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Nonlinear Eigenproblems in Image Processing and Computer Vision pp 15–37Cite as

Variational Methods in Image Processing

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Part of the book series: Advances in Computer Vision and Pattern Recognition ((ACVPR))

Abstract

A short review is given on the rationale for using cost functions and optimization methods for modeling image processing and computer vision problems. Classical examples of various costs and functionals are given, illustrating this highly effective algorithmic approach. We examine different mathematical models Sects. (2.1, 2.2 and 2.5), as well as image processing tasks Sects. (2.3, 2.4).

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References

  1. A. Tikhonov, Solution of incorrectly formulated problems and the regularization method. Sov. Math. Dokl. 4, 1035–1038 (1963)

    Google Scholar 

  2. L. Rudin, S. Osher, E. Fatemi, Nonlinear total variation based noise removal algorithms. Physica D 60, 259–268 (1992)

    Google Scholar 

  3. M. Nikolova, A variational approach to remove outliers and impulse noise. JMIV 20(1–2), 99–120 (2004)

    Google Scholar 

  4. T.F. Chan, S. Esedoglu, Aspects of total variation regularized l 1 function approximation. SIAM J. Appl. Math. 65(5), 1817–1837 (2005)

    Google Scholar 

  5. B.M. Ter Haar Romeny, Introduction to scale-space theory: multiscale geometric image analysis. in First International Conference on Scale-Space theory (Citeseer, 1996)

    Google Scholar 

  6. P. Perona, J. Malik, Scale-space and edge detection using anisotropic diffusion. PAMI 12(7), 629–639 (1990)

    Google Scholar 

  7. B.M. Ter Haar Romeny (ed.), Geometry Driven Diffusion in Computer Vision (Kluwer Academic Publishers, Dordrecht, 1994)

    Google Scholar 

  8. F. Catté, P.-L. Lions, J.-M. Morel, T. Coll, Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 29(1), 182–193 (1992)

    Google Scholar 

  9. R.T. Whitaker, S.M. Pizer, A multi-scale approach to nonuniform diffusion. CVGIP Image Underst. 57(1), 99–110 (1993)

    Google Scholar 

  10. J. Weickert, B. Benhamouda, A semidiscrete nonlinear scale-space theory and its relation to the perona malik paradox, in Advances in Computer Vision (Springer, Berlin, 1997), pp. 1–10

    Google Scholar 

  11. E. Radmoser, O. Scherzer, J. Weickert, Scale-space properties of nonstationary iterative regularization methods. J. Vis. Commun. Image Represent. 8, 96–114 (2000)

    Google Scholar 

  12. P. Charbonnier, L. Blanc-Feraud, G. Aubert, M. Barlaud, Two deterministic half-quadratic regularization algorithms for computed imaging. in Proceedings of the IEEE International Conference ICIP ’94, vol. 2 (1994), pp. 168–172

    Google Scholar 

  13. F. Andreu, C. Ballester, V. Caselles, J.M. Mazón, Minimizing total variation flow. Differ. Integral Equ. 14(3), 321–360 (2001)

    Google Scholar 

  14. J. Weickert, Coherence-enhancing diffusion filtering. Int. J. Comput. Vis. 31(2–3), 111–127 (1999)

    Google Scholar 

  15. J. Weickert, Coherence-enhancing diffusion of colour images. IVC 17, 201–212 (1999)

    Google Scholar 

  16. J. Weickert, Anisotropic Diffusion in Image Processing (Teubner-Verlag, Germany, 1998)

    Google Scholar 

  17. S. Osher, M. Burger, D. Goldfarb, J. Xu, W. Yin, An iterative regularization method for total variation based image restoration. SIAM J. Multiscale Model. Simul. 4, 460–489 (2005)

    Google Scholar 

  18. B.D. Lucas, T. Kanade et al., An iterative image registration technique with an application to stereo vision. IJCAI 81, 674–679 (1981)

    Google Scholar 

  19. B.K.P. Horn, B.G. Schunck. Determining optical flow. Artif. Intell. 17(1), 185–203 (1981)

    Google Scholar 

  20. A. Bruhn, J. Weickert, C. Schnörr, Lucas/kanade meets horn/schunck: combining local and global optic flow methods. Int. J. Comput. Vis. 61(3), 211–231 (2005)

    Google Scholar 

  21. G. Aubert, R. Deriche, P. Kornprobst, Computing optical flow via variational techniques. SIAM J. Appl. Math. 60(1), 156–182 (1999)

    Google Scholar 

  22. C. Zach, T. Pock, H. Bischof, A duality based approach for realtime TV-L 1 optical flow. Pattern Recognit. 214–223 (2007)

    Google Scholar 

  23. S. Baker, D. Scharstein, J.P. Lewis, S. Roth, M.J. Black, R. Szeliski. A database and evaluation methodology for optical flow. Int. J. Comput. Vis. 92(1), 1–31 (2011)

    Google Scholar 

  24. D. Sun, S. Roth, M.J. Black, Secrets of optical flow estimation and their principles. in IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2010 (IEEE, 2010), pp. 2432–2439

    Google Scholar 

  25. D. Cremers, S. Soatto, Motion competition: a variational approach to piecewise parametric motion segmentation. Int. J. Comput. Vis. 62(3), 249–265 (2005)

    Google Scholar 

  26. T. Brox, A. Bruhn, N. Papenberg, J. Weickert, High accuracy optical flow estimation based on a theory for warping, in Computer Vision-ECCV 2004 (Springer, Berlin, 2004), pp. 25–36

    Google Scholar 

  27. D. Mumford, J. Shah, Optimal approximations by piece-wise smooth functions and assosiated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989)

