# Multiscale Higher Order TV Operators for L1 Regularization

## TL;DR

This paper introduces a multiscale higher order total variation (MHOTV) method for signal and image denoising that improves upon traditional $ ext{L}_1$ regularization by reducing artifacts and better promoting sparsity, leveraging connections to wavelets.

## Contribution

The authors develop a novel MHOTV approach related to multiscale wavelets, with efficient computational tools, demonstrating improvements over existing wavelet and HOTV methods.

## Key findings

- MHOTV reduces artifacts compared to traditional $ ext{L}_1$ methods.
- MHOTV shows better sparsity promotion and denoising performance.
- Numerical results confirm the advantages of MHOTV over wavelets and HOTV.

## Abstract

In the realm of signal and image denoising and reconstruction, $\ell_1$ regularization techniques have generated a great deal of attention with a multitude of variants. A key component for their success is that under certain assumptions, the solution of minimum $\ell_1$ norm is a good approximation to the solution of minimum $\ell_0$ norm. In this work, we demonstrate that this approximation can result in artifacts that are inconsistent with desired sparsity promoting $\ell_0$ properties, resulting in subpar results in {some} instances. With this as our motivation, we develop a multiscale higher order total variation (MHOTV) approach, which we show is related to the use of multiscale Daubechies wavelets. We also develop the tools necessary for MHOTV computations to be performed efficiently, via operator decomposition and alternatively converting the problem into Fourier space. The relationship of higher order regularization methods with wavelets, which we believe has generally gone unrecognized, is shown to hold in several numerical results, although notable improvements are seen with our approach over both wavelets and classical HOTV.