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Christopher Batty

Publications and source records attributed to Christopher Batty.

18 recordsLinked to original sources
arXiv2026

Greed for the Spheres: A Signed Distance Interpolation Method

We propose a method to interpolate Signed Distance Function (SDF) data from a discrete set of samples. Unlike prior work, our approach ensures that the new SDF data values are fully consistent with the input and each other, such that the augmented data still corresponds to a geometrically realizable surface. We express the theoretical properties of SDFs as hard geometric constraints, and construct an efficient greedy algorithm for consistent SDF interpolation that is made even faster with powerful parallelized GPU preprocessing. We exemplify the usefulness of our method by evaluating it on three practical applications: global SDF refinement, in which the SDF data is upsampled without knowledge of the ground truth; mesh reconstruction, where our method can reconstruct highly detailed surfaces using global information from coarse input SDFs; and repair of pseudo-SDFs, which result from many pipelines such as CSG Boolean operations and must be turned into valid SDFs for downstream processing tasks. Our refined SDFs are guaranteed to be consistent with the input, where previous methods have no such guarantee.

cs.GR↗
arXiv2026

SpUDD: Superpower Contouring of Unsigned Distance Data

Unsigned distance functions offer a powerful and flexible implicit surface representation that, unlike their signed counterparts, allow for surfaces that are open, non-orientable, or non-manifold. We consider the problem of reconstructing arbitrary surfaces from a finite set of samples of unsigned distance data. Existing methods for mesh reconstruction from distance data rely on sign information, accurate gradients, a corresponding continuous distance function, or extensive data-dependent training. However, they fail when applied to input that is both discrete and unsigned. Inspired by this challenge, we study the power diagram generated by the distance samples and propose a novel theoretical concept, the superpower contour, which we prove converges to the true surface in the limit of sampling density. We use this superpower contour as an initial surface proxy and design an algorithm that leverages it to produce a polygonal mesh approximating the unknown true geometry. Our method vastly outperforms other conceivable strategies for the discrete unsigned distance reconstruction task, and sets the stage for future work on this mathematically rich problem.

cs.GR↗
arXiv2026

Dual Contouring of Signed Distance Data

We propose an algorithm to reconstruct explicit polygonal meshes from discretely sampled Signed Distance Function (SDF) data, which is especially effective at recovering sharp features. Building on the traditional Dual Contouring of Hermite Data method, we design and solve a quadratic optimization problem to decide the optimal placement of the mesh's vertices within each cell of a regular grid. Critically, this optimization relies solely on discretely sampled SDF data, without requiring arbitrary access to the function, gradient information, or training on large-scale datasets. Our method sets a new state of the art in surface reconstruction from SDFs at medium and high resolutions, and opens the door for applications in 3D modeling and design.

cs.GR↗
arXiv2026

Monte Carlo Rendering to Diffusion Curves with Differential BEM

We present a method for generating vector graphics, in the form of diffusion curves, directly from noisy samples produced by a Monte Carlo renderer. While generating raster images from 3D geometry via Monte Carlo raytracing is commonplace, there is no corresponding practical approach for robustly and directly extracting editable vector images with shading information from 3D geometry. To fill this gap, we formulate the problem as a stochastic optimization problem over the space of geometries and colors of diffusion curve handles, and solve it with the Levenberg-Marquardt algorithm. At the core of our method is a novel differential boundary element method (BEM) framework that reconstructs colors from diffusion curve handles and computes gradients with respect to their parameters, requiring the expensive matrix factorization only once at the beginning of the optimization. Unlike triangulation-based techniques that require a clean domain decomposition, our method is robust to geometrically challenging scenarios, such as intersecting diffusion curves, and to color noise in the target image, enabling the direct use of noisy Monte Carlo samples without requiring a converged, error-free input image. We demonstrate the robustness and broad applicability of our approach across several test cases. Finally, we highlight several open questions raised by our work, which spans both theory and applications.

cs.GR↗
arXiv2024

Optimizing Parameters for Static Equilibrium of Discrete Elastic Rods with Active-Set Cholesky

We propose a parameter optimization method for achieving static equilibrium of discrete elastic rods. Our method simultaneously optimizes material stiffness and rest shape parameters under box constraints to exactly enforce zero net force while avoiding stability issues and violations of physical laws. For efficiency, we split our constrained optimization problem into primal and dual subproblems via the augmented Lagrangian method, while handling the dual subproblem via simple vector updates. To efficiently solve the box-constrained primal subproblem, we propose a new active-set Cholesky preconditioner. Our method surpasses prior work in generality, robustness, and speed.

