cs.CGMar 9, 2026

Topologically Stable Hough Transform

Authors: Stefan Huber, Kristóf Huszár, Michael Kerber, Martin Uray

Organizations: Josef Ressel Centre for Intelligent and Secure Industrial Automation, Salzburg University of Applied Sciences, Austria · Institute of Geometry, Graz University of Technology, Austria

Read PDF
Ask AI
Cite

Copy citation

DOINot available for this paper

Abstract

We propose an alternative formulation of the well-known Hough transform to detect lines in point clouds. Replacing the discretized voting scheme of the classical Hough transform by a continuous score function, its persistent features in the sense of persistent homology give a set of candidate lines. We also devise and implement an algorithm to efficiently compute these candidate lines.

Figures & tables

Figure 1: Left: Ten points sampled from two lines. Middle: The classical Hough transform passes to the space of lines in which every input point dualizes into a sinusoidal curve. The blue point on the left, for instance, is transformed to the blue curve in the middle. The pixels that the blue curve votes for are marked. In that example, there are two pixels (red) that receive five votes. Right: The centers of the high-count pixels are two lines in primal space that correspond to the two drawn lines.
Figure 2: Top: the subdivided S~P\tilde{\mathrm{S}}_{P} . Bottom: The super-levelsets for threshold 0.80.8 , 0.70.7 , and 0.60.6 , and the nerve of the boxes (the graph structure in blue). Vertices and edges that enter the graph at this threshold are drawn thicker. We observe that for 0.80.8 , a new connected component is formed in the upper region. For 0.70.7 , another isolated vertex is formed; moreover, the upper-right box connects with the lower-left box because of the Möbius geometry, hence, the component formed at 0.80.8 merges with another one, yielding a persistence of 0.10.1 for the component. At 0.60.6 , the component formed at 0.70.7 also gets merged, yielding a persistence of 0.10.1 as well. Note that the graph is connected for 0.60.6 .
Figure 3: Illustration of the proposed method.
Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Illustration of the admissible threshold ranges ΔVote\Delta_{\text{Vote}} and ΔPers\Delta_{\text{Pers}} in Hough space for persistence-based thresholding (our method) and discrete, vote-based thresholding ( OpenCV ).
Figure 5: Results of the experiment evaluating detection quality using the Euclidean norm in normalized parameter space and the absolute errors in rr and Θ\Theta relative to the ground-truth lines, computed from the best pairwise matches.
Figure 6: Analysis of the optimal kernel width σ\sigma . We plot the error as a function of the ratio between the noise level (in pixels) and σ\sigma . Each configuration is repeated 20 times and the mean error is reported. The minimum for each configuration is marked by a red dot and a vertical dashed line. The distribution of all minima is shown in the bottom boxplot.
Figure 7: Detection performance and runtime (ms) as a function of ϵ\epsilon , evaluated over 20 trials per configuration on a single random line with 18 sample points on a 32×3232\times 32 image.
More in this topic

Point Cloud Understanding

Point cloud understanding turns sparse 3D measurements into object, scene, shape, and motion information for machines.

See all 45 papers

Explore similar work

CardsList

  1. Topological data analysis using persistent discrete homology

    Jun 17, 2025Chris Kapulkin, Nathan KershawNoise RobustnessPersistent Homology

  2. Hough-SIFT: Robust Image Registration for Linear Structures via Hough Space

    Jul 16, 2026Masaki SatohImage MatchingLocal Feature Matching

    Morpho, Inc. Tokyo, Japan

  3. A Persistent Homology Design Space for 3D Point Cloud Deep Learning

    Apr 5, 2026Prachi Kudeshia, Jiju Poovvancheri, Amr Ghoneim +1Persistent HomologyTopological Representation Learning

    Saint Mary’s University, Halifax, Canada · Nanjing Forestry University, China