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Modern geometric measure theory, developed largely to solve the Plateau problem, has generated a great deal of technical machinery which is unfortunately regarded as inaccessible by outsiders. Some of its tools (e.g., flat norm distance and…
The Intrinsic Surface Finite Element Method (ISFEM) was recently proposed to solve Partial Differential Equations (PDEs) on surfaces. ISFEM proceeds by writing the PDE with respect to a local coordinate system anchored to the surface and…
Point containment queries on trimmed surfaces are fundamental to CAD modeling, solid geometry processing, and surface tessellation. Existing approaches such as ray casting and generalized winding numbers often face limitations in robustness…
In this work, we study the perception problem for sampled surfaces (possibly with boundary) using tools from computational topology, specifically, how to identify their underlying topology starting from point-cloud samples in space, such as…
Generic event boundary detection aims to localize the generic, taxonomy-free event boundaries that segment videos into chunks. Existing methods typically require video frames to be decoded before feeding into the network, which contains…
In this paper, we consider a class of optimization problems constrained to the generalized Stiefel manifold. Such problems are fundamental to a wide range of real-world applications, including generalized canonical correlation analysis,…
In this paper, the generalized finite element method (GFEM) for solving second order elliptic equations with rough coefficients is studied. New optimal local approximation spaces for GFEMs based on local eigenvalue problems involving a…
We propose a new shape analysis approach based on the non-local analysis of local shape variations. Our method relies on a novel description of shape variations, called Local Probing Field (LPF), which describes how a local probing operator…
While texture analysis is largely addressed for images, the comparison of the geometric reliefs on surfaces embedded in the 3D space is still an open challenge. Starting from the Local Binary Pattern (LBP) description originally defined for…
This paper presents a new spatial-temporal nonlocal traffic flow model formulated to overcome the boundedness limitations inherent in classical local formulations. The model introduces an adaptive kernel that captures both spatial and…
We propose an efficient and accurate parametric finite element method (PFEM) for solving sharp-interface continuum models for solid-state dewetting of thin films with anisotropic surface energies. The governing equations of the…
The local analysis is an established approach to the study of singularities and mobility of linkages. Key result of such analyses is a local picture of the finite motion through a configuration. This reveals the finite mobility at that…
Collision detection is one of the most time-consuming operations during motion planning. Thus, there is an increasing interest in exploring machine learning techniques to speed up collision detection and sampling-based motion planning. A…
Networks are ubiquitous in various fields, representing systems where nodes and their interconnections constitute their intricate structures. We introduce a network decomposition scheme to reveal multiscale core-periphery structures lurking…
Traditional problems in computational geometry involve aspects that are both discrete and continuous. One such example is nearest-neighbor searching, where the input is discrete, but the result depends on distances, which vary continuously.…
We propose a parametric finite element method (PFEM) for efficiently solving the morphological evolution of solid-state dewetting of thin films on a flat rigid substrate in three dimensions (3D). The interface evolution of the dewetting…
Boundary representation (B-rep) is the industry standard for computer-aided design (CAD). While deep learning shows promise in processing B-rep models, existing methods suffer from a representation gap: continuous approaches offer…
Understanding the flow of deformable particles such as liquid drops, synthetic capsules and vesicles, and biological cells confined in a small channel is essential to a wide range of potential chemical and biomedical engineering…
The paper studies a finite element method for computing transport and diffusion along evolving surfaces. The method does not require a parametrization of a surface or an extension of a PDE from a surface into a bulk outer domain. The…
We construct global curves of rotational traveling wave solutions to the $2D$ water wave equations on a compact domain. The real analytic interface is subject to surface tension, while gravitational effects are ignored. In contrast to the…