Problem In Geometry Processing Research Articles

PCDNF: Revisiting Learning-Based Point Cloud Denoising via Joint Normal Filtering.

Point cloud denoising is a fundamental and challenging problem in geometry processing. Existing methods typically involve direct denoising of noisy input or filtering raw normals followed by point position updates. Recognizing the crucial relationship between point cloud denoising and normal filtering, we re-examine this problem from a multitask perspective and propose an end-to-end network called PCDNF for joint normal filtering-based point cloud denoising. We introduce an auxiliary normal filtering task to enhance the network's ability to remove noise while preserving geometric features more accurately. Our network incorporates two novel modules. First, we design a shape-aware selector to improve noise removal performance by constructing latent tangent space representations for specific points, taking into account learned point and normal features as well as geometric priors. Second, we develop a feature refinement module to fuse point and normal features, capitalizing on the strengths of point features in describing geometric details and normal features in representing geometric structures, such as sharp edges and corners. This combination overcomes the limitations of each feature type and better recovers geometric information. Extensive evaluations, comparisons, and ablation studies demonstrate that the proposed method outperforms state-of-the-art approaches in both point cloud denoising and normal filtering.

Differentiable Geodesic Distance for Intrinsic Minimization on Triangle Meshes

Computing intrinsic distances on discrete surfaces is at the heart of many minimization problems in geometry processing and beyond. Solving these problems is extremely challenging as it demands the computation of on-surface distances along with their derivatives. We present a novel approach for intrinsic minimization of distance-based objectives defined on triangle meshes. Using a variational formulation of shortest-path geodesics, we compute first and second-order distance derivatives based on the implicit function theorem, thus opening the door to efficient Newton-type minimization solvers. We demonstrate our differentiable geodesic distance framework on a wide range of examples, including geodesic networks and membranes on surfaces of arbitrary genus, two-way coupling between hosting surface and embedded system, differentiable geodesic Voronoi diagrams, and efficient computation of Karcher means on complex shapes. Our analysis shows that second-order descent methods based on our differentiable geodesics outperform existing first-order and quasi-Newton methods by large margins.

A Closest Point Method for PDEs on Manifolds with Interior Boundary Conditions for Geometry Processing

Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in \(\mathbb {R}^2 \) or \(\mathbb {R}^3 \) , such as curves or surfaces. Such manifold PDEs often involve boundary conditions (e.g., Dirichlet or Neumann) prescribed at points or curves on the manifold’s interior or along the geometric (exterior) boundary of an open manifold. However, input manifolds can take many forms (e.g., triangle meshes, parametrizations, point clouds, implicit functions, etc.). Typically, one must generate a mesh to apply finite element-type techniques or derive specialized discretization procedures for each distinct manifold representation. We propose instead to address such problems in a unified manner through a novel extension of the closest point method (CPM) to handle interior boundary conditions. CPM solves the manifold PDE by solving a volumetric PDE defined over the Cartesian embedding space containing the manifold, and requires only a closest point representation of the manifold. Hence, CPM supports objects that are open or closed, orientable or not, and of any codimension. To enable support for interior boundary conditions we derive a method that implicitly partitions the embedding space across interior boundaries. CPM’s finite difference and interpolation stencils are adapted to respect this partition while preserving second-order accuracy. Additionally, we develop an efficient sparse-grid implementation and numerical solver that can scale to tens of millions of degrees of freedom, allowing PDEs to be solved on more complex manifolds. We demonstrate our method’s convergence behaviour on selected model PDEs and explore several geometry processing problems: diffusion curves on surfaces, geodesic distance, tangent vector field design, harmonic map construction, and reaction-diffusion textures. Our proposed approach thus offers a powerful and flexible new tool for a range of geometry processing tasks on general manifold representations.

