Abstract
Image enhancement with forward-and-backward (FAB) diffusion is numerically very challenging due to its negative diffusivities. As a remedy, we first extend the explicit nonstandard scheme by Welk et al. (2009) from the 1D scenario to the practically relevant two-dimensional setting. We prove that under a fairly severe time step restriction, this 2D scheme preserves a maximum–minimum principle. Moreover, we find an interesting Lyapunov sequence which guarantees convergence to a flat steady state. Since a global application of the time step size restriction leads to very slow algorithms and is more restrictive than necessary for most pixels, we introduce a much more efficient scheme with locally adapted time step sizes. It applies diffusive two-pixel interactions in a randomised order and adapts the time step size to the specific pixel pair. These space-variant time steps are synchronised at sync times. Our experiments show that our novel two-pixel scheme allows to compute FAB diffusion with guaranteed \(L^\infty \)-stability at a speed that can be three orders of magnitude larger than its explicit counterpart with a global time step size.
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
Similar content being viewed by others
References
Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Applied Mathematical Sciences, vol. 147, 2nd edn. Springer, New York (2006)
Burgeth, B., Weickert, J., Tari, S.: Minimally stochastic schemes for singular diffusion equations. In: Tai, X.C., Lie, K.A., Chan, T.F., Osher, S. (eds.) Image Processing Based on Partial Differential Equations. (MATHVISUAL), pp. 325–339. Springer, Berlin (2007). doi:10.1007/978-3-540-33267-1_18
Gilboa, G., Sochen, N., Zeevi, Y.Y.: Image sharpening by flows based on triple well potentials. J. Math. Imaging Vis. 20, 121–131 (2004)
Gilboa, G., Sochen, N.A., Zeevi, Y.Y.: Forward-and-backward diffusion processes for adaptive image enhancement and denoising. IEEE Trans. Image Process. 11, 689–703 (2002)
Kramer, H.P., Bruckner, J.B.: Iterations of a non-linear transformation for enhancement of digital images. Pattern Recogn. 7, 53–58 (1975)
Mrázek, P., Weickert, J., Steidl, G.: Diffusion-inspired shrinkage functions and stability results for wavelet denoising. Int. J. Comput. Vis. 64, 171–186 (2005)
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 J. Numer. Anal. 27, 919–940 (1990)
Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)
Pollak, I., Willsky, A.S., Krim, H.: Image segmentation and edge enhancement with stabilized inverse diffusion equations. IEEE Trans. Image Process. 9, 256–266 (2000)
Smolka, B.: Combined forward and backward anisotropic diffusion filtering of color images. In: Van Gool, L. (ed.) DAGM 2002. LNCS, vol. 2449, pp. 314–322. Springer, Heidelberg (2002). doi:10.1007/3-540-45783-6_38
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. (ACS), pp. 1–10. Springer, Vienna (1997). doi:10.1007/978-3-7091-6867-7_1
Welk, M., Gilboa, G., Weickert, J.: Theoretical foundations for discrete forward-and-backward diffusion filtering. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) SSVM 2009. LNCS, vol. 5567, pp. 527–538. Springer, Heidelberg (2009). doi:10.1007/978-3-642-02256-2_44
Welk, M., Weickert, J., Galić, I.: Theoretical foundations for spatially discrete 1-D shock filtering. Image Vis. Comput. 25, 455–463 (2007)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Welk, M., Weickert, J. (2017). An Efficient and Stable Two-Pixel Scheme for 2D Forward-and-Backward Diffusion. In: Lauze, F., Dong, Y., Dahl, A. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2017. Lecture Notes in Computer Science(), vol 10302. Springer, Cham. https://doi.org/10.1007/978-3-319-58771-4_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-58771-4_8
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-58770-7
Online ISBN: 978-3-319-58771-4
eBook Packages: Computer ScienceComputer Science (R0)