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Denoising of Sphere- and SO(3)-Valued Data by Relaxed Tikhonov Regularization

Robert Beinert, Jonas Bresch, Gabriele Steidl

TL;DR

The paper tackles denoising of manifold-valued signals on graphs by convexifying the nonconvex quadratic Tikhonov regularization through PSD-encoded relaxations. It develops parallel complex, real, and simplified real formulations for circle-valued data and extends the framework to sphere- and ${\mathrm{SO}}(3)$-valued data via quaternion representations, showing that a Schur-complement-based simplification preserves the minimizers. An ADMM solver is derived for the simplified real model, yielding faster convergence than existing relaxations and enabling scalable denoising of circle-, sphere-, and ${\mathrm{SO}}(3)$-valued data on graphs. Numerical experiments across 1D and 2D settings, including hue, chromaticity, and rotations, demonstrate accurate restoration to the underlying manifold with high computational efficiency. The approach provides a unified, tractable pipeline for manifold-valued denoising with potential impact on imaging, computer vision, and 3D data processing where data live on spheres or rotation groups.

Abstract

Manifold-valued signal- and image processing has received attention due to modern image acquisition techniques. Recently, a convex relaxation of the otherwise nonconvex Tikhonov-regularization for denoising circle-valued data has been proposed by Condat (2022). The circle constraints are here encoded in a series of low-dimensional, positive semi-definite matrices. Using Schur complement arguments, we show that the resulting variational model can be simplified while leading to the same solution. The simplified model can be generalized to higher dimensional spheres and to SO(3)-valued data, where we rely on the quaternion representation of the latter. Standard algorithms from convex analysis can be applied to solve the resulting convex minimization problem. As proof-of-the-concept, we use the alternating direction method of multipliers to demonstrate the denoising behavior of the proposed method. In a series of experiments, we demonstrate the numerical convergence of the signal- or image values to the underlying manifold.

Denoising of Sphere- and SO(3)-Valued Data by Relaxed Tikhonov Regularization

TL;DR

The paper tackles denoising of manifold-valued signals on graphs by convexifying the nonconvex quadratic Tikhonov regularization through PSD-encoded relaxations. It develops parallel complex, real, and simplified real formulations for circle-valued data and extends the framework to sphere- and -valued data via quaternion representations, showing that a Schur-complement-based simplification preserves the minimizers. An ADMM solver is derived for the simplified real model, yielding faster convergence than existing relaxations and enabling scalable denoising of circle-, sphere-, and -valued data on graphs. Numerical experiments across 1D and 2D settings, including hue, chromaticity, and rotations, demonstrate accurate restoration to the underlying manifold with high computational efficiency. The approach provides a unified, tractable pipeline for manifold-valued denoising with potential impact on imaging, computer vision, and 3D data processing where data live on spheres or rotation groups.

Abstract

Manifold-valued signal- and image processing has received attention due to modern image acquisition techniques. Recently, a convex relaxation of the otherwise nonconvex Tikhonov-regularization for denoising circle-valued data has been proposed by Condat (2022). The circle constraints are here encoded in a series of low-dimensional, positive semi-definite matrices. Using Schur complement arguments, we show that the resulting variational model can be simplified while leading to the same solution. The simplified model can be generalized to higher dimensional spheres and to SO(3)-valued data, where we rely on the quaternion representation of the latter. Standard algorithms from convex analysis can be applied to solve the resulting convex minimization problem. As proof-of-the-concept, we use the alternating direction method of multipliers to demonstrate the denoising behavior of the proposed method. In a series of experiments, we demonstrate the numerical convergence of the signal- or image values to the underlying manifold.
Paper Structure (13 sections, 11 theorems, 69 equations, 5 figures, 3 tables, 1 algorithm)

This paper contains 13 sections, 11 theorems, 69 equations, 5 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Denoising of Circle-Valued Data
  3. $\mathbb C$-Valued Model
  4. Related $\mathbb R^2$-Valued Model
  5. Simplified $\mathbb R^2$-Valued Model
  6. Generalization to Sphere-Valued Data
  7. Generalization to ${\mathrm{SO}}(3)$-Valued Data
  8. Algorithm
  9. Numerical Experiments
  10. $\mathbb S_1$-Valued Data
  11. Denoising Colour Values
  12. ${\mathrm{SO}}(3)$-Valued Data
  13. Proof of Theorem \ref{['the:equiv-quad']}.

