A variational approach to nonlocal image restoration flows

TL;DR

This paper establishes the existence and uniqueness of variational solutions for nonlocal total variation flows used in image denoising and deblurring, employing a variational approach that handles fractional BV spaces and general fidelity terms.

Contribution

It introduces a variational framework for nonlocal total variation flows, proving existence and uniqueness of solutions without smoothness assumptions and accommodating various fractional BV definitions.

Findings

Proved existence of parabolic minimisers for nonlocal TV flows.

Established uniqueness of solutions without strict convexity.

Provided a new method for constructing solutions to fractional 1-Laplace equations.

Abstract

We prove existence, uniqueness and initial time regularity for variational solutions to nonlocal total variation flows associated with image denoising and deblurring. In particular, we prove existence of parabolic minimisers $u$, that is, $$\int_0^T\int_\Omega u\partial_t\phi\,dx + \textbf{F}(u(t))\,dt\leq \int_0^T \textbf{F}(u+\phi)(t)\,dt,$$ for $\phi\in C^\infty_c(\Omega\times (0,T))$. The prototypical functional $\textbf{F}(u)$ is $\textbf{F}(u)=\textbf{TV}^{\alpha}_{\cdot}(u)+\frac{\kappa}{\zeta}\int_\Omega|u(x)-u_0(x)|^\zeta\,dx$ for $\zeta\geq 1$. Here $\textbf{TV}^{\alpha}_{\cdot}$ is a fractional total variation of either the Riesz or the Gagliardo type and the second term is a regression term. These models are based on different definitions of fractional $\textbf{BV}$ spaces that have been proposed in the literature. The notion of solution is completely variational and based on the weighted dissipation method. We demonstrate existence without smoothness assumptions on the domain and exhibit uniqueness without using strict convexity. We can also deal with fairly general fidelity or regression terms in the model. Furthermore, the method also provides a novel route to constructing solutions of the parabolic fractional $1$-Laplace equation.