几何变分理论在图像处理中的应用
|
摘要
本文的研究基于几何变分理论和偏微分方程理论,主要讨论了图像处理中的几个常见问题,如图像去噪,图像分割,图像去模糊,图像增强等。基本的思路是根据不同的背景提出相应的能量泛函,并通过求解Euler-Lagrange方程极小化能量泛函。对于求解Euler-Lagrange方程的方法,本文也给出了一些快速算法,使得数值计算更加稳定。具体地,本文的研究结果包括下面几个方面：
1.基于对Nonlocal—BV函数的分析,以ROF模型为例,讨论了包含Nonlocal-TV正则项的泛函极小解的存在唯一性及其相应的热扩散方程解的存在唯一性。应用方面,我们将量子集BV函数推广到Nonlocal的情形,从而提高了图像分割的效果。
2.讨论了基于frame的图像去模糊问题,通过分析未知函数在空间域和频率域上正则项的不同选取,提出了三种不同的盲去模糊模型进行对比研究,并引进快速算法,详细介绍了模型的数值实现过程。
3.在经典的图像去噪模型-ROF模型和图像分割模型-测地活动环路模型的基础上,引进张量投票技术,分别对平衡参数和停止函数作了修改,使得它们在保留纹理的去噪和对比度较低的图像分割方面更加有效。
4.基于Retinex理论,我们提出了一种分解光照函数和反射函数的新模型,使用快速算法求解达到了很好的图像增强效果,并讨论了模型解的存在唯一性,算法的收敛性等理论。
This paper is based on variational method and partial differential equations. We focus on some common problems in image processing, such as denoising, segmentation, deblurring and image enhancement. The mean idea is to deal with an energy minimization problem by looking for the solution of the related Euler-Lagrange equation. We also give some fast algorithms which are stable and efficient to solve the equations. The mean results are as follows:
1. We study the Nonlocal-BV function space and give some properties. Then the nonlocal-ROF model is discussed about the following topics:existence and uniqueness of the solution in nonlocal-BV space, existence and uniqueness of the solution to the related heat equation. We give the definition of the nonlocal quantum BV function, and show the applications in image segmentation numerically.
2. We discuss the frame-based image delurring, including different choice of the regu-larization term from space domain and frequency domain. In order to compare the choice, we give three different models, then we explain them in detail including the numerical implement.
3. We introduce some applications of Tensor Voting method by combining it with clas-sical models. we modify Geodesic Active Contour model by adding influence of Tensor Voting in the stopping function during the detecting procedure, and we im-prove ROF model with the same idea. Numerical results show the effectiveness.
4. Based on the Retinex theory, we give a new model to decompose the image into illu-mination function and reflection function. We also give some existence and conver-gence theory about the new model. Some fast algorithms help us get good numerical results.
1.基于对Nonlocal—BV函数的分析,以ROF模型为例,讨论了包含Nonlocal-TV正则项的泛函极小解的存在唯一性及其相应的热扩散方程解的存在唯一性。应用方面,我们将量子集BV函数推广到Nonlocal的情形,从而提高了图像分割的效果。
2.讨论了基于frame的图像去模糊问题,通过分析未知函数在空间域和频率域上正则项的不同选取,提出了三种不同的盲去模糊模型进行对比研究,并引进快速算法,详细介绍了模型的数值实现过程。
3.在经典的图像去噪模型-ROF模型和图像分割模型-测地活动环路模型的基础上,引进张量投票技术,分别对平衡参数和停止函数作了修改,使得它们在保留纹理的去噪和对比度较低的图像分割方面更加有效。
4.基于Retinex理论,我们提出了一种分解光照函数和反射函数的新模型,使用快速算法求解达到了很好的图像增强效果,并讨论了模型解的存在唯一性,算法的收敛性等理论。
This paper is based on variational method and partial differential equations. We focus on some common problems in image processing, such as denoising, segmentation, deblurring and image enhancement. The mean idea is to deal with an energy minimization problem by looking for the solution of the related Euler-Lagrange equation. We also give some fast algorithms which are stable and efficient to solve the equations. The mean results are as follows:
1. We study the Nonlocal-BV function space and give some properties. Then the nonlocal-ROF model is discussed about the following topics:existence and uniqueness of the solution in nonlocal-BV space, existence and uniqueness of the solution to the related heat equation. We give the definition of the nonlocal quantum BV function, and show the applications in image segmentation numerically.
