Inpainting via High-dimensional Universal Shearlet Systems
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Inpainting via High-dimensional Universal Shearlet Systems Z. Amiri1,2 · R.A. Kamyabi-Gol1,2
Received: 18 October 2016 / Accepted: 15 January 2018 © Springer Science+Business Media B.V., part of Springer Nature 2018
Abstract Thresholding and compressed sensing in combination with both wavelet and shearlet transforms have been very successful in inpainting tasks. Recent results have demonstrated that shearlets outperform wavelets in the problem of image inpainting. In this paper, we provide a general framework for universal shearlet systems in high dimensions. This theoretical framework is used to analyze the recovery of missing data via 1 minimization in an abstract model situation. In addition, we set up a particular model inspired by seismic data and a box mask to model missing data. Finally, the results of numerical experiments comparing various inpainting methods are presented. Keywords Inpainting · 1 Minimization · Compressed sensing · Cluster coherence · Shearlets Mathematics Subject Classification (2000) 42C40 · 42C15 · 65J22 · 65T60
1 Introduction Reconstructing missing data is a popular challenge in both analog and digital fields. Also known as inpainting, this activity is the process of filling in a missing region or for making undetectable modifications to images, modifying the corrupted regions which are not consistent with the original images. Applications of inpainting range from restoring of missing blocks in video data to removal of occlusions such as text from images and repairing of scratched photos [1, 2, 13, 14].
B R.A. Kamyabi-Gol
[email protected] Z. Amiri [email protected]
1
Department of Mathematics, Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran
2
Center of Excellence in Analysis on Algebraic Structures (CEAAS), Ferdowsi University of Mashhad, P.O. Box 1159-91775, Mashhad, Iran
Z. Amiri, R.A. Kamyabi-Gol
Due to the vast interest on this topic, there exist several excellent works on inpainting via compressed sensing which is a fundamental method to recover sparsified data by 1 minimization [7, 16]. Previous works have focused on the concept of clustered sparsity which have led to theoretical bounds and results. In this setting, directional representation systems such as shearlets have been shown to outperform not only wavelets, but also other directional systems [5, 6, 18, 23]. In addition, the superiority of shearlets over wavelets for a basic thresholding algorithm and geometric separation was shown in [12, 16]. In [9], Genzel and Kutyniok introduced the more flexible universal shearlet systems, which are associated with an arbitrary scaling sequence. The performance for inpainting of this novel construction shearlet system in two dimensions was also analyzed. In this paper, we extend the framework of asymptotic analysis of inpainting by Genzel and Kutyniok to the higher-dimensional setting by generalizing the concept of universal shearlet systems. Using a method based on the original construction of Guo and Labate in [11], it is possible to provide a gener
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