Metric Optimization in Penner Coordinates

Many parametrization and mapping-related problems in geometry processing can be viewed as metric optimization problems, i.e., computing a metric minimizing a functional and satisfying a set of constraints, such as flatness. Penner coordinates are global coordinates on the space of metrics on meshes with a fixed vertex set and topology, but varying connectivity, making it homeomorphic to the Euclidean space of dimension equal to the number of edges in the mesh, without any additional constraints imposed. These coordinates play an important role in the theory of discrete conformal maps, enabling recent development of highly robust algorithms with convergence and solution existence guarantees for computing such maps. We demonstrate how Penner coordinates can be used to solve a general class of optimization problems involving metrics, including optimization and interpolation, while retaining the key solution existence guarantees available for discrete conformal maps.

Expansion Cones: A Progressive Volumetric Mapping Framework

Volumetric mapping is a ubiquitous and difficult problem in Geometry Processing and has been the subject of research in numerous and various directions. While several methods show encouraging results, the field still lacks a general approach with guarantees regarding map bijectivity. Through this work, we aim at opening the door to a new family of methods by providing a novel framework based on the concept of progressive expansion. Starting from an initial map of a tetrahedral mesh whose image may contain degeneracies but no inversions, we incrementally adjust vertex images to expand degenerate elements. By restricting movement to so-called expansion cones , it is done in such a way that the number of degenerate elements decreases in a strictly monotonic manner, without ever introducing any inversion. Adaptive local refinement of the mesh is performed to facilitate this process. We describe a prototype algorithm in the realm of this framework for the computation of maps from ball-topology tetrahedral meshes to convex or star-shaped domains. This algorithm is evaluated and compared to state-of-the-art methods, demonstrating its benefits in terms of bijectivity. We also discuss the associated cost in terms of sometimes significant mesh refinement to obtain the necessary degrees of freedom required for establishing a valid mapping. Our conclusions include that while this algorithm is only of limited immediate practical utility due to efficiency concerns, the general framework has the potential to inspire a range of novel methods improving on the efficiency aspect.

Symmetric Volume Maps: Order-invariant Volumetric Mesh Correspondence with Free Boundary

Although shape correspondence is a central problem in geometry processing, most methods for this task apply only to two-dimensional surfaces. The neglected task of volumetric correspondence—a natural extension relevant to shapes extracted from simulation, medical imaging, and volume rendering—presents unique challenges that do not appear in the two-dimensional case. In this work, we propose a method for mapping between volumes represented as tetrahedral meshes. Our formulation minimizes a distortion energy designed to extract maps symmetrically, i.e., without dependence on the ordering of the source and target domains. We accompany our method with theoretical discussion describing the consequences of this symmetry assumption, leading us to select a symmetrized ARAP energy that favors isometric correspondences. Our final formulation optimizes for near-isometry while matching the boundary. We demonstrate our method on a diverse geometric dataset, producing low-distortion matchings that align closely to the boundary.

Multiscale Mesh Deformation Component Analysis With Attention-Based Autoencoders.

Deformation component analysis is a fundamental problem in geometry processing and shape understanding. Existing approaches mainly extract deformation components in local regions at a similar scale while deformations of real-world objects are usually distributed in a multi-scale manner. In this article, we propose a novel method to exact multiscale deformation components automatically with a stacked attention-based autoencoder. The attention mechanism is designed to learn to softly weight multi-scale deformation components in active deformation regions, and the stacked attention-based autoencoder is learned to represent the deformation components at different scales. Quantitative and qualitative evaluations show that our method outperforms state-of-the-art methods. Furthermore, with the multiscale deformation components extracted by our method, the user can edit shapes in a coarse-to-fine fashion which facilitates effective modeling of new shapes.

Sum-of-squares geometry processing

Geometry processing presents a variety of difficult numerical problems, each seeming to require its own tailored solution. This breadth is largely due to the expansive list of geometric primitives , e.g., splines, triangles, and hexahedra, joined with an ever-expanding variety of objectives one might want to achieve with them. With the recent increase in attention toward higher-order surfaces , we can expect a variety of challenges porting existing solutions that work on triangle meshes to work on these more complex geometry types. In this paper, we present a framework for solving many core geometry processing problems on higher-order surfaces. We achieve this goal through sum-of-squares optimization, which transforms nonlinear polynomial optimization problems into sequences of convex problems whose complexity is captured by a single degree parameter. This allows us to solve a suite of problems on higher-order surfaces, such as continuous collision detection and closest point queries on curved patches, with only minor changes between formulations and geometries.

Point cloud decomposition by internal and external critical points

Decomposing a complex object into simple components is a fundamental problem in geometry processing. Existing methods for decomposing point clouds rely on local or global features of an object, which leads to over-segmentation or unnatural component boundaries. In this paper, we propose a novel method for decomposing the point cloud by using internal and external critical points. First, we propose a novel shrinking strategy to build the global skeleton topology, from which we can extract internal critical points for locating components. External critical points are selected from the ridge and valley points for component segmentation. Then, we apply the constraint of internal and external critical points to decompose an object into semantic components by skeleton-based piecewise labeling. Experimental results demonstrate that our method is effective in decomposing 3D point clouds and is robust to limited noise and incomplete data.

Orienting point clouds with dipole propagation

Establishing a consistent normal orientation for point clouds is a notoriously difficult problem in geometry processing, requiring attention to both local and global shape characteristics. The normal direction of a point is a function of the local surface neighborhood; yet, point clouds do not disclose the full underlying surface structure. Even assuming known geodesic proximity, calculating a consistent normal orientation requires the global context. In this work, we introduce a novel approach for establishing a globally consistent normal orientation for point clouds. Our solution separates the local and global components into two different sub-problems. In the local phase, we train a neural network to learn a coherent normal direction per patch ( i.e. , consistently oriented normals within a single patch). In the global phase, we propagate the orientation across all coherent patches using a dipole propagation. Our dipole propagation decides to orient each patch using the electric field defined by all previously orientated patches. This gives rise to a global propagation that is stable, as well as being robust to nearby surfaces, holes, sharp features and noise.

Foldover-free maps in 50 lines of code

Mapping a triangulated surface to 2D space (or a tetrahedral mesh to 3D space) is an important problem in geometry processing. In computational physics, untangling plays an important role in mesh generation: it takes a mesh as an input, and moves the vertices to get rid of foldovers. In fact, mesh untangling can be considered as a special case of mapping where the geometry of the object is to be defined in the map space and the geometric domain is not explicit, supposing that each element is regular. In this paper, we propose a mapping method inspired by the untangling problem and compare its performance to the state of the art. The main advantage of our method is that the untangling aims at producing locally injective maps, which is the major challenge of mapping. In practice, our method produces locally injective maps in very difficult settings, both in 2D and 3D. We demonstrate it on a large reference database as well as on more difficult stress tests. For a better reproducibility, we publish the code in Python for a basic evaluation, and in C++ for more advanced applications.

Shape-aware Mesh Normal Filtering

Mesh denoising is a fundamental yet open problem in geometry processing. The main challenge is to remove noise while recovering the shape of the underlying surface as accurately as possible. In this paper, we propose a novel joint bilateral filter on the face normal field. The key is to estimate a reliable guidance normal field by constructing a shape-aware consistent patch for accurately describing the local shape of each face. To this end, we first select a candidate patch for each face by using a newly defined consistent metric considering both patch flatness and face-to-patch orientation similarity. Then, spectral analysis is used in combination with ℓ0 minimization to refine the candidate patches in a shape-aware manner. The refined patches do not contain any features, and therefore they can accurately describe the local shape of the underlying surface. After smoothing the face normal field, vertex positions are reconstructed to match the filtered face normals. Our mesh denoising method is theoretically rooted and practical for dealing with the meshes containing corners with low sampling rates, multi-scale features, or narrow structure regions. Extensive experimental results demonstrate that our method can significantly improve the feature preserving capability of joint normal filter and outperforms state-of-the-art methods visually and quantitatively.