2. We discuss the frame-based image delurring, including different choice of the regu-larization term from space domain and frequency domain. In order to compare the choice, we give three different models, then we explain them in detail including the numerical implement.
3. We introduce some applications of Tensor Voting method by combining it with clas-sical models. we modify Geodesic Active Contour model by adding influence of Tensor Voting in the stopping function during the detecting procedure, and we im-prove ROF model with the same idea. Numerical results show the effectiveness.
4. Based on the Retinex theory, we give a new model to decompose the image into illu-mination function and reflection function. We also give some existence and conver-gence theory about the new model. Some fast algorithms help us get good numerical results.
引文
[1]L. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise remove algorithms, Physica D,60:259-268,1992.
[2]Y. Meyer, Oscillating patterns in image processing and nonlinear evolution equations,大学演讲系列,Vol.22,AMs 2002.
[3]L. Vese and S. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, Journal of Scientific Computing,19:553-572,2003.
[4]E. Tadmor, S. Nezzar, and L. Vese, Multiscale hierarchical decomposition of images with applications to deblurring, denoising and segmentation, Commun. Math. Sci.6:281-307, 2008.
[5]S. Osher, A. Sole and L. Vese, Image decomposition and restoration using total variation minimiation and the H-1 norm, Mutiscale Modeling and Simulation,1:349-370,2003.
[6]S. Kindermann, S. Osher and P. Jones, Deblurring and denoising of images by nonlocal functionals, SIAM Mutiscale Modeling and Simulation,4(4):1091-1115,2005.
[7]G. Gilboa, J. Darbon, S. Osher and T. F. Chan, Nonlocal convex functional for image regu-larization,2006. UCLA CAM Report 06-57.
[8]G. Gilboa and S. Osher, Nonlocal operators with applications to image processing,2007. UCLA CAM Report 07-23.
[9]G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, volume 147 of Applied Mathematical Sciences, Springer-Verlag,2002.
[10]W. Rudin, Real and Complex Analysis, McGraw-Hill, Inc.1987.
[11]L. Vese, A Study in the BV Space of a Denoising-Deblurring Variational Problem, Applied Mathematics Optimization,44:131-161,2001.
[12]L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford University Press, New York,2000.
[13]L.C.Evans, Partial Differential Equations, American Mathematical Society,1998.
[14]F. Demengel and R. Temam, Convex functions of a measure and applications, Indiana Uni-versity Mathematics Journal,33:673-709,1984.
[15]L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, CRC Press, 1992.
[16]E.Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhauser Boston, Inc. 1984.
[17]L. I. Rudin, P. L. Lions and S. Osher, Mutiplicative denoising and deblurring: theory and algrithms, In. Osher and N. Paragios, editors, Geometric Level Sets in Image Vision, and Graphics.103-119,2003.
[18]T. Le, R. Chartrand and T. T. Asaki, A variatioal approach for reconstructing images cor-rupted by poisson noise, UCLA CAM Report 05-49.
[19]G.Aubert and J.F.Aujol, A Variational Approach to Remove Multiplicative Noise, CMLA 2006-11.
[20]A. Chambolle, An algrithm for total variation minimization and applications, JMIV,20:89-97,2004.
[21]X. Zhang, M. Burger, X. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, UCLA CAM Report 09-03.
[22]G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, SIAM Mutiscale Modeling and Simulation,6(2):595-630,2007.
[23]J. Shen and S. H. Kang, Quantum tv and applications in image processing, Inverse Problems and Imaging,00(2):1-18,2007.
