3D shapes from scanning, reconstruction, or AI-generated content often lack simple quad mesh layouts—critical for efficient editing and modeling. Existing quad-remeshing techniques typically produce complex layouts with irregular loops, leading to tedious manual cleanup and extensive algorithm tuning. We introduce SQUADGEN, a diffusion-based generative framework that leverages Chart Distance Fields (CDF) to synthesize simple quad layouts on 3D shapes. Our approach addresses two key challenges: (1) the discrete nature of mesh connectivity, which hinders learning, and (2) the scarcity of large-scale datasets with simple quad meshes. To overcome the first, we propose CDF, a continuous surface-based representation enabling effective learning and synthesis of quad layouts. To address the second, we define loop-aware simplicity metrics and construct a large-scale dataset of high-quality quad layouts recovered from public 3D repositories through a robust quad-recovery pipeline. Extensive evaluations across diverse 3D inputs show that SQUADGEN consistently outperforms existing methods, producing robust, artist-friendly simple quad layouts.

Synthesizing realistic 3D indoor scenes remains challenging due to data scarcity and the difficulty of simultaneously enforcing global architectural constraints and local semantic consistency. Existing approaches often overlook structural boundaries or rely on fully connected relation graphs that introduce redundant generation errors. Inspired by human design cognition, we present CasLayout, a cascaded diffusion framework that decomposes the joint scene generation task into four conditional sub-stages with explicit physical and semantic roles: (1) predicting furniture quantity and categories, (2) refining object sizes and feature embeddings, (3) modeling spatial relationships in a latent space, and (4) generating Oriented Bounding Boxes (OBBs). This decoupled architecture reduces data requirements and enables flexible integration of Large Language Models (LLMs) and Vision Language Models (VLMs) for zero-shot tasks such as image-to-scene generation. To maintain physical validity within complex floor plans, we explicitly model building elements ( e.g. , walls, doors, and windows) as conditional constraints. Furthermore, to address the high entropy of dense relation graphs, we introduce a sparse relation graph formulation aligned with human spatial descriptions. By encoding these sparse graphs into a compact latent space using a bidirectional Variational Autoencoder (VAE), the proposed framework provides enhanced relational controllability, allowing generated layouts to better respect functional organization. Experiments demonstrate that CasLayout achieves state-of-the-art performance in fidelity and diversity while enabling improved controllability in practical applications.

Lattice metamaterials enable lightweight, multifunctional structures, yet homogenization-based evaluation of their effective properties remains computationally expensive. Neural surrogates offer speed but often lack the accuracy and stability required for engineering-grade simulations. We introduce GMT, a G eometric M ultigrid T ransformer - a neural solver with high numerical fidelity for fast and reliable lattice homogenization. GMT achieves architectural alignment with Geometric Multigrid (GMG) by restructuring Point Transformer V3 to operate across sparse GMG hierarchies, capturing long-range dependencies and cross-level interactions essential for multigrid convergence. To enforce physical consistency, GMT incorporates physics-aware positional encoding for strict enforcement of periodicity and predicts both the finest-level solution and multi-level residual corrections. These predictions deliver a spectrally-aligned initialization , enabling end-to-end training under physics-informed and solver-aware losses and requiring only a single GMG V-cycle refinement to reach convergence. This fusion of neural prediction and numerical rigor achieves relative residual errors of 10 -5 with a 160× speedup over state-of-the-art GPU-based solvers at equivalent accuracy - particularly at high resolutions (e.g. , 512 3 ), where traditional methods become most costly. We validate GMT across mechanical and thermal domains, demonstrate robust generalization to unseen geometries and non-periodic settings, and showcase scalability to high resolutions - enabling real-time design iteration, multi-scale simulations, high-throughput material discovery, and inverse design.

In this paper, we study an under-explored but important factor of diffusion generative models, i.e., the combinatorial complexity. Data samples are generally high-dimensional, and for various structured generation tasks, additional attributes are combined to associate with data samples. We show that the space spanned by the combination of dimensions and attributes can be insufficiently covered by existing training schemes of diffusion generative models, potentially limiting test time performance. We present a simple fix to this problem by constructing stochastic processes that fully exploit the combinatorial structures, hence the name ComboStoc. Using this simple strategy, we show that network training is significantly accelerated across diverse data modalities, including images and 3D structured shapes. Moreover, ComboStoc enables a new way of test time generation which uses asynchronous time steps for different dimensions and attributes, thus allowing for varying degrees of control over them. Our code is available at: https://github.com/Xrvitd/ComboStoc.

We introduce SAMPro3D for zero-shot instance segmentation of 3D scenes. Given the 3D point cloud and multiple posed RGB-D frames of 3D scenes, our approach segments 3D instances by applying the pretrained Segment Anything Model (SAM) to 2D frames. Our key idea in-volves locating SAM prompts in 3D to align their projected pixel prompts across frames, ensuring the view consistency of SAM-predicted masks. Moreover, we suggest selecting prompts from the initial set guided by the information of SAM-predicted masks across all views, which enhances the overall performance. We further propose to consolidate different prompts if they are segmenting different surface parts of the same 3D instance, bringing a more comprehensive segmentation. Notably, our method does not require any additional training. Extensive experiments on diverse benchmarks show that our method achieves comparable or better performance compared to previous zero-shot or fully supervised approaches, and in many cases surpasses human annotations. Furthermore, since our fine-grained predictions often lack annotations in available datasets, we present ScanNet200-Fine50 test data which provides fine-grained annotations on 50 scenes from ScanNet200 dataset.

Data diversity and abundance are essential for improving the performance and generalization of models in natural language processing and 2D vision. However, the 3D vision domain suffers from a lack of 3D data, and simply combining multiple 3D datasets for pretraining a 3D backbone does not yield significant improvement, due to the domain discrepancies among different 3D datasets that impede effective feature learning. In this work, we identify the main sources of the domain discrepancies between 3D indoor scene datasets, and propose Swin3D++, an enhanced architecture based on Swin3D for efficient pretraining on multi-source 3D point clouds. Swin3D++ introduces domain-specific mechanisms to Swin3D's modules to address domain discrepancies and enhance the network capability on multi-source pretraining. Moreover, we devise a simple sourceaugmentation strategy to increase the pretraining data scale and facilitate supervised pretraining. We validate the effectiveness of our design, and demonstrate that Swin3D++ surpasses the state-of-the-art 3D pretraining methods on typical indoor scene understanding tasks.

The use of pretrained backbones with fine-tuning has shown success for 2D vision and natural language processing tasks, with advantages over task-specific networks. In this paper, we introduce a pretrained 3D backbone, called Swin3d, for 3D indoor scene understanding. We designed a 3D Swin Transformer as our backbone network, which enables efficient self-attention on sparse voxels with linear memory complexity, making the backbone scalable to large models and datasets. We also introduce a generalized contextual relative positional embedding scheme to capture various irregularities of point signals for improved network performance. We pretrained a large Swin3d model on a synthetic Structured3D dataset, which is an order of magnitude larger than the ScanNet dataset. Our model pretrained on the synthetic dataset not only generalizes well to downstream segmentation and detection on real 3D point datasets but also outperforms state-of-the-art methods on downstream tasks with +2.3 mIoU and +2.2 mIoU on S3DIS Area5 and 6-fold semantic segmentation, respectively, +1.8 mIoU on ScanNet segmentation (val), +1.9 [email protected] on ScanNet detection, and +8.1 [email protected] on S3DIS detection. A series of extensive ablation studies further validated the scalability, generality, and superior performance enabled by our approach.

