Mark Gillespie
Research
Winding Numbers on Discrete Surfaces
ACM TOG (SIGGRAPH 2023)
Winding numbers are a basic component of geometric algorithms such as point-in-polygon tests, and their generalization to data with noise or topological errors has proven invaluable in robust geometry processing. However, standard definitions do not immediately apply on surfaces, where not all curves bound regions. We develop a meaningful generalization, starting with the well-known relationship between winding numbers and harmonic functions. Ultimately, our algorithm yields (i) a closed completion the input curves, (ii) integer labels for regions that are meaningfully bounded by these curves, and (iii) the complementary curves that do not bound any region.
Surface Simplification using Intrinsic Error Metrics
ACM TOG (SIGGRAPH 2023)
This paper describes a method for fast simplification of surface meshes. 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. The overall payoff is a “black box” approach to geometry processing, which decouples mesh resolution from the size of matrices used to solve equations.
Integer Coordinates for Intrinsic Geometry Processing
ACM TOG (SIGGRAPH Asia 2021)
This paper describes a numerically robust data structure for encoding intrinsic triangulations of polyhedral surfaces. Our starting point is the framework of normal coordinates from geometric topology, which we extend to a broader set of operations needed for mesh processing. As a stress test, we successfully compute an intrinsic Delaunay refinement and associated subdivision for all manifold meshes in the Thingi10k dataset.
project
pdf (7.6 mb)
arxiv
video (20 min)
video (5 min)
video (45 s)
code (C++ demo app)
code (C++ library)
bibtex
Geometry Processing with Intrinsic Triangulations
ACM SIGGRAPH Courses (2021), IMR 2021 Courses
Intrinsic triangulations de-couple the mesh used to encode geometry from the one used for computation. The basic shift in perspective is to encode the geometry of a mesh not in terms of ordinary vertex positions, but instead only in terms of edge lengths. This course provides a first introduction to intrinsic triangulations and their use in mesh processing algorithms, covering the theory, practical details, and some cutting-edge research in the area.
Discrete Conformal Equivalence of Polyhedral Surfaces
ACM TOG (SIGGRAPH 2021)
We present a numerical method for surface parameterization, leveraging hyperbolic geometry to yield maps that are locally injective and discretely conformal in an exact sense. Stress tests involving difficult cone configurations and near-degenerate triangulations indicate that the method is extremely robust in practice, and provides high-quality interpolation even on meshes with poor elements.
About Me
I'm a fifth year Computer Science PhD student at Carnegie Mellon University, advised by Keenan Crane. My research interests are in computer graphics and differential geometry. Recently, I have worked on computing discrete conformal maps and data structures for intrinsic triangulations.
Here is a blog that I write with some friends. And here is my CV.
In my spare time, I like to fold origami and knit. You can find more here.
Email: mgillesp@cs.cmu.eduAwards
2019 | NSF Graduate Research Fellowship |
2017 | SIGGRAPH ACM Turing Award Celebration Grant |
I was one of 10 students sponsored by SIGGRAPH to attend the ACM Turing Award Celebration. | |
2017, 2016 | Arthur R. Adams SURF Fellowship |
Fellowship to fund my summer research. | |
2016 | William Lowell Putnam Mathematics Competition |
31 points (rank: 365/3214) |
Education
Schools
- Carnegie Mellon University
- 2018-present
- PhD student in the Computer Science Department
- California Institute of Technology
- 2014-2018
- Majors: Computer Science, Mathematics
- GPA: 4.1
Talks
Apr. 2022 | Discrete Conformal Equivalence of Polyhedral Surfaces |
UCSD Pixel Cafe | |
45 minute talk about my paper of the same name. The slides are available here. | |
Mar. 2022 | Discrete Conformal Equivalence of Polyhedral Surfaces |
Toronto Geometry Colloquium | |
10 minute talk about my paper of the same name. The slides are available here, and the beautiful poster designed by Rachel Joan Wallis is available here. | |
Dec. 2021 | Integer Coordinates for Intrinsic Geometry Processing |
Siggraph Asia Technical Papers | |
20 minute talk about my paper of the same name. The full talk is available on YouTube here, and a 5-minute version is available here. | |
Aug. 2021 | Discrete Conformal Equivalence of Polyhedral Surfaces |
Siggraph Technical Papers | |
20 minute talk about my paper of the same name. The full talk is available on YouTube here, and a 5-minute version is available here. | |
Oct. 2019 | Origami and Geometry |
CMU Graphics Lab | |
1 hour talk overview of assorted algorithms for origami design, and how they relate to geometetry. The slides are available here. | |
Jun. 2019 | Hyperbolic Geometry and Discrete Conformal Maps |
CMU Graphics Lab | |
1 hour talk about discrete conformal maps, length cross ratios, and hyperbolic geometry. The slides are available here. | |
Jan. 2019 | Magnetohydrodynamics and the Geometry of Conservation Laws |
CMU Graphics Lab | |
1 hour talk on my previous summer research. Focused on general background about geometric mechanics and MHD. The slides are available here. | |
Oct. 2017 | 2D Plasma Simulation via Discrete Exterior Calculus |
Caltech Summer Research Seminar Day | |
15 minute presentation on the results of my summer research. The slides are available here. | |
Sept. 2017 | Combinatorics and the Probabilistic Method |
Westfield High School Seminar in College Mathematics | |
30 minute presentation to a high school math class. Gave an introduction to elementary combinatorics and presented some simple applications of the probabilistic method. My notes are available here. | |
Mar. 2017 | Continuous and Discrete Mechanics for Variational Integrators |
Caltech CS 177b | |
1.5 hour final presentation for a computer graphics class. Gave an overview of Hamiltonian and Lagrangian mechanics, and discussed how to discretize them to produce variational time integrators. Extended notes are available here. |
Miscellanea
Web demo which traces out geodesics on a mesh. The code is available here.
Notes on hyperbolic geometry and discrete conformal maps.
A shader to make meshes tartan.
Web demo computing the Karcher mean of points on a sphere.
Visualization of the phase space of a pendulum as a cylinder.
Comparison of explicit, implicit, and symplectic Euler integrators for a pendulum.
Online tool to check if a word is in the dictionary.