[24]J. Cai, H. Ji, C. Liu and Z. Shen, Blind motion deblurring using multiple images, UCLA CAM Report 09-59.
[25]J. Cai, H. Ji, C. Liu and Z. Shen, High quality curvelet based motion deblurring from an image pair, UCLA CAM Report 09-60.
[26]J. Cai, H. Ji, C. Liu and Z. Shen, Blind motion deblurring from a single image using sparse approximation, UCLA CAM Report 09-61.
[27]A. Ron and Z. Shen, Affine system in L2(Rd):the analysis of the analysis operator, J. of Func. Anal.,148:408-477,1997.
[28]A. Chai and Z. Shen, Deconvolution:a wavelet frame approach, Numer. Math.,106:529-587, 2007.
[29]J. Cai, R. H. Chan and Z. Shen, A framelet based image inpainting algorithm, Appl. Comput. Harmon. Anal.,24:131-149,2008.
[30]S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, SIAM Multiscale Model. and Simu,4:460-489, 2005.
[31]W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for 11 minimiza-tion with applications to compressed sensing, SIAM J. Imaging Sci.,1:143-168,2008.
[32]T. Goldstein and S. Osher, The split bregman method for 11 regularized problems, UCLA CAM Report 08-29.
[33]J. Cai, S. Osher and Z. Shen, Linearized bregman iterations for compressed sensing, UCLA CAM Report 08-06.
[34]J. Cai, S. Osher and Z. Shen, Convergence of the linearized bregman iterations for 11-norm minimization, UCLA CAM Report 08-52.
[35]Y. Wang, W. Yin and Y. Zhang, A fast algorithm for image deblurring with total variation regularization, CAAM Technical Report TR0710.
[36]G. Guy and G. Medioni, Inferring global perceptual contours from local features, Interna-tional Journal of Computer Vision,20:113-133,1996.
[37]G. Guy and G. Medioni, Inference of surfaces,3-D curves and Junctions from sparse, noisy 3-D data, IEEE trans. Pattern Analysis and Machine Intelligence,19(11):1265-1277, Nov, 1997.
[38]G. Medioni, Mi-Suen Lee, and Chi-Keung Tang, A Computational Framework for Segmen-tation and Grouping, Elsevier Science,2000.
[39]G. Medioni, Chi-Keung Tang, and Mi-Suen Lee, Tensor voting: theory and applications, In Proceedings of RFIA, Paris, France,2000.
[40]M. Kass, A. Witkin and D. Terzopoulos, Snakes: active contour models, International Journal of Computer Vision,1:321-331,1987.
[41]G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge Univer-sity Press 2001.
[42]N. Paragios, Yunmei. Chen and O. Faugeras, Handbook of Mathematical Models in Com-puter Vision.
[43]S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi, Conformal curvature flows: from phase transitions to active vision, Archive for Rational Mechanics and Analysis, 134:275-301,1996.
[44]S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi, Gradient flows and ge-ometric active contour models, In Proceedings of the 5th International Conference on Com-puter Vision, pages 810-815, Boston. MA, June 1995. IEEE Computer Society Press.
[45]V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, In Proceedings of the 5th International Conference on Computer Vision, pages 694-699, Boston, MA, June 1995. IEEE Computer Society Press.
[46]S. Osher and R. Fedkiw, geometric level set methods in imaging, vision, and graphics, Springer-Verlag,2003.
[47]Chun-ming Li, Rui Huang, Zhao-hua Ding, Chris Gatenby, Dimitris Metaxas, and John Gore, A variational level set approach to segmentation and bias correction of images with inten- sity inhomogeneity, Medical Image Computing and Computer-Assisted Intervention,5242: 1083-1091,2008.
[48]X. Bresson, S. Esedoglu, P. Vandergheynst, J. Thiran and S. Osher, Fast global minimization of active contour/snake model, Journal of Mathematical Imaging and Vision,28:151-167, 2007.
