enables accurate recognition and classification of point clouds. Experimental results demonstrate the effectiveness of our proposed method.

Surface registration between cortical surfaces is crucial in medical imaging for performing systematic comparisons between brains. Landmark-matching registration that matches anatomical features, called the sulcal landmarks, is often required to obtain a meaningful 1-1 correspondence between brain surfaces. This is commonly done by parameterizing the surface onto a simple parameter domain, such as the unit sphere, in which the sulcal landmarks are consistently aligned. Landmark-matching surface registration can then be obtained from the landmark aligned parameterizations. For genus-0 closed brain surfaces, the optimized spherical harmonic parameterization, which aligns landmarks to consistent locations on the sphere, has been widely used. This approach is limited by the loss of bijectivity under large deformations and the slow computation. In this paper, we propose FLASH, a fast algorithm to compute the optimized spherical harmonic parameterization with consistent landmark alignment. This is achieved by formulating the optimization problem to $\overline{\mathbb{C}}$ and thereby linearizing the problem. Errors introduced near the pole are corrected using quasi-conformal theories. Also, by adjusting the Beltrami differential of the mapping, a diffeomorphic (1-1, onto) spherical parameterization can be effectively obtained. The proposed algorithm has been tested on 38 human brain surfaces. Experimental results demonstrate that the computation of the landmark aligned spherical harmonic parameterization is significantly accelerated using the proposed 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.

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.

We present a new approach to obtain diffeomorphic registrations with large deformations using landmark and intensity information via quasi-conformal maps. The basic idea is to minimize an energy functional involving a Beltrami coefficient term, which measures the distortion of the quasi-conformal map. The Beltrami coefficient effectively controls the bijectivity and smoothness of the registration. In this paper, we first propose the quasi-conformal landmark registration (QCLR) algorithm to obtain diffeomorphic (1-1 and onto) registrations between images or surfaces. Using QCLR, landmark-aligned diffeomorphisms between images or surfaces can be obtained, even with a large geometric difference or a large number of landmark constraints. This algorithm is then extended to the quasi-conformal hybrid registration (QCHR) algorithm, which combines landmark and intensity (such as image intensity or surface curvature) information to achieve a more accurate registration result. Experiments have been carried out on both synthetic and real data. Results demonstrate the stability and efficacy of the proposed algorithm to obtain diffeomorphic registrations between images or surfaces.

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.

This paper presents a method of obtaining geometric registrations between high-genus ($g\geq 1$) surfaces. Surface registration between simple surfaces, such as simply connected open surfaces, has been well studied. However, very few works have been carried out for the registration of high-genus surfaces. The high-genus topology of the surface poses a great challenge for surface registration. A possible approach is to partition surfaces into simply connected patches and registration can be done in a patch-by-patch manner. Consistent cuts are required, which are usually difficult to obtain and prone to error. In this work, we propose an effective way to obtain geometric registration between high-genus surfaces without introducing consistent cuts. The key idea is to conformally parameterize the surface into its universal covering space, which is either the Euclidean plane or the hyperbolic disk embedded in $\mathbb{R}^2$. Registration can then be done on the universal covering space by iteratively minimizing a shape mismatching energy measuring the geometric dissimilarity between the two surfaces. The Beltrami coefficient of the mapping is considered and adjusted in order to control the bijectivity of the mappings in each iteration. Our proposed algorithm effectively computes a smooth registration between high-genus surfaces that matches geometric information as much as possible. The algorithm can also be applied to find a smooth registration minimizing any general energy functionals. Numerical experiments on high-genus surface data show that our proposed method is effective for registering high-genus surfaces with geometric matching. We also applied the method to register anatomical structures for medical imaging, which demonstrates the usefulness of the proposed algorithm.

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.

This paper proposes a novel approach for extracting two intrinsic feature curves on hippocampal (HC) surfaces. The hippocampus is a key target of study in medical imaging, as it degenerates in conditions such as epilepsy and Alzheimer's disease (AD), but its structure is complex. To facilitate HC morphometry, we generate two intrinsic feature curves that describe their global geometries. For example, the separation of them captures thickness changes in HC surfaces, which can be used to effectively measure HC atrophy found in patients with AD. They also separate HC surfaces into upper and lower surface patches where intrinsic shape analysis using conformal modules can be carried out. Based on these curves, we further propose a parameterization of HC surfaces called the eigen-harmonic parameterization (EHP). EHP maps each HC surface onto a parameter domain and imposes longitudinal and azimuthal coordinates on each surface, which follow the gradient and level sets of its first nontrivial Laplace--Beltrami eigenfunction, respectively. Each tubular domain is constructed according to the geometry of an individual HC surface. This gives a parameter domain with much less geometric distortion compared to spherical parameterization. With EHP, all HC surfaces are automatically registered with intrinsic feature curves preserved and geometric distortions minimized. This allows shape analysis on any number of HC surfaces to be performed consistently. We studied geometric changes over time in 138 HC surfaces of patients with AD and normal subjects scanned at two different times. We successfully located areas with significantly different shape changes over time between the two groups.

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.

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.

Surface registration, which transforms different sets of surface data into one common reference space, is an important process which allows us to compare or integrate the surface data effectively. If a nonrigid transformation is required, surface registration is commonly done by parameterizing the surfaces onto a simple parameter domain, such as the unit square or sphere. In this work, we are interested in looking for meaningful registrations between surfaces through parameterizations, using prior features in the form of landmark curves on the surfaces. In particular, we generate optimized conformal parameterizations which match landmark curves exactly with shape-based correspondences between them. We propose a variational method to minimize a compound energy functional that measures the harmonic energy of the parameterization maps and the shape dissimilarity between mapped points on the landmark curves. The novelty is that the computed maps are guaranteed to align the landmark features consistently and give a shape-based diffeomorphism between the landmark curves. We achieve this by intrinsically modeling our search space of maps as flows of smooth vector fields that do not flow across the landmark curves. By using the local surface geometry on the curves to define a shape measure, we compute registrations that ensure consistent correspondences between anatomical features. We test our algorithm on synthetic surface data. An application of our model to medical imaging research is shown, using experiments on brain cortical surfaces, with anatomical (sulcal) landmarks delineated, which show that our computed maps give a shape-based alignment of the sulcal curves without significantly impairing conformality. This ensures correct averaging and comparison of data across subjects.

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.

Anatomical features on cortical surfaces are usually represented by landmark curves, called sulci/gyri curves. These landmark curves are important information for neuroscientists to study brain diseases and to match different cortical surfaces. Manual labelling of these landmark curves is time-consuming, especially when there is a large set of data. In this paper, we proposed to trace the landmark curves on cortical surfaces automatically based on the principal directions. Suppose we are given the global conformal parametrization of a cortical surface, By fixing two endpoints, the anchor points, we propose to trace the landmark curves iteratively on the spherical/rectangular parameter domain along the principal direction. Consequently, the landmark curves can be mapped onto the cortical surface. To speed up the iterative scheme, a good initial guess of the landmark curve is necessary. We proposed a method to get a good initialization by extracting the high curvature region on the cortical surface using the Chan-Vese segmentation. This involves solving a PDE on the manifold using our global conformal parametrization technique. Experimental results show that the landmark curves detected by our algorithm closely resemble to those manually labelled curves. As an application, we used these automatically labelled landmark curves to build average cortical surfaces with an optimized brain conformal mapping method. Experimental results show our method can help automatically matching brain cortical surfaces.