cs.GR↗
arXiv2024

Projected Walk on Spheres: A Monte Carlo Closest Point Method for Surface PDEs

We present projected walk on spheres (PWoS), a novel pointwise and discretization-free Monte Carlo solver for surface PDEs with Dirichlet boundaries, as a generalization of the walk on spheres method (WoS) [Muller 1956; Sawhney and Crane 2020]. We adapt the recursive relationship of WoS designed for PDEs in volumetric domains to a volumetric neighborhood around the surface, and at the end of each recursion step, we project the sample point on the sphere back to the surface. We motivate this simple modification to WoS with the theory of the closest point extension used in the closest point method. To define the valid volumetric neighborhood domain for PWoS, we develop strategies to estimate the local feature size of the surface and to compute the distance to the Dirichlet boundaries on the surface extended in their normal directions. We also design a mean value filtering method for PWoS to improve the method's efficiency when the surface is represented as a polygonal mesh or a point cloud. Finally, we study the convergence of PWoS and demonstrate its application to graphics tasks, including diffusion curves, geodesic distance computation, and wave propagation animation. We show that our method works with various types of surfaces, including a surface of mixed codimension.

math.NA↗
arXiv2024

Galerkin Method of Regularized Stokeslets for Procedural Fluid Flow with Control Curves

We present a new procedural incompressible velocity field authoring tool, which lets users design a volumetric flow by directly specifying velocity along control curves. Our method combines the Method of Regularized Stokeslets with Galerkin discretization. Based on the highly viscous Stokes flow assumption, we find the force along a given set of curves that satisfies the velocity constraints along them. We can then evaluate the velocity anywhere inside the surrounding infinite 2D or 3D domain. We also show the extension of our method to control the angular velocity along control curves. Compared to a collocation discretization, our method is not very sensitive to the vertex sampling rate along control curves and only requires a small linear system solve.

cs.GR↗
arXiv2024

Rest Shape Optimization for Sag-Free Discrete Elastic Rods

We propose a new rest shape optimization framework to achieve sag-free simulations of discrete elastic rods. To optimize rest shape parameters, we formulate a minimization problem based on the kinetic energy with a regularizer while imposing box constraints on these parameters to ensure the system's stability. Our method solves the resulting constrained minimization problem via the Gauss-Newton algorithm augmented with penalty methods. We demonstrate that the optimized rest shape parameters enable discrete elastic rods to achieve static equilibrium for a wide range of strand geometries and material parameters.

cs.GR↗
arXiv2024

Velocity-Based Monte Carlo Fluids

We present a velocity-based Monte Carlo fluid solver that overcomes the limitations of its existing vorticity-based counterpart. Because the velocity-based formulation is more commonly used in graphics, our Monte Carlo solver can be readily extended with various techniques from the fluid simulation literature. We derive our method by solving the Navier-Stokes equations via operator splitting and designing a pointwise Monte Carlo estimator for each substep. We reformulate the projection and diffusion steps as integration problems based on the recently introduced walk-on-boundary technique [Sugimoto et al. 2023]. We transform the volume integral arising from the source term of the pressure Poisson equation into a form more amenable to practical numerical evaluation. Our resulting velocity-based formulation allows for the proper simulation of scenes that the prior vorticity-based Monte Carlo method [Rioux-Lavoie and Sugimoto et al. 2022] either simulates incorrectly or cannot support. We demonstrate that our method can easily incorporate advancements drawn from conventional non-Monte Carlo methods by showing how one can straightforwardly add buoyancy effects, divergence control capabilities, and numerical dissipation reduction methods, such as advection-reflection and PIC/FLIP methods.

cs.GR↗
arXiv2023

Reach For the Spheres: Tangency-Aware Surface Reconstruction of SDFs

Signed distance fields (SDFs) are a widely used implicit surface representation, with broad applications in computer graphics, computer vision, and applied mathematics. To reconstruct an explicit triangle mesh surface corresponding to an SDF, traditional isosurfacing methods, such as Marching Cubes and and its variants, are typically used. However, these methods overlook fundamental properties of SDFs, resulting in reconstructions that exhibit severe oversmoothing and feature loss. To address this shortcoming, we propose a novel method based on a key insight: each SDF sample corresponds to a spherical region that must lie fully inside or outside the surface, depending on its sign, and that must be tangent to the surface at some point. Leveraging this understanding, we formulate an energy that gauges the degree of violation of tangency constraints by a proposed surface. We then employ a gradient flow that minimizes our energy, starting from an initial triangle mesh that encapsulates the surface. This algorithm yields superior reconstructions to previous methods, even with sparsely sampled SDFs. Our approach provides a more nuanced understanding of SDFs and offers significant improvements in surface reconstruction.

