The topological gradient method for semi-linear problems and application to edge detection and noise removal

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February 2016, 10(1): 51-86. doi: 10.3934/ipi.2016.10.51

The topological gradient method for semi-linear problems and application to edge detection and noise removal

Audric Drogoul ^1, and Gilles Aubert ^1,

1.	Univ. Nice Sophia Antipolis, CNRS, LJAD, UMR 7351, 06100 Nice, France, France

Received December 2014 Revised July 2015 Published February 2016

Abstract
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The goal of this paper is to apply the topological gradient method to edge detection and noise removal for images degraded by various noises and blurs. First applied to edge detection for images degraded by a Gaussian noise, we propose here to extend the method to blurred images contaminated either by a multiplicative noise of gamma law or to blurred Poissonian images. We compute, both for perforated and cracked domains, the topological gradient for each noise model. Then we present an edge detection/restoration algorithm based on this notion and we apply it to the two degradation models previously described. We compare our method with other classical variational approaches such that the Mumford-Shah and TV restoration models and with the classical Canny edge detector. Some experimental results showing the efficiency, the robustness and the rapidity of the method are presented.

Keywords: speckle noise, semi-linear problem, Poisson noise, Topological gradient, restoration., segmentation.

Mathematics Subject Classification: 35J91, 49Q10, 49Q12, 94A08, 94A1.

Citation: Audric Drogoul, Gilles Aubert. The topological gradient method for semi-linear problems and application to edge detection and noise removal. Inverse Problems and Imaging, 2016, 10 (1) : 51-86. doi: 10.3934/ipi.2016.10.51

References:

