arXiv subjects

Mirela Ben-Chen

Publications and source records attributed to Mirela Ben-Chen.

16 recordsLinked to original sources

arXiv2026

Phong-Rodrigues Extrinsic Vector-Field Processing

We introduce a new extrinsic discretization of tangent vector fields on triangle meshes that is continuous, with bounded derivatives that are continuous almost everywhere, supporting pointwise evaluation and integration of differential operators. We achieve this by building a continuous normal field over the mesh via Phong interpolation and using minimal Rodrigues rotations to transport vertex-based tangent vectors into triangle interiors. Unlike most existing discretizations, which typically sacrifice either continuity or the ability to evaluate derivatives pointwise, our approach supports both. Because it is pointwise evaluatable, and using the fact that the covariant derivative can be decomposed into its symmetric, antisymmetric, and scalar components, our discretization supports the construction of standard vector-field processing operators including the connection and Hodge Laplacians, Killing energy, divergence, curl, and the Lie bracket. This framework provides a simple and practical finite-element formulation for vector-field processing on meshes, supporting both integration-based operators and pointwise queries. To our knowledge, ours is the first discretization that jointly enables extrinsic continuous vector fields, bounded derivatives, and pointwise evaluation of this collection of operators.

cs.GR↗

arXiv2025

SDFs from Unoriented Point Clouds using Neural Variational Heat Distances

We propose a novel variational approach for computing neural Signed Distance Fields (SDF) from unoriented point clouds. To this end, we replace the commonly used eikonal equation with the heat method, carrying over to the neural domain what has long been standard practice for computing distances on discrete surfaces. This yields two convex optimization problems for whose solution we employ neural networks: We first compute a neural approximation of the gradients of the unsigned distance field through a small time step of heat flow with weighted point cloud densities as initial data. Then we use it to compute a neural approximation of the SDF. We prove that the underlying variational problems are well-posed. Through numerical experiments, we demonstrate that our method provides state-of-the-art surface reconstruction and consistent SDF gradients. Furthermore, we show in a proof-of-concept that it is accurate enough for solving a PDE on the zero-level set.

math.NA↗

arXiv2025

FRIDU: Functional Map Refinement with Guided Image Diffusion

We propose a novel approach for refining a given correspondence map between two shapes. A correspondence map represented as a functional map, namely a change of basis matrix, can be additionally treated as a 2D image. With this perspective, we train an image diffusion model directly in the space of functional maps, enabling it to generate accurate maps conditioned on an inaccurate initial map. The training is done purely in the functional space, and thus is highly efficient. At inference time, we use the pointwise map corresponding to the current functional map as guidance during the diffusion process. The guidance can additionally encourage different functional map objectives, such as orthogonality and commutativity with the Laplace-Beltrami operator. We show that our approach is competitive with state-of-the-art methods of map refinement and that guided diffusion models provide a promising pathway to functional map processing.

cs.CV↗

arXiv2025

CageNet: A Meta-Framework for Learning on Wild Meshes

Learning on triangle meshes has recently proven to be instrumental to a myriad of tasks, from shape classification, to segmentation, to deformation and animation, to mention just a few. While some of these applications are tackled through neural network architectures which are tailored to the application at hand, many others use generic frameworks for triangle meshes where the only customization required is the modification of the input features and the loss function. Our goal in this paper is to broaden the applicability of these generic frameworks to "wild", i.e. meshes in-the-wild which often have multiple components, non-manifold elements, disrupted connectivity, or a combination of these. We propose a configurable meta-framework based on the concept of caged geometry: Given a mesh, a cage is a single component manifold triangle mesh that envelopes it closely. Generalized barycentric coordinates map between functions on the cage, and functions on the mesh, allowing us to learn and test on a variety of data, in different applications. We demonstrate this concept by learning segmentation and skinning weights on difficult data, achieving better performance to state of the art techniques on wild meshes.

cs.GR↗

arXiv2023

BPM: Blended Piecewise Moebius Maps

We propose a novel Moebius interpolator that takes as an input a discrete map between the vertices of two planar triangle meshes, and outputs a smooth map on the input domain. The output map interpolates the discrete map, is continuous between triangles, and has low quasi-conformal distortion when the input map is discrete conformal. Our map leads to considerably smoother texture transfer compared to the alternatives, even on very coarse triangulations. Furthermore, our approach has a closed-form expression, is local, applicable to any discrete map, and leads to smooth results even for extreme deformations. Finally, by working with local intrinsic coordinates, our approach is easily generalizable to discrete maps between a surface triangle mesh and a planar mesh, i.e., a planar parameterization. We compare our method with existing approaches, and demonstrate better texture transfer results, and lower quasi-conformal errors.

