Institute for Mathematical Sciences                                        Programs & Activities



Combining PDE and wavelet techniques for image processing
Tony Chan, University of California at Los Angeles

(Joint work with Haomin Zhou, Math Dept, Georgia Tech)


Standard wavelet linear approximations (truncating high frequency coefficients) generate oscillations (Gibbs' phenomenon) near singularities in piecewise smooth functions. Nonlinear and data dependent methods are often used to overcome this problem. Recently, partial differential equation (PDE) and variational techniques have been introduced into wavelet transforms for the same purpose. In this talk, I will present our work on two different approaches that we have been working on in this direction. One is to use PDE ideas to directly change the standard wavelet transform algorithms so as to generate wavelet coefficients which can avoid oscillations in reconstructions when the high frequency coefficients are truncated. We have designed an adaptive ENO wavelet transform by using ideas from Essentially Non-Oscillatory (ENO) schemes for numerical shock capturing. ENO-wavelet transforms retains the essential properties and advantages of standard wavelet transforms without any edge artifacts. We have shown the stability and a rigorous error bound which depends only on the size of the derivative of the function away from the discontinuities. The second one is to stay with standard wavelet transforms and use variational PDE techniques to modify the coefficients in the truncation process so that the oscillations are reduced in the reconstruction processes. In particular, we use minimization of total variation (TV), to select and modify the retained standard wavelet coefficients so that the reconstructed images have fewer oscillations near edges. Examples in applications including image compression, denoising, inpainting will be presented.

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An urn model of Diaconis
David O. Siegmund, Stanford University

An urn model of Diaconis and some generalizations are discussed. As an application of the almost supermartingale convergence theorem of Robbins and Siegmund (1972), a convergence theorem is proved that implies for Diaconis' model that the empirical distribution of balls in the urn converges with probability one to the uniform distribution. Related results are discussed.

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Is there a mystery behind the Ricci curvature?
Jean-Pierre Bourguignon, Institut des Hautes Etudes Scientifiques and Centre National de la Recherche Scientifique


The way Gregorio Ricci-Curbastro isolated the part of the curvature that now bears his name in Riemannian Geometry (and generalizes to Metric Geometry, and to the pseudo-Riemannian context in particular) is through an analogy with the second fundamental form in submanifold theory. It is interesting to note that, although completely legitimate from a geometric point of view, this idea has not born much fruit... when studies involving the Ricci curvature have thrived.


There are several other paths that lead to the Ricci curvature, each providing interesting insights. The most penetrating definition of the Ricci curvature may still lie ahead of us.

The lecture proposes a tour of the various definitions and outstanding problems and results connected with them.

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