[1]	L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Gamma-convergence, Communications on Pure and Applied Mathematics, 43 (1990), 999-1036. doi: 10.1002/cpa.3160430805.
[2]	S. Amstutz, Topological sensitivity analysis for some nonlinear PDE system, J. Math. Pures Appl., 85 (2006), 540-557. doi: 10.1016/j.matpur.2005.10.008.
[3]	S. Amstutz, The topological asymptotic for the Navier-Stokes equations, ESAIM: Control, Optimisation and Calculus of Variations, 11 (2005), 401-425. doi: 10.1051/cocv:2005012.
[4]	S. Amstutz and J. Fehrenbach, Edge detection using topological gradients: A scale-space approach, J. Math. Imaging Vision, 52 (2015), 249-266. doi: 10.1007/s10851-015-0558-z.
[5]	S. Amstutz, I. Horchani and M. Masmoudi, Crack detection by the toplogical gradient method, Control and Cybernetics, 34 (2005), 81-101.
[6]	G. Aubert and J.-F. Aujol, A variational approach to removing multiplicative noise, SIAM Journal of Applied Mathematics, 68 (2008), 925-946. doi: 10.1137/060671814.
[7]	G. Aubert and A. Drogoul, Topological gradient for a fourth order operator used in image analysis, ESAIM Control Optim. Calc. Var., 21 (2015), 1120-1149. doi: 10.1051/cocv/2014061.
[8]	G. Aubert and A. Drogoul, Topological gradient for fourth order pde and application to the detection of fine structures in 2d images, C. R. Math. Acad. Sci. Paris, 352 (2014), 609-613. doi: 10.1016/j.crma.2014.06.005.
[9]	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-Verlag, 2006.
[10]	D. Auroux, From restoration by topological gradient to medical image segmentation via an asymptotic expansion, Math. Comput. Model., 49 (2009), 2191-2205. doi: 10.1016/j.mcm.2008.07.002.
[11]	D. Auroux and M. Masmoudi, A one-shot inpainting algorithm based on the topological asymptotic analysis, Computational & Applied Mathematics, 25 (2006), 251-267. doi: 10.1590/S0101-82052006000200008.
[12]	D. Auroux and M. Masmoudi, Image processing by topological asymptotic expansion, Journal of Mathematical Imaging and Vision, 33 (2009), 122-134. doi: 10.1007/s10851-008-0121-2.
[13]	D. Auroux, M. Masmoudi and L. Jaafar Belaid, Image restoration and classification by topological asymptotic expansion, Variational Formulations in Mechanics: Theory and Applications (eds. E. Taroco, E. A. de Souza Neto and A. A. Novotny), CIMNE, Barcelona, Spain, 2007, 23-42.
[14]	H. Ayasso and A. Mohammad-Djafari, Joint image restorationand segmentation using Gauss-Markov-Potts prior models and variational Bayesian computation, IEEE Trans. Image Process, 19 (2010), 2265-2277. doi: 10.1109/TIP.2010.2047902.
[15]	L. J. Belaid, M. Jaoua, M. Masmoudi and L. Siala, Image restoration and edge detection by topological asymptotic expansion, C. R. Acad. Sci. Paris, 342 (2006), 313-318. doi: 10.1016/j.crma.2005.12.009.
[16]	L. J. Belaid, M. Jaoua, M. Masmoudi and L. Siala, Application of the topological gradient to image restoration and edge detection, Engineering Analysis with Boundary Elements, 32 (2008), 891-899.
[17]	S. Ben Hadj, L. Blanc Féraud and G. Aubert, Space Variant Blind Image Restoration, SIAM Journal on Imaging Sciences, 7 (2014), 2196-2225. doi: 10.1137/130945776.
[18]	S. Bonettini, R. Zanella and L. Zanni, A scaled gradient projection method for constrained image deblurring, Inverse Problems, 25 (2009), 015002, 23pp. doi: 10.1088/0266-5611/25/1/015002.
[19]	T. Chan and J. Shen, Image Processing And Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, 2005. doi: 10.1137/1.9780898717877.
[20]	N. Dey, L. Blanc-Féraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin and J. Zerubia, 3D Microscopy Deconvolution using Richardson-Lucy Algorithm with Total Variation Regularization, Rapport de recherche RR-5272, INRIA, 2004.
[21]	A. Drogoul, Numerical analysis of the topological gradient method for fourth order models and applications to the detection of fine structures in 2D imaging, SIAM J. Imaging Sci., 7 (2014), 2700-2731. doi: 10.1137/140967374.
[22]	A. Drogoul, Topological Gradient Method Applied to the Detection of Edges and Fine Structures in Imaging, Ph.D Thesis, University of Nice Sophia Antipolis, 2014.
[23]	S. Geman and D. Geman, Stochastic relaxation, gibbs distributions, and the bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (1984), 721-741. doi: 10.1109/TPAMI.1984.4767596.
[24]	T. Hebert and R. Leahy, A generalized EM algorithm for 3D Bayesian recontruction from Poisson data using Gibbs prior, IEEE Trans. Medical imaging, 8 (1989), 194-202.
[25]	F. M. Henderson, A. J. Lewis and R. A. Ryerson, eds., Principles and Applications of Imaging Radar. Manual of Remote Sensing, 3rd edition, Wiley, New York, 1998. doi: 10.1029/99EO00047.
[26]	M. Hintermüller, Fast-set based algorithms using shape and topological sensitivity information, Control and Cybernetics, 34 (2005), 305-324.
[27]	M. Iguernane, S. A. Nazarov, J.-R. Roche, J. Sokolowski and K. Szulc, Topological derivatives for semilinear elliptic equations, Applied Mathematics and Computer Science, 19 (2009), 191-205. doi: 10.2478/v10006-009-0016-4.
[28]	K. Krissian, R. Kikinis, C.-F. Westin and K. G. Vosburgh, Speckle-constrained filtering of ultrasound images, in Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on, IEEE Computer Society, 2, 2005, 547-552. doi: 10.1109/CVPR.2005.331.
[29]	S. Larnier and J. Fehrenbach, Edge detection and image restoration with anisotropic topological gradient, in Proceedings ICASSP , International Conference on Acoustic Speech and Signal Processing, 2010, 1362-1365. doi: 10.1109/ICASSP.2010.5495448.
[30]	S. Larnier, J. Fehrenbach and M. Masmoudi, The topological gradient method: From optimal design to image processing, Milan Journal of Mathematics, 80 (2012), 411-441. doi: 10.1007/s00032-012-0196-5.
[31]	I. Larrabide, A. A. Novotny, R. A. Feijo and E. Taroco, A medical image enhancement algorithms based on topological derivative and anisotropic diffusion, in Proceedings of the XXVI Iberian Latin-American Congess on Computational Methods in Engineering, 2005.
[32]	S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1998.
[33]	M. Masmoudi, The topological asymptotic, in Computational Methods for Control Applications, GAKUTO Internat. Ser. Math. Appl., 16, Tokyo, Japan, 2001.
[34]	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.
[35]	J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Applied Mathematical Sciences, Springer, New York, 2001. doi: 10.1007/978-1-4757-4393-7.
[36]	J. Pawley, Handbook of Biological Confocal Microscopy, Springer, Berlin, 2006.
[37]	A. Sawatzky, D. Tenbrinck, X. Jiang and M. Burger, A variational framework for region-based segmentation incorporating physical noise models, Journal of Mathematical Imaging and Vision, 47 (2013), 179-209. doi: 10.1007/s10851-013-0419-6.
[38]	J. Serra, Image Analysis and Mathematical Morphology, Academic Press, Inc., Orlando, FL, USA, 1984.
[39]	J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization, SIAM J. Control Optim., 37 (1999), 1251-1272. doi: 10.1137/S0363012997323230.
[40]	M. Tur, K. C. Chin and J. W. Goodman, When is speckle noise multiplicative?, Applied Optics, 21 (1982), 1157-1159. doi: 10.1364/AO.21.001157.
[41]	M. N. Wernick and J. N. Aarsvold, eds., Emission Tomography: The Fundamental of PET and SPECT, Elsevier, Amsterdam, 2004.

