Homogeneous Penalizers and Constraints in Convex I

Authors: R Ciak B Shafei G Steidl

Publish Date: 2012/10/27

Volume: 47, Issue: 3, Pages: 210-230

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Abstract

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 operatornameargmin varPhix mbox subject to varPsix letau and their penalized counterparts operatornameargminvarPhix + lambdavarPsix We recall general results on the topic by the help of an epigraphical projection Then we deal with the special setting Ψ=∥L⋅∥ with L∈ℝ mn and Φ=φH⋅ where H∈ℝ nn and φℝ n →ℝ∪+∞ 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 τ and λ are in the graph of this function We illustrate the relation between τ and λ 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 Idivergence as data fitting term φ 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

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