American Institute of Mathematical Sciences

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

A texture model based on a concentration of measure

Hayden Schaeffer 1, , John Garnett 1, and  Luminita A. Vese 2,

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.
Keywords: deblurring., concentration of measure, Texture, dual spaces.
Mathematics Subject Classification: 49M27, 28E99, 35K6.
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, Applied Mathematical Sciences, 147, Springer, New York, 2006.  Google Scholar

[2]

Gilles Aubert and Jean-François Aujol, Modeling very oscillating signals. Application to image processing, Applied Mathematics and Optimization, 51 (2005), 163-182. 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, June, 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), 1004-1023. 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), 916-928. 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), 71-88. 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, IEEE Transactions on, 15 (2006), 483-493. 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), 882-889. 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), 123-139.  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), 1-16. 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), 1401-1428. 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), 25-56. 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), 1005-1028. 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), 285-295. 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), 2793-2800. 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), 43-68. 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), 135-148. 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), 390-423. 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), 167-193. doi: 10.1007/s00245-008-9047-8.  Google Scholar

[21]

G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Advances in Mathematics, 217 (2008), 1838-1868. 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, American Mathematical Soc., Providence, RI, 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), 577-685. 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), 349-370. doi: 10.1137/S1540345902416247.  Google Scholar

[25]

R. J. Renka, A simple explanation of the sobolev gradient method, (2006). Google Scholar

[26]

W. B. Richardson, Sobolev gradient preconditioning for image-processing PDEs, Communications in Numerical Methods in Engineering, 24 (2006), 493-504. 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), 259-268. 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), 226-262. 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), 2005. Google Scholar

[30]

G. Sundaramoorthi, A. Yezzi and A. C. Mennucci, Sobolev active contours, International Journal of Computer Vision, 73 (2007), 345-366. 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), 553-572. 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), 7-18. 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, Applied Mathematical Sciences, 147, Springer, New York, 2006.  Google Scholar

[2]

Gilles Aubert and Jean-François Aujol, Modeling very oscillating signals. Application to image processing, Applied Mathematics and Optimization, 51 (2005), 163-182. 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, June, 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), 1004-1023. 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), 916-928. 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), 71-88. 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, IEEE Transactions on, 15 (2006), 483-493. 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), 882-889. 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), 123-139.  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), 1-16. 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), 1401-1428. 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), 25-56. 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), 1005-1028. 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), 285-295. 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), 2793-2800. 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), 43-68. 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), 135-148. 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), 390-423. 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), 167-193. doi: 10.1007/s00245-008-9047-8.  Google Scholar

[21]

G. Lu and P. Wang, Inhomogeneous infinity Laplace equation, Advances in Mathematics, 217 (2008), 1838-1868. 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, American Mathematical Soc., Providence, RI, 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), 577-685. 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), 349-370. doi: 10.1137/S1540345902416247.  Google Scholar

[25]

R. J. Renka, A simple explanation of the sobolev gradient method, (2006). Google Scholar

[26]

W. B. Richardson, Sobolev gradient preconditioning for image-processing PDEs, Communications in Numerical Methods in Engineering, 24 (2006), 493-504. 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), 259-268. 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), 226-262. 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), 2005. Google Scholar

[30]

G. Sundaramoorthi, A. Yezzi and A. C. Mennucci, Sobolev active contours, International Journal of Computer Vision, 73 (2007), 345-366. 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), 553-572. 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), 7-18. doi: 10.1023/B:JMIV.0000011316.54027.6a.  Google Scholar

[1]

Ugo Bessi. The stochastic value function in metric measure spaces. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 1819-1839. doi: 10.3934/dcds.2017076
[2]

Thomas Honold, Ivan Landjev. The dual construction for arcs in projective Hjelmslev spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 11-21. doi: 10.3934/amc.2011.5.11
[3]

Ferenc Weisz. Dual spaces of mixed-norm martingale Hardy spaces. Communications on Pure & Applied Analysis, 2021, 20 (2) : 681-695. doi: 10.3934/cpaa.2020285
[4]

