Top

Published in:

2021 | OriginalPaper | Chapter

Diffusion, Pre-smoothing and Gradient Descent

Author : Martin Welk

Published in: Scale Space and Variational Methods in Computer Vision

Publisher: Springer International Publishing

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

Nonlinear diffusion of images, both isotropic and anisotropic, has become a well-established and well-understood denoising tool during the last three decades. Moreover, it is a component of partial differential equation methods for various further tasks in image analysis. For the analysis of such methods, their understanding as gradient descents of energy functionals often plays an important role. Often the diffusivity or diffusion tensor field for nonlinear diffusion is computed from pre-smoothed image gradients. What was not clear so far was whether nonlinear diffusion with this pre-smoothing step still is the gradient descent for some energy functional. This question is answered to the negative in the present paper. Triggered by this result, possible modifications of the pre-smoothing step to retain the gradient descent property of diffusion are discussed.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Fast Morphological Dilation and Erosion for Grey Scale Images Using the Fourier Transform

next chapter Local Culprits of Shape Complexity

Andreu-Vaillo, F., Caselles, V., Mazon, J.M.: Parabolic Quasilinear Equations Minimizing Linear Growth Functionals, Progress in Mathematics, vol. 223. Birkhäuser, Basel (2004)CrossRef

Bellettini, G., Caselles, V., Novaga, M.: The total variation flow in \(R^N\). J. Differ. Equ. 184(2), 475–525 (2002)CrossRef

Bellettini, G., Novaga, M., Paolini, M.: Convergence for long-times of a semidiscrete Perona-Malik equation in one dimension. Math. Models Methods Appl. Sci. 21(2), 241–265 (2011)MathSciNetCrossRef

Bellettini, G., Novaga, M., Paolini, M., Tornese, C.: Convergence of discrete schemes for the Perona-Malik equation. J. Differ. Equ. 245, 892–924 (2008)MathSciNetCrossRef

Catté, F., Lions, P.L., Morel, J.M., Coll, T.: Image selective smoothing and edge detection by nonlinear diffusion. SIAM J. Numer. Anal. 32, 1895–1909 (1992)MathSciNetCrossRef

Dibos, F., Koepfler, G.: Global total variation minimization. SIAM J. Numer. Anal. 37(2), 646–664 (2000)MathSciNetCrossRef

Ghisi, M., Gobbino, M.: A class of local classical solutions for the one-dimensional Perona-Malik equation. Trans. Am. Math. Soc. 361(12), 6429–6446 (2009)MathSciNetCrossRef

Ghisi, M., Gobbino, M.: An example of global classical solution for the Perona-Malik equation. Commun. Partial. Differ. Equ. 36(8), 1318–1352 (2011)MathSciNetCrossRef

Guidotti, P.: Anisotropic diffusions of image processing from Perona-Malik on. In: Ambrosio, L., Giga, Y., Rybka, P., Tonegawa, Y. (eds.) Variational Methods for Evolving Objects. Advanced Studies in Pure Mathematics, vol. 67, pp. 131–156. Mathematical Society of Japan, Tokyo (2015)CrossRef

10.

Kawohl, B., Kutev, N.: Maximum and comparison principle for one-dimensional anisotropic diffusion. Mathematische Annalen 311, 107–123 (1998)MathSciNetCrossRef

11.

Nitzberg, M., Shiota, T.: Nonlinear image filtering with edge and corner enhancement. IEEE Trans. Pattern Anal. Mach. Intell. 14, 826–833 (1992)CrossRef

12.

Nordström, N.: Biased anisotropic diffusion - a unified regularization and diffusion approach to edge detection. Image Vis. Comput. 8, 318–327 (1990)CrossRef

13.

Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)CrossRef

14.

Weickert, J.: Theoretical foundations of anisotropic diffusion in image processing. Comput. Suppl. 11, 221–236 (1996)CrossRef

15.

Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)MATH

16.

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

17.

Weickert, J., Benhamouda, B.: A semidiscrete nonlinear scale-space theory and its relation to the Perona-Malik paradox. In: Solina, F., Kropatsch, W.G., Klette, R., Bajcsy, R. (eds.) Advances in Computer Vision, pp. 1–10. Springer, Wien (1997). https://doi.org/10.1007/978-3-7091-6867-7CrossRef

18.

Weickert, J., Schnörr, C.: A theoretical framework for convex regularizers in PDE-based computation of image motion. Int. J. Comput. Vis. 45(3), 245–264 (2001)CrossRef

19.

Welk, M., Steidl, G., Weickert, J.: Locally analytic schemes: a link between diffusion filtering and wavelet shrinkage. Appl. Comput. Harmon. Anal. 24, 195–224 (2008)MathSciNetCrossRef

20.

Welk, M., Weickert, J.: PDE evolutions for M-smoothers: from common myths to robust numerics. In: Lellmann, J., Burger, M., Modersitzki, J. (eds.) SSVM 2019. LNCS, vol. 11603, pp. 236–248. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-22368-7_19CrossRef

21.

Welk, M., Weickert, J.: PDE evolutions for M-smoothers in one, two, and three dimensions. J. Math. Imaging Vis. 63(2), 157–185 (2021)MathSciNetCrossRef

22.

Welk, M., Weickert, J., Gilboa, G.: A discrete theory and efficient algorithms for forward-and-backward diffusion filtering. J. Math. Imaging Vis. 60(9), 1399–1426 (2018)MathSciNetCrossRef

23.

Zhang, K.: Existence of infinitely many solutions for the one-dimensional Perona-Malik model. Calc. Var. Partial. Differ. Equ. 26(2), 126–171 (2006)MathSciNetCrossRef

Title: Diffusion, Pre-smoothing and Gradient Descent
Author: Martin Welk
Publisher: Springer International Publishing
Book: Scale Space and Variational Methods in Computer Vision
Print ISBN: 978-3-030-75548-5

Electronic ISBN: 978-3-030-75549-2

Copyright Year: 2021
DOI: https://doi.org/10.1007/978-3-030-75549-2_7

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner