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Motivated by geometry processing for surfaces with non-trivial topology, we study discrete harmonic maps between closed surfaces of genus at least two. Harmonic maps provide a natural framework for comparing surfaces by minimizing…

Generating realistic 3D point clouds is a fundamental problem in computer vision with applications in remote sensing, robotics, and digital object modeling. Existing generative approaches primarily capture geometry, and when semantics are…

We introduce a continuous global optimization method to the field of surface reconstruction from discrete noisy cloud of points with weak information on orientation. The proposed method uses an energy functional combining flux-based…

Nanoparticles with "sticky patches" have long been proposed as building blocks for the self-assembly of complex structures. The synthetic realizability of such patchy particles, however, greatly lags behind predictions of patterns they…

Object parsing and segmentation from point clouds are challenging tasks because the relevant data is available only as thin structures along object boundaries or other features, and is corrupted by large amounts of noise. To handle this…

Generative models show great promise for the inverse design of molecules and inorganic crystals, but remain largely ineffective within more complex structures such as amorphous materials. Here, we present a diffusion model that reliably…

In this work, we explore links between natural homology and persistent homology for the classification of directed spaces. The former is an algebraic invariant of directed spaces, a semantic model of concurrent programs. The latter was…

Ring polymers are characterized by topology-specific entanglements called threadings. In the limit of large rings, it is conjectured that a "topological glass" should emerge due to the proliferation of threadings. In this study, we used…

In this work, we construct an effective continuum model for architected sheets that are composed of bulky tiles connected by slender elastic joints. Due to their mesostructure, these sheets feature quasi-mechanisms -- low-energy local…

Visualization of implicit surfaces is an actively researched topic. While raytracing can produce high quality images, it is not well suited for creating a quick preview of the surface. Indirect algorithms (e.g. Marching Cubes) create an…

Recent years have witnessed the surge of learned representations that directly build upon point clouds. Though becoming increasingly expressive, most existing representations still struggle to generate ordered point sets. Inspired by…

Persistent homology is a technique recently developed in algebraic and computational topology well-suited to analysing structure in complex, high-dimensional data. In this paper, we exposit the theory of persistent homology from first…

In this paper we introduce a statistic, the persistent homology transform (PHT), to model surfaces in $\mathbb{R}^3$ and shapes in $\mathbb{R}^2$. This statistic is a collection of persistence diagrams - multiscale topological summaries…

We propose to explore the properties of raw point clouds through the \emph{winding clearness}, a concept we first introduce for measuring the clarity of the interior/exterior relationships represented by the winding number field of the…

A method is presented for the distributed computation of persistent homology, based on an extension of the generalized Mayer-Vietoris principle to filtered spaces. Cellular cosheaves and spectral sequences are used to compute global…

Persistent homology is a popular computational tool for analyzing the topology of point clouds, such as the presence of loops or voids. However, many real-world datasets with low intrinsic dimensionality reside in an ambient space of much…

Dense point cloud generation from a sparse or incomplete point cloud is a crucial and challenging problem in 3D computer vision and computer graphics. So far, the existing methods are either computationally too expensive, suffer from…

A new method is given for computing generators of the homology groups with integer coefficients for any finite $T_0$-space. An important role in this method is played by irreducible cycles which are defined here and give rise to continuous…

We propose a new mechanism to create self-assembled porous media with highly tunable geometrical properties and permeabilities: We first allow a particle-stabilized emulsion to form from a mixture of two fluids and colloidal particles.…

Topological insulators show important properties, such as topological phase transitions and topological edge states. Although these properties and phenomena can be simulated by well-designed circuits, it is undoubtedly difficult to design…