# American Institute of Mathematical Sciences

August  2013, 7(3): 927-946. doi: 10.3934/ipi.2013.7.927

## A texture model based on a concentration of measure

 1 Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, United States, United States 2 Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095-1555, United States

Received  July 2012 Revised  May 2013 Published  September 2013

Cartoon-texture regularization is a technique for reconstructing fine-scale details from ill-posed imaging problems, which are often ill-posed because of some non-invertible convolution kernel. Here we propose a cartoon-texture regularization for deblurring problems with semi-known kernel. The cartoon component is modeled by a function of bounded variation, while the texture component is measured by an approximate duality with Lipschitz functions ($W^{1,\infty}$). To approximate the dual Lipschitz norm, which is difficult to calculate, we propose an approach using concentration of measure. This provides an accurate and differentiable expression. We also present numerical results for our cartoon-texture decomposition, both in the case of a semi-known deblurring kernel and the case of a known kernel. The texture norm enables one to numerically reconstruct fine scale details that are typically difficult to recover from blurred images, and for semi-known deblurring the method quickly leads to the correct kernel, even when that kernel contains noise.
Citation: Hayden Schaeffer, John Garnett, Luminita A. Vese. A texture model based on a concentration of measure. Inverse Problems & Imaging, 2013, 7 (3) : 927-946. doi: 10.3934/ipi.2013.7.927
##### References:
 [1] G. Aubert and P. Kornprobst, "Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations,", Second edition, 147 (2006).   Google Scholar [2] Gilles Aubert and Jean-François Aujol, Modeling very oscillating signals. Application to image processing,, Applied Mathematics and Optimization, 51 (2005), 163.  doi: 10.1007/s00245-004-0812-z.  Google Scholar [3] J.-F. Aujol, "Contribution à l'Analyse de Textures en Traitement d'Images par Méthodes Variationnelles et Équations aux Dérivées Partielles,", Ph.D thesis, (2004).   Google Scholar [4] J.-F. Aujol and T. F. Chan, Combining geometrical and textured information to perform image classification,, Journal of Visual Communication and Image Representation, 17 (2006), 1004.  doi: 10.1016/j.jvcir.2006.02.001.  Google Scholar [5] J.-F. Aujol and S. H. Kang, Color image decomposition and restoration,, Journal of Visual Communication and Image Representation, 17 (2006), 916.  doi: 10.1016/j.jvcir.2005.02.001.  Google Scholar [6] Jean-François Aujol, Gilles Aubert, Laure Blanc-Féraud and Antonin Chambolle, Image decomposition into a bounded variation component and an oscillating component,, Journal of Mathematical Imaging and Vision, 22 (2005), 71.  doi: 10.1007/s10851-005-4783-8.  Google Scholar [7] L. Bar, N. Sochen and N. Kiryati, Semi-blind image restoration via Mumford-Shah regularization,, Image Processing, 15 (2006), 483.  doi: 10.1109/TIP.2005.863120.  Google Scholar [8] M. Bertalmio, L. Vese, G. Sapiro and S. Osher, Simultaneous structure and texture image inpainting,, IEEE Transactions on Image Processing, 12 (2003), 882.   Google Scholar [9] M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian,, Calculus of Variations and Partial Differential Equations, 13 (2001), 123.   Google Scholar [10] I. Daubechies and G. Teschke, Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising,, Applied and Computational Harmonic Analysis, 19 (2005), 1.  doi: 10.1016/j.acha.2004.12.004.  Google Scholar [11] I. Ekeland and R. Témam, "Convex Analysis and Variational Problems,", SIAM, (1999).   Google Scholar [12] L. C. Evans and Y. Yu, Various properties of solutions of the infinity-Laplacian equation,, Communications in Partial Differential Equations, 30 (2005), 1401.  