René Ciak, Behrang Shafei, Gabriele Steidl

• Recently convex optimization models were successfully applied for solving various problems in image analysis and restoration. In this paper, we are interested in relations between convex constrained optimization problems of the form \({\rm argmin} \{ \Phi(x)\) subject to \(\Psi(x) \le \tau \}\) and their penalized counterparts \({\rm argmin} \{\Phi(x) + \lambda \Psi(x)\}\). We recall general results on the topic by the help of an epigraphical projection. Then we deal with the special setting \(\Psi := \| L \cdot\|\) with \(L \in \mathbb{R}^{m,n}\) and \(\Phi := \varphi(H \cdot)\), where \(H \in \mathbb{R}^{n,n}\) and \(\varphi: \mathbb R^n \rightarrow \mathbb{R} \cup \{+\infty\} \) meet certain requirements which are often fulfilled in image processing models. In this case we prove by incorporating the dual problems that there exists a bijective function such that the solutions of the constrained problem coincide with those of the penalized problem if and only if \(\tau\) and \(\lambda\) are in the graph of this function. We illustrate the relation between \(\tau\) and \(\lambda\) for various problems arising in image processing. In particular, we point out the relation to the Pareto frontier for joint sparsity problems. We demonstrate the performance of the constrained model in restoration tasks of images corrupted by Poisson noise with the \(I\)-divergence as data fitting term \(\varphi\) and in inpainting models with the constrained nuclear norm. Such models can be useful if we have a priori knowledge on the image rather than on the noise level.