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Denoising Hyperbolic-Valued Data by Relaxed Regularizations

Robert Beinert, Jonas Bresch

TL;DR

A novel relaxation strategy for denoising hyperbolic-valued data by exploiting the Euclidean embedding and encoding the hyperbolic sheet using a novel matrix representation to allow the utilization of well-established convex optimization procedures like the alternating directions method of multipliers (ADMM).

Abstract

We introduce a novel relaxation strategy for denoising hyperbolic-valued data. The main challenge is here the non-convexity of the hyperbolic sheet. Instead of considering the denoising problem directly on the hyperbolic space, we exploit the Euclidean embedding and encode the hyperbolic sheet using a novel matrix representation. For denoising, we employ the Euclidean Tikhonov and total variation (TV) model, where we incorporate our matrix representation. The major contribution is then a convex relaxation of the variational ansätze allowing the utilization of well-established convex optimization procedures like the alternating directions method of multipliers (ADMM). The resulting denoisers are applied to a real-world Gaussian image processing task, where we simultaneously restore the pixelwise mean and standard deviation of a retina scan series.

Denoising Hyperbolic-Valued Data by Relaxed Regularizations

TL;DR

A novel relaxation strategy for denoising hyperbolic-valued data by exploiting the Euclidean embedding and encoding the hyperbolic sheet using a novel matrix representation to allow the utilization of well-established convex optimization procedures like the alternating directions method of multipliers (ADMM).

Abstract

We introduce a novel relaxation strategy for denoising hyperbolic-valued data. The main challenge is here the non-convexity of the hyperbolic sheet. Instead of considering the denoising problem directly on the hyperbolic space, we exploit the Euclidean embedding and encode the hyperbolic sheet using a novel matrix representation. For denoising, we employ the Euclidean Tikhonov and total variation (TV) model, where we incorporate our matrix representation. The major contribution is then a convex relaxation of the variational ansätze allowing the utilization of well-established convex optimization procedures like the alternating directions method of multipliers (ADMM). The resulting denoisers are applied to a real-world Gaussian image processing task, where we simultaneously restore the pixelwise mean and standard deviation of a retina scan series.

Paper Structure

This paper contains 11 sections, 4 theorems, 38 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Denoising of Hyperbolic-Valued Data
  3. Tikhonov-like Regularization
  4. TV Regularization
  5. Denoising Models and Algorithms
  6. Solving the Relaxed Tikhonov-like denoising model
  7. Solving the Relaxed TV denoising model
  8. Numerical Experiments
  9. Synthetic Image Processing.
  10. Gaussian Image Processing.
  11. Acknowledgements

Key Result

proposition thmcounterproposition

Let $\bm{x}_n, \bm{x}_m \in \mathbb A_{d+1}$. Then $\bm{x}_n, \bm{x}_m \in \mathbb H_d$, $\bm{v}_n = \lVert\bm{x}_n\rVert_2^2$, $\bm{v}_m = \lVert\bm{x}_m\rVert_2^2$, $\bm{\ell}_{(n,m)} = \eta(\bm{x}_n, \bm{x}_m)$, and $\bm{f}_{(n,m)} = \langle \bm{x}_n, \bm{x}_m\rangle$ if and only if $\bm{Q}_{(n,m

Figures (3)

  • Figure 1: Restoration of smooth synthetic line signals (blue) from noisy measurements (black/gray). For the Tikhonov model (red), we choose $\rho = 10^{-1}$, $\lambda = 6$ (top) and $\lambda = 5$ (right). For the TV model (green), we choose $\rho = 1$, $\mu = 0.75$ (top), and $\mu = 0.1$ (right). Note that the $\mathbb H_1$ signal (top) is visualized using the parametrization in \ref{['eq:paraH1']}. The $\mathbb H_2$ signal (right) is directly visualized on the hyperbolic sheet in $\mathbb A_3 \subset \mathbb R^3$.
  • Figure 2: Empirical mean (top) and standard deviation (bottom) for the Brandenburg Gate image. Here, $K = 20$ shots are randomly sampled by adding white noise with $\sigma = 0.15$. The empirical quantities are shown on the left, the Tikhonov denoised images (Thm. \ref{['thm:solADMM_Tik']}, $\rho = 10$, $\lambda = 4$) in the middle, and the TV denoised images (Thm. \ref{['thm:solADMM_TV']}, $\rho = 1$, $\mu=0.6$) on the right.
  • Figure 3: Clearing the estimated mean and standard deviation of $K = 20$ retina scans. Leftmost: the 1st (top) and 20th (bottom) image of the retina scans. From left to right: the empirical mean (top) and standard deviation (bottom); Tikhonov denoised images (Thm. \ref{['thm:solADMM_Tik']}, $\lambda = 1.5$, $\rho = 10$); TV denoised images (Thm. \ref{['thm:solADMM_TV']}, $\mu = 0.15$, $\rho = 1$).

Theorems & Definitions (7)

  • proposition thmcounterproposition
  • proof
  • proposition thmcounterproposition
  • theorem thmcountertheorem
  • proof
  • theorem thmcountertheorem
  • proof