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We show how to compute the circular area invariant of planar curves, and the spherical volume invariant of surfaces, in terms of line and surface integrals, respectively. We use the Divergence Theorem to express the area and volume…
We introduce a novel two-step approach for estimating a probability density function (pdf) given its samples, with the second and important step coming from a geometric formulation. The procedure involves obtaining an initial estimate of…
Localized features such as singularities, sharp gradients, discontinuities, and moving sources require adaptive finite element discretizations. Conventional refinement strategies introduce significant computational overhead through…
Omnidirectional depth sensing has its advantage over the conventional stereo systems since it enables us to recognize the objects of interest in all directions without any blind regions. In this paper, we propose a novel wide-baseline…
We develop a universal distributional calculus for regulated volumes of metrics that are singular along hypersurfaces. When the hypersurface is a conformal infinity we give simple integrated distribution expressions for the divergences and…
We describe a numerical framework that uses random sampling to efficiently capture low-rank local solution spaces of multiscale PDE problems arising in domain decomposition. In contrast to existing techniques, our method does not rely on…
Automated surface segmentation is important and challenging in many medical image analysis applications. Recent deep learning based methods have been developed for various object segmentation tasks. Most of them are a classification based…
This paper describes a method for fast simplification of surface meshes. Whereas past methods focus on visual appearance, our goal is to solve equations on the surface. Hence, rather than approximate the extrinsic geometry, we construct a…
This paper concerns a theoretical approach that combines topological data analysis (TDA) and sheaf theory. Topological data analysis, a rising field in mathematics and computer science, concerns the shape of the data and has been proven…
The aim of this paper is to provide a review of structural decomposition methods in discrete optimization and to give a unified framework in the form of local elimination algorithms (LEA). This paper is organized as follows. Local…
We present results of direct numerical simulation of incompressible fluid flow over a thick bed of mobile, spherically-shaped particles. The algorithm is based upon the immersed boundary technique for fluid-solid coupling and uses a…
We present a numerical method to efficiently solve optimization problems governed by large-scale nonlinear systems of equations, including discretized partial differential equations, using projection-based reduced-order models accelerated…
Knowledge of shape geometry plays a pivotal role in many shape analysis applications. In this paper we introduce a local geometry-inclusive global representation of 3D shapes based on computation of the shortest quasi-geodesic paths between…
We address the problem of mesh reconstruction from live RGB-D video, assuming a calibrated camera and poses provided externally (e.g., by a SLAM system). In contrast to most existing approaches, we do not fuse depth measurements in a volume…
An algorithm for the computation of global discrete conformal parametrizations with prescribed global holonomy signatures for triangle meshes was recently described in [Campen and Zorin 2017]. In this paper we provide a detailed analysis of…
Algorithmicists are well-aware that fast dynamic programming algorithms are very often the correct choice when computing on compositional (or even recursive) graphs. Here we initiate the study of how to generalize this folklore intuition to…
Semantic segmentation, which aims to classify every pixel in an image, is a key task in machine perception, with many applications across robotics and autonomous driving. Due to the high dimensionality of this task, most existing approaches…
Polytopes are the basic finite data structures for convex sets: they appear as feasible regions in linear optimization, as geometric summaries in algorithms, and as random objects in stochastic geometry. A natural geometric question is…
In this paper, we define and evaluate a weighting scheme for neighborhoods in point sets. Our weighting takes the shape of the geometry, i.e., the normal information, into account. This causes the obtained neighborhoods to be more reliable…
We present an immersed boundary method to simulate the creeping motion of a rigid particle in a fluid described by the Stokes equations discretized thanks to a finite element strategy on unfitted meshes, called Phi-FEM, that uses the…