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Two Models for Surface Segmentation using the Total Variation of the Normal Vector

Lukas Baumgärtner, Ronny Bergmann, Roland Herzog, Stephan Schmidt, Manuel Weiß

TL;DR

This paper compares two variational models for segmenting triangulated surfaces based on the surface normal field. It contrasts assignment-space TV (A-TV), which regularizes the assignment φ on the simplex, with label-space TV (L-TV), which regularizes the nonlinear mixture of labels via the sphere using a Riemannian center of mass. A CP-based method solves A-TV efficiently, while a specialized ADMM framework solves L-TV, including subproblems on triangles, edges, and the global center of mass. Experiments on unit-sphere and fandisk meshes show that L-TV achieves higher labeling accuracy and more complete use of available labels, particularly in regions of constant curvature, but at substantially higher computational cost. The work highlights a trade-off between segmentation quality and computational burden and points to future acceleration of the Riemannian center-of-mass subproblem.

Abstract

We consider the problem of surface segmentation, where the goal is to partition a surface represented by a triangular mesh. The segmentation is based on the similarity of the normal vector field to a given set of label vectors. We propose a variational approach and compare two different regularizers, both based on a total variation measure. The first regularizer penalizes the total variation of the assignment function directly, while the second regularizer penalizes the total variation in the label space. In order to solve the resulting optimization problems, we use variations of the split Bregman (ADMM) iteration adapted to the problem at hand. While computationally more expensive, the second regularizer yields better results in our experiments, in particular it removes noise more reliably in regions of constant curvature.

Two Models for Surface Segmentation using the Total Variation of the Normal Vector

TL;DR

This paper compares two variational models for segmenting triangulated surfaces based on the surface normal field. It contrasts assignment-space TV (A-TV), which regularizes the assignment φ on the simplex, with label-space TV (L-TV), which regularizes the nonlinear mixture of labels via the sphere using a Riemannian center of mass. A CP-based method solves A-TV efficiently, while a specialized ADMM framework solves L-TV, including subproblems on triangles, edges, and the global center of mass. Experiments on unit-sphere and fandisk meshes show that L-TV achieves higher labeling accuracy and more complete use of available labels, particularly in regions of constant curvature, but at substantially higher computational cost. The work highlights a trade-off between segmentation quality and computational burden and points to future acceleration of the Riemannian center-of-mass subproblem.

Abstract

We consider the problem of surface segmentation, where the goal is to partition a surface represented by a triangular mesh. The segmentation is based on the similarity of the normal vector field to a given set of label vectors. We propose a variational approach and compare two different regularizers, both based on a total variation measure. The first regularizer penalizes the total variation of the assignment function directly, while the second regularizer penalizes the total variation in the label space. In order to solve the resulting optimization problems, we use variations of the split Bregman (ADMM) iteration adapted to the problem at hand. While computationally more expensive, the second regularizer yields better results in our experiments, in particular it removes noise more reliably in regions of constant curvature.

Paper Structure

This paper contains 20 sections, 35 equations, 6 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Discrete Surfaces
  4. Differential Geometry on the Sphere
  5. Differential Geometry on the Simplex
  6. Two Total Variation Regularizers
  7. Assignment Space Total Variation
  8. Label Space Total Variation
  9. Numerical Algorithms
  10. Chambolle-Pock Algorithm for Assignment Space Total Variation
  11. ADMM for Label Space Total Variation
  12. The Y-Subproblem
  13. The X-Subproblem
  14. The phi-Subproblem
  15. The m-Subproblem
  16. ...and 5 more sections

Figures (6)

  • Figure 3.1: Visualization of the assignments ${\boldsymbol{\varphi}}, \widetilde{{\boldsymbol{\varphi}}}$ as described in example:comparison-of-the-regularizers.
  • Figure 5.1: Visualization of the label set \ref{['eq:unit-sphere-mesh:labels']} for the sphere mesh example (subsection:numerical-examples:unit-sphere-mesh).
  • Figure 5.2: Assignments for the noisy sphere mesh with different values of $\beta$ and the label set depicted in figure:unit-sphere-mesh:labels. Triangles are colored according to their assigned label. One of the poles is shown in light pink. \ref{['figure:sphere-mesh:atv:small', 'figure:sphere-mesh:atv:optimal', 'figure:sphere-mesh:atv:large']} show the results using \ref{['eq:atv:model-problem']}, while figure:sphere-mesh:ltv:small,figure:sphere-mesh:ltv:optimal,figure:sphere-mesh:ltv:large show the results using the new \ref{['eq:ltv:model-problem']} model.
  • Figure 5.3: Visualization of the two label sets for the fandisk mesh example (subsection:numerical-examples:fandisk-mesh).
  • Figure 5.4: Assignments for the noisy fandisk mesh with the first label set of $29$ labels (figure:fandisk-mesh:labels:nonuniform) and different values of $\beta$. Triangles are colored according to their assigned label. \ref{['figure:fandisk-mesh:nonuniform:atv:small', 'figure:fandisk-mesh:nonuniform:atv:optimal', 'figure:fandisk-mesh:nonuniform:atv:large']} show the results using \ref{['eq:atv:model-problem']}, while figure:fandisk-mesh:nonuniform:ltv:small,figure:fandisk-mesh:nonuniform:ltv:optimal,figure:fandisk-mesh:nonuniform:ltv:large show the results using \ref{['eq:ltv:model-problem']}.
  • ...and 1 more figures