Man-made 3D shapes are naturally organized in parts and hierarchies; such structures provide important constraints for shape reconstruction and generation. Modeling shape structures is difficult, because there can be multiple hierarchies for a given shape, causing ambiguity, and across different categories, the shape structures are correlated with semantics, limiting generalization. We present StructRe , a structure rewriting system, as a novel approach to structured shape modeling. Given a 3D object represented by points and components, StructRe can rewrite it upward into more concise structures, or downward into more detailed structures; by iterating the rewriting process, hierarchies are obtained. Such a localized rewriting process enables probabilistic modeling of ambiguous structures and robust generalization across object categories. We train StructRe on PartNet data and show its generalization to cross-category and multiple object hierarchies, and test its extension to ShapeNet. We also demonstrate the benefits of probabilistic and generalizable structure modeling for shape reconstruction, generation and editing tasks.

Masked autoencoders (MAE) have recently been introduced to 3D self-supervised pretraining for point clouds due to their great success in NLP and computer vision. Unlike MAEs used in the image domain, where the pretext task is to restore features at the masked pixels, such as colors, the existing 3D MAE works reconstruct the missing geometry only, i.e, the location of the masked points. In contrast to previous studies, we advocate that point location recovery is inessential and restoring intrinsic point features is much superior. To this end, we propose to ignore point position reconstruction and recover high-order features at masked points including surface normals and surface variations, through a novel attention-based decoder which is independent of the encoder design. We validate the effectiveness of our pretext task and decoder design using different encoder structures for 3D training and demonstrate the advantages of our pretrained networks on various point cloud analysis tasks.

Recognizing 3D part instances from a 3D point cloud is crucial for 3D structure and scene understanding. Several learning-based approaches use semantic segmentation and instance center prediction as training tasks and fail to further exploit the inherent relationship between shape semantics and part instances. In this paper, we present a new method for 3D part instance segmentation. Our method exploits semantic segmentation to fuse nonlocal instance features, such as center prediction, and further enhances the fusion scheme in a multi- and cross-level way. We also propose a semantic region center prediction task to train and leverage the prediction results to improve the clustering of instance points. Our method outperforms existing methods with a large-margin improvement in the PartNet benchmark. We also demonstrate that our feature fusion scheme can be applied to other existing methods to improve their performance in indoor scene instance segmentation tasks.

The lack of fine-grained 3D shape segmentation data is the main obstacle to developing learning-based 3D segmentation techniques. We propose an effective semi-supervised method for learning 3D segmentations from a few labeled 3D shapes and a large amount of unlabeled 3D data. For the unlabeled data, we present a novel multilevel consistency loss to enforce consistency of network predictions between perturbed copies of a 3D shape at multiple levels: point level, part level, and hierarchical level. For the labeled data, we develop a simple yet effective part substitution scheme to augment the labeled 3D shapes with more structural variations to enhance training. Our method has been extensively validated on the task of 3D object semantic segmentation on PartNet and ShapeNetPart, and indoor scene semantic segmentation on ScanNet. It exhibits superior performance to existing semi-supervised and unsupervised pre-training 3D approaches.

Although the recent rapid evolution of 3D generative neural networks greatly improves 3D shape generation, it is still not convenient for ordinary users to create 3D shapes and control the local geometry of generated shapes. To address these challenges, we propose a diffusion-based 3D generation framework --- locally attentional SDF diffusion , to model plausible 3D shapes, via 2D sketch image input. Our method is built on a two-stage diffusion model. The first stage, named occupancy-diffusion , aims to generate a low-resolution occupancy field to approximate the shape shell. The second stage, named SDF-diffusion , synthesizes a high-resolution signed distance field within the occupied voxels determined by the first stage to extract fine geometry. Our model is empowered by a novel view-aware local attention mechanism for image-conditioned shape generation, which takes advantage of 2D image patch features to guide 3D voxel feature learning, greatly improving local controllability and model generalizability. Through extensive experiments in sketch-conditioned and category-conditioned 3D shape generation tasks, we validate and demonstrate the ability of our method to provide plausible and diverse 3D shapes, as well as its superior controllability and generalizability over existing work.

The freeform architectural modeling process often involves two important stages: concept design and digital modeling. In the first stage, architects usually sketch the overall 3D shape and the panel layout on a physical or digital paper briefly. In the second stage, a digital 3D model is created using the sketch as a reference. The digital model needs to incorporate geometric requirements for its components, such as the planarity of panels due to consideration of construction costs, which can make the modeling process more challenging. In this work, we present a novel sketch-based system to bridge the concept design and digital modeling of freeform roof-like shapes represented as planar quadrilateral (PQ) meshes. Our system allows the user to sketch the surface boundary and contour lines under axonometric projection and supports the sketching of occluded regions. In addition, the user can sketch feature lines to provide directional guidance to the PQ mesh layout. Given the 2D sketch input, we propose a deep neural network to infer in real-time the underlying surface shape along with a dense conjugate direction field, both of which are used to extract the final PQ mesh. To train and validate our network, we generate a large synthetic dataset that mimics architect sketching of freeform quadrilateral patches. The effectiveness and usability of our system are demonstrated with quantitative and qualitative evaluation as well as user studies.

Abstract We present a StyleGAN2‐based deep learning approach for 3D shape generation, called SDF‐StyleGAN, with the aim of reducing visual and geometric dissimilarity between generated shapes and a shape collection. We extend StyleGAN2 to 3D generation and utilize the implicit signed distance function (SDF) as the 3D shape representation, and introduce two novel global and local shape discriminators that distinguish real and fake SDF values and gradients to significantly improve shape geometry and visual quality. We further complement the evaluation metrics of 3D generative models with the shading‐image‐based Fréchet inception distance (FID) scores to better assess visual quality and shape distribution of the generated shapes. Experiments on shape generation demonstrate the superior performance of SDF‐StyleGAN over the state‐of‐the‐art. We further demonstrate the efficacy of SDF‐StyleGAN in various tasks based on GAN inversion, including shape reconstruction, shape completion from partial point clouds, single‐view image‐based shape generation, and shape style editing. Extensive ablation studies justify the efficacy of our framework design. Our code and trained models are available at https://github.com/Zhengxinyang/SDF‐StyleGAN .

