Frame Soft Shrinkage as Proximity Operator

Abstract

Let $\mathcal{H}$ and $\mathcal{K}$ be real Hilbert spaces and $T\in \mathcal{B}(\mathcal{H},\mathcal{K})$ an injective operator with closed range and Moore-Penrose inverse $T^{†}$. Based on the well-known characterization of proximity operators by Moreau, we prove that for any proximity operator $\text{Prox}:\mathcal{K}\to \mathcal{K}$ the operator $T^{†}\text{Prox}T$ is a proximity operator on the linear space $\mathcal{H}$ equipped with a suitable norm. In particular, it follows for the frequently applied soft shrinkage operator $\text{Prox}=S_{\lambda }:{\ell }_2\to {\ell }_2$ and any frame analysis operator $T:\mathcal{H}\to {\ell }_2$, that the frame shrinkage operator $T^{†}{S}_{\lambda }T$ is a proximity operator in a suitable Hilbert space.

References in corpus (1)

• Scene-Adapted Plug-and-Play Algorithm with Guaranteed Convergence: Applications to Data Fusion in Imaging