Multiscale Decompositions and Optimization

TL;DR

This paper systematically studies multiscale decompositions based on Tikhonov regularization involving BV and Sobolev spaces, characterizing solutions, improving convergence results, and providing numerical implementations for image processing and inverse problems.

Contribution

It generalizes existing results on multiscale decompositions, characterizes solutions for specific function space pairs, and enhances convergence analysis with numerical demonstrations.

Findings

Characterization of minimizers for (L2, BV) spaces.

Improved L2 convergence results for multiscale schemes.

Numerical implementation demonstrating practical effectiveness.

Abstract

In this paper, the following type Tikhonov regularization problem will be systematically studied: [(u_t,v_t):=\argmin_{u+v=f} {|v|_X+t|u|_Y},] where $Y$ is a smooth space such as a $\BV$ space or a Sobolev space and $X$ is the pace in which we measure distortion. Examples of the above problem occur in denoising in image processing, in numerically treating inverse problems, and in the sparse recovery problem of compressed sensing. It is also at the heart of interpolation of linear operators by the real method of interpolation. We shall characterize of the minimizing pair $(u_t,v_t)$ for $(X,Y)=(L_2(\Omega),\BV(\Omega))$ as a primary example and generalize Yves Meyer's result in [11] and Antonin Chambolle's result in [6]. After that, the following multiscale decomposition scheme will be studied: [u_{k+1}:=\argmin_{u\in \BV(\Omega)\cap L_2(\Omega)} {1/2|f-u|^2_{L_2}+t_{k}|u-u_k|_{\BV}},] where $u_0=0$ and $\Omega$ is a bounded Lipschitz domain in $\R^d$. This method was introduced by Eitan Tadmor et al. and we will improve the $L_2$ convergence result in \cite{Tadmor}. Other pairs such as $(X,Y)=(L_p,W^{1}(L_\tau))$ and $(X,Y)=(\ell_2,\ell_p)$ will also be mentioned. In the end, the numerical implementation for $(X,Y)=(L_2(\Omega),\BV(\Omega))$ and the corresponding convergence results will be given.