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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 |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
I. Ekeland and R. Témam, "Convex Analysis and Variational Problems," SIAM, 1999. |
[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. |
[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. |
[14] |
G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2009), 1005-1028.
doi: 10.1137/070698592. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
R. J. Renka, A simple explanation of the sobolev gradient method, (2006). |
[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. |
[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. |
[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. |
[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. |
[30] |
G. Sundaramoorthi, A. Yezzi and A. C. Mennucci, Sobolev active contours, International Journal of Computer Vision, 73 (2007), 345-366. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[11] |
I. Ekeland and R. Témam, "Convex Analysis and Variational Problems," SIAM, 1999. |
[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. |
[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. |
[14] |
G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling & Simulation, 7 (2009), 1005-1028.
doi: 10.1137/070698592. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[25] |
R. J. Renka, A simple explanation of the sobolev gradient method, (2006). |
[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. |
[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. |
[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. |
[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. |
[30] |
G. Sundaramoorthi, A. Yezzi and A. C. Mennucci, Sobolev active contours, International Journal of Computer Vision, 73 (2007), 345-366. |
[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. |
[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. |
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