Key Result

Lemma 1

Let $n,m \in V$ and $(n,m) \in E$. Then $x_n,x_m \in \mathbb S_{\mathbb C}$ and $r_{(n,m)} = \bar{x}_m x_n$ if and only if is positive semi-definite and has rank one.

Figures (5)

  • Figure 1: Ground truth (blue), noisy observation with $\kappa = 10$ (black), numerical solution by Alg. \ref{['alg:1']} (red) with $\lambda \equiv 25$ and $\rho = 3$ of the line graph signal of length $N = 1000$ in Section \ref{['sec:s1-data']} in comparison with the solution obtained by CPPA-TV (green) with regularization parameter $\lambda \equiv 0.8$ and $\lambda_0 = \pi$. The $\mathbb S_1$-values are represented by their angles in $[-\pi,\pi)$.
  • Figure 2: Top row: Ground truth (left) and noisy observation with $\kappa = 20$ (right). Bottom row: Numerical solution (left) using Alg. \ref{['alg:1']} on the $90 \times 90$ pixel image graph signal in Section \ref{['sec:s1-data']} with $w_n \equiv 1, \lambda \equiv 1$ and $\rho = 3$. The solution comes with an restoration error (RMSE) $7.627\cdot10^{-2}$, mean distance to the sphere $7.834\cdot10^{-5}$. Comparison with CPPA-TV (right) with $\lambda \equiv 0.3$ and $\lambda_0 = \pi$. The solution comes with an restoration error (RMSE) $8.702\cdot10^{-2}$. The colour map corresponds to the angles of the $\mathbb S_1$-values.
  • Figure 3: Left: Ground truth of size $200 \times 200$ for the colour denoising in Section \ref{['sec:denoise-col']}. Below: Hue ground truth (left), noisy observation with $\kappa = 10$ (middle), and denoised version (right) of the hue experiment in Section \ref{['sec:denoise-col']} with respect to Figure \ref{['fig:hue_corals']}. ADMM has been employed with $w_n \coloneqq 1$, $\lambda_{(n,m)} \coloneqq 1$ and $\rho \coloneqq 3$. The solution comes with an mean distance to the sphere $5.874\cdot10^{-4}$.
  • Figure 4: Ground truth (left), noisy observation with $\kappa = 100$ (middle), and denoised version (right) of the chromaticity experiment in Section \ref{['sec:denoise-col']} with respect to Figure \ref{['fig:hue_corals']}. ADMM has been employed with $w_n \coloneqq 1$, $\lambda_{(n,m)} \coloneqq 3$, and $\rho \coloneqq 3$. The solution comes with an mean distance to the sphere $2.343\cdot10^{-10}$.
  • Figure 5: Ground truth (left), noisy observation with $\kappa_1 \coloneqq 30$ and $\kappa_2 \coloneqq 5$ (middle), and denoised version (right) of the ${\mathrm{SO}}(3)$-valued signal on the image graph in Section \ref{['sec:so3-data']}. The rotations are visualized by operating on a colored 3d cone. The entire signal consists of $90 \times 90$ pixel, where only every third pixel from the 25th to the 75th pixel is visualized. The parameters have been $w_n \coloneqq 1$, $\lambda_{(n,m)} \coloneqq 1$, and $\rho \coloneqq 3$. The solution comes with an mean distance to the unit quaternions $1.667\cdot10^{-10}$.

Theorems & Definitions (19)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • Proposition 4: HJ13
  • Theorem 5
  • proof
  • Corollary 6
  • Lemma 7
  • ...and 9 more