We present a simple yet effective method for constructing 3D self-supporting surfaces with planar quadrilateral (PQ) elements. Starting with a triangular discretization of a self-supporting surface, we first compute the principal curvatures and directions of each triangular face using a new discrete differential geometry approach, yielding more accurate results than existing methods. Then, we smooth the principal direction field to reduce the number of singularities. Next, we partition all faces into two groups in terms of principal curvature difference. For each face with small curvature difference, we compute a stretch matrix that turns the principal directions into a pair of conjugate directions. For the remaining triangular faces, we simply keep their smoothed principal directions. Finally, applying a mixed-integer programming solver to the mixed principal and conjugate direction field, we obtain a planar quadrilateral mesh. Experimental results show that our method is computationally efficient and can yield high-quality PQ meshes that well approximate the geometry of the input surfaces and maintain their self-supporting properties.

We present a novel implicit representation ---neural halfspace representation(NH-Rep), to convert manifold B-Rep solids to implicit representations. NH-Rep is a Boolean tree built on a set of implicit functions represented by the neural network, and the composite Boolean function is capable of representing solid geometry while preserving sharp features. We propose an efficient algorithm to extract the Boolean tree from a manifold B-Rep solid and devise a neural network-based optimization approach to compute the implicit functions. We demonstrate the high quality offered by our conversion algorithm on ten thousand manifold B-Rep CAD models that contain various curved patches including NURBS, and the superiority of our learning approach over other representative implicit conversion algorithms in terms of surface reconstruction, sharp feature preservation, signed distance field approximation, and robustness to various surface geometry, as well as a set of applications supported by NH-Rep.

We view the reconstruction of CAD models in the boundary representation (B-Rep) as the detection of geometric primitives of different orders, i.e. , vertices, edges and surface patches, and the correspondence of primitives, which are holistically modeled as a chain complex, and show that by modeling such comprehensive structures more complete and regularized reconstructions can be achieved. We solve the complex generation problem in two steps. First, we propose a novel neural framework that consists of a sparse CNN encoder for input point cloud processing and a tri-path transformer decoder for generating geometric primitives and their mutual relationships with estimated probabilities. Second, given the probabilistic structure predicted by the neural network, we recover a definite B-Rep chain complex by solving a global optimization maximizing the likelihood under structural validness constraints and applying geometric refinements. Extensive tests on large scale CAD datasets demonstrate that the modeling of B-Rep chain complex structure enables more accurate detection for learning and more constrained reconstruction for optimization, leading to structurally more faithful and complete CAD B-Rep models than previous results.

We present an adaptive deep representation of volumetric fields of 3D shapes and an efficient approach to learn this deep representation for high-quality 3D shape reconstruction and auto-encoding. Our method encodes the volumetric field of a 3D shape with an adaptive feature volume organized by an octree and applies a compact multilayer perceptron network for mapping the features to the field value at each 3D position. An encoder-decoder network is designed to learn the adaptive feature volume based on the graph convolutions over the dual graph of octree nodes. The core of our network is a new graph convolution operator defined over a regular grid of features fused from irregular neighboring octree nodes at different levels, which not only reduces the computational and memory cost of the convolutions over irregular neighboring octree nodes, but also improves the performance of feature learning. Our method effectively encodes shape details, enables fast 3D shape reconstruction, and exhibits good generality for modeling 3D shapes out of training categories. We evaluate our method on a set of reconstruction tasks of 3D shapes and scenes and validate its superiority over other existing approaches. Our code, data, and trained models are available at https://wang-ps.github.io/dualocnn.

We propose an unsupervised method for learning a generic and efficient shape encoding network for different shape analysis tasks. Our key idea is to jointly encode and learn shape and point features from unlabeled 3D point clouds. For this purpose, we adapt HRNet to octree-based convolutional neural networks for jointly encoding shape and point features with fused multiresolution subnetworks and design a simple-yet-efficient Multiresolution Instance Discrimination (MID) loss for jointly learning the shape and point features. Our network takes a 3D point cloud as input and output both shape and point features. After training, Our network is concatenated with simple task-specific back-ends and fine-tuned for different shape analysis tasks. We evaluate the efficacy and generality of our method with a set of shape analysis tasks, including shape classification, semantic shape segmentation, as well as shape registration tasks. With simple back-ends, our network demonstrates the best performance among all unsupervised methods and achieves competitive performance to supervised methods. For fine-grained shape segmentation on the PartNet dataset, our method even surpasses existing supervised methods by a large margin.

Point set is a flexible and lightweight representation widely used for 3D deep learning. However, their discrete nature prevents them from representing continuous and fine geometry, posing a major issue for learning-based shape generation. In this work, we turn the discrete point sets into smooth surfaces by introducing the well-known implicit moving least-squares (IMLS) surface formulation, which naturally defines locally implicit functions on point sets. We incorporate IMLS surface generation into deep neural networks for inheriting both the flexibility of point sets and the high quality of implicit surfaces. Our IMLSNet predicts an octree structure as a scaffold for generating MLS points where needed and characterizes shape geometry with learned local priors. Furthermore, our implicit function evaluation is independent of the neural network once the MLS points are predicted, thus enabling fast runtime evaluation. Our experiments on 3D object reconstruction demonstrate that IMLSNets outperform state-of-the-art learning-based methods in terms of reconstruction quality and computational efficiency. Extensive ablation tests also validate our network design and loss functions.

Sparse voxel-based 3D convolutional neural networks (CNNs) are widely used for various 3D vision tasks. Sparse voxel-based 3D CNNs create sparse non-empty voxels from the 3D input and perform 3D convolution operations on them only. We propose a simple yet effective padding scheme — interpolation-aware padding to pad a few empty voxels adjacent to the non-empty voxels and involve them in the 3D CNN computation so that all neighboring voxels exist when computing point-wise features via the trilinear interpolation. For fine-grained 3D vision tasks where point-wise features are essential, like semantic segmentation and 3D detection, our network achieves higher prediction accuracy than the existing networks using the nearest neighbor interpolation or the normalized trilinear interpolation with the zero-padding or the octree-padding scheme. Through extensive compar-isons on various 3D segmentation and detection tasks, we demonstrate the superiority of 3D sparse CNNs with our padding scheme in conjunction with feature interpolation.

Multilayer perceptrons (MLPs) have been successfully used to represent 3D shapes implicitly and compactly, by mapping 3D coordinates to the corresponding signed distance values or occupancy values. In this paper, we propose a novel positional encoding scheme, called Spline Positional Encoding, to map the input coordinates to a high dimensional space before passing them to MLPs, which help recover 3D signed distance fields with fine-scale geometric details from unorganized 3D point clouds. We verified the superiority of our approach over other positional encoding schemes on tasks of 3D shape reconstruction and 3D shape space learning from input point clouds. The efficacy of our approach extended to image reconstruction is also demonstrated and evaluated.

We present a novel and efficient approach to estimate 6D object poses of known objects in complex scenes represented by point clouds. Our approach is based on the well-known point pair feature (PPF) matching, which utilizes self-similar point pairs to compute potential matches and thereby cast votes for the object pose by a voting scheme. The main contribution of this paper is to present an improved PPF-based recognition framework, especially a new center voting strategy based on the relative geometric relationship between the object center and point pair features. Using this geometric relationship, we first generate votes to object centers resulting in vote clusters near real object centers. Then we group and aggregate these votes to generate a set of pose hypotheses. Finally, a pose verification operator is performed to filter out false positives and predict appropriate 6D poses of the target object. Our approach is also suitable to solve the multi-instance and multi-object detection tasks. Extensive experiments on a variety of challenging benchmark datasets demonstrate that the proposed algorithm is discriminative and robust towards similar-looking distractors, sensor noise, and geometrically simple shapes. The advantage of our work is further verified by comparing to the state-of-the-art approaches.