[49]Chun-ming Li, Chen-yang Xu, Chang-feng Gui and Martin D. Fox, Level set evolution with-out re-initialization:a new variational formulation, In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition,1:430-436,2005.
[50]A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision,20:89-97,2004.
[51]E. H. Land, The retinex theory of color vision, Sci. Amer.,237:108-128,1977.
[52]E. H. Land, Recent advances in the retinex theory and some implications for cortical com-putation:color vision and natural image, Proc. Nat. Acad. Sci. USA,80:5163-5169,1983.
[53]E. H. Land and J. J. McCann, Lightness and the retinex theory, J. Opt. Soc. Am.,61:1-11, 1971.
[54]D. J. Jobson, Z. Rahman and G. A. Woodell, Properties and performance of the cen-ter/surround retinex, IEEE Trans, on Image Proc.,6:451-462,1997.
[55]D. J. Jobson, Z. Rahman and G. A. Woodell, A multiscale retinex for bridging the gap be-tween color images and the human observation of scenes, IEEE Trans. on Image Proc., Vol. 6,1997.
[56]E. H. Land, An alternative technique for the computation of the designator in the retinex theory of color vision, Proc. Nat. Acad. Sci. USA,83:3078-3080,1986.
[57]A. Blake, Boundary conditions of lightness computation in mondrian world, Computer Vi-sion Graphics and Image Processing,32:314-327,1985.
[58]B. K. P. Horn, Determining lightness from an image, Computer Graphics and Image Pro-cessing,3:277-299,1974.
[59]D. Terzopoulos, Image analysis using multigrid relaxation methods, IEEE,Trans. on PAMI, 8:129-139,1986.
[60]R. Kimmel, M. Elad, D. Shaked, R. Keshet and I. Sobel, A variational framework for retinex, Int'l J. Computer Vision,52(1):7-23,2003.
[61]I. Ekeland and R. Temam, Analyse convexe et problemes variationnels, volume 224 of Grundlehrender mathematischen Wissenschaften, Dunod, second edition,1983.
[62]S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Tran. Pattern Anal. Machine Intel,6:721-741,1984.
[63]D. Mumford, and J. Shah, Optimal approximations by piecewise smooth functions and asso-ciated variational problems, Comm. Pure Appl. Math,42(5):577-685,1989.
[2]Y. Meyer, Oscillating patterns in image processing and nonlinear evolution equations,大学演讲系列,Vol.22,AMs 2002.
[3]L. Vese and S. Osher, Modeling textures with total variation minimization and oscillating patterns in image processing, Journal of Scientific Computing,19:553-572,2003.
[4]E. Tadmor, S. Nezzar, and L. Vese, Multiscale hierarchical decomposition of images with applications to deblurring, denoising and segmentation, Commun. Math. Sci.6:281-307, 2008.
[5]S. Osher, A. Sole and L. Vese, Image decomposition and restoration using total variation minimiation and the H-1 norm, Mutiscale Modeling and Simulation,1:349-370,2003.
[6]S. Kindermann, S. Osher and P. Jones, Deblurring and denoising of images by nonlocal functionals, SIAM Mutiscale Modeling and Simulation,4(4):1091-1115,2005.
[7]G. Gilboa, J. Darbon, S. Osher and T. F. Chan, Nonlocal convex functional for image regu-larization,2006. UCLA CAM Report 06-57.
[8]G. Gilboa and S. Osher, Nonlocal operators with applications to image processing,2007. UCLA CAM Report 07-23.
[9]G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, volume 147 of Applied Mathematical Sciences, Springer-Verlag,2002.
[10]W. Rudin, Real and Complex Analysis, McGraw-Hill, Inc.1987.
[11]L. Vese, A Study in the BV Space of a Denoising-Deblurring Variational Problem, Applied Mathematics Optimization,44:131-161,2001.
[12]L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford University Press, New York,2000.
[13]L.C.Evans, Partial Differential Equations, American Mathematical Society,1998.
[14]F. Demengel and R. Temam, Convex functions of a measure and applications, Indiana Uni-versity Mathematics Journal,33:673-709,1984.
[15]L. C. Evans and R. F. Gariepy, Measure theory and fine properties of functions, CRC Press, 1992.
[16]E.Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhauser Boston, Inc. 1984.
[17]L. I. Rudin, P. L. Lions and S. Osher, Mutiplicative denoising and deblurring: theory and algrithms, In. Osher and N. Paragios, editors, Geometric Level Sets in Image Vision, and Graphics.103-119,2003.
[18]T. Le, R. Chartrand and T. T. Asaki, A variatioal approach for reconstructing images cor-rupted by poisson noise, UCLA CAM Report 05-49.
[19]G.Aubert and J.F.Aujol, A Variational Approach to Remove Multiplicative Noise, CMLA 2006-11.
[20]A. Chambolle, An algrithm for total variation minimization and applications, JMIV,20:89-97,2004.
[21]X. Zhang, M. Burger, X. Bresson and S. Osher, Bregmanized nonlocal regularization for deconvolution and sparse reconstruction, UCLA CAM Report 09-03.
[22]G. Gilboa and S. Osher, Nonlocal linear image regularization and supervised segmentation, SIAM Mutiscale Modeling and Simulation,6(2):595-630,2007.
[23]J. Shen and S. H. Kang, Quantum tv and applications in image processing, Inverse Problems and Imaging,00(2):1-18,2007.
[24]J. Cai, H. Ji, C. Liu and Z. Shen, Blind motion deblurring using multiple images, UCLA CAM Report 09-59.
[25]J. Cai, H. Ji, C. Liu and Z. Shen, High quality curvelet based motion deblurring from an image pair, UCLA CAM Report 09-60.
[26]J. Cai, H. Ji, C. Liu and Z. Shen, Blind motion deblurring from a single image using sparse approximation, UCLA CAM Report 09-61.
[27]A. Ron and Z. Shen, Affine system in L2(Rd):the analysis of the analysis operator, J. of Func. Anal.,148:408-477,1997.
[28]A. Chai and Z. Shen, Deconvolution:a wavelet frame approach, Numer. Math.,106:529-587, 2007.
[29]J. Cai, R. H. Chan and Z. Shen, A framelet based image inpainting algorithm, Appl. Comput. Harmon. Anal.,24:131-149,2008.
[30]S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, SIAM Multiscale Model. and Simu,4:460-489, 2005.
[31]W. Yin, S. Osher, D. Goldfarb and J. Darbon, Bregman iterative algorithms for 11 minimiza-tion with applications to compressed sensing, SIAM J. Imaging Sci.,1:143-168,2008.
[32]T. Goldstein and S. Osher, The split bregman method for 11 regularized problems, UCLA CAM Report 08-29.
[33]J. Cai, S. Osher and Z. Shen, Linearized bregman iterations for compressed sensing, UCLA CAM Report 08-06.
[34]J. Cai, S. Osher and Z. Shen, Convergence of the linearized bregman iterations for 11-norm minimization, UCLA CAM Report 08-52.
[35]Y. Wang, W. Yin and Y. Zhang, A fast algorithm for image deblurring with total variation regularization, CAAM Technical Report TR0710.
[36]G. Guy and G. Medioni, Inferring global perceptual contours from local features, Interna-tional Journal of Computer Vision,20:113-133,1996.
[37]G. Guy and G. Medioni, Inference of surfaces,3-D curves and Junctions from sparse, noisy 3-D data, IEEE trans. Pattern Analysis and Machine Intelligence,19(11):1265-1277, Nov, 1997.
[38]G. Medioni, Mi-Suen Lee, and Chi-Keung Tang, A Computational Framework for Segmen-tation and Grouping, Elsevier Science,2000.
[39]G. Medioni, Chi-Keung Tang, and Mi-Suen Lee, Tensor voting: theory and applications, In Proceedings of RFIA, Paris, France,2000.
