Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Geometry Denoising with Preferred Normal Vectors

Manuel Weiß, Lukas Baumgärtner, Roland Herzog, Stephan Schmidt

TL;DR

This paper introduces a variational framework for simultaneous geometry denoising and segmentation of triangulated surfaces by leveraging a user-specified set of preferred normal vectors. The method couples a fidelity term with a total-variation regularization on a triangle-wise label assignment, enforcing alignment of triangle normals with the label set and creating coherent regions of constant normal direction. An ADMM scheme decouples non-smooth terms, yielding closed-form updates for the u-, v-, and w-subproblems, a CG-based solution for the label-coupled phi-update, and a shape-Newton-based, preconditioned vertex update to adjust the mesh geometry. Numerical experiments on spheres, platonic solids, city skylines, and the Stanford bunny demonstrate how the prior normals improve denoising quality and controllably sculpt geometry toward desired directional features, with clear advantages over baseline TV-based denoising in preserving sharp features and achieving label-consistent regions.

Abstract

We introduce a new paradigm for geometry denoising using prior knowledge about the surface normal vector. This prior knowledge comes in the form of a set of preferred normal vectors, which we refer to as label vectors. A segmentation problem is naturally embedded in the denoising process. The segmentation is based on the similarity of the normal vector to the elements of the set of label vectors. Regularization is achieved by a total variation term. We formulate a split Bregman (ADMM) approach to solve the resulting optimization problem. The vertex update step is based on second-order shape calculus.

Geometry Denoising with Preferred Normal Vectors

TL;DR

This paper introduces a variational framework for simultaneous geometry denoising and segmentation of triangulated surfaces by leveraging a user-specified set of preferred normal vectors. The method couples a fidelity term with a total-variation regularization on a triangle-wise label assignment, enforcing alignment of triangle normals with the label set and creating coherent regions of constant normal direction. An ADMM scheme decouples non-smooth terms, yielding closed-form updates for the u-, v-, and w-subproblems, a CG-based solution for the label-coupled phi-update, and a shape-Newton-based, preconditioned vertex update to adjust the mesh geometry. Numerical experiments on spheres, platonic solids, city skylines, and the Stanford bunny demonstrate how the prior normals improve denoising quality and controllably sculpt geometry toward desired directional features, with clear advantages over baseline TV-based denoising in preserving sharp features and achieving label-consistent regions.

Abstract

We introduce a new paradigm for geometry denoising using prior knowledge about the surface normal vector. This prior knowledge comes in the form of a set of preferred normal vectors, which we refer to as label vectors. A segmentation problem is naturally embedded in the denoising process. The segmentation is based on the similarity of the normal vector to the elements of the set of label vectors. Regularization is achieved by a total variation term. We formulate a split Bregman (ADMM) approach to solve the resulting optimization problem. The vertex update step is based on second-order shape calculus.

Paper Structure

This paper contains 16 sections, 1 theorem, 30 equations, 6 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Introduction
  3. Preliminaries
  4. ADMM Scheme
  5. The u-Problem
  6. The v-Problem
  7. The w-Problem
  8. The phi-Problem
  9. The Shape Optimization Problem
  10. Overall ADMM Scheme
  11. Numerical Experiments
  12. Sphere Example
  13. Platonic Solids
  14. City Skyline Denoising
  15. Stanford Bunny
  16. ...and 1 more sections

Key Result

Lemma 4.1

Suppose that $\gamma \in \mathbb{R}$ and ${\boldsymbol{c}} \in \mathbb{R}^n$, $n \in \mathbb{N}$, are given. Then the global solutions ${\boldsymbol{u}}^*$ of are given by the soft-thresholding operation where ${\boldsymbol{e}} \in \mathbb{R}^n$ is an arbitrary vector of length ${\boldsymbol{e}} _2 = 1$.

Figures (6)

  • Figure 5.1: Noisy input data for the sphere geometry experiment (subsection:example:sphere).
  • Figure 5.2: Solutions for problem \ref{['eq:general-model-repeated']} for the sphere example (subsection:example:sphere, $L = 20$ labels) with noise as shown in figure:sphere:noisy-data for different values of the assignment weight~$\alpha$ and the TV weight $\beta$. Cells are colored according to the assigned label.
  • Figure 5.3: Iteration history of algorithm:ADMM for the tetrahedron platonic solid problem (subsection:example:platonic-solids, $L = 4$ labels) with a sphere as initial guess and an assignment weight $\alpha = 20$. Cells are colored according to the assigned label.
  • Figure 5.4: Iteration history of algorithm:ADMM for the dodecahedron platonic solid problem (subsection:example:platonic-solids, $L = 12$ labels) with a sphere as initial guess and an assignment weight $\alpha = 20$. Cells are colored according to the assigned label.
  • Figure 5.5: Denoising of the city skyline geometry (subsection:example:city-skyline) with the proposed model \ref{['eq:general-model-repeated']} (bottom left) and the baseline total variation-based model \ref{['eq:problem:baseline-tv-based-denoising']} (bottom right), which is not informed by preferred normal directions.
  • ...and 1 more figures

Theorems & Definitions (2)

  • Lemma 4.1
  • proof