Abstract
We present in this paper a unifying generalization of the Mumford-Shah functional, in the Ambrosio-Totorelli set up, and the Beltrami framework. The generalization of the Ambrosio-Tortorelli is in using a diffusion tensor as an indicator of the edge set instead of a function. The generalization of the Beltrami framework is in adding a penalty term on the metric such that it is defined dynamically from minimization of the functional.
We show that we are able, in this way, to have the benefits of true anisotropic diffusion together with a dynamically tuned metric/diffusion tensor. The functional is naturally defined in terms of the vielbein-the metric’s square root. Preliminary results show improvement on both the Beltrami flow and the Mumford-Shah flow.
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
Ambrosio, L., Tortorelli, V.M.: Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math. 43, 999–1036 (1990)
Armijo, L.: Minimization of functions having Lipschitz continuous first partial derivatives. Pacific J. Math. 16(1), 13 (1966)
Bar, L., Sochen, N., Kiryati, N.: Image Deblurring in the Presence of Impulsive Noise. International Journal of Computer Vision 70(3), 279–298 (2006)
Bar, L., Sochen, N., Kiryati, N.: Semi-Blind Image Restoration via Mumford-Shah Regularization. IEEE Trans. on Image Processing 15(2), 483–493 (2006)
Deriche, R., Faugeras, O.: Les EDP en Traitement des Images et Vision par Ordinateur. Traitement du Signal 13(6) (1996)
Mumford, D., Shah, J.: Optimal Approximations by Piecewise Smooth Functions and Associated Variational Problems. Comm. on Pure and Applied Math. XLII(5), 577–684 (1989)
Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Transactions on Pattern Analysis and Machine Intelligence 12(7), 629–639 (1990)
Rosenau, P.: Free energy functionals at the high-gradient limit. Phys. Rev. A (Rapid Communications) 41(4), 2227–2230 (1990)
Rudin, L., Osher, S., Fatemi, E.: Nonlinear Total Variation based noise removal algorithms. Physica D 60, 259–268 (1992)
Sochen, N., Kimmel, R., Malladi, R.: A general framework for low level vision. IEEE Transactions on Image Processing 7, 310–318 (1998)
Weickert, J.: Coherence-Enhancing Diffusion Filtering. International Journal of Computer Vision 31(2/3), 111–127 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Sochen, N., Bar, L. (2012). The Beltrami-Mumford-Shah Functional. In: Bruckstein, A.M., ter Haar Romeny, B.M., Bronstein, A.M., Bronstein, M.M. (eds) Scale Space and Variational Methods in Computer Vision. SSVM 2011. Lecture Notes in Computer Science, vol 6667. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24785-9_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-24785-9_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24784-2
Online ISBN: 978-3-642-24785-9
eBook Packages: Computer ScienceComputer Science (R0)