4.3 Article

LOCALLY SPARSE RECONSTRUCTION USING THE l1,∞-NORM

Journal

INVERSE PROBLEMS AND IMAGING
Volume 9, Issue 4, Pages 1093-1137

Publisher

AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/ipi.2015.9.1093

Keywords

Local sparsity; l(1,infinity)-D-regularization; inverse problems; variational methods; mixed norms; compressed sensing

Funding

  1. German Ministry for Science and Education (BMBF) [05M13PMC]
  2. ERC via Grant EU FP 7 - ERC Consolidator Grant [615216]
  3. ERC Starting Grant ConvexVision
  4. European Research Council (ERC) [615216] Funding Source: European Research Council (ERC)

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This paper discusses the incorporation of local sparsity information, e.g. in each pixel of an image, via minimization of the l(1,infinity)-norm. We discuss the basic properties of this norm when used as a regularization functional and associated optimization problems, for which we derive equivalent reformulations either more amenable to theory or to numerical computation. Further focus of the analysis is put on the locally 1-sparse case, which is well motivated by some biomedical imaging applications. Our computational approaches are based on alternating direction methods of multipliers (ADMM) and appropriate splittings with augmented Lagrangians. Those are tested for a model scenario related to dynamic positron emission tomography (PET), which is a functional imaging technique in nuclear medicine. The results of this paper provide insight into the potential impact of regularization with the l(1,infinity)-norm for local sparsity in appropriate settings. However, it also indicates several shortcomings, possibly related to the non-tightness of the functional as a relaxation of the l(1,infinity)-norm.

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