cs.GR↗

arXiv2023

A Convex Optimization Framework for Regularized Geodesic Distances

We propose a general convex optimization problem for computing regularized geodesic distances. We show that under mild conditions on the regularizer the problem is well posed. We propose three different regularizers and provide analytical solutions in special cases, as well as corresponding efficient optimization algorithms. Additionally, we show how to generalize the approach to the all pairs case by formulating the problem on the product manifold, which leads to symmetric distances. Our regularized distances compare favorably to existing methods, in terms of robustness and ease of calibration.

cs.GR↗

arXiv2022

AmiGo: Computational Design of Amigurumi Crochet Patterns

We propose an approach for generating crochet instructions (patterns) from an input 3D model. We focus on Amigurumi, which are knitted stuffed toys. Given a closed triangle mesh, and a single point specified by the user, we generate crochet instructions, which when knitted and stuffed result in a toy similar to the input geometry. Our approach relies on constructing the geometry and connectivity of a Crochet Graph, which is then translated into a crochet pattern. We segment the shape automatically into chrochetable components, which are connected using the join-as-you-go method, requiring no additional sewing. We demonstrate that our method is applicable to a large variety of shapes and geometries, and yields easily crochetable patterns.

cs.GR↗

arXiv2021

PH-CPF: Planar Hexagonal Meshing using Coordinate Power Fields

We present a new approach for computing planar hexagonal meshes that approximate a given surface, represented as a triangle mesh. Our method is based on two novel technical contributions. First, we introduce Coordinate Power Fields, which are a pair of tangent vector fields on the surface that fulfill a certain continuity constraint. We prove that the fulfillment of this constraint guarantees the existence of a seamless parameterization with quantized rotational jumps, which we then use to regularly remesh the surface. We additionally propose an optimization framework for finding Coordinate Power Fields, which also fulfill additional constraints, such as alignment, sizing and bijectivity. Second, we build upon this framework to address a challenging meshing problem: planar hexagonal meshing. To this end, we suggest a combination of conjugacy, scaling and alignment constraints, which together lead to planarizable hexagons. We demonstrate our approach on a variety of surfaces, automatically generating planar hexagonal meshes on complicated meshes, which were not achievable with existing methods.

cs.GR↗

arXiv2021

Surface Multigrid via Intrinsic Prolongation

This paper introduces a novel geometric multigrid solver for unstructured curved surfaces. Multigrid methods are highly efficient iterative methods for solving systems of linear equations. Despite the success in solving problems defined on structured domains, generalizing multigrid to unstructured curved domains remains a challenging problem. The critical missing ingredient is a prolongation operator to transfer functions across different multigrid levels. We propose a novel method for computing the prolongation for triangulated surfaces based on intrinsic geometry, enabling an efficient geometric multigrid solver for curved surfaces. Our surface multigrid solver achieves better convergence than existing multigrid methods. Compared to direct solvers, our solver is orders of magnitude faster. We evaluate our method on many geometry processing applications and a wide variety of complex shapes with and without boundaries. By simply replacing the direct solver, we upgrade existing algorithms to interactive frame rates, and shift the computational bottleneck away from solving linear systems.

cs.GR↗

arXiv2021

A Differential Geometry Perspective on Orthogonal Recurrent Models

Recently, orthogonal recurrent neural networks (RNNs) have emerged as state-of-the-art models for learning long-term dependencies. This class of models mitigates the exploding and vanishing gradients problem by design. In this work, we employ tools and insights from differential geometry to offer a novel perspective on orthogonal RNNs. We show that orthogonal RNNs may be viewed as optimizing in the space of divergence-free vector fields. Specifically, based on a well-known result in differential geometry that relates vector fields and linear operators, we prove that every divergence-free vector field is related to a skew-symmetric matrix. Motivated by this observation, we study a new recurrent model, which spans the entire space of vector fields. Our method parameterizes vector fields via the directional derivatives of scalar functions. This requires the construction of latent inner product, gradient, and divergence operators. In comparison to state-of-the-art orthogonal RNNs, our approach achieves comparable or better results on a variety of benchmark tasks.