Detecting 3D Points of Interest Using Projective Neural Networks

Detecting points of interest on 3D shapes is a fundamental research problem in geometry processing. Due to the complicated relationship between points of interest and their geometric features, detecting points of interest on any given 3D shape remains challenging. Due to the lack of training data, previous data-driven methods for detecting 3D points of interest mainly focus on utilizing hand-crafted geometric features to predict the probabilities of each point being a POI, which greatly limits detection performance. In this paper, we propose a novel algorithm for detecting 3D points of interest by using projective neural networks. Our method first projects the labeled training 3D shapes into multiple 2D views and then learns the required features from the 2D views in an end-to-end fashion. The points of interest on test 3D shapes are then automatically detected by applying the learned neural network and our improved density peak clustering. Our method relies neither on hand-crafted feature descriptors nor a large quantity of expensive 3D training data to obtain satisfactory results. Experimental results show significantly superior detection performance of our method over the state-of-the-art methods.

Nonisometric Surface Registration via Conformal Laplace–Beltrami Basis Pursuit

Surface registration is one of the most fundamental problems in geometry processing. Many approaches have been developed to tackle this problem in cases where the surfaces are nearly isometric. However, it is much more challenging to compute correspondence between surfaces which are intrinsically less similar. In this paper, we propose a variational model to align the Laplace-Beltrami (LB) eigensytems of two non-isometric genus zero shapes via conformal deformations. This method enables us to compute geometrically meaningful point-to-point maps between non-isometric shapes. Our model is based on a novel basis pursuit scheme whereby we simultaneously compute a conformal deformation of a ’target shape’ and its deformed LB eigensystem. We solve the model using a proximal alternating minimization algorithm hybridized with the augmented Lagrangian method which produces accurate correspondences given only a few landmark points. We also propose a re-initialization scheme to overcome some of the difficulties caused by the non-convexity of the variational problem. Intensive numerical experiments illustrate the effectiveness and robustness of the proposed method to handle non-isometric surfaces with large deformation with respect to both noises on the underlying manifolds and errors within the given landmarks or feature functions.

SHREC’20: Shape correspondence with non-isometric deformations

Estimating correspondence between two shapes continues to be a challenging problem in geometry processing. Most current methods assume deformation to be near-isometric, however this is often not the case. For this paper, a collection of shapes of different animals has been curated, where parts of the animals (e.g., mouths, tails & ears) correspond yet are naturally non-isometric. Ground-truth correspondences were established by asking three specialists to independently label corresponding points on each of the models with respect to a previously labelled reference model. We employ an algorithmic strategy to select a single point for each correspondence that is representative of the proposed labels. A novel technique that characterises the sparsity and distribution of correspondences is employed to measure the performance of ten shape correspondence methods.

Interacting with Self-Similarity

Shape similarity is a fundamental problem in geometry processing, enabling applications such as surface correspondence, segmentation, and edit propagation. For example, a user may paint a stroke on one finger of a model and desire the edit to propagate to all fingers. Automatic approaches have difficulty matching user expectations, either due to an algorithm’s inability to guess the scale at which the user is intending to edit or due to underlying deficiencies in the similarity metric (e.g., semantic information not present in the geometry).We propose an approach to interactively design self-similarity maps. We investigate two primitive operations, useful in a variety of scenarios: region and curve similarity. Users select example similar and dissimilar regions. Starting with an automatically generated multi-scale shape signature, our approach solves for a scale parameter and thresholds that group the example regions as specified. We propose a new Smooth Shape Diameter Signature (SSDS) as a more efficient alternative to the Heat or Wave Kernel Signature. If no such parameters can be found, our approach modifies the shape signature itself. Given a curve drawn on the surface, we perform hybrid discrete/continuous optimization to find similar curves elsewhere.We apply our approach for interactive editing scenarios: propagating mesh geometry, patterns duplication, and segmentation.