    Google Scholar 

  28. T. Chan, L. Vese, Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)

    Google Scholar 

  29. V. Caselles, R. Kimmel, G. Sapiro, Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–79 (1997)

    Google Scholar 

  30. L. Ambrosio, V.M. Tortorelli, Approximation of functional depending on jumps by elliptic functional via t-convergence. Commun. Pure Appl. Math. 43(8), 999–1036 (1990)

    Google Scholar 

  31. G. Aubert, P. Kornprobst, in Mathematical Problems in Image Processing, Applied Mathematical Sciences, vol 147 (Springer, Berlin, 2002)

    Google Scholar 

  32. A. Braides, Approximation of Free-Discontinuity Problems (Springer, Berlin, 1998)

    Google Scholar 

  33. T. Pock, D. Cremers, H. Bischof, A. Chambolle, An algorithm for minimizing the mumford-shah functional. in IEEE 12th International Conference on Computer Vision, 2009 (IEEE, 2009), pp. 1133–1140

    Google Scholar 

  34. T.F. Chan, S. Esedoglu, M. Nikolova, Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 1632–1648 (2006)

    Google Scholar 

  35. M. Kass, A. Witkin, D. Terzopoulos, Snakes: active contour models. Int. J. Comput. Vis. 1(4), 321–331 (1987)

    Google Scholar 

  36. J.A. Sethian, Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer vision, and Materials Science (Cambridge university press, Cambridge, 1999)

    Google Scholar 

  37. S. Osher, R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, Berlin, 2002)

    Google Scholar 

  38. R. Kimmel, Numerical Geometry of Images: Theory, Algorithms, and Applications (Springer, Berlin, 2012)

    Google Scholar 

  39. S. Osher, J.A. Sethian, Fronts propagating with curvature dependent speed-algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)

    Google Scholar 

  40. S. Osher, N. Paragios (eds.), Geometric Level Set Methods in Imaging, Vision, and Graphics (Springer, Berlin, 2003)

    Google Scholar 

  41. D. Cremers, M. Rousson, R. Deriche, A review of statistical approaches to level set segmentation: integrating color, texture, motion and shape. Int. J. Comput. Vis. 72(2), 195–215 (2007)

    Google Scholar 

  42. J. Lie, M. Lysaker, X.-C. Tai, A binary level set model and some applications to mumford-shah image segmentation. IEEE Trans. Image Process. 15(5), 1171–1181 (2006)

    Google Scholar 

  43. N. Paragios, R. Deriche, Geodesic active regions and level set methods for supervised texture segmentation. Int. J. Comput. Vis. 46(3), 223–247 (2002)

    Google Scholar 

  44. A.A. Efros, T.K. Leung, Texture synthesis by non-parametric sampling. ICCV 2, 1033–1038 (1999)

    Google Scholar 

  45. A. Buades, B. Coll, J.-M. Morel, A review of image denoising algorithms, with a new one. SIAM Multiscale Model. Simul. 4(2), 490–530 (2005)

    Google Scholar 

  46. S. Kindermann, S. Osher, P. Jones, Deblurring and denoising of images by nonlocal functionals. SIAM Multiscale Model. Simul. 4(4), 1091–1115 (2005)

    Google Scholar 

  47. G. Gilboa, S. Osher, Nonlocal linear image regularization and supervised segmentation. SIAM Multiscale Model. Simul. 6(2), 595–630 (2007)

    Google Scholar 

  48. G. Gilboa, S. Osher, Nonlocal operators with applications to image processing. SIAM Multiscale Model. Simul. 7(3), 1005–1028 (2008)

    Google Scholar 

  49. F. Chung, Spectral Graph Theory, vol. 92, CBMS Regional Conference Series in Mathematics (American Mathematical Society, Providence, 1997)

    Google Scholar 

  50. S. Bougleux, A. Elmoataz, M. Melkemi, Discrete regularization on weighted graphs for image and mesh filtering. in 1st International Conference on Scale Space and Variational Methods in Computer Vision (SSVM). Lecture Notes in Computer Science, vol. 4485 (2007), pp. 128–139

    Google Scholar 

  51. D. Zhou, B. Scholkopf, Regularization on discrete spaces. in Pattern Recognition, Proceedings of the 27th DAGM Symposium (Berlin, Germany, 2005), pp. 361–368,

    Google Scholar 

  52. A.D. Szlam, M. Maggioni, Jr. J.C. Bremer, R.R. Coifman, Diffusion-driven multiscale analysis on manifolds and graphs: top-down and bottom-up constructions. in SPIE (2005)

    Google Scholar 

  53. C. Kervrann, J. Boulanger, Optimal spatial adaptation for patch-based image denoising. IEEE Trans. Image Process. 15(10), 2866–2878 (2006)

    Google Scholar 

  54. G. Peyré, S. Bougleux, L. Cohen, Non-local regularization of inverse problems, Computer Vision–ECCV 2008 (Springer, Berlin, 2008), pp. 57–68

    Google Scholar 

  55. Y. Lou, X. Zhang, S. Osher, A. Bertozzi, Image recovery via nonlocal operators. J. Sci. Comput. 42(2), 185–197 (2010)

    Google Scholar 

  56. T.F. Chan, J. Shen, Mathematical models of local non-texture inpaintings. SIAM J. Appl. Math. 62(3), 1019–1043 (2001)

    Google Scholar 

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  1. Technion—Israel Institute of Technology, Haifa, Israel

    Guy Gilboa

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  1. Guy Gilboa
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Correspondence to Guy Gilboa .

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Gilboa, G. (2018). Variational Methods in Image Processing. In: Nonlinear Eigenproblems in Image Processing and Computer Vision. Advances in Computer Vision and Pattern Recognition. Springer, Cham. https://doi.org/10.1007/978-3-319-75847-3_2

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  • DOI: https://doi.org/10.1007/978-3-319-75847-3_2

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