Abstract
Forward-and-backward (FAB) diffusion is a method for sharpening blurry images (Gilboa et al. 2002). It combines forward diffusion with a positive diffusivity and backward diffusion where negative diffusivities are used. The well-posedness properties of FAB diffusion are unknown, and it has been observed that standard discretisations can violate a maximum-minimum principle. We show that for a novel nonstandard space discretisation which pays specific attention to image extrema, one can apply a modification of the space-discrete well-posedness and scale-space framework of Weickert (1998). This allows to establish well-posedness and a maximum-minimum principle for the resulting dynamical system. In the fully discrete 1-D case with an explicit time discretisation, a maximum-minimum principle and total variation reduction are proven in spite of the fact that negative diffusivities may appear. This provides a theoretical justification for applying FAB diffusion to digital images.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, 2nd edn. Applied Mathematical Sciences, vol. 147. Springer, New York (2006)
Breuß, M., Welk, M.: Staircasing in semidiscrete stabilised inverse diffusion algorithms. Journal of Computational and Applied Mathematics 206, 520–533 (2007)
Gilboa, G., Sochen, N.A., Zeevi, Y.Y.: Forward-and-backward diffusion processes for adaptive image enhancement and denoising. IEEE Transactions on Image Processing 11(7), 689–703 (2002)
Gilboa, G., Sochen, N.A., Zeevi, Y.Y.: Image sharpening by flows based on triple well potentials. Journal of Mathematical Imaging and Vision 20, 121–131 (2004)
Kramer, H.P., Bruckner, J.B.: Iterations of a non-linear transformation for enhancement of digital images. Pattern Recognition 7, 53–58 (1975)
Mrázek, P., Weickert, J., Steidl, G.: Diffusion-inspired shrinkage functions and stability results for wavelet denoising. International Journal of Computer Vision 64(2/3), 171–186 (2005)
Nikolova, M.: Local strong homogeneity of a regularized estimator. SIAM Journal on Applied Mathematics 61(2), 633–658 (2000)
Nikolova, M.: Minimizers of cost-functions involving nonsmooth data fidelity terms. Application to the processing of outliers. SIAM Journal on Numerical Analysis 40(3), 965–994 (2002)
Osher, S., Rudin, L.: Shocks and other nonlinear filtering applied to image processing. In: Tescher, A.G. (ed.) Applications of Digital Image Processing XIV. Proceedings of SPIE, vol. 1567, pp. 414–431. SPIE Press, Bellingham (1991)
Osher, S., Rudin, L.I.: Feature-oriented image enhancement using shock filters. SIAM Journal on Numerical Analysis 27, 919–940 (1990)
Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12, 629–639 (1990)
Pollak, I., Willsky, A.S., Krim, H.: Image segmentation and edge enhancement with stabilized inverse diffusion equations. IEEE Transactions on Image Processing 9(2), 256–266 (2000)
Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)
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)
Welk, M., Weickert, J., Galić, I.: Theoretical foundations for spatially discrete 1-D shock filtering. Image and Vision Computing 25(4), 455–463 (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Welk, M., Gilboa, G., Weickert, J. (2009). Theoretical Foundations for Discrete Forward-and-Backward Diffusion Filtering. In: Tai, XC., Mørken, K., Lysaker, M., Lie, KA. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2009. Lecture Notes in Computer Science, vol 5567. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02256-2_44
Download citation
DOI: https://doi.org/10.1007/978-3-642-02256-2_44
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02255-5
Online ISBN: 978-3-642-02256-2
eBook Packages: Computer ScienceComputer Science (R0)