Tim Ophelders

Thijs van der Horst, Tim Ophelders

The Fréchet distance is a well-studied distance measure for paths in a metric space. It is mostly studied for paths in $d$-dimensional Euclidean space. Here, computing the Fréchet distance between two polylines takes time roughly quadratic in the number of vertices. Assuming the strong exponential time hypothesis (SETH), it cannot be approximated to within a factor less than $3$ in strongly-subquadratic time. Recently, it was shown that for any $\varepsilon>0$, there exists a randomized algorithm that can compute a $(7+\varepsilon)$-approximation in strongly-subquadratic expected time [Cheng, Huang, and Zhang; STOC'25]. For polylines with $n$ and $m$ vertices in a Euclidean space of constant dimension, where $n \geq m$, their algorithm takes $O(nm^{0.99} \log(n/\varepsilon))$ time in expectation. We present a deterministic approximation algorithm that significantly improves upon the approximation factor and running time. Specifically, our algorithm computes a $(3+\varepsilon)$-approximation in $O(nm^{2/3} \log n \cdot \log (\frac{1}{\varepsilon} \log n))$ time. Our algorithm nearly matches the conditional lower bound on the approximation factor implied by SETH. For polylines in $\mathbb{R}$, we present a $3$-approximation algorithm that runs in $O(nm^{2/3} \log^{5/3} n)$ time, and exactly matches the conditional lower bound. For our results, we introduce a general strongly-subquadratic time $3$-approximate decision algorithm. This algorithm makes no assumptions on the ambient metric space, and relies only on standard assumptions on the so-called free space of the input paths. Under some mild assumptions, our decision algorithm leads to a $(3+\varepsilon)$-approximation algorithm in general metric spaces. These assumptions hold automatically for polylines in any metric space $(\mathbb{R}^d, L_p)$ with $p \geq 1$.

Jette Gutzeit, Kalani Kistler, Tim Ophelders et al.

The verbose persistent homology transform (VPHT) is a topological summary of shapes in Euclidean space. Assuming general position, the VPHT is injective, meaning shapes can be reconstructed using only the VPHT. In this work, we investigate cases in which the VPHT is not injective, focusing on a simple setting of degeneracy; graphs whose vertices are all collinear. We identify both necessary properties and sufficient properties for non-reconstructibility of such graphs, bringing us closer to a complete classification.

Kayla Makela, Tim Ophelders, Michelle Quigley et al.

Tree ring widths are an important source of climatic and historical data, but measuring these widths typically requires extensive manual work. Computer vision techniques provide promising directions towards the automation of tree ring detection, but most automated methods still require a substantial amount of user interaction to obtain high accuracy. We perform analysis on 3D X-ray CT images of a cross-section of a tree trunk, known as a tree disk. We present novel automated methods for locating the pith (center) of a tree disk, and ring boundaries. Our methods use a combination of standard image processing techniques and tools from topological data analysis. We evaluate the efficacy of our method for two different CT scans by comparing its results to manually located rings and centers and show that it is better than current automatic methods in terms of correctly counting each ring and its location. Our methods have several parameters, which we optimize experimentally by minimizing edit distances to the manually obtained locations.