cs.LG↗

arXiv2020

ENIGMA: Evolutionary Non-Isometric Geometry Matching

In this paper we propose a fully automatic method for shape correspondence that is widely applicable, and especially effective for non isometric shapes and shapes of different topology. We observe that fully-automatic shape correspondence can be decomposed as a hybrid discrete/continuous optimization problem, and we find the best sparse landmark correspondence, whose sparse-to-dense extension minimizes a local metric distortion. To tackle the combinatorial task of landmark correspondence we use an evolutionary genetic algorithm, where the local distortion of the sparse-to-dense extension is used as the objective function. We design novel geometrically guided genetic operators, which, when combined with our objective, are highly effective for non isometric shape matching. Our method outperforms state of the art methods for automatic shape correspondence both quantitatively and qualitatively on challenging datasets.

cs.GR↗

arXiv2018

Recovering Hidden Components in Multimodal Data with Composite Diffusion Operators

Finding appropriate low dimensional representations of high-dimensional multi-modal data can be challenging, since each modality embodies unique deformations and interferences. In this paper, we address the problem using manifold learning, where the data from each modality is assumed to lie on some manifold. In this context, the goal is to characterize the relations between the different modalities by studying their underlying manifolds. We propose two new diffusion operators that allow to isolate, enhance and attenuate the hidden components of multi-modal data in a data-driven manner. Based on these new operators, efficient low-dimensional representations can be constructed for such data, which characterize the common structures and the differences between the manifolds underlying the different modalities. The capabilities of the proposed operators are demonstrated on 3D shapes and on a fetal heart rate monitoring application.

eess.SP↗

arXiv2018

Parallel Transport on the Cone Manifold of SPD Matrices for Domain Adaptation

In this paper, we consider the problem of domain adaptation. We propose to view the data through the lens of covariance matrices and present a method for domain adaptation using parallel transport on the cone manifold of symmetric positive-definite matrices. We provide rigorous analysis using Riemanninan geometry, illuminating the theoretical guarantees and benefits of the presented method. In addition, we demonstrate these benefits using experimental results on simulations and real-measured data.

eess.SP↗

arXiv2018

Steklov Spectral Geometry for Extrinsic Shape Analysis

We propose using the Dirichlet-to-Neumann operator as an extrinsic alternative to the Laplacian for spectral geometry processing and shape analysis. Intrinsic approaches, usually based on the Laplace-Beltrami operator, cannot capture the spatial embedding of a shape up to rigid motion, and many previous extrinsic methods lack theoretical justification. Instead, we consider the Steklov eigenvalue problem, computing the spectrum of the Dirichlet-to-Neumann operator of a surface bounding a volume. A remarkable property of this operator is that it completely encodes volumetric geometry. We use the boundary element method (BEM) to discretize the operator, accelerated by hierarchical numerical schemes and preconditioning; this pipeline allows us to solve eigenvalue and linear problems on large-scale meshes despite the density of the Dirichlet-to-Neumann discretization. We further demonstrate that our operators naturally fit into existing frameworks for geometry processing, making a shift from intrinsic to extrinsic geometry as simple as substituting the Laplace-Beltrami operator with the Dirichlet-to-Neumann operator.

cs.GR↗

arXiv2018

Reversible Harmonic Maps between Discrete Surfaces

Information transfer between triangle meshes is of great importance in computer graphics and geometry processing. To facilitate this process, a smooth and accurate map is typically required between the two meshes. While such maps can sometimes be computed between nearly-isometric meshes, the more general case of meshes with diverse geometries remains challenging. We propose a novel approach for direct map computation between triangle meshes without mapping to an intermediate domain, which optimizes for the harmonicity and reversibility of the forward and backward maps. Our method is general both in the information it can receive as input, e.g. point landmarks, a dense map or a functional map, and in the diversity of the geometries to which it can be applied. We demonstrate that our maps exhibit lower conformal distortion than the state-of-the-art, while succeeding in correctly mapping key features of the input shapes.

cs.GR↗

arXiv2012

Can Mean-Curvature Flow Be Made Non-Singular?

This work considers the question of whether mean-curvature flow can be modified to avoid the formation of singularities. We analyze the finite-elements discretization and demonstrate why the original flow can result in numerical instability due to division by zero. We propose a variation on the flow that removes the numerical instability in the discretization and show that this modification results in a simpler formulation for both the discretized and continuous formulations. We discuss the properties of the modified flow and present empirical evidence that not only does it define a stable surface evolution for genus-zero surfaces, but that the evolution converges to a conformal parameterization of the surface onto the sphere.

math.DG↗