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We study the joint image of lines incident to points, meaning the set of image tuples obtained from fixed cameras observing a varying 3D point-line incidence. We prove a formula for the number of complex critical points of the triangulation…
Computation fundamentally separates time from space: nondeterministic search is exponential in time but polynomially simulable in space (Savitch's Theorem). We propose that the brain physically instantiates a biological variant of this…
Reconstructing models from unorganized point clouds presents a significant challenge, especially when the models consist of multiple components represented by their surface point clouds. Such models often involve point clouds with noise…
In the presence of occlusions and measurement noise, geometrically accurate scene reconstructions -- which fit the sensor data -- can still be physically incorrect. For instance, when estimating the poses and shapes of objects in the scene…
We prove that for a given flat surface with conical singularities, any pair of geometric triangulations can be connected by a chain of flips.
In a previous work, we proved that each minimal symplectic filling of any oriented lens space, viewed as the singularity link of some cyclic quotient singularity and equipped with its canonical contact structure, can be obtained from the…
Many computations in robotics can be dramatically accelerated if the robot configuration space is described as a collection of simple sets. For example, recently developed motion planners rely on a convex decomposition of the free space to…
For the past decades we have experienced an enormous expansion of the accumulated data that humanity produces. Daily a numerous number of smart devices, usually interconnected over internet, produce vast, real-values datasets. Time series…
We show that $O(n^2)$ exchanging flips suffice to transform any edge-labelled pointed pseudo-triangulation into any other with the same set of labels. By using insertion, deletion and exchanging flips, we can transform any edge-labelled…
In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one…
Algorithms for persistent homology and zigzag persistent homology are well-studied for persistence modules where homomorphisms are induced by inclusion maps. In this paper, we propose a practical algorithm for computing persistence under…
Given a metric space X and a distance threshold r>0, the Vietoris-Rips simplicial complex has as its simplices the finite subsets of X of diameter less than r. A theorem of Jean-Claude Hausmann states that if X is a Riemannian manifold and…
Finite systems in confining potentials are known to undergo structural transitions similar to phase transitions. However, these systems are inhomogeneous, and their "melting" point may depend on the position in the trap and vary with the…
This paper develops a complete foundational treatment of simplicial complexes from Euclidean spaces through geometric realizations, emphasizing concrete computations, examples, and practical verification methods. Beginning with finite point…
State-of-the-art image segmentation algorithms generally consist of at least two successive and distinct computations: a boundary detection process that uses local image information to classify image locations as boundaries between objects,…
When three-dimensional bodies contain thin features, non-trivial topology, or scan-derived surfaces, volumetric meshing can become the dominant bottleneck in simulation workflows. We replace this step with a learned geometric…
We introduce a family of copulas which are locally piecewise uniform in the interior of the unit cube of any given dimension. Within that family, the simultaneous control of tail dependencies of all projections to faces of the cube is…
A $d$-dimensional simplicial complex is balanced if the underlying graph is $(d+1)$-colorable. We present an implementation of cross-flips, a set of local moves introduced by Izmestiev, Klee and Novik which connect any two PL-homeomorphic…
We present two-dimensional simulations of finite, self-gravitating gaseous sheets. Unlike the case of infinite sheets, such configurations do not constitute equilibrium states but instead are subject to global collapse unless countered by…
We propose the CliPS procedure when fitting Bayesian mixture models in the context of model-based clustering to identify the cluster distributions while simultaneously assessing the suitability of a cluster solution and validating the…