When using transform coders, we consider a given digitized image to be a vector in a very high dimensional space. The goal is to find a new basis (from some restricted class) in which the information in the image is concentrated in a small number of coefficients. One can then store the image by saving just a few coefficients. This process of choosing an efficient basis representation for functions has a variety of other important applications which I will discuss briefly as well.
In the second half of the talk, I'll examine "fractal" image compression, an unusual (and somewhat controversial) compression technique which arose from the study of iterated function systems (IFS). Fractal schemes operate on a completely different principle than transform coders: the encoding of an image entails the construction of a contraction map of which the image is an approximate fixed point. Compressed images are saved by storing this map, and the images are recovered by iterating the map until it converges to its fixed point.
Fractal schemes have proved difficult to analyze using conventional tools and are poorly understood (though this lack of understanding has not proved to be a barrier to the successful commercialization of the technique). I'll introduce a wavelet-based framework for analyzing fractal coders which simplifies matters considerably. Using this framework, we see that fractal coders are not so different from conventional transform coders as has been thought. I'll show that the central mechanism of fractal schemes is an extrapolation of fine-scale Haar wavelet coefficients from coarse-scale coefficients. I'll describe a new compression algorithm using this wavelet framework which outperforms the best fractal schemes in the literature by a substantial margin and which has performance comparable to some of the best conventional schemes.
Tea. High tea will be served at 3:30pm in the Lounge.
Emmy's. Certain refreshments will be available at the Emmy's after the talk.