One major cause of performance degradation in predictive models is that the test samples are not well covered by the training data. Such not well-represented samples are called OoD samples. In this article, we propose OoDAnalyzer, a visual analysis approach for interactively identifying OoD samples and explaining them in context. Our approach integrates an ensemble OoD detection method and a grid-based visualization. The detection method is improved from deep ensembles by combining more features with algorithms in the same family. To better analyze and understand the OoD samples in context, we have developed a novelkNN-based grid layout algorithm motivated by Hall's theorem. The algorithm approximates the optimal layout and has O(kN2)O(kN2) time complexity, faster than the grid layout algorithm with overall best performance but O(N3)O(N3) time complexity. Quantitative evaluation and case studies were performed on several datasets to demonstrate the effectiveness and usefulness of OoDAnalyzer.

Surface meshes are widely used shape representations and capture finer geometry data than point clouds or volumetric grids, but are challenging to apply CNNs directly due to their non-Euclidean structure. We use parallel frames on surface to define PFCNNs that enable effective feature learning on surface meshes by mimicking standard convolutions faithfully. In particular, the convolution of PFCNN not only maps local surface patches onto flat tangent planes, but also aligns the tangent planes such that they locally form a flat Euclidean structure, thus enabling recovery of standard convolutions. The alignment is achieved by the tool of locally flat connections borrowed from discrete differential geometry, which can be efficiently encoded and computed by parallel frame fields. In addition, the lack of canonical axis on surface is handled by sampling with the frame directions. Experiments show that for tasks including classification, segmentation and registration on deformable geometric domains, as well as semantic scene segmentation on rigid domains, PFCNNs achieve robust and superior performances without using sophisticated input features than state-of-the-art surface based CNNs.

Abstract We present a novel data‐driven skinning model—rigidity‐aware skinning (RAS) model, for simulating both active and passive 3D facial animation of different identities in real time. Our model builds upon a linear blend skinning (LBS) scheme, where the bone set and skinning weights are shared for diverse identities and learned from the data via a sparse and localized skinning decomposition algorithm. Our model characterizes the animated face into the active expression and the passive deformation: The former is represented by an LBS‐based multi‐linear model learned from the FaceWareHouse data set, and the latter is represented by a spatially varying as‐rigid‐as‐possible deformation applied to the LBS‐based multi‐linear model, whose rigidity parameters are learned from the data by a novel rigidity estimation algorithm. Our RAS model is not only generic and expressive for faithfully modelling medium‐scale facial deformation, but also compact and lightweight for generating vivid facial animation in real time. We validate the efficiency and effectiveness of our RAS model for real‐time 3D facial animation and expression editing.

Volumetric PolyCube-Map-based methods offer automatic ways to construct all-hexahedral meshes for closed 3D polyhedral domains, but their meshing quality is limited by the lack of interior singularities and feature alignment. In the presented work, we propose cut-enhanced PolyCube-Maps , to introduce essential interior singularities and preserve most input features. Our main idea is simple and intuitive: by inserting proper parameterization seams into the initial PolyCube-Map via novel PolyCube cutting operations, the mapping distortion can be reduced significantly. The cut-enhanced PolyCube-Map computation includes feature-aware PolyCube-Map construction and cut-enhanced PolyCube deformation. The former aims to preserve input feature edges during the initial PolyCube-Map construction. The latter introduces seams into the volumetric PolyCube shape by cutting it through selective PolyCube edges and deforms the modified PolyCube under the seamless constraints to compute a low-distortion PolyCube-Map. The hexahedral mesh induced by the final PolyCube-Map can be further enhanced by our mesh improvement algorithm. We validate the efficacy of our method on a collection of more than one hundred CAD models and demonstrate its advantages over other automatic all-hex meshing methods and padding strategies. The limitations of cut-enhanced PolyCube-Maps are also discussed thoroughly.

Abstract Freeform surfaces whose principal curvature line network is regularly distributed, are essential to many real applications like CAD modeling, architecture design, and industrial fabrication. However, most designed surfaces do not hold this nice property because it is hard to enforce such constraints in the design process. In this paper, we present a novel method for surface fairing which takes a regular distribution of the principal curvature line network on a surface as an objective. Our method first removes the high‐frequency signals from the curvature tensor field of an input freeform surface by a novel rolling guidance tensor filter, which results in a more regular and smooth curvature tensor field, then deforms the input surface to match the smoothed field as much as possible. As an application, we solve the problem of approximating freeform surfaces with regular principal curvature line networks, discretized by quadrilateral meshes. By introducing the circular or conical conditions on the quadrilateral mesh to guarantee the existence of discrete principal curvature line networks, and minimizing the approximate error to the original surface and improving the fairness of the quad mesh, we obtain a regular discrete principal curvature line network that approximates the original surface. We evaluate the efficacy of our method on various freeform surfaces and demonstrate the superiority of the rolling guidance tensor filter over other tensor smoothing techniques. We also utilize our method to generate high‐quality circular/conical meshes for architecture design and cyclide spline surfaces for CAD modeling.

3D mesh models created by human users and shared through online platforms and datasets flourish recently. While the creators generally have spent large efforts in modeling the visually appealing shapes with both large scale structures and intricate details, a majority of the meshes are unfortunately flawed in terms of having duplicate faces, mis-oriented regions, disconnected patches, etc., due to multiple factors involving both human errors and software inconsistencies. All these artifacts have severely limited the possible low-level and high-level processing tasks that can be applied to the rich datasets. In this work, we present a novel approach to fix these man-made meshes such that the outputs are guaranteed to be oriented manifold meshes that preserve the original structures, big and small, as much as possible. Our key observation is that the models all visually look meaningful, which leads to our strategy of repairing the flaws while always preserving the visual quality. We apply local refinements and removals only where necessary to achieve minimal intrusion of the original meshes, and global adjustments through robust optimization to ensure the outputs are valid manifold meshes with optimal connections. We test the approach on large-scale 3D datasets, and obtain quality meshes that are more readily usable for further geometry processing tasks.

