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Shen, Jianhong

On the foundations of vision modeling. I: Weber’s law and Weberized TV restoration. (English) Zbl 1006.91057

Physica D 175, No. 3-4, 241-251 (2003).
Summary: Most conventional image processors consider little the influence of human vision psychology. Weber’s law in psychology and psychophysics claims that humans’ perception and response to the intensity fluctuation \(\delta u\) of visual signals are weighted by the background stimulus \(u\), instead of being plainly uniform. This paper attempts to integrate this well known perceptual law into the classical total variation (TV) image restoration model of L. I. Rudin et al. [Physica D 60, No. 1-4, 259–268 (1992; Zbl 0780.49028)]. We study the issues of existence and uniqueness for the proposed Weberized nonlinear TV restoration model, making use of the direct method in the space of functions with bounded variations. We also propose an iterative algorithm based on the linearization technique for the associated nonlinear Euler-Lagrange equation.

Cited in 1 Review
Cited in 8 Documents

MSC:

91E30 Psychophysics and psychophysiology; perception
92C99 Physiological, cellular and medical topics
68U10 Computing methodologies for image processing
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory

Keywords:

nonlinear Euler-Lagrange equation; human perception; nonlinear total variation image restoration model

Citations:

Zbl 0780.49028
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Full Text: DOI

References:

[1] Chan, T. F.; Golub, G. H.; Mulet, P., A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20, 6, 1964-1977 (1999) · Zbl 0929.68118
[2] Chambolle, A.; Lions, P. L., Image recovery via total variational minimization and related problems, Numer. Math., 76, 167-188 (1997) · Zbl 0874.68299
[3] Chan, T. F.; Osher, S.; Shen, J., The digital TV filter and nonlinear denoising, IEEE Trans. Image Process., 10, 2, 231-241 (2001) · Zbl 1039.68778
[4] Chan, T. F.; Shen, J., Variational restoration of non-flat image features: models and algorithms, SIAM J. Appl. Math., 61, 4, 1338-1361 (2000) · Zbl 1011.94001
[5] Chan, T. F.; Shen, J., Mathematical models for local nontexture inpaintings, SIAM J. Appl. Math., 62, 3, 1019-1043 (2001) · Zbl 1050.68157
[6] T.F. Chan, J. Shen, On the role of the BV image model in image restoration, UCLA’s Mathematics Department CAM Report 02-14, AMS Contemp. Math., in press. http://www.math.ucla.edu/∼imagers; T.F. Chan, J. Shen, On the role of the BV image model in image restoration, UCLA’s Mathematics Department CAM Report 02-14, AMS Contemp. Math., in press. http://www.math.ucla.edu/∼imagers · Zbl 1035.94501
[7] Fechner, G. T., Über ein wichtiges psychophysiches Grundgesetz und dessen Beziehung zur Schäzung der Sterngrössen. Abk. k. Ges. Wissensch., Math.-Phys., 1, 4 (1858)
[8] Geman, S.; Geman, D., Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images, IEEE Trans. Pattern Anal. Mach. Intell., 6, 721-741 (1984) · Zbl 0573.62030
[9] E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhauser, Boston, 1984.; E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Birkhauser, Boston, 1984. · Zbl 0545.49018
[10] Marr, D.; Hildreth, E., Theory of edge detection, Proc. Roy. Soc. Lond. B, 207, 187-217 (1980)
[11] Marquina, A.; Osher, S., Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal, SIAM. J. Sci. Comput., 22, 387-405 (2000) · Zbl 0969.65081
[12] D. Mumford, Geometry Driven Diffusion in Computer Vision, The Bayesian Rationale for Energy Functionals, Kluwer Academic Publishers, Dordrecht, 1994, pp. 141-153.; D. Mumford, Geometry Driven Diffusion in Computer Vision, The Bayesian Rationale for Energy Functionals, Kluwer Academic Publishers, Dordrecht, 1994, pp. 141-153. · Zbl 0832.68111
[13] L. Rudin, S. Osher, Total variation based image restoration with free local constraints, in: Proceedings of the First IEEE ICIP Conference, vol. 1, 1994, pp. 31-35.; L. Rudin, S. Osher, Total variation based image restoration with free local constraints, in: Proceedings of the First IEEE ICIP Conference, vol. 1, 1994, pp. 31-35.
[14] Rudin, L.; Osher, S.; Fatemi, E., Nonlinear total variation based noise removal algorithms, Physica D, 60, 259-268 (1992) · Zbl 0780.49028
[15] G. Strang, Introduction to Applied Mathematics, Wellesley-Cambridge Press, Massachusetts, 1993.; G. Strang, Introduction to Applied Mathematics, Wellesley-Cambridge Press, Massachusetts, 1993. · Zbl 0618.00015
[16] Vogel, C. R.; Oman, M. E., Iterative methods for total variation denoising, SIAM J. Sci. Comput., 17, 1, 227-238 (1996) · Zbl 0847.65083
[17] E.H. Weber, De pulsu, resorptione, audita et tactu, Annotationes anatomicae et physiologicae, Koehler, Leipzig, 1834.; E.H. Weber, De pulsu, resorptione, audita et tactu, Annotationes anatomicae et physiologicae, Koehler, Leipzig, 1834.
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