On Minimum Entropy Segmentation

We describe segmented multiresolution analyses of [0; 1]. Such multiresolution analyses lead to segmented wavelet bases which are adapted to discontinuities, cusps, etc., at a given location 2 [0; 1