Sketching provides an intuitive user interface for communicating free form shapes. While human observers can easily envision the shapes they intend to communicate, replicating this process algorithmically requires resolving numerous ambiguities. Existing sketch-based modeling methods resolve these ambiguities by either relying on expensive user annotations or by restricting the modeled shapes to specific narrow categories. We present an approach for modeling generic freeform 3D surfaces from sparse, expressive 2D sketches that overcomes both limitations by incorporating convolution neural networks (CNN) into the sketch processing workflow. Given a 2D sketch of a 3D surface, we use CNNs to infer the depth and normal maps representing the surface. To combat ambiguity we introduce an intermediate CNN layer that models the dense curvature direction, or flow, field of the surface, and produce an additional output confidence map along with depth and normal. The flow field guides our subsequent surface reconstruction for improved regularity; the confidence map trained unsupervised measures ambiguity and provides a robust estimator for data fitting. To reduce ambiguities in input sketches users can refine their input by providing optional depth values at sparse points and curvature hints for strokes. Our CNN is trained on a large dataset generated by rendering sketches of various 3D shapes using non-photo-realistic line rendering (NPR) method that mimics human sketching of free-form shapes. We use the CNN model to process both single- and multi-view sketches. Using our multi-view framework users progressively complete the shape by sketching in different views, generating complete closed shapes. For each new view, the modeling is assisted by partial sketches and depth cues provided by surfaces generated in earlier views. The partial surfaces are fused into a complete shape using predicted confidence levels as weights. We validate our approach, compare it with previous methods and alternative structures, and evaluate its performance with various modeling tasks. The results demonstrate our method is a new approach for efficiently modeling freeform shapes with succinct but expressive 2D sketches.

We present an Adaptive Octree-based Convolutional Neural Network (Adaptive O-CNN) for efficient 3D shape encoding and decoding. Different from volumetric-based or octree-based CNN methods that represent a 3D shape with voxels in the same resolution, our method represents a 3D shape adaptively with octants at different levels and models the 3D shape within each octant with a planar patch. Based on this adaptive patch-based representation, we propose an Adaptive O-CNN encoder and decoder for encoding and decoding 3D shapes. The Adaptive O-CNN encoder takes the planar patch normal and displacement as input and performs 3D convolutions only at the octants at each level, while the Adaptive O-CNN decoder infers the shape occupancy and subdivision status of octants at each level and estimates the best plane normal and displacement for each leaf octant. As a general framework for 3D shape analysis and generation, the Adaptive O-CNN not only reduces the memory and computational cost, but also offers better shape generation capability than the existing 3D-CNN approaches. We validate Adaptive O-CNN in terms of efficiency and effectiveness on different shape analysis and generation tasks, including shape classification, 3D autoencoding, shape prediction from a single image, and shape completion for noisy and incomplete point clouds.

Sketch-based modeling provides a powerful paradigm for geometric modeling. Recent research had shown, sketch based modeling methods are most effective when targeting a specific family of surfaces. A large and growing arsenal of sketching tools is available for different types of geometries and different target user populations. Our work augments this arsenal with a new and powerful tool for modeling complex freeform shapes by sketching sparse 2D strokes; our method complements existing approaches in enabling the generation of surfaces with complex curvature patterns that are challenging to produce with existing methods. To model a desired surface patch with our technique, the user sketches the patch boundary as well as a small number of strokes representing the major bending directions of the shape. Our method uses this input to generate a curvature field that conforms to the user strokes and then uses this field to derive a freeform surface with the desired curvature pattern. To infer the surface from the strokes we first disambiguate the convex versus concave bending directions indicated by the strokes and estimate the surface bending magnitude along the strokes. We subsequently construct a curvature field based on these estimates, using a non-orthogonal 4-direction field coupled with a scalar magnitude field, and finally construct a surface whose curvature pattern reflects this field through an iterative sequence of simple linear optimizations. Our framework is well suited for single-view modeling, but also supports multi-view interaction, necessary to model complex shapes portions of which can be occluded in many views. It effectively combines multi-view inputs to obtain a coherent 3D shape. It runs at interactive speed allowing for immediate user feedback. We demonstrate the effectiveness of the proposed method through a large collection of complex examples created by both artists and amateurs. Our framework provides a useful complement to the existing sketch-based modeling methods.

We present an end-to-end solution for design and fabrication of soft pneumatic objects with desired deformations. Given a 3D object with its rest and deformed target shapes, our method automatically optimizes the chamber structure and material distribution inside the object volume so that the fabricated object can deform to all the target deformed poses with controlled air injection. To this end, our method models the object volume with a set of chambers separated by material shells. Each chamber has individual channels connected to the object surface and thus can be separately controlled with a pneumatic system, while the shell is comprised of base material with an embedded frame structure. A two-step algorithm is developed to compute the geometric layout of the chambers and frame structure as well as the material properties of the frame structure from the input. The design results can be fabricated with 3D printing and deformed by a controlled pneumatic system. We validate and demonstrate the efficacy of our method with soft pneumatic objects that have different shapes and deformation behaviors.

We present O-CNN , an Octree-based Convolutional Neural Network (CNN) for 3D shape analysis. Built upon the octree representation of 3D shapes, our method takes the average normal vectors of a 3D model sampled in the finest leaf octants as input and performs 3D CNN operations on the octants occupied by the 3D shape surface. We design a novel octree data structure to efficiently store the octant information and CNN features into the graphics memory and execute the entire O-CNN training and evaluation on the GPU. O-CNN supports various CNN structures and works for 3D shapes in different representations. By restraining the computations on the octants occupied by 3D surfaces, the memory and computational costs of the O-CNN grow quadratically as the depth of the octree increases, which makes the 3D CNN feasible for high-resolution 3D models. We compare the performance of the O-CNN with other existing 3D CNN solutions and demonstrate the efficiency and efficacy of O-CNN in three shape analysis tasks, including object classification, shape retrieval, and shape segmentation.

In this paper, we present a novel method on surface partition from the perspective of approximation theory. Different from previous shape proxies, the ellipsoidal variance proxy is proposed to penalize the partition results falling into disconnected parts. On its support, the Principle Component Analysis (PCA) based energy is developed for asymptotic cluster aspect ratio and size control. We provide the theoretical explanation on how the minimization of the PCA-based energy leads to the optimal asymptotic behavior for approximation. Moreover, we show the partitions on densely sampled triangular meshes converge to the theoretic expectations. To evaluate the effectiveness of surface approximation, polygonal/triangular surface remeshing results are generated. The experimental results demonstrate the high approximation quality of our method.

Abstract PolyCubes provide compact representations for closed complex shapes and are essential to many computer graphics applications. Existing automatic PolyCube construction methods usually suffer from poor quality or time‐consuming computation. In this paper, we provide a highly efficient method to compute volumetric PolyCube‐maps. Given an input tetrahedral mesh, we utilize two novel normal‐driven volumetric deformation schemes and a polycube‐allowable mesh segmentation to drive the input to a volumetric PolyCube structure. Our method can robustly generate foldover‐free and low‐distortion PolyCube‐maps in practice, and provide a flexible control on the number of corners of Polycubes. Compared with state‐of‐the‐art methods, our method is at least one order of magnitude faster and has better mapping qualities. We demonstrate the efficiency and efficacy of our method in PolyCube construction and all‐hexahedral meshing on various complex models.

