# Total variation regularization with variable Lebesgue prior

## TL;DR

This paper introduces a convex total variation regularization method using a variable Lebesgue exponent, enabling local adaptation between edge preservation and smoothing, with efficient computation and promising results in denoising and tomography.

## Contribution

It proposes a convex TV regularization with a fixed, spatially varying exponent function, improving flexibility and computational efficiency over previous variable exponent methods.

## Key findings

- Favorable comparison to TGV and TV in denoising tasks
- Efficient evaluation of proximal operators
- Convex formulation with fixed variable exponent

## Abstract

This work proposes the variable exponent Lebesgue modular as a replacement for the 1-norm in total variation (TV) regularization. It allows the exponent to vary with spatial location and thus enables users to locally select whether to preserve edges or smooth intensity variations. In contrast to earlier work using TV-like methods with variable exponents, the exponent function is here computed offline as a fixed parameter of the final optimization problem, resulting in a convex goal functional. The obtained formulas for the convex conjugate and the proximal operators are simple in structure and can be evaluated very efficiently, an important property for practical usability. Numerical results with variable $L^p$ TV prior in denoising and tomography problems on synthetic data compare favorably to total generalized variation (TGV) and TV.