cs.GR↗
arXiv2023

A Practical Walk-on-Boundary Method for Boundary Value Problems

We introduce the walk-on-boundary (WoB) method for solving boundary value problems to computer graphics. WoB is a grid-free Monte Carlo solver for certain classes of second order partial differential equations. A similar Monte Carlo solver, the walk-on-spheres (WoS) method, has been recently popularized in computer graphics due to its advantages over traditional spatial discretization-based alternatives. We show that WoB's intrinsic properties yield further advantages beyond those of WoS. Unlike WoS, WoB naturally supports various boundary conditions (Dirichlet, Neumann, Robin, and mixed) for both interior and exterior domains. WoB builds upon boundary integral formulations, and it is mathematically more similar to light transport simulation in rendering than the random walk formulation of WoS. This similarity between WoB and rendering allows us to implement WoB on top of Monte Carlo ray tracing, and to incorporate advanced rendering techniques (e.g., bidirectional estimators with multiple importance sampling, the virtual point lights method, and Markov chain Monte Carlo) into WoB. WoB does not suffer from the intrinsic bias of WoS near the boundary and can estimate solutions precisely on the boundary. Our numerical results highlight the advantages of WoB over WoS as an attractive alternative to solve boundary value problems based on Monte Carlo.

cs.GR↗
arXiv2023

A Closest Point Method for PDEs on Manifolds with Interior Boundary Conditions for Geometry Processing

Many geometry processing techniques require the solution of partial differential equations (PDEs) on manifolds embedded in $\mathbb{R}^2$ or $\mathbb{R}^3$, such as curves or surfaces. Such manifold PDEs often involve boundary conditions (e.g., Dirichlet or Neumann) prescribed at points or curves on the manifold's interior or along the geometric (exterior) boundary of an open manifold. However, input manifolds can take many forms (e.g., triangle meshes, parametrizations, point clouds, implicit functions, etc.). Typically, one must generate a mesh to apply finite element-type techniques or derive specialized discretization procedures for each distinct manifold representation. We propose instead to address such problems in a unified manner through a novel extension of the closest point method (CPM) to handle interior boundary conditions. CPM solves the manifold PDE by solving a volumetric PDE defined over the Cartesian embedding space containing the manifold, and requires only a closest point representation of the manifold. Hence, CPM supports objects that are open or closed, orientable or not, and of any codimension. To enable support for interior boundary conditions we derive a method that implicitly partitions the embedding space across interior boundaries. CPM's finite difference and interpolation stencils are adapted to respect this partition while preserving second-order accuracy. Additionally, we develop an efficient sparse-grid implementation and numerical solver. We demonstrate our method's convergence behaviour on selected model PDEs and explore several geometry processing problems: diffusion curves on surfaces, geodesic distance, tangent vector field design, harmonic map construction, and reaction-diffusion textures. Our proposed approach thus offers a powerful and flexible new tool for a range of geometry processing tasks on general manifold representations.

cs.GR↗
arXiv2021

Curl-Flow: Boundary-Respecting Pointwise Incompressible Velocity Interpolation for Grid-Based Fluids

We propose to augment standard grid-based fluid solvers with pointwise divergence-free velocity interpolation, thereby ensuring exact incompressibility down to the sub-cell level. Our method takes as input a discretely divergence-free velocity field generated by a staggered grid pressure projection, and first recovers a corresponding discrete vector potential. Instead of solving a costly vector Poisson problem for the potential, we develop a fast parallel sweeping strategy to find a candidate potential and apply a gauge transformation to enforce the Coulomb gauge condition and thereby make it numerically smooth. Interpolating this discrete potential generates a pointwise vector potential whose analytical curl is a pointwise incompressible velocity field. Our method further supports irregular solid geometry through the use of level set-based cut-cells and a novel Curl-Noise-inspired potential ramping procedure that simultaneously offers strictly non-penetrating velocities and incompressibility. Experimental comparisons demonstrate that the vector potential reconstruction procedure at the heart of our approach is consistently faster than prior such reconstruction schemes, especially those that solve vector Poisson problems. Moreover, in exchange for its modest extra cost, our overall Curl-Flow framework produces significantly improved particle trajectories that closely respect irregular obstacles, do not suffer from spurious sources or sinks, and yield superior particle distributions over time.