We present a novel method, called Simplex Assembly , to compute inversion-free mappings with low or bounded distortion on simplicial meshes. Our method involves two steps: simplex disassembly and simplex assembly. Given a simplicial mesh and its initial piecewise affine mapping, we project the affine transformation associated with each simplex into the inversion-free and distortion-bounded space. The projection disassembles the input mesh into disjoint simplices. The disjoint simplices are then assembled to recover the original connectivity by minimizing the mapping distortion and the difference of the disjoint vertices with respect to the piecewise affine transformations, while the piecewise affine mapping is restricted inside the feasible space. Due to the use of affine transformations as variables, our method explicitly guarantees that no inverted simplex occurs, and that the mapping distortion is below the bound during the optimization. Compared with existing methods, our method is robust to an initialization with many inverted elements and positional constraints. We demonstrate the efficiency and robustness of our method through a variety of geometric processing tasks.

We present a data-driven approach for mesh denoising. Our key idea is to formulate the denoising process with cascaded non-linear regression functions and learn them from a set of noisy meshes and their ground-truth counterparts. Each regression function infers the normal of a denoised output mesh facet from geometry features extracted from its neighborhood facets on the input mesh and sends the result as the input of the next regression function. Specifically, we develop afiltered facet normal descriptor (FND)for modeling the geometry features around each facet on the noisy mesh and model a regression function with neural networks for mapping the FNDs to the facet normals of the denoised mesh. To handle meshes with different geometry features and reduce the training difficulty, we cluster the input mesh facets according to their FNDs and train neural networks for each cluster separately in an offline learning stage. At runtime, our method applies the learned cascaded regression functions to a noisy input mesh and reconstructs the denoised mesh from the output facet normals. Our method learns the non-linear denoising process from the training data and makes no specific assumptions about the noise distribution and geometry features in the input. The runtime denoising process is fully automatic for different input meshes. Our method can be easily adapted to meshes with arbitrary noise patterns by training a dedicated regression scheme with mesh data and the particular noise pattern. We evaluate our method on meshes with both synthetic and real scanned noise, and compare it to other mesh denoising algorithms. Results demonstrate that our method outperforms the state-of-the-art mesh denoising methods and successfully removes different kinds of noise for meshes with various geometry features.

Computing locally injective mappings with low distortion in an efficient way is a fundamental task in computer graphics. By revisiting the well-known MIPS (Most-Isometric ParameterizationS) method, we introduce an advanced MIPS method that inherits the local injectivity of MIPS, achieves as low as possible distortions compared to the state-of-the-art locally injective mapping techniques, and performs one to two orders of magnitude faster in computing a mesh-based mapping. The success of our method relies on two key components. The first one is an enhanced MIPS energy function that penalizes the maximal distortion significantly and distributes the distortion evenly over the domain for both mesh-based and meshless mappings. The second is a use of the inexact block coordinate descent method in mesh-based mapping in a way that efficiently minimizes the distortion with the capability not to be trapped early by the local minimum. We demonstrate the capability and superiority of our method in various applications including mesh parameterization, mesh-based and meshless deformation, and mesh improvement.

We propose a new approach for automatic surfacing of 3D curve networks, a long standing computer graphics problem which has garnered new attention with the emergence of sketch based modeling systems capable of producing such networks. Our approach is motivated by recent studies suggesting that artist-designed curve networks consist of descriptive curves that convey intrinsic shape properties, and are dominated byrepresentative flow linesdesigned to convey the principal curvature lines on the surface. Studies indicate that viewers complete the intended surface shape by envisioning a surface whose curvature lines smoothly blend these flow-line curves. Following these observations we design a surfacing framework that automatically aligns the curvature lines of the constructed surface with the representative flow lines and smoothly interpolates these representative flow, or curvature directions while minimizing undesired curvature variation. Starting with an initial triangle mesh of the network, we dynamically adapt the mesh to maximize the agreement between the principal curvature direction field on the surface and a smoothflow fieldsuggested by the representative flow-line curves. Our main technical contribution is a framework for curvature-based surface modeling, that facilitates the creation of surfaces with prescribed curvature characteristics. We validate our method via visual inspection, via comparison to artist created and ground truth surfaces, as well as comparison to prior art, and confirm that our results are well aligned with the computed flow fields and with viewer perception of the input networks.

3D geometric features constitute rich details of polygonal meshes. Their analysis and editing can lead to vivid appearance of shapes and better understanding of the underlying geometry for shape processing and analysis. Traditional mesh smoothing techniques mainly focus on noise filtering and they cannot distinguish different scales of features well, even mixing them up. We present an efficient method to process different scale geometric features based on a novel rolling-guidance normal filter. Given a 3D mesh, our method iteratively applies a joint bilateral filter to face normals at a specified scale, which empirically smooths small-scale geometric features while preserving large-scale features. Our method recovers the mesh from the filtered face normals by a modified Poisson-based gradient deformation that yields better surface quality than existing methods. We demonstrate the effectiveness and superiority of our method on a series of geometry processing tasks, including geometry texture removal and enhancement, coating transfer, mesh segmentation and level-of-detail meshing.

Mesh surfaces with planar hexagonal faces, what we refer to as PH meshes, offer an elegant way of paneling freeform architectural surfaces due to their node simplicity (i.e., valence-3 nodes) and naturally appealing layout. We investigate PH meshes to understand how the shape, size, and pattern of PH faces are constrained by surface geometry. This understanding enables us to develop an effective method for paneling freeform architectural surfaces with PH meshes. Our method first constructs an ideal triangulation of a given smooth surface, guided by surface geometry. We show that such an ideal triangulation leads to a Dupin-regular PH mesh via tangent duality on the surface. We have developed several novel and effective techniques for improving undesirable mesh layouts caused by singular behaviors of surface curvature. We compute support structures associated with PH meshes, including exact vertex offsets and approximate edge offsets, as demanded in panel manufacturing. The efficacy of our method is validated by a number of architectural examples.

We present a novel method to generate high-quality simplicial meshes with specified anisotropy. Given a surface or volumetric domain equipped with a Riemannian metric that encodes the desired anisotropy, we transform the problem to one of functional approximation. We construct a convex function over each mesh simplex whose Hessian locally matches the Riemannian metric, and iteratively adapt vertex positions and mesh connectivity to minimize the difference between the target convex functions and their piecewise-linear interpolation over the mesh. Our method generalizes optimal Delaunay triangulation and leads to a simple and efficient algorithm. We demonstrate its quality and speed compared to state-of-the-art methods on a variety of domains and metrics.

Self-supporting structures are prominent in historical and contemporary architecture due to advantageous structural properties and efficient use of material. Computer graphics research has recently contributed new design tools that allow creating and interactively exploring self-supporting freeform designs. However, the physical construction of such freeform structures remains challenging, even on small scales. Current construction processes require extensive formwork during assembly, which quickly leads to prohibitively high construction costs for realizations on a building scale. This greatly limits the practical impact of the existing freeform design tools. We propose to replace the commonly used dense formwork with a sparse set of temporary chains. Our method enables gradual construction of the masonry model in stable sections and drastically reduces the material requirements and construction costs. We analyze the input using a variational method to find stable sections, and devise a computationally tractable divide-and-conquer strategy for the combinatorial problem of finding an optimal construction sequence. We validate our method on 3D printed models, demonstrate an application to the restoration of historical models, and create designs of recreational, collaborative self-supporting puzzles.

