arXiv subjects

Abhishek Madan

Publications and source records attributed to Abhishek Madan.

8 recordsLinked to original sources

arXiv2026

The Configurational Element Method for Nonconvex Granular Media

Granular media surround us, comprising everything from the ground we walk on to the foods we eat. Owing to their ubiquity our ability to understand and predict the mechanical evolution of grains is not only of key scientific importance, but is also a key component to synthesize believable animations of our world. Despite their importance, shortcomings persist in our ability to simulate granular media. In particular, simulating grains with non-convex shapes remains a challenging and computationally expensive task. We propose a method to simulate non-convex rigid grains by posing geometric contact in configuration space and learning the resulting contact map with a neural network. Our formulation reduces the complex task of modeling and simulating non-convex shapes to simple network evaluations that are easily run on standard compute hardware, allowing us to quickly and robustly simulate large scale systems of non-convex grains.

cs.GR↗

arXiv2025

Local Surface Parameterizations via Smoothed Geodesic Splines

We present a general method for computing local parameterizations rooted at a point on a surface, where the surface is described only through a signed implicit function and a corresponding projection function. Using a two-stage process, we compute several points radially emanating from the map origin, and interpolate between them with a spline surface. The narrow interface of our method allows it to support several kinds of geometry such as signed distance functions, general analytic implicit functions, triangle meshes, neural implicits, and point clouds. We demonstrate the high quality of our generated parameterizations on a variety of examples, and show applications in local texturing and surface curve drawing.

cs.GR↗

arXiv2025

Uniform Sampling of Surfaces by Casting Rays

Randomly sampling points on surfaces is an essential operation in geometry processing. This sampling is computationally straightforward on explicit meshes, but it is much more difficult on other shape representations, such as widely-used implicit surfaces. This work studies a simple and general scheme for sampling points on a surface, which is derived from a connection to the intersections of random rays with the surface. Concretely, given a subroutine to cast a ray against a surface and find all intersections, we can use that subroutine to uniformly sample white noise points on the surface. This approach is particularly effective in the context of implicit signed distance functions, where sphere marching allows us to efficiently cast rays and sample points, without needing to extract an intermediate mesh. We analyze the basic method to show that it guarantees uniformity, and find experimentally that it is significantly more efficient than alternative strategies on a variety of representations. Furthermore, we show extensions to blue noise sampling and stratified sampling, and applications to deform neural implicit surfaces as well as moment estimation.

cs.GR↗

arXiv2025

Stochastic Barnes-Hut Approximation for Fast Summation on the GPU

We present a novel stochastic version of the Barnes-Hut approximation. Regarding the level-of-detail (LOD) family of approximations as control variates, we construct an unbiased estimator of the kernel sum being approximated. Through several examples in graphics applications such as winding number computation and smooth distance evaluation, we demonstrate that our method is well-suited for GPU computation, capable of outperforming a GPU-optimized implementation of the deterministic Barnes-Hut approximation by achieving equal median error in up to 9.4x less time.

cs.GR↗

arXiv2022

Fast Evaluation of Smooth Distance Constraints on Co-Dimensional Geometry

We present a new method for computing a smooth minimum distance function based on the LogSumExp function for point clouds, edge meshes, triangle meshes, and combinations of all three. We derive blending weights and a modified Barnes-Hut acceleration approach that ensure our method approximates the true distance, and is conservative (points outside the zero isosurface are guaranteed to be outside the surface) and efficient to evaluate for all the above data types. This, in combination with its ability to smooth sparsely sampled and noisy data, like point clouds, shortens the gap between data acquisition and simulation, and thereby enables new applications such as direct, co-dimensional rigid body simulation using unprocessed lidar data.

cs.GR↗

arXiv2022

Multi-Agent Path Planning with Asymmetric Interactions In Tight Spaces

By starting with the assumption that motion is fundamentally a decision making problem, we use the world-line concept from Special Relativity as the inspiration for a novel multi-agent path planning method. We have identified a particular set of problems that have so far been overlooked by previous works. We present our solution for the global path planning problem for each agent and ensure smooth local collision avoidance for each pair of agents in the scene. We accomplish this by modeling the trajectories of the agents through 2D space and time as curves in 3D. Global path planning is solved using a modified Djikstra's algorithm to ensure that initial trajectories for agents do not intersect. We then solve for smooth local trajectories using a quasi-Newton interior point solver, providing the trajectory curves with a radius to turn them into rods. Subsequently, resolving collision of the rods ensures that no two agents are in the same spatial position at the same time. This space-time formulation allows us to simulate previously ignored phenomena such as highly asymmetric interactions in very constrained environments. It also provides a solution for scenes with unnaturally symmetric agent alignments without the need for jittering agent positions or velocities.

cs.RO↗

arXiv2020

Diffusion Structures for Architectural Stripe Pattern Generation

We present Diffusion Structures, a family of resilient shell structures from the eigenfunctions of a pair of novel diffusion operators. This approach is based on Michell's theorem but avoids expensive non-linear optimization with computation that amounts to constructing and solving two generalized eigenvalue problems to generate two sets of stripe patterns. This structure family can be generated quickly, and navigated in real-time using a small number of tuneable parameters.

cs.GR↗

arXiv2018

State Complexity of Pattern Matching in Regular Languages

In a simple pattern matching problem one has a pattern $w$ and a text $t$, which are words over a finite alphabet $Σ$. One may ask whether $w$ occurs in $t$, and if so, where? More generally, we may have a set $P$ of patterns and a set $T$ of texts, where $P$ and $T$ are regular languages. We are interested whether any word of $T$ begins with a word of $P$, ends with a word of $P$, has a word of $P$ as a factor, or has a word of $P$ as a subsequence. Thus we are interested in the languages $(PΣ^*)\cap T$, $(Σ^*P)\cap T$, $(Σ^* PΣ^*)\cap T$, and $(Σ^* \mathbin{\operatorname{shu}} P)\cap T$, where $\operatorname{shu}$ is the shuffle operation. The state complexity $κ(L)$ of a regular language $L$ is the number of states in the minimal deterministic finite automaton recognizing $L$. We derive the following upper bounds on the state complexities of our pattern-matching languages, where $κ(P)\le m$, and $κ(T)\le n$: $κ((PΣ^*)\cap T) \le mn$; $κ((Σ^*P)\cap T) \le 2^{m-1}n$; $κ((Σ^*PΣ^*)\cap T) \le (2^{m-2}+1)n$; and $κ((Σ^*\mathbin{\operatorname{shu}} P)\cap T) \le (2^{m-2}+1)n$. We prove that these bounds are tight, and that to meet them, the alphabet must have at least two letters in the first three cases, and at least $m-1$ letters in the last case. We also consider the special case where $P$ is a single word $w$, and obtain the following tight upper bounds: $κ((wΣ^*)\cap T_n) \le m+n-1$; $κ((Σ^*w)\cap T_n) \le (m-1)n-(m-2)$; $κ((Σ^*wΣ^*)\cap T_n) \le (m-1)n$; and $κ((Σ^*\mathbin{\operatorname{shu}} w)\cap T_n) \le (m-1)n$. For unary languages, we have a tight upper bound of $m+n-2$ in all eight of the aforementioned cases.

cs.FL↗