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Given a set of realizations of a certain stochastic processes, finding a basis that sparsifies their representations is important for data compression. There is another competing strategy, which tries to find a basis that makes its coordinates statistically independent for that stochastic process. These two strategies are often confusing, and in this talk, I will clarify their relationship using very simple synthetic stochastic processes, namely, "spike" and "ramp" processes, as well as databases of natural scenes. For these processes, both the sparsity and the statistical independence criteria produced essentially the same bases. However, to reach to those bases via the sparsity is much simpler computationally and conceptually than the statistical independence.

The second part of my talk is about how to extract rotation and scale invaraint features of images for classification and recognition of various geometric objects in images. We use the partial Radon transform and the Fourier transform to extract such features. We further extract discriminant features from them using the so-called local discriminant basis (LDB) developed by myself and Raphy Coifman some time ago. I will compare the classification performance of our approach with the standard approach using linear discriminant functions.

This is a joint work with Jean-Marie Aubry, Bertrand Benichou, and Brons Larson.