show all references

References:

[1]	L. Ambrosio and V. M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Gamma-convergence, Communications on Pure and Applied Mathematics, 43 (1990), 999-1036. doi: 10.1002/cpa.3160430805.
[2]	S. Amstutz, Topological sensitivity analysis for some nonlinear PDE system, J. Math. Pures Appl., 85 (2006), 540-557. doi: 10.1016/j.matpur.2005.10.008.
[3]	S. Amstutz, The topological asymptotic for the Navier-Stokes equations, ESAIM: Control, Optimisation and Calculus of Variations, 11 (2005), 401-425. doi: 10.1051/cocv:2005012.
[4]	S. Amstutz and J. Fehrenbach, Edge detection using topological gradients: A scale-space approach, J. Math. Imaging Vision, 52 (2015), 249-266. doi: 10.1007/s10851-015-0558-z.
[5]	S. Amstutz, I. Horchani and M. Masmoudi, Crack detection by the toplogical gradient method, Control and Cybernetics, 34 (2005), 81-101.
[6]	G. Aubert and J.-F. Aujol, A variational approach to removing multiplicative noise, SIAM Journal of Applied Mathematics, 68 (2008), 925-946. doi: 10.1137/060671814.
[7]	G. Aubert and A. Drogoul, Topological gradient for a fourth order operator used in image analysis, ESAIM Control Optim. Calc. Var., 21 (2015), 1120-1149. doi: 10.1051/cocv/2014061.
[8]	G. Aubert and A. Drogoul, Topological gradient for fourth order pde and application to the detection of fine structures in 2d images, C. R. Math. Acad. Sci. Paris, 352 (2014), 609-613. doi: 10.1016/j.crma.2014.06.005.
[9]	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-Verlag, 2006.
[10]	D. Auroux, From restoration by topological gradient to medical image segmentation via an asymptotic expansion, Math. Comput. Model., 49 (2009), 2191-2205. doi: 10.1016/j.mcm.2008.07.002.
[11]	D. Auroux and M. Masmoudi, A one-shot inpainting algorithm based on the topological asymptotic analysis, Computational & Applied Mathematics, 25 (2006), 251-267. doi: 10.1590/S0101-82052006000200008.
[12]	D. Auroux and M. Masmoudi, Image processing by topological asymptotic expansion, Journal of Mathematical Imaging and Vision, 33 (2009), 122-134. doi: 10.1007/s10851-008-0121-2.
[13]	D. Auroux, M. Masmoudi and L. Jaafar Belaid, Image restoration and classification by topological asymptotic expansion, Variational Formulations in Mechanics: Theory and Applications (eds. E. Taroco, E. A. de Souza Neto and A. A. Novotny), CIMNE, Barcelona, Spain, 2007, 23-42.
[14]	H. Ayasso and A. Mohammad-Djafari, Joint image restorationand segmentation using Gauss-Markov-Potts prior models and variational Bayesian computation, IEEE Trans. Image Process, 19 (2010), 2265-2277. doi: 10.1109/TIP.2010.2047902.
[15]	L. J. Belaid, M. Jaoua, M. Masmoudi and L. Siala, Image restoration and edge detection by topological asymptotic expansion, C. R. Acad. Sci. Paris, 342 (2006), 313-318. doi: 10.1016/j.crma.2005.12.009.
[16]	L. J. Belaid, M. Jaoua, M. Masmoudi and L. Siala, Application of the topological gradient to image restoration and edge detection, Engineering Analysis with Boundary Elements, 32 (2008), 891-899.
[17]	S. Ben Hadj, L. Blanc Féraud and G. Aubert, Space Variant Blind Image Restoration, SIAM Journal on Imaging Sciences, 7 (2014), 2196-2225. doi: 10.1137/130945776.
[18]	S. Bonettini, R. Zanella and L. Zanni, A scaled gradient projection method for constrained image deblurring, Inverse Problems, 25 (2009), 015002, 23pp. doi: 10.1088/0266-5611/25/1/015002.
[19]	T. Chan and J. Shen, Image Processing And Analysis: Variational, PDE, Wavelet, and Stochastic Methods, SIAM, 2005. doi: 10.1137/1.9780898717877.
[20]	N. Dey, L. Blanc-Féraud, C. Zimmer, P. Roux, Z. Kam, J.-C. Olivo-Marin and J. Zerubia, 3D Microscopy Deconvolution using Richardson-Lucy Algorithm with Total Variation Regularization, Rapport de recherche RR-5272, INRIA, 2004.
