Harmonic Analysis Perspective on Geometric Diffusions and Low level Vision

Wednesday, October 18, 2000 - 11:00am - 12:00pm
Keller 3-180
David Donoho (Stanford University)
In recent years, two very important trends have emerged which are of compelling interest to the mathematically-trained who are thinking of applications in image processing. On the one hand, there is the widespread use of PDEs to process images, for example with the use of geometry-driven diffusions to remove noise from images and perform segmentation. On the other hand, there are intensive studies of computational vision, and interesting speculations and investigations about mathematical structures which might be involved in biological vision.

In my talk, I will take a completely different discipline -- Harmonic Analysis -- and consider some recent developments in this field. With constructions such as wavelets, time-frequency analysis, and other more exotic schemes, there is a wealth of ideas which can be compared and contrasted with recent developments in both geometry-driven diffusions and in computational vision.

In my talk I will focus on two topics:

[1] Existing geometry-driven diffusions go in the right direction -- smoothing anisotropically in the vicinity of edges. But this is only qualitatively correct. Do they really do the quantitatively correct thing? Does it matter?

[2] Many existing studies relating phenomena in natural images to computational structures that might be relevant to the visual cortex and computational analogs take Fourier and Gabor analysis, and more recently wavelet analysis, as models for the possible underlying structures which are best-adapted for image analysis. Are these the right ideas? How do other developments in harmonic analysis (e.g. brushlets, beamlets) compare?

I hope to convey both the spirit and some of the specific mathematical ideas of recent developments in applied harmonic analysis.

Parts of my talk will describe joint work with Emmanuel Candes (CalTech), with Drs. Georgina Flesia and Arne Stoschek (Stanford).