doi: 10.1080/03605300500258956.  Google Scholar [13] J. B. Garnett, T. M. Le, Y. Meyer and L. A. Vese, Image decompositions using bounded variation and generalized homogeneous besov spaces,, Applied and Computational Harmonic Analysis, 23 (2007), 25.  doi: 10.1016/j.acha.2007.01.005.  Google Scholar [14] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing,, Multiscale Modeling & Simulation, 7 (2009), 1005.  doi: 10.1137/070698592.  Google Scholar [15] J. Gilles, Noisy image decomposition: A new structure, texture and noise model based on local adaptivity,, Journal of Mathematical Imaging and Vision, 28 (2007), 285.  doi: 10.1007/s10851-007-0020-y.  Google Scholar [16] J. Gilles and Y. Meyer, Properties of BV-G structures + textures decomposition models. Application to road detection in satellite images,, IEEE Transactions on Image Processing}, 19 (2010), 2793.  doi: 10.1109/TIP.2010.2049946.  Google Scholar [17] Y. Kim and L. Vese, Image recovery using functions of bounded variation and sobolev spaces of negative differentiability,, Inverse Problems and Imaging, 3 (2009), 43.  doi: 10.3934/ipi.2009.3.43.  Google Scholar [18] T. M. Le, L. H. Lieu and L. A. Vese, $(\phi,\phi*)$ image decomposition models and minimization algorithms,, Journal of Mathematical Imaging and Vision, 33 (2009), 135.  doi: 10.1007/s10851-008-0130-1.  Google Scholar [19] T. M. Le and L. A. Vese, Image decomposition using total variation and div (bmo),, Multiscale Modeling and Simulation, 4 (2005), 390.  doi: 10.1137/040610052.  Google Scholar [20] L. H. Lieu and L. A. Vese, Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev spaces,, Applied Mathematics and Optimization, 58 (2008), 167.  doi: 10.1007/s00245-008-9047-8.  Google Scholar [21] G. Lu and P. Wang, Inhomogeneous infinity Laplace equation,, Advances in Mathematics, 217 (2008), 1838.  doi: 10.1016/j.aim.2007.11.020.  Google Scholar [22] Y. Meyer, "Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures,", University Lecture Series, 22 (2001).   Google Scholar [23] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems,, Communications on Pure and Applied Mathematics, 42 (1989), 577.  doi: 10.1002/cpa.3160420503.  Google Scholar [24] S. Osher, A. Solé and L. Vese, Image decomposition and restoration using total variation minimization and the h1,, Multiscale Modeling & Simulation, 1 (2003), 349.  doi: 10.1137/S1540345902416247.  Google Scholar [25] R. J. Renka, A simple explanation of the sobolev gradient method,, (2006)., (2006).   Google Scholar [26] W. B. Richardson, Sobolev gradient preconditioning for image-processing PDEs,, Communications in Numerical Methods in Engineering, 24 (2006), 493.  doi: 10.1002/cnm.951.  Google Scholar [27] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D: Nonlinear Phenomena, 60 (1992), 259.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [28] Hayden Schaeffer and Stanley Osher, A low patch-rank interpretation of texture,, SIAM Journal on Imaging Sciences, 6 (2013), 226.  doi: 10.1137/110854989.  Google Scholar [29] J. Shen, Piecewise $H^{-1} -H^0 - H^1$ images and the Mumford-Shah-Sobolev model for segmented image decomposition,, APPL. MATH. RES. EXP, 4 (2005).   Google Scholar [30] G. Sundaramoorthi, A. Yezzi and A. C. Mennucci, Sobolev active contours,, International Journal of Computer Vision, 73 (2007), 345.   Google Scholar [31] L. A. Vese and S. J. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing,, Journal of Scientific Computing, 19 (2002), 553.  doi: 10.1023/A:1025384832106.  Google Scholar [32] L. A. Vese and S. J. Osher, Image denoising and decomposition with total variation minimization and oscillatory functions,, Journal of Mathematical Imaging and Vision, 20 (2004), 7.  doi: 10.1023/B:JMIV.0000011316.54027.6a.  Google Scholar

