arXiv subjects

Chandrajit L. Bajaj

Publications and source records attributed to Chandrajit L. Bajaj.

3 recordsLinked to original sources

arXiv2023

VoroCrust: Voronoi Meshing Without Clipping

Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrarily curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations. The VoroCrust algorithm is the first provably-correct algorithm for conforming polyhedral Voronoi meshing for non-convex and non-manifold domains with guarantees on the quality of both surface and volume elements. A robust refinement process estimates a suitable sizing field that enables the careful placement of Voronoi seeds across the surface circumventing the need for clipping and avoiding its many drawbacks. The algorithm has the flexibility of filling the interior by either structured or random samples, while preserving all sharp features in the output mesh. We demonstrate the capabilities of the algorithm on a variety of models and compare against state-of-the-art polyhedral meshing methods based on clipped Voronoi cells establishing the clear advantage of VoroCrust output.

cs.GR↗

arXiv2022

Probabilistic PolarGMM: Unsupervised Cluster Learning of Very Noisy Projection Images of Unknown Pose

A crucial step in single particle analysis (SPA) of cryogenic electron microscopy (Cryo-EM), 2D classification and alignment takes a collection of noisy particle images to infer orientations and group similar images together. Averaging these aligned and clustered noisy images produces a set of clean images, ready for further analysis such as 3D reconstruction. Fourier-Bessel steerable principal component analysis (FBsPCA) enables an efficient, adaptable, low-rank rotation operator. We extend the FBsPCA to additionally handle translations. In this extended FBsPCA representation, we use a probabilistic polar-coordinate Gaussian mixture model to learn soft clusters in an unsupervised fashion using an expectation maximization (EM) algorithm. The obtained rotational clusters are thus additionally robust to the presence of pairwise alignment imperfections. Multiple benchmarks from simulated Cryo-EM datasets show probabilistic PolarGMM's improved performance in comparisons with standard single-particle Cryo-EM tools, EMAN2 and RELION, in terms of various clustering metrics and alignment errors.

cs.CV↗

arXiv2018

Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm

We study the problem of decomposing a volume bounded by a smooth surface into a collection of Voronoi cells. Unlike the dual problem of conforming Delaunay meshing, a principled solution to this problem for generic smooth surfaces remained elusive. VoroCrust leverages ideas from $α$-shapes and the power crust algorithm to produce unweighted Voronoi cells conforming to the surface, yielding the first provably-correct algorithm for this problem. Given an $ε$-sample on the bounding surface, with a weak $σ$-sparsity condition, we work with the balls of radius $δ$ times the local feature size centered at each sample. The corners of this union of balls are the Voronoi sites, on both sides of the surface. The facets common to cells on opposite sides reconstruct the surface. For appropriate values of $ε$, $σ$ and $δ$, we prove that the surface reconstruction is isotopic to the bounding surface. With the surface protected, the enclosed volume can be further decomposed into an isotopic volume mesh of fat Voronoi cells by generating a bounded number of sites in its interior. Compared to state-of-the-art methods based on clipping, VoroCrust cells are full Voronoi cells, with convexity and fatness guarantees. Compared to the power crust algorithm, VoroCrust cells are not filtered, are unweighted, and offer greater flexibility in meshing the enclosed volume by either structured grids or random samples.

cs.CG↗