Jian-Feng Cai, Raymond H. Chan, Zuowei Shen. Simultaneous cartoon and texture inpainting. Inverse Problems & Imaging, 2010, 4 (3) : 379-395. doi: 10.3934/ipi.2010.4.379
[5]

Luigi Ambrosio, Michele Miranda jr., Diego Pallara. Sets with finite perimeter in Wiener spaces, perimeter measure and boundary rectifiability. Discrete & Continuous Dynamical Systems, 2010, 28 (2) : 591-606. doi: 10.3934/dcds.2010.28.591
[6]

Piotr Gwiazda, Piotr Orlinski, Agnieszka Ulikowska. Finite range method of approximation for balance laws in measure spaces. Kinetic & Related Models, 2017, 10 (3) : 669-688. doi: 10.3934/krm.2017027
[7]

Giulia Luise, Giuseppe Savaré. Contraction and regularizing properties of heat flows in metric measure spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 273-297. doi: 10.3934/dcdss.2020327
[8]

Bang-Xian Han. New characterizations of Ricci curvature on RCD metric measure spaces. Discrete & Continuous Dynamical Systems, 2018, 38 (10) : 4915-4927. doi: 10.3934/dcds.2018214
[9]

Alberto Fiorenza, Anna Mercaldo, Jean Michel Rakotoson. Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 893-906. doi: 10.3934/dcds.2002.8.893
[10]

Xiangtuan Xiong, Jinmei Li, Jin Wen. Some novel linear regularization methods for a deblurring problem. Inverse Problems & Imaging, 2017, 11 (2) : 403-426. doi: 10.3934/ipi.2017019
[11]

Zhichang Guo, Wenjuan Yao, Jiebao Sun, Boying Wu. Nonlinear fractional diffusion model for deblurring images with textures. Inverse Problems & Imaging, 2019, 13 (6) : 1161-1188. doi: 10.3934/ipi.2019052
[12]

Jie Huang, Marco Donatelli, Raymond H. Chan. Nonstationary iterated thresholding algorithms for image deblurring. Inverse Problems & Imaging, 2013, 7 (3) : 717-736. doi: 10.3934/ipi.2013.7.717
[13]

Giuseppe Savaré. Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in $RCD (K, \infty)$ metric measure spaces. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1641-1661. doi: 10.3934/dcds.2014.34.1641
[14]

Pierre-Étienne Druet. Higher $L^p$ regularity for vector fields that satisfy divergence and rotation constraints in dual Sobolev spaces, and application to some low-frequency Maxwell equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 475-496. doi: 10.3934/dcdss.2015.8.475
[15]

Grégory Faye, Pascal Chossat. A spatialized model of visual texture perception using the structure tensor formalism. Networks & Heterogeneous Media, 2013, 8 (1) : 211-260. doi: 10.3934/nhm.2013.8.211
[16]

Hong Zhou, M. Gregory Forest, Qi Wang. Anchoring-induced texture & shear banding of nematic polymers in shear cells. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 707-733. doi: 10.3934/dcdsb.2007.8.707
[17]

Nam-Yong Lee, Bradley J. Lucier. Preconditioned conjugate gradient method for boundary artifact-free image deblurring. Inverse Problems & Imaging, 2016, 10 (1) : 195-225. doi: 10.3934/ipi.2016.10.195
[18]

Zhongyuan Liu. Concentration of solutions for the fractional Nirenberg problem. Communications on Pure & Applied Analysis, 2016, 15 (2) : 563-576. doi: 10.3934/cpaa.2016.15.563
[19]

Stefano Bianchini, Elio Marconi. On the concentration of entropy for scalar conservation laws. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 73-88. doi: 10.3934/dcdss.2016.9.73
[20]

Shuangjie Peng, Jing Zhou. Concentration of solutions for a Paneitz type problem. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 1055-1072. doi: 10.3934/dcds.2010.26.1055

2020 Impact Factor: 1.639

Tools

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences