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Journal of Mathematical Imaging and Vision

, Volume 48, Issue 2, pp 205–234 | Cite as

Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis

  • Emily J. King
  • Gitta Kutyniok
  • Xiaosheng Zhuang
Article

Abstract

Recently, compressed sensing techniques in combination with both wavelet and directional representation systems have been very effectively applied to the problem of image inpainting. However, a mathematical analysis of these techniques which reveals the underlying geometrical content is missing. In this paper, we provide the first comprehensive analysis in the continuum domain utilizing the novel concept of clustered sparsity, which besides leading to asymptotic error bounds also makes the superior behavior of directional representation systems over wavelets precise. First, we propose an abstract model for problems of data recovery and derive error bounds for two different recovery schemes, namely 1 minimization and thresholding. Second, we set up a particular microlocal model for an image governed by edges inspired by seismic data as well as a particular mask to model the missing data, namely a linear singularity masked by a horizontal strip. Applying the abstract estimate in the case of wavelets and of shearlets we prove that—provided the size of the missing part is asymptotic to the size of the analyzing functions—asymptotically precise inpainting can be obtained for this model. Finally, we show that shearlets can fill strictly larger gaps than wavelets in this model.

Keywords

1 minimization Cluster coherence Inpainting Parseval frames Sparse representation Data recovery Shearlets Meyer wavelets 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Emily J. King
    • 1
  • Gitta Kutyniok
    • 1
  • Xiaosheng Zhuang
    • 2
  1. 1.Department of MathematicsTechnische Universität BerlinBerlinGermany
  2. 2.Department of MathematicsCity University of Hong KongHong KongChina

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