Analysis of Inpainting via Clustered Sparsity and Microlocal Analysis

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 ℓ1 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.

[1]  Gitta Kutyniok,et al.  Shearlets: Multiscale Analysis for Multivariate Data , 2012 .

[2]  Gitta Kutyniok,et al.  Geometric Separation by Single-Pass Alternating Thresholding , 2012, ArXiv.

[3]  Wang-Q Lim,et al.  Shearlets and Optimally Sparse Approximations , 2011, ArXiv.

[4]  Gitta Kutyniok,et al.  Microlocal Analysis of the Geometric Separation Problem , 2010, ArXiv.

[5]  S. Osher,et al.  Image restoration: Total variation, wavelet frames, and beyond , 2012 .

[6]  L. Hörmander.  The analysis of linear partial differential operators , 1990 .

[7]  Iddo Drori,et al.  Fast Minimization by Iterative Thresholding for Multidimensional NMR Spectroscopy , 2007, EURASIP J. Adv. Signal Process..

[8]  Michael Elad,et al.  Cosparse analysis modeling - uniqueness and algorithms , 2011, 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP).

[9]  Michael Elad,et al.  Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[10]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[11]  Michael Elad,et al.  A generalized uncertainty principle and sparse representation in pairs of bases , 2002, IEEE Trans. Inf. Theory.

[12]  Yves Meyer,et al.  Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures , 2001 .

[13]  Gitta Kutyniok,et al.  Parabolic Molecules , 2012, Found. Comput. Math..

[14]  C. Chui,et al.  Inequalities of Littlewood-Paley type for frames and wavelets , 1993 .

[15]  M. Elad,et al.  $rm K$-SVD: An Algorithm for Designing Overcomplete Dictionaries for Sparse Representation , 2006, IEEE Transactions on Signal Processing.

[16]  Tony F. Chan,et al.  Mathematical Models for Local Nontexture Inpaintings , 2002, SIAM J. Appl. Math..

[17]  D. Labate,et al.  The Construction of Smooth Parseval Frames of Shearlets , 2013 .

[18]  A. Cohen.  Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61, I. Daubechies, SIAM, 1992, xix + 357 pp. , 1994 .

[19]  Wang-Q Lim,et al.  Compactly supported shearlets are optimally sparse , 2010, J. Approx. Theory.

[20]  Philipp Grohs,et al.  Continuous shearlet frames and resolution of the wavefront set , 2009, 0909.3712.

[21]  D. Donoho,et al.  Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA) , 2005 .

[22]  E. Candès,et al.  Continuous curvelet transform: II. Discretization and frames , 2005 .

[23]  Wang-Q Lim,et al.  Compactly Supported Shearlets , 2010, 1009.4359.

[24]  Demetrio Labate,et al.  Analysis and detection of surface discontinuities using the 3D continuous shearlet transform , 2011 .

[25]  Wang-Q Lim,et al.  Optimally Sparse Approximations of 3D Functions by Compactly Supported Shearlet Frames , 2011, SIAM J. Math. Anal..

[26]  Guillermo Sapiro,et al.  Filling-in by joint interpolation of vector fields and gray levels , 2001, IEEE Trans. Image Process..

[27]  Hui Ji,et al.  Wavelet frame based blind image inpainting , 2012 .

[28]  E. King,et al.  Wavelet and frame theory: frame bound gaps, generalized shearlets, Grassmannian fusion frames, and p-adic wavelets , 2009 .

[29]  D. Labate,et al.  Resolution of the wavefront set using continuous shearlets , 2006, math/0605375.

[30]  Kjersti Engan,et al.  Method of optimal directions for frame design , 1999, 1999 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings. ICASSP99 (Cat. No.99CH36258).

[31]  Guillermo Sapiro,et al.  Navier-stokes, fluid dynamics, and image and video inpainting , 2001, Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition. CVPR 2001.

[32]  David J. Field,et al.  Sparse coding with an overcomplete basis set: A strategy employed by V1? , 1997, Vision Research.

[33]  Tony F. Chan,et al.  Euler's Elastica and Curvature-Based Inpainting , 2003, SIAM J. Appl. Math..

[34]  Xiaoming Huo,et al.  Uncertainty principles and ideal atomic decomposition , 2001, IEEE Trans. Inf. Theory.

[35]  Felix J. Herrmann,et al.  Non-parametric seismic data recovery with curvelet frames , 2008 .

[36]  Jian-Feng Cai,et al.  Simultaneous cartoon and texture inpainting , 2010 .

[37]  Bruno Torrésani,et al.  Sparsity and persistence: mixed norms provide simple signal models with dependent coefficients , 2009, Signal Image Video Process..

[38]  A. Ron.  Review of An introduction to Frames and Riesz bases, applied and numerical Harmonic analysis by Ole Christensen Birkhäuser, Basel, 2003 , 2005 .

[39]  Zhang Jing,et al.  On the stability of wavelet and Gabor frames (Riesz bases) , 1999 .

[40]  E. Candès,et al.  Continuous curvelet transform , 2003 .

[41]  Steven A. Orszag,et al.  CBMS-NSF REGIONAL CONFERENCE SERIES IN APPLIED MATHEMATICS , 1978 .

[42]  A. Bruckstein,et al.  K-SVD : An Algorithm for Designing of Overcomplete Dictionaries for Sparse Representation , 2005 .

[43]  F. Herrmann,et al.  Nonequispaced curvelet transform for seismic data reconstruction: A sparsity-promoting approach , 2010 .

[44]  Felix J. Herrmann,et al.  Application of Stable Signal Recovery to Seismic Data Interpolation , 2006 .

[45]  E. Candès,et al.  Continuous Curvelet Transform : I . Resolution of the Wavefront Set , 2003 .

[46]  Guillermo Sapiro,et al.  Image inpainting , 2000, SIGGRAPH.

[47]  Tony F. Chan,et al.  Error Analysis for Image Inpainting , 2006, Journal of Mathematical Imaging and Vision.

[48]  Y. Meyer.  Principe d'incertitude, bases hilbertiennes et algèbres d'opérateurs , 1986 .

[49]  Simon Foucart,et al.  RECOVERING JOINTLY SPARSE VECTORS VIA HARD THRESHOLDING PURSUIT , 2011 .

[50]  Gitta Kutyniok,et al.  Analysis of data separation and recovery problems using clustered sparsity , 2011, Optical Engineering + Applications.

[51]  O. Christensen.  An introduction to frames and Riesz bases , 2002 .

[52]  Michael Elad,et al.  The Cosparse Analysis Model and Algorithms , 2011, ArXiv.

[53]  E. Candès,et al.  New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities , 2004 .