Denoising Hyperbolic-Valued Data by Relaxed Regularizations

TL;DR

This paper presents a convex relaxation approach for denoising hyperbolic-valued data by embedding hyperbolic space into Euclidean space and applying convex optimization techniques, demonstrated on retinal image data.

Contribution

It introduces a novel matrix representation for hyperbolic space and develops a convex relaxation method for denoising hyperbolic-valued data, enabling the use of standard convex optimization algorithms.

Findings

Effective denoising of hyperbolic data using convex relaxation.

Successful application to real-world retinal image series.

Simultaneous restoration of pixelwise mean and standard deviation.

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 ans\"atze 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.