BodyAvatar is a Kinect-based interactive system that allows users without professional skills to create freeform 3D avatars using body gestures. Unlike existing gesture-based 3D modeling tools, BodyAvatar centers around a first-person "you're the avatar" metaphor, where the user treats their own body as a physical proxy of the virtual avatar. Based on an intuitive body-centric mapping, the user performs gestures to their own body as if wanting to modify it, which in turn results in corresponding modifications to the avatar. BodyAvatar provides an intuitive, immersive, and playful creation experience for the user. We present a formative study that leads to the design of BodyAvatar, the system's interactions and underlying algorithms, and results from initial user trials.

Masonry structures must be compressively self-supporting; designing such surfaces forms an important topic in architecture as well as a challenging problem in geometric modeling. Under certain conditions, a surjective mapping exists between a power diagram , defined by a set of 2D vertices and associated weights, and the reciprocal diagram that characterizes the force diagram of a discrete self-supporting network. This observation lets us define a new and convenient parameterization for the space of self-supporting networks. Based on it and the discrete geometry of this design space, we present novel geometry processing methods including surface smoothing and remeshing which significantly reduce the magnitude of force densities and homogenize their distribution.

Storyline visualizations, which are useful in many applications, aim to illustrate the dynamic relationships between entities in a story. However, the growing complexity and scalability of stories pose great challenges for existing approaches. In this paper, we propose an efficient optimization approach to generating an aesthetically appealing storyline visualization, which effectively handles the hierarchical relationships between entities over time. The approach formulates the storyline layout as a novel hybrid optimization approach that combines discrete and continuous optimization. The discrete method generates an initial layout through the ordering and alignment of entities, and the continuous method optimizes the initial layout to produce the optimal one. The efficient approach makes real-time interactions (e.g., bundling and straightening) possible, thus enabling users to better understand and track how the story evolves. Experiments and case studies are conducted to demonstrate the effectiveness and usefulness of the optimization approach.

Decomposing a volume into high-quality hexahedral cells is a challenging task in geometric modeling and computational geometry. Inspired by the use of cross field in quad meshing and the CubeCover approach in hex meshing, we present a complete all-hex meshing framework based on singularity-restricted field that is essential to induce a valid all-hex structure. Given a volume represented by a tetrahedral mesh, we first compute a boundary-aligned 3D frame field inside it, then convert the frame field to be singularity-restricted by our effective topological operations. In our all-hex meshing framework, we apply the CubeCover method to achieve the volume parametrization. For reducing degenerate elements appearing in the volume parametrization, we also propose novel tetrahedral split operations to preprocess singularity-restricted frame fields. Experimental results show that our algorithm generates high-quality all-hex meshes from a variety of 3D volumes robustly and efficiently.

We present a new method for modeling discrete constant mean curvature (CMC) surfaces, which arise frequently in nature and are highly demanded in architecture and other engineering applications. Our method is based on a novel use of the CVT ( centroidal Voronoi tessellation ) optimization framework. We devise a CVT-CMC energy function defined as a combination of an extended CVT energy and a volume functional. We show that minimizing the CVT-CMC energy is asymptotically equivalent to minimizing mesh surface area with a fixed volume, thus defining a discrete CMC surface. The CVT term in the energy function ensures high mesh quality throughout the evolution of a CMC surface in an interactive design process for form finding. Our method is capable of modeling CMC surfaces with fixed or free boundaries and is robust with respect to input mesh quality and topology changes. Experiments show that the new method generates discrete CMC surfaces of improved mesh quality over existing methods.

We introduce a new method to synthesize mechanical toys solely from the motion of their features. The designer specifies the geometry and a time-varying rotation and translation of each rigid feature component. Our algorithm automatically generates a mechanism assembly located in a box below the feature base that produces the specified motion. Parts in the assembly are selected from a parameterized set including belt-pulleys, gears, crank-sliders, quick-returns, and various cams (snail, ellipse, and double-ellipse). Positions and parameters for these parts are optimized to generate the specified motion, minimize a simple measure of complexity, and yield a well-distributed layout of parts over the driving axes. Our solution uses a special initialization procedure followed by simulated annealing to efficiently search the complex configuration space for an optimal assembly.

We present a novel method to approximate a freeform shape with a planar quadrilateral (PQ) mesh for modeling architectural glass structures. Our method is based on the study of conjugate direction fields (CDF) which allow the presence of ±κ/4(κ ε Z) singularities. Starting with a triangle discretization of a freeform shape, we first compute an as smooth as possible conjugate direction field satisfying the user's directional and angular constraints, then apply mixed-integer quadrangulation and planarization techniques to generate a PQ mesh which approximates the input shape faithfully. We demonstrate that our method is effective and robust on various 3D 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.

This paper introduces L p -Centroidal Voronoi Tessellation ( L p -CVT), a generalization of CVT that minimizes a higher-order moment of the coordinates on the Voronoi cells. This generalization allows for aligning the axes of the Voronoi cells with a predefined background tensor field (anisotropy). L p -CVT is computed by a quasi-Newton optimization framework, based on closed-form derivations of the objective function and its gradient. The derivations are given for both surface meshing (Ω is a triangulated mesh with per-facet anisotropy) and volume meshing (Ω is the interior of a closed triangulated mesh with a 3D anisotropy field). Applications to anisotropic, quad-dominant surface remeshing and to hexdominant volume meshing are presented. Unlike previous work, L p -CVT captures sharp features and intersections without requiring any pre-tagging.

Abstract We propose a new isotropic remeshing method, based onCentroidal Voronoi Tessellation (CVT). Constructing CVT requires to repeatedly computeRestricted Voronoi Diagram (RVD), defined as the intersection between a 3D Voronoi diagram and an input mesh surface. Existing methods use some approximations of RVD. In this paper, we introduce an efficient algorithm that computes RVD exactly and robustly. As a consequence, we achieve better remeshing quality than approximation‐based approaches, without sacrificing efficiency. Our method for RVD computation uses a simple procedure and akd‐tree to quickly identify and compute the intersection of each triangle face with its incident Voronoi cells. Its time complexity isO(mlogn), wherenis the number of seed points andmis the number of triangles of the input mesh. Fast convergence of CVT is achieved using a quasi‐Newton method, which proved much faster than Lloyd's iteration. Examples are presented to demonstrate the better quality of remeshing results with our method than with the state‐of‐art approaches.

Centroidal Voronoi tessellation (CVT) is a particular type of Voronoi tessellation that has many applications in computational sciences and engineering, including computer graphics. The prevailing method for computing CVT is Lloyd's method, which has linear convergence and is inefficient in practice. We develop new efficient methods for CVT computation and demonstrate the fast convergence of these methods. Specifically, we show that the CVT energy function has 2nd order smoothness for convex domains with smooth density, as well as in most situations encountered in optimization. Due to the 2nd order smoothness, it is possible to minimize the CVT energy functions using Newton-like optimization methods and expect fast convergence. We propose a quasi-Newton method to compute CVT and demonstrate its faster convergence than Lloyd's method with various numerical examples. It is also significantly faster and more robust than the Lloyd-Newton method, a previous attempt to accelerate CVT. We also demonstrate surface remeshing as a possible application.

