arXiv subjects

Mark Gillespie

Publications and source records attributed to Mark Gillespie.

4 recordsLinked to original sources

arXiv2026

Subgrid Marching Tetrahedra

We describe a method for recovering a manifold, intersection-free triangle mesh from the points where edges of a tetrahedral grid pierce a continuous surface. Unlike classic marching cubes or tets, our subgrid marching scheme allows arbitrarily many surface patches within a single cell, capturing fine features and thin sheets. Moreover, it requires neither a well-defined inside/outside (allowing surfaces with boundary), nor consistently-oriented input geometry. Yet we retain the local, parallel nature of classic marching: reconstruction is performed independently per tet, yielding a conforming mesh across tet boundaries. Our key innovation is a generalization of normal coordinates from geometric topology, which encode surface connectivity via arbitrary integer intersection counts along each grid edge. This encoding sidesteps the usual Nyquist--Shannon limit, putting no lower bound on the size of features that can be resolved on a fixed grid. In practice, for similar compute time and equal grid resolution -- or even an equal number of output triangles -- meshes produced by subgrid marching are far more accurate than those from classic marching. Beyond standard contouring, our method can be used to convert polygon soup into a manifold, intersection-free mesh.

cs.GR↗

arXiv2026

Implicit Minimal Surfaces for Bijective Correspondences

We introduce an implicit representation of continuous, bijective, orientation-preserving maps between genus zero surfaces with or without boundary. The distortion of these maps can easily be minimized by optimizing the Ginzburg-Landau functional - a ubiquitous model in physics and differential geometry - leading to a simple algorithm for computing bijective correspondences using only standard tools of the tangent vector field toolbox. The method avoids combinatorial mesh modifications and does not require barrier functions to enforce bijectivity making it more robust to noise and simpler to implement. Moreover, the algorithm does not assume a bijective initialization and can untangle non-bijective correspondences generated by computationally cheaper methods such as functional maps. It supports the use of both landmark points and landmark curves to guide the correspondence. The key idea is that a bijection between surfaces defines a two-dimensional mapping surface sitting inside the four-dimensional product space of the two inputs, and this mapping surface can be stored implicitly as the zero set of a complex section - essentially a complex function defined on the product space. Now the distortion of the map can be optimized by minimizing the area of this mapping surface, which amounts to minimizing the Ginzburg-Landau functional of the complex section. We demonstrate the practical benefits of our method by comparing to state-of-the-art correspondence algorithms and show that our implicit representation offers improved stability and naturally supports constraints that are difficult to enforce with explicit map representations.

cs.GR↗

arXiv2023

Surface Simplification using Intrinsic Error Metrics

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 coarse intrinsic triangulation of the input domain. In the spirit of the quadric error metric (QEM), we perform greedy decimation while agglomerating global information about approximation error. In lieu of extrinsic quadrics, however, we store intrinsic tangent vectors that track how far curvature "drifts" during simplification. This process also yields a bijective map between the fine and coarse mesh, and prolongation operators for both scalar- and vector-valued data. Moreover, we obtain hard guarantees on element quality via intrinsic retriangulation - a feature unique to the intrinsic setting. The overall payoff is a "black box" approach to geometry processing, which decouples mesh resolution from the size of matrices used to solve equations. We show how our method benefits several fundamental tasks, including geometric multigrid, all-pairs geodesic distance, mean curvature flow, geodesic Voronoi diagrams, and the discrete exponential map.

cs.GR↗

arXiv2021

Integer Coordinates for Intrinsic Geometry Processing

In this work, we present a general, efficient, and provably robust representation for intrinsic triangulations. These triangulations have emerged as a powerful tool for robust geometry processing of surface meshes, taking a low-quality mesh and retriangulating it with high-quality intrinsic triangles. However, existing representations either support only edge flips, or do not offer a robust procedure to recover the common subdivision, that is, how the intrinsic triangulation sits along the original surface. To build a general-purpose robust structure, we extend the framework of normal coordinates, which have been deeply studied in topology, as well as the more recent idea of roundabouts from geometry processing, to support a variety of mesh processing operations like vertex insertions, edge splits, etc. The basic idea is to store an integer per mesh edge counting the number of times a curve crosses that edge. We show that this paradigm offers a highly effective representation for intrinsic triangulations with strong robustness guarantees. The resulting data structure is general and efficient, while offering a guarantee of always encoding a valid subdivision. Among other things, this allows us to generate a high-quality intrinsic Delaunay refinement of all manifold meshes in the challenging Thingi10k dataset for the first time. This enables a broad class of existing surface geometry algorithms to be applied out-of-the-box to low-quality triangulations.

cs.GR↗