Uniqueness of Minimizers of Some Variational Problems Arising in Image Processing

We will study an open problem pertaining to the uniqueness of minimizers for a class of variational problems emanating from Meyer's model for the decomposition of an image into a geometric part and a texture part. Mainly, we are interested in the uniqueness of minimizers of the problem: \begin{equation} \label{eq:abs-1} (0.1)\qquad \inf \left\{J(u)+J^*\left(\frac{v}{\mu}\right): (u,v)\in L^2(\Omega)\times L^2( \Omega){,}\; f=u+v \right\} \end{equation} where the image $f$ is a square integrable function on the domain $\Omega$, the number $\mu$ is a parameter, the functional $J$ stands for the total variation and the functional $J^*$ is its Legendre transform. We will consider Problem (0.1) as a special case of the problem: \begin{equation} \label{eq:abs-2} (0.2)\qquad \inf\{s(f-u)+s^*(u): u\in \mathcal{X}\} \end{equation} where $\mathcal X$ is a Hilbert space containing $f$ and $s$ is a continuous semi-norm on $\mathcal X$. In finite dimensions, we will prove that Problem (0.2) reduces to a projection problem onto the polar of the unit ball associated to a given norm on an appropriate Euclidean space. We will also provide a characterization for the uniqueness of minimizers for a more general projection problem defined by using any norm and any nonempty, closed, bounded and convex set of an Euclidean space. Finally, we will provide numerical evidence in favor of the uniqueness of minimizers for the decomposition problem.