cs.GR↗
arXiv2020

Asynchronous Liquids: Regional Time Stepping for Faster SPH and PCISPH

This paper presents novel and efficient strategies to spatially adapt the amount of computational effort applied based on the local dynamics of a free surface flow, for both classic weakly compressible SPH (WCSPH) and predictive-corrective incompressible SPH (PCISPH). Using a convenient and readily parallelizable block-based approach, different regions of the fluid are assigned differing time steps and solved at different rates to minimize computational cost. Our approach for WCSPH scheme extends an asynchronous SPH technique from compressible flow of astrophysical phenomena to the incompressible free surface setting, and further accelerates it by entirely decoupling the time steps of widely spaced particles. Similarly, our approach to PCISPH adjusts the the number of iterations of density correction applied to different regions, and asynchronously updates the neighborhood regions used to perform these corrections; this sharply reduces the computational cost of slowly deforming regions while preserving the standard density invariant. We demonstrate our approaches on a number of highly dynamic scenarios, demonstrating that they can typically double the speed of a simulation compared to standard methods while achieving visually consistent results.

cs.GR↗
arXiv2017

Constraint Bubbles: Adding Efficient Zero-Density Bubbles to Incompressible Free Surface Flow

Liquid simulations for computer animation often avoid simulating the air phase to reduce computational costs and ensure good conditioning of the linear systems required to enforce incompressibility. However, this free surface assumption leads to an inability to realistically treat bubbles: submerged gaps in the liquid are interpreted as empty voids that immediately collapse. To address this shortcoming, we present an efficient, practical, and conceptually simple approach to augment free surface flows with negligible density bubbles. Our method adds a new constraint to each disconnected air region that guarantees zero net flux across its entire surface, and requires neither simulating both phases nor reformulating into stream function variables. Implementation of the method requires only minor modifications to the pressure solve of a standard grid-based fluid solver, and yields linear systems that remain sparse and symmetric positive definite. In our evaluations, solving the modified pressure projection system took no more than 10% longer than the corresponding free surface solve. We demonstrate the method's effectiveness and flexibility by incorporating it into commercial fluid animation software and using it to generate a variety of dynamic bubble scenarios showcasing glugging effects, viscous and inviscid bubbles, interactions with irregularly-shaped and moving solid boundaries, and surface tension effects.

cs.GR↗
arXiv2013

Multimaterial Front Tracking

We present the first triangle mesh-based technique for tracking the evolution of general three-dimensional multimaterial interfaces undergoing complex topology changes induced by deformations and collisions. Our core representation is a non-manifold triangle surface mesh with material labels assigned to each half-face to distinguish volumetric regions. We advect the vertices of the mesh in a Lagrangian manner, and employ a complete set of collision-safe mesh improvement and topological operations that track and update material labels. In particular, we develop a unified, collision-safe strategy for handling complex topological operations acting on evolving triple- and higher-valence junctions, and a flexible method to merge colliding multimaterial meshes. We demonstrate our approach with a number of challenging geometric flows, including passive advection, normal flow, and mean curvature flow.

cs.GR↗
arXiv2010

A simple finite difference method for time-dependent, variable coefficient Stokes flow on irregular domains

We present a simple and efficient variational finite difference method for simulating time-dependent Stokes flow in the presence of irregular free surfaces and moving solid boundaries. The method uses an embedded boundary approach on staggered Cartesian grids, avoiding the need for expensive remeshing operations, and can be applied to flows in both two and three dimensions. It uses fully implicit backwards Euler integration to provide stability and supports spatially varying density and viscosity, while requiring the solution of just a single sparse, symmetric positive-definite linear system per time step. By expressing the problem in a variational form, challenging irregular domains are supported implicitly through the use of natural boundary conditions. In practice, the discretization requires only centred finite difference stencils and per-cell volume fractions, and is straightforward to implement. The variational form further permits generalizations to coupling other mechanics, all the while reducing to a sparse symmetric positive definite matrix. We demonstrate consistent first order convergence of velocity in L1 and Linf norms on a range of analytical test cases in two dimensions. Furthermore, we apply our method as part of a simple Navier-Stokes solver to illustrate that it can reproduce the characteristic jet buckling phenomenon of Newtonian liquids at moderate viscosities, in both two and three dimensions.

physics.comp-ph↗