ZerNet: Convolutional Neural Networks on Arbitrary Surfaces Via Zernike Local Tangent Space Estimation

AbstractIn this paper, we propose a novel formulation extending convolutional neural networks (CNN) to arbitrary two‐dimensional manifolds using orthogonal basis functions called Zernike polynomials. In many areas, geometric features play a key role in understanding scientific trends and phenomena, where accurate numerical quantification of geometric features is critical. Recently, CNNs have demonstrated a substantial improvement in extracting and codifying geometric features. However, the progress is mostly centred around computer vision and its applications where an inherent grid‐like data representation is naturally present. In contrast, many geometry processing problems deal with curved surfaces and the application of CNNs is not trivial due to the lack of canonical grid‐like representation, the absence of globally consistent orientation and the incompatible local discretizations. In this paper, we show that the Zernike polynomials allow rigourous yet practical mathematical generalization of CNNs to arbitrary surfaces. We prove that the convolution of two functions can be represented as a simple dot product between Zernike coefficients and the rotation of a convolution kernel is essentially a set of 2 × 2 rotation matrices applied to the coefficients. The key contribution of this work is in such a computationally efficient but rigorous generalization of the major CNN building blocks.

A feature-preserving framework for point cloud denoising

Point cloud denoising has been an attractive problem in geometry processing. The main challenge is to eliminate noise while preserving different levels of features and preventing unnatural effects (such as over-sharpened artifacts on smoothly curved faces and cross artifacts on sharp edges). In this paper, we propose a novel feature-preserving framework to achieve these goals. Firstly, we newly define some discrete operators on point clouds, which can be used to construct a second order regularization for restoring a point normal field. Then, based on the filtered normals, we perform a feature detection step by a bi-tensor voting scheme. As will be seen, it is robust against noise and can locate underlying geometric features accurately. Finally, we reposition points with a multi-normal strategy by using a simple yet effective RANSAC-based algorithm. Intensive experimental results show that the proposed method performs favorably compared to other state-of-the-art approaches.

Geometry Processing Problems Using the Total Variation of the Normal Vector Field

AbstractThis paper presents a novel approach to the formulation and solution of discrete geometry processing problems including mesh denoising. The main quantity of interest is the piecewise constant unit normal vector field, which we consider on piecewise flat, triangulated surfaces. In a similar fashion as one does for images defined on ‘flat’ domains, our goal is to remove noise while preserving shape features such as sharp edges. To this end, we model a denoising problem via a quadratic vertex tracking term and a regularizer based on the total variation of the normal vector field. Since the latter has values on the unit sphere, its total variation reduces to a finite sum of the geodesic distances of neighboring normals. We solve the model numerically by applying split‐Bregmann iterations and provide results for the ‘fandisk’ benchmark.

A minimalistic approach for fast computation of geodesic distances on triangular meshes

The computation of geodesic distances is an important research topic in Geometry Processing and 3D Shape Analysis as it is a basic component of many methods used in these areas. In this work, we present a minimalistic parallel algorithm based on front propagation to compute approximate geodesic distances on meshes. Our method is practical and simple to implement, and does not require any heavy pre-processing. The convergence of our algorithm depends on the number of discrete level sets around the source points from which distance information propagates. To appropriately implement our method on GPUs taking into account memory coalescence problems, we take advantage of a graph representation based on a breadth-first search traversal that works harmoniously with our parallel front propagation approach. We report experiments that show how our method scales with the size of the problem. We compare the mean error and processing time obtained by our method with such measures computed using other methods. Our method produces results in competitive times with almost the same accuracy, especially for large meshes. We also demonstrate its use for solving two classical geometry processing problems: the regular sampling problem and the Voronoi tessellation on meshes.