(with Guillermo Sapiro, David Marimont and David Heeger)
This work explores the relationships between anisotropic diffusion and robust statistics. Specifically, we show that anisotropic diffusion can be seen as a robust estimation procedure that estimates a piecewise smooth image from a noisy input image. The "edge-stopping" function in the anisotropic diffusion equation is closely related to the error norm and influence function in the robust estimation framework. This connection leads to a new "edge-stopping" function based on Tukey's biweight robust estimator, that preserves sharper boundaries than previous formulations and improves the automatic stopping of the diffusion. The robust statistical interpretation also provides a means for detecting the boundaries (edges) between the piecewise smooth regions in an image that has been smoothed with anisotropic diffusion. Additionally, we derive a relationship between anisotropic diffusion and regularization with line processes. Adding constraints on the spatial organization of the line processes allows us to develop new anisotropic diffusion equations that result in a qualitative improvement in the continuity of edges.
Black, M. J. and Sapiro, G., Edges as outliers: Anisotropic smoothing using local image statistics, Scale-Space Theories in Computer Vision, Second Int. Conf., Scale-Space '99, Corfu, Greece, LNCS 1682, Springer, Sept. 1999, pp. 259-270. (postscript, 1.3MB)(pdf, 0.3MB).
Black, M. J., Sapiro, G., Marimont, D., Heeger, D., Robust anisotropic diffusion, IEEE Transactions on Image Processing, Special issue on Partial Differential Equations and Geometry Driven Diffusion in Image Processing and Analysis, 7(3), pp. 421-432, March 1998. (pdf), (postscript)
Black, M. J., Sapiro, G., Marimont, D., Heeger, D., Robust anisotropic diffusion and sharpening of scalar and vector images. Int. Conf. on Image Processing, ICIP, Vol. 1, Santa Barbara, CA, Oct. 1997, pp. 263-266.
Black, M. J., Sapiro, G., Marimont, D., Heeger, D., Robust anisotropic diffusion: Connections between robust statistics, line processing, and anisotropic diffusion, in Scale-Space Theory in Computer Vision, Scale-Space'97, B. ter Haar Romeny, L. Florack, J. Koenderink, and M. Viergever (Eds.), Springer Verlag, LNCS 1252, Utrecht, the Netherlands, July 1997, pp. 323-326. (postscript)