[21]	A. Drogoul, Numerical analysis of the topological gradient method for fourth order models and applications to the detection of fine structures in 2D imaging, SIAM J. Imaging Sci., 7 (2014), 2700-2731. doi: 10.1137/140967374.
[22]	A. Drogoul, Topological Gradient Method Applied to the Detection of Edges and Fine Structures in Imaging, Ph.D Thesis, University of Nice Sophia Antipolis, 2014.
[23]	S. Geman and D. Geman, Stochastic relaxation, gibbs distributions, and the bayesian restoration of images, IEEE Transactions on Pattern Analysis and Machine Intelligence, 6 (1984), 721-741. doi: 10.1109/TPAMI.1984.4767596.
[24]	T. Hebert and R. Leahy, A generalized EM algorithm for 3D Bayesian recontruction from Poisson data using Gibbs prior, IEEE Trans. Medical imaging, 8 (1989), 194-202.
[25]	F. M. Henderson, A. J. Lewis and R. A. Ryerson, eds., Principles and Applications of Imaging Radar. Manual of Remote Sensing, 3rd edition, Wiley, New York, 1998. doi: 10.1029/99EO00047.
[26]	M. Hintermüller, Fast-set based algorithms using shape and topological sensitivity information, Control and Cybernetics, 34 (2005), 305-324.
[27]	M. Iguernane, S. A. Nazarov, J.-R. Roche, J. Sokolowski and K. Szulc, Topological derivatives for semilinear elliptic equations, Applied Mathematics and Computer Science, 19 (2009), 191-205. doi: 10.2478/v10006-009-0016-4.
[28]	K. Krissian, R. Kikinis, C.-F. Westin and K. G. Vosburgh, Speckle-constrained filtering of ultrasound images, in Computer Vision and Pattern Recognition, 2005. CVPR 2005. IEEE Computer Society Conference on, IEEE Computer Society, 2, 2005, 547-552. doi: 10.1109/CVPR.2005.331.
[29]	S. Larnier and J. Fehrenbach, Edge detection and image restoration with anisotropic topological gradient, in Proceedings ICASSP , International Conference on Acoustic Speech and Signal Processing, 2010, 1362-1365. doi: 10.1109/ICASSP.2010.5495448.
[30]	S. Larnier, J. Fehrenbach and M. Masmoudi, The topological gradient method: From optimal design to image processing, Milan Journal of Mathematics, 80 (2012), 411-441. doi: 10.1007/s00032-012-0196-5.
[31]	I. Larrabide, A. A. Novotny, R. A. Feijo and E. Taroco, A medical image enhancement algorithms based on topological derivative and anisotropic diffusion, in Proceedings of the XXVI Iberian Latin-American Congess on Computational Methods in Engineering, 2005.
[32]	S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, 1998.
[33]	M. Masmoudi, The topological asymptotic, in Computational Methods for Control Applications, GAKUTO Internat. Ser. Math. Appl., 16, Tokyo, Japan, 2001.
[34]	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.
[35]	J.-C. Nédélec, Acoustic and Electromagnetic Equations: Integral Representations for Harmonic Problems, Applied Mathematical Sciences, Springer, New York, 2001. doi: 10.1007/978-1-4757-4393-7.
[36]	J. Pawley, Handbook of Biological Confocal Microscopy, Springer, Berlin, 2006.
[37]	A. Sawatzky, D. Tenbrinck, X. Jiang and M. Burger, A variational framework for region-based segmentation incorporating physical noise models, Journal of Mathematical Imaging and Vision, 47 (2013), 179-209. doi: 10.1007/s10851-013-0419-6.
[38]	J. Serra, Image Analysis and Mathematical Morphology, Academic Press, Inc., Orlando, FL, USA, 1984.
[39]	J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization, SIAM J. Control Optim., 37 (1999), 1251-1272. doi: 10.1137/S0363012997323230.
[40]	M. Tur, K. C. Chin and J. W. Goodman, When is speckle noise multiplicative?, Applied Optics, 21 (1982), 1157-1159. doi: 10.1364/AO.21.001157.
[41]	M. N. Wernick and J. N. Aarsvold, eds., Emission Tomography: The Fundamental of PET and SPECT, Elsevier, Amsterdam, 2004.

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