show all references

##### References:
 [1] G. Aubert and P. Kornprobst, "Mathematical Problems in Image Processing. Partial Differential Equations and the Calculus of Variations,", Second edition, 147 (2006).   Google Scholar [2] Gilles Aubert and Jean-François Aujol, Modeling very oscillating signals. Application to image processing,, Applied Mathematics and Optimization, 51 (2005), 163.  doi: 10.1007/s00245-004-0812-z.  Google Scholar [3] J.-F. Aujol, "Contribution à l'Analyse de Textures en Traitement d'Images par Méthodes Variationnelles et Équations aux Dérivées Partielles,", Ph.D thesis, (2004).   Google Scholar [4] J.-F. Aujol and T. F. Chan, Combining geometrical and textured information to perform image classification,, Journal of Visual Communication and Image Representation, 17 (2006), 1004.  doi: 10.1016/j.jvcir.2006.02.001.  Google Scholar [5] J.-F. Aujol and S. H. Kang, Color image decomposition and restoration,, Journal of Visual Communication and Image Representation, 17 (2006), 916.  doi: 10.1016/j.jvcir.2005.02.001.  Google Scholar [6] Jean-François Aujol, Gilles Aubert, Laure Blanc-Féraud and Antonin Chambolle, Image decomposition into a bounded variation component and an oscillating component,, Journal of Mathematical Imaging and Vision, 22 (2005), 71.  doi: 10.1007/s10851-005-4783-8.  Google Scholar [7] L. Bar, N. Sochen and N. Kiryati, Semi-blind image restoration via Mumford-Shah regularization,, Image Processing, 15 (2006), 483.  doi: 10.1109/TIP.2005.863120.  Google Scholar [8] M. Bertalmio, L. Vese, G. Sapiro and S. Osher, Simultaneous structure and texture image inpainting,, IEEE Transactions on Image Processing, 12 (2003), 882.   Google Scholar [9] M. G. Crandall, L. C. Evans and R. F. Gariepy, Optimal Lipschitz extensions and the infinity Laplacian,, Calculus of Variations and Partial Differential Equations, 13 (2001), 123.   Google Scholar [10] I. Daubechies and G. Teschke, Variational image restoration by means of wavelets: Simultaneous decomposition, deblurring, and denoising,, Applied and Computational Harmonic Analysis, 19 (2005), 1.  doi: 10.1016/j.acha.2004.12.004.  Google Scholar [11] I. Ekeland and R. Témam, "Convex Analysis and Variational Problems,", SIAM, (1999).   Google Scholar [12] L. C. Evans and Y. Yu, Various properties of solutions of the infinity-Laplacian equation,, Communications in Partial Differential Equations, 30 (2005), 1401.  doi: 10.1080/03605300500258956.  Google Scholar [13] J. B. Garnett, T. M. Le, Y. Meyer and L. A. Vese, Image decompositions using bounded variation and generalized homogeneous besov spaces,, Applied and Computational Harmonic Analysis, 23 (2007), 25.  doi: 10.1016/j.acha.2007.01.005.  Google Scholar [14] G. Gilboa and S. Osher, Nonlocal operators with applications to image processing,, Multiscale Modeling & Simulation, 7 (2009), 1005.  doi: 10.1137/070698592.  Google Scholar [15] J. Gilles, Noisy image decomposition: A new structure, texture and noise model based on local adaptivity,, Journal of Mathematical Imaging and Vision, 28 (2007), 285.  doi: 10.1007/s10851-007-0020-y.  Google Scholar [16] J. Gilles and Y. Meyer, Properties of BV-G structures + textures decomposition models. Application to road detection in satellite images,, IEEE Transactions on Image Processing}, 19 (2010), 2793.  doi: 10.1109/TIP.2010.2049946.  Google Scholar [17] Y. Kim and L. Vese, Image recovery using functions of bounded variation and sobolev spaces of negative differentiability,, Inverse Problems and Imaging, 3 (2009), 43.  doi: 10.3934/ipi.2009.3.43.  Google Scholar [18] T. M. Le, L. H. Lieu and L. A. Vese, $(\phi,\phi*)$ image decomposition models and minimization algorithms,, Journal of Mathematical Imaging and Vision, 33 (2009), 135.  doi: 10.1007/s10851-008-0130-1.  Google Scholar [19] T. M. Le and L. A. Vese, Image decomposition using total variation and div (bmo),, Multiscale Modeling and Simulation, 4 (2005), 390.  doi: 10.1137/040610052.  Google Scholar [20] L. H. Lieu and L. A. Vese, Image restoration and decomposition via bounded total variation and negative Hilbert-Sobolev spaces,, Applied Mathematics and Optimization, 58 (2008), 167.  doi: 10.1007/s00245-008-9047-8.  Google Scholar [21] G. Lu and P. Wang, Inhomogeneous infinity Laplace equation,, Advances in Mathematics, 217 (2008), 1838.  doi: 10.1016/j.aim.2007.11.020.  Google Scholar [22] Y. Meyer, "Oscillating Patterns in Image Processing and Nonlinear Evolution Equations. The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures,", University Lecture Series, 22 (2001).   Google Scholar [23] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems,, Communications on Pure and Applied Mathematics, 42 (1989), 577.  doi: 10.1002/cpa.3160420503.  Google Scholar [24] S. Osher, A. Solé and L. Vese, Image decomposition and restoration using total variation minimization and the h1,, Multiscale Modeling & Simulation, 1 (2003), 349.  doi: 10.1137/S1540345902416247.  Google Scholar [25] R. J. Renka, A simple explanation of the sobolev gradient method,, (2006)., (2006).   Google Scholar [26] W. B. Richardson, Sobolev gradient preconditioning for image-processing PDEs,, Communications in Numerical Methods in Engineering, 24 (2006), 493.  doi: 10.1002/cnm.951.  Google Scholar [27] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms,, Physica D: Nonlinear Phenomena, 60 (1992), 259.  doi: 10.1016/0167-2789(92)90242-F.  Google Scholar [28] Hayden Schaeffer and Stanley Osher, A low patch-rank interpretation of texture,, SIAM Journal on Imaging Sciences, 6 (2013), 226.  doi: 10.1137/110854989.  Google Scholar [29] J. Shen, Piecewise $H^{-1} -H^0 - H^1$ images and the Mumford-Shah-Sobolev model for segmented image decomposition,, APPL. MATH. RES. EXP, 4 (2005).   Google Scholar [30] G. Sundaramoorthi, A. Yezzi and A. C. Mennucci, Sobolev active contours,, International Journal of Computer Vision, 73 (2007), 345.   Google Scholar [31] L. A. Vese and S. J. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing,, Journal of Scientific Computing, 19 (2002), 553.  doi: 10.1023/A:1025384832106.  Google Scholar [32] L. A. Vese and S. J. Osher, Image denoising and decomposition with total variation minimization and oscillatory functions,, Journal of Mathematical Imaging and Vision, 20 (2004), 7.  doi: 10.1023/B:JMIV.0000011316.54027.6a.  Google Scholar
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