Due to its minimal twist, the rotation minimizing frame (RMF) is widely used in computer graphics, including sweep or blending surface modeling, motion design and control in computer animation and robotics, streamline visualization, and tool path planning in CAD/CAM. We present a novel simple and efficient method for accurate and stable computation of RMF of a curve in 3D. This method, called the double reflection method , uses two reflections to compute each frame from its preceding one to yield a sequence of frames to approximate an exact RMF. The double reflection method has the fourth order global approximation error, thus it is much more accurate than the two currently prevailing methods with the second order approximation error—the projection method by Klok and the rotation method by Bloomenthal, while all these methods have nearly the same per-frame computational cost. Furthermore, the double reflection method is much simpler and faster than using the standard fourth order Runge-Kutta method to integrate the defining ODE of the RMF, though they have the same accuracy. We also investigate further properties and extensions of the double reflection method, and discuss the variational principles in design moving frames with boundary conditions, based on RMF.

The geometric challenges in the architectural design of freeform shapes come mainly from the physical realization of beams and nodes. We approach them via the concept of parallel meshes, and present methods of computation and optimization. We discuss planar faces, beams of controlled height, node geometry, and multilayer constructions. Beams of constant height are achieved with the new type of edge offset meshes. Mesh parallelism is also the main ingredient in a novel discrete theory of curvatures. These methods are applied to the construction of quadrilateral, pentagonal and hexagonal meshes, discrete minimal surfaces, discrete constant mean curvature surfaces, and their geometric transforms. We show how to design geometrically optimal shapes, and how to find a meaningful meshing and beam layout for existing shapes.

We present a complete framework for computing a subdivision surface to approximate unorganized point sample data, which is a separable nonlinear least squares problem. We study the convergence and stability of three geometrically-motivated optimization schemes and reveal their intrinsic relations with standard methods for constrained nonlinear optimization. A commonly-used method in graphics, called point distance minimization, is shown to use a variant of the gradient descent step and thus has only linear convergence. The second method, called tangent distance minimization, which is well-known in computer vision, is shown to use the Gauss-Newton step, and thus demonstrates near quadratic convergence for zero residual problems but may not converge otherwise. Finally, we show that an optimization scheme called squared distance minimization, recently proposed by Pottmann et al., can be derived from the Newton method. Hence, with proper regularization, tangent distance minimization and squared distance minimization are more efficient than point distance minimization. We also investigate the effects of two step size control methods -- Levenberg-Marquardt regularization and the Armijo rule -- on the convergence stability and efficiency of the above optimization schemes.

In architectural freeform design, the relation between shape and fabrication poses new challenges and requires more sophistication from the underlying geometry. The new concept of conical meshes satisfies central requirements for this application: They are quadrilateral meshes with planar faces, and therefore particularly suitable for the design of freeform glass structures. Moreover, they possess a natural offsetting operation and provide a support structure orthogonal to the mesh. Being a discrete analogue of the network of principal curvature lines, they represent fundamental shape characteristics. We show how to optimize a quad mesh such that its faces become planar, or the mesh becomes even conical. Combining this perturbation with subdivision yields a powerful new modeling tool for all types of quad meshes with planar faces, making subdivision attractive for architecture design and providing an elegant way of modeling developable surfaces.

Computing a curve to approximate data points is a problem encountered frequently in many applications in computer graphics, computer vision, CAD/CAM, and image processing. We present a novel and efficient method, called squared distance minimization (SDM), for computing a planar B-spline curve, closed or open, to approximate a target shape defined by a point cloud , that is, a set of unorganized, possibly noisy data points. We show that SDM significantly outperforms other optimization methods used currently in common practice of curve fitting. In SDM, a B-spline curve starts from some properly specified initial shape and converges towards the target shape through iterative quadratic minimization of the fitting error. Our contribution is the introduction of a new fitting error term, called the squared distance (SD) error term , defined by a curvature-based quadratic approximant of squared distances from data points to a fitting curve. The SD error term faithfully measures the geometric distance between a fitting curve and a target shape, thus leading to faster and more stable convergence than the point distance (PD) error term, which is commonly used in computer graphics and CAGD, and the tangent distance (TD) error term, which is often adopted in the computer vision community. To provide a theoretical explanation of the superior performance of SDM, we formulate the B-spline curve fitting problem as a nonlinear least squares problem and conclude that SDM is a quasi-Newton method which employs a curvature-based positive definite approximant to the true Hessian of the objective function. Furthermore, we show that the method based on the TD error term is a Gauss-Newton iteration, which is unstable for target shapes with high curvature variations, whereas optimization based on the PD error term is the alternating method that is known to have linear convergence.

Collision detection and avoidance are important in robotics. Compared with commonly used circular disks, elliptic disks provide a more compact shape representation for robots or other vehicles confined to move in the plane. Furthermore, elliptic disks allow a simpler analytic representation than rectangular boxes, which makes it easier to perform continuous collision detection (CCD). We shall present a fast and accurate method for CCD between two moving elliptic disks, which avoids any need to sample the time domain of the motion, thus avoiding the possibility of missing collisions between time samples. Based on some new algebraic conditions on the separation of two ellipses, we reduce collision detection for two moving ellipses to the problem of detecting real roots of a univariate equation, which is the discriminant of the characteristic polynomial of the two ellipses. Several techniques are investigated for robust and accurate processing of this univariate equation for two classes of commonly used motions: planar cycloidal motions and planar rational motions. Experimental results demonstrate the efficiency, accuracy, and robustness of our method.

We present a novel algorithm based on least-squares minimization to approximate point cloud data in 2D plane with a smooth B-spline curve. The point cloud data may represent an open curve with self intersection and sharp corner. Unlike other existing methods, such as the moving least-squares method and the principle curve method, our algorithm does not need a thinning process. The idea of our algorithm is intuitive and simple - we make a B-spline curve grow along the tangential directions at its two end-points following local geometry of point clouds. Our algorithm generates appropriate control points of the fitting B-spline curve in the least squares sense. Although presented for the 2D case, our method can be extended in a straightforward manner to fitting data points by a B-spline curve in higher dimensions

We study the reconstruction of smooth surfaces from point clouds. We use a new squared distance error term in optimization to fit a subdivision surface to a set of unorganized points, which defines a closed target surface of arbitrary topology. The resulting method is based on the framework of squared distance minimization (SDM) proposed by Pottmann et al. Specifically, with an initial subdivision surface having a coarse control mesh as input, we adjust the control points by optimizing an objective function through iterative minimization of a quadratic approximant of the squared distance function of the target shape. Our experiments show that the new method (SDM) converges much faster than the commonly used optimization method using the point distance error function, which is known to have only linear convergence. This observation is further supported by our recent result that SDM can be derived from the Newton method with necessary modifications to make the Hessian positive definite and the fact that the Newton method has quadratic convergence.