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Vertex characterization via second-order topological derivatives

Peter Gangl, Bochra Mejri, Otmar Scherzer

TL;DR

The paper addresses vertex characterization in 2D images by leveraging a second-order topological derivative $d^2\mathcal{J}$ of a $Mumford$-$Shah$-type functional with respect to polygonal inclusion shapes encoding vertex types. It derives explicit formulas for $d\mathcal{J}$ and $d^2\mathcal{J}$ using exterior corrector problems and polarization matrices, and then develops a one-shot numerical algorithm that detects vertex location and type by evaluating $d^2\mathcal{J}$ across a library of inclusion shapes. Numerical experiments on cube-based scenes demonstrate accurate localization and classification of vertices (e.g., L-corners, Forks, Arrows, and T-junctions), with a precomputation step that accelerates shape evaluations. The approach provides a principled, data-efficient means to extract geometric vertex information from 2D projections, with potential benefits for downstream 3D scene interpretation and edge-structure analysis.

Abstract

This paper focuses on identifying vertex characteristics in 2D images using topological asymptotic analysis. Vertex characteristics include both the location and the type of the vertex, with the latter defined by the number of lines forming it and the corresponding angles. This problem is crucial for computer vision tasks, such as distinguishing between fore- and background objects in 3D scenes. We compute the second-order topological derivative of a Mumford-Shah type functional with respect to inclusion shapes representing various vertex types. This derivative assigns a likelihood to each pixel that a particular vertex type appears there. Numerical tests demonstrate the effectiveness of the proposed approach.

Vertex characterization via second-order topological derivatives

TL;DR

The paper addresses vertex characterization in 2D images by leveraging a second-order topological derivative of a --type functional with respect to polygonal inclusion shapes encoding vertex types. It derives explicit formulas for and using exterior corrector problems and polarization matrices, and then develops a one-shot numerical algorithm that detects vertex location and type by evaluating across a library of inclusion shapes. Numerical experiments on cube-based scenes demonstrate accurate localization and classification of vertices (e.g., L-corners, Forks, Arrows, and T-junctions), with a precomputation step that accelerates shape evaluations. The approach provides a principled, data-efficient means to extract geometric vertex information from 2D projections, with potential benefits for downstream 3D scene interpretation and edge-structure analysis.

Abstract

This paper focuses on identifying vertex characteristics in 2D images using topological asymptotic analysis. Vertex characteristics include both the location and the type of the vertex, with the latter defined by the number of lines forming it and the corresponding angles. This problem is crucial for computer vision tasks, such as distinguishing between fore- and background objects in 3D scenes. We compute the second-order topological derivative of a Mumford-Shah type functional with respect to inclusion shapes representing various vertex types. This derivative assigns a likelihood to each pixel that a particular vertex type appears there. Numerical tests demonstrate the effectiveness of the proposed approach.
Paper Structure (17 sections, 5 theorems, 52 equations, 15 figures, 1 table, 1 algorithm)

This paper contains 17 sections, 5 theorems, 52 equations, 15 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model problem
  3. Asymptotic expansion of $\mathcal{J}$
  4. Preliminaries
  5. Analysis of the perturbed state equation
  6. Topological derivatives of the cost function
  7. A numerical vertex detection method
  8. Numerical algorithm
  9. Choice of inclusion shapes
  10. One-shot detection process for given inclusion shape
  11. Detection of unknown inclusion shape
  12. Numerical experiments
  13. Cube: 'L-corner', 'Fork' and 'Arrow' detection
  14. Localization of given inclusion shapes.
  15. Simultaneous determination of location and inclusion shape.
  16. ...and 2 more sections

Key Result

Lemma 3.1

The function $K_{\varepsilon, \omega}^{(1)} \in H^1(D_\varepsilon)$ defined in Definition def_Keps is the unique solution to for all $v \in H^1(D_\varepsilon)$. Moreover, $\nabla K_{\varepsilon, \omega}^{(1)}$ is bounded, i.e., there exists $C>0$ such that

Figures (15)

  • Figure 1: Classification of vertices Guz68.
  • Figure 2: Examples of classification of vertices. (a) - Cube: 'L-corner', 'Fork' and 'Arrow' vertices. (b) - Overlapping cubes: 'T-junction' vertex.
  • Figure 3: (a) Unperturbed configuration. (b) Perturbed configuration where the domain is perturbed in an inclusion shape $\omega_\varepsilon$ whose center is the point $z$.
  • Figure 4: Original perturbed domain $D$ with subdomains $\Omega$ and $\omega_\varepsilon$ and rescaled perturbed domain $D_\varepsilon = \phi_\varepsilon^{-1}(D)$ with subdomains $\phi_\varepsilon^{-1}(\Omega)$ and $\omega = \phi_\varepsilon^{-1}(\omega_\varepsilon)$.
  • Figure 5: Delineation of a vertices configuration and the corresponding enlargement set.
  • ...and 10 more figures

Theorems & Definitions (19)

  • Remark 1
  • Definition 1
  • Remark 2
  • Definition 2
  • Lemma 3.1
  • Proof 1
  • Definition 3
  • Remark 3
  • Lemma 3.2
  • Definition 4
  • ...and 9 more