Monte Carlo geometry processing

Abstract

This paper explores how core problems in PDE-based geometry processing can be efficiently and reliably solved via grid-free Monte Carlo methods. Modern geometric algorithms often need to solve Poisson-like equations on geometrically intricate domains. Conventional methods most often mesh the domain, which is both challenging and expensive for geometry with fine details or imperfections (holes, self-intersections, etc. ). In contrast, grid-free Monte Carlo methods avoid mesh generation entirely, and instead just evaluate closest point queries. They hence do not discretize space, time, nor even function spaces, and provide the exact solution (in expectation) even on extremely challenging models. More broadly, they share many benefits with Monte Carlo methods from photorealistic rendering: excellent scaling, trivial parallel implementation, view-dependent evaluation, and the ability to work with any kind of geometry (including implicit or procedural descriptions). We develop a complete "black box" solver that encompasses integration, variance reduction, and visualization, and explore how it can be used for various geometry processing tasks. In particular, we consider several fundamental linear elliptic PDEs with constant coefficients on solid regions of R n. Overall we find that Monte Carlo methods significantly broaden the horizons of geometry processing, since they easily handle problems of size and complexity that are essentially hopeless for conventional methods.

Walk on Stars: A Grid-Free Monte Carlo Method for PDEs with Neumann Boundary Conditions

“…Monte Carlo methods. Monte Carlo estimators for linear elliptic PDEs with Dirichlet boundaries, such as WoS, date back to Muller [1956], and have recently received renewed interest following their introduction to graphics by Sawhney and Crane [2020]. This has resulted in rapid advances along two main thrusts: First, increasing efficiency through optimized implementations [Mossberg 2021], bidirectional formulations [Qi et al 2022], and sample caching techniques [Miller et al 2023].…”

“…Prompted by the disparity between rendering and simulation methods, Sawhney and Crane [2020] advocate the use of grid-free Monte Carlo methods to solve partial differential equations (PDEs) on domains of extreme geometric complexity. Such methods need not discretize the problem domain (as in finite difference methods), nor even pick a finite basis of functions (as in finite element and boundary element methods).…”

A Hybrid Boundary Element and Boundary Integral Equation Method for Accurate Diffusion Curves

“…Diffusion curves can also be evaluated using stochastic methods. The fully meshless Walk on Spheres (WoS) [Sawhney and Crane 2020], on the other hand, does diffuse colors around obstacles. However, WoS has difficulties with Neumann boundary conditions, and this turns out to be a major limitation, since such boundary conditions turn out to be exceedingly useful in practice.…”

Drift-Diffusion Dynamics and Phase Separation in Curved Cell Membranes and Dendritic Spines: Hybrid Discrete-Continuum Methods

“…Methods modeling at the level of continuum fields and partial differential equation (PDE) descriptions include continuum concentration and phase fields in vesicles in [17], protein aggregation in [18], and phase separation in [19]. Methods modeling at the level of particles include Monte-Carlo (MC) Methods and Kinetic MC (KMC) in [20][21][22][23][24][25][26], Molecular Dynamics (MD) studies in [27][28][29], and Coarse-Grained (CG) Models in [30,31]. Some work has been done on hybrid discrete-continuum approaches for membranes in [32][33][34][35][36][37][38], and taking into account geometric effects in [35,39], and through point-cloud representations in [26,[40][41][42][43].…”

“…Methods modeling at the level of particles include Monte-Carlo (MC) Methods and Kinetic MC (KMC) in [20][21][22][23][24][25][26], Molecular Dynamics (MD) studies in [27][28][29], and Coarse-Grained (CG) Models in [30,31]. Some work has been done on hybrid discrete-continuum approaches for membranes in [32][33][34][35][36][37][38], and taking into account geometric effects in [35,39], and through point-cloud representations in [26,[40][41][42][43].…”