[40]M. Kass, A. Witkin and D. Terzopoulos, Snakes: active contour models, International Journal of Computer Vision,1:321-331,1987.
[41]G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge Univer-sity Press 2001.
[42]N. Paragios, Yunmei. Chen and O. Faugeras, Handbook of Mathematical Models in Com-puter Vision.
[43]S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi, Conformal curvature flows: from phase transitions to active vision, Archive for Rational Mechanics and Analysis, 134:275-301,1996.
[44]S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum and A. Yezzi, Gradient flows and ge-ometric active contour models, In Proceedings of the 5th International Conference on Com-puter Vision, pages 810-815, Boston. MA, June 1995. IEEE Computer Society Press.
[45]V. Caselles, R. Kimmel and G. Sapiro, Geodesic active contours, In Proceedings of the 5th International Conference on Computer Vision, pages 694-699, Boston, MA, June 1995. IEEE Computer Society Press.
[46]S. Osher and R. Fedkiw, geometric level set methods in imaging, vision, and graphics, Springer-Verlag,2003.
[47]Chun-ming Li, Rui Huang, Zhao-hua Ding, Chris Gatenby, Dimitris Metaxas, and John Gore, A variational level set approach to segmentation and bias correction of images with inten- sity inhomogeneity, Medical Image Computing and Computer-Assisted Intervention,5242: 1083-1091,2008.
[48]X. Bresson, S. Esedoglu, P. Vandergheynst, J. Thiran and S. Osher, Fast global minimization of active contour/snake model, Journal of Mathematical Imaging and Vision,28:151-167, 2007.
[49]Chun-ming Li, Chen-yang Xu, Chang-feng Gui and Martin D. Fox, Level set evolution with-out re-initialization:a new variational formulation, In Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition,1:430-436,2005.
[50]A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision,20:89-97,2004.
[51]E. H. Land, The retinex theory of color vision, Sci. Amer.,237:108-128,1977.
[52]E. H. Land, Recent advances in the retinex theory and some implications for cortical com-putation:color vision and natural image, Proc. Nat. Acad. Sci. USA,80:5163-5169,1983.
[53]E. H. Land and J. J. McCann, Lightness and the retinex theory, J. Opt. Soc. Am.,61:1-11, 1971.
[54]D. J. Jobson, Z. Rahman and G. A. Woodell, Properties and performance of the cen-ter/surround retinex, IEEE Trans, on Image Proc.,6:451-462,1997.
[55]D. J. Jobson, Z. Rahman and G. A. Woodell, A multiscale retinex for bridging the gap be-tween color images and the human observation of scenes, IEEE Trans. on Image Proc., Vol. 6,1997.
[56]E. H. Land, An alternative technique for the computation of the designator in the retinex theory of color vision, Proc. Nat. Acad. Sci. USA,83:3078-3080,1986.
[57]A. Blake, Boundary conditions of lightness computation in mondrian world, Computer Vi-sion Graphics and Image Processing,32:314-327,1985.
[58]B. K. P. Horn, Determining lightness from an image, Computer Graphics and Image Pro-cessing,3:277-299,1974.
[59]D. Terzopoulos, Image analysis using multigrid relaxation methods, IEEE,Trans. on PAMI, 8:129-139,1986.
[60]R. Kimmel, M. Elad, D. Shaked, R. Keshet and I. Sobel, A variational framework for retinex, Int'l J. Computer Vision,52(1):7-23,2003.
[61]I. Ekeland and R. Temam, Analyse convexe et problemes variationnels, volume 224 of Grundlehrender mathematischen Wissenschaften, Dunod, second edition,1983.
[62]S. Geman and D. Geman, Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Tran. Pattern Anal. Machine Intel,6:721-741,1984.
[63]D. Mumford, and J. Shah, Optimal approximations by piecewise smooth functions and asso-ciated variational problems, Comm. Pure Appl. Math,42(5):577-685,1989.