223Accesses
75Citations
Abstract
In this paper we link, through simple examples, between three basic approaches for signal and image denoising and segmentation: (1) PDE axiomatics, (2) energy minimization and (3) adaptive filtering. We show the relation between PDE's that are derived from a master energy functional, i.e. the Polyakov harmonic action, and non-linear filters of robust statistics. This relation gives a simple and intuitive way of understanding geometric differential filters like the Beltrami flow. The relation between PDE's and filters is mediated through the short time kernel.
This is a preview of subscription content,log in via an institution to check access.
Access this article
Subscribe and save
- Get 10 units per month
- Download Article/Chapter or eBook
- 1 Unit = 1 Article or 1 Chapter
- Cancel anytime
Buy Now
Price includes VAT (Japan)
Instant access to the full article PDF.
Similar content being viewed by others
References
L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel, “Axioms and fundamental equations of image processing,”Arch. Rational Mechanics, Vol. 123, 1993.
D. Barash, “Bilateral filtering and anisotropic diffusion: towards a unified viewpoint,” HP Labs., Technical Report HPL-18-2000, 2000.
M. Black, G. Sapiro, D. Marimont, and D. Heeger, “Robust anisotropic diffusion,”IEEE Trans. on Image Processing,Vol. 7, No. 3, pp. 421–432, 1998.
A.M. Bruckstein and D. Shaked, “On projective invariant smoothing and evolutions of planar curves and polygons,”Journal of Mathematical Imaging and Vision, Vol. 7, pp. 225–240, 1997.
E. Calabi, P.J. Olver, C. Shakiban, A. Tannenbaum, and S. Haker, “Differential and numerically invariant signature curves applied to object recognition,”Int. J. Computer Vision, Vol. 26, pp. 107–135, 1998.
D. Comaniciu, and P. Meer, “Mean shift analysis and applications,” inProc. of the 7th IEEE Int. Conf. on Computer Vision CA, USA, 1999, Vol. 2, pp. 1197–203.
F.C. Chan, S. Osher, and J. Shen, “The digital TV filter and nonlinear denoising,” UCLA-Technical Report, 2000.
T. Chan and J. Shen, “Variational restoration of non-flat image features: Models and algorithms,” Technical Report, Math-UCLA, 1999.
J.K. Cohen, F.G. Hagin, and J.B. Keller, “Short time asymptotic expansions of solutions of parabolic equations,”Journal of Mathematical Analysis and Applications, Vol. 38, pp. 82–91, 1972.
M. Elad and D. Shaked, Personal communication. InHP Labs Israel, 1999.
D. Geman and G. Reynolds,IEEE Trans. on PAMI, Vol. 14, pp. 376–383, 1992.
R. Kimmel, R. Malladi, and N. Sochen, “Images as embedded maps and minimal surfaces:Movies, color, texture, and volumetric medical images,”International Journal of Computer Vision, Vol. 39, No. 2, pp. 111–129, 2000.
R. Kimmel and J.A. Sethian, “Computing geodesic paths on manifolds,”Proceedings of National Academy of Sciences, USA, Vol. 95, No. 15, pp. 8431–8435, 1998.
R. Kimmel and N. Sochen, “Orientation diffusion or how to comb a porcupine?,”Journal of Visual Communication and Image Representation, 2000.
D. Mumford and J. Shah, “Boundary detection by minimizing functionals,” inProc. of CVPR, Computer Vision and Pattern Recognition, San Francisco, 1985.
P. Perona, “Orientation diffusions,”IEEE Trans. on Image Processing, Vol. 7, No. 3, pp. 457–467, 1998.
P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,”IEEE-PAMI, Vol. 12, pp. 629–639, 1990.
L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,”Physica D, Vol. 60, pp. 259–268, 1992.
S. Sapiro and A. Tannenbaum, “Affine invariant scale space,”Int. Journal of Computer Vision, Vol. 11, No. 1, pp. 25–44, 1993.
N. Sochen, “Stochastic processes in vision, I: From Langevin to Beltrami,”CC Pub #285 June 1999, Technion, Israel.
N. Sochen, R. Kimmel, and R. Malladi, “A geometrical framework for low level vision,”IEEE Trans. on Image Processing, Vol. 7, No. 3, pp. 310–318, 1998.
B. Tang, G. Sapiro, and V. Caselles, “Direction diffusion,” inInternational Conference on Computer Vision, 1999.
C. Tomasi and R. Manduchi, “Bilateral filtering for gray and color images,” inProc. of the IEEE International Conference on Computer Vision, 1998, pp. 839–846.
R. Vogel, and M.E. Oman, “Iterative methods for total variation denoising,”SIAM. J. Sci. Statist. Comput., Vol. 17, No. 1, pp. 227–238, 1996.
J. Weickert,Anisotropic Diffusion in Image Processing, Teubner: Stuttgart, 1998.
Author information
Authors and Affiliations
Department of Applied Mathematics, School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 69978, Israel
N. Sochen
Department of Computer Science, Technion-Israel Institute of Technology, Technion City, Haifa, 32000, Israel
R. Kimmel & A.M. Bruckstein
- N. Sochen
You can also search for this author inPubMed Google Scholar
- R. Kimmel
You can also search for this author inPubMed Google Scholar
- A.M. Bruckstein
You can also search for this author inPubMed Google Scholar
Rights and permissions
About this article
Cite this article
Sochen, N., Kimmel, R. & Bruckstein, A. Diffusions and Confusions in Signal and Image Processing.Journal of Mathematical Imaging and Vision14, 195–209 (2001). https://doi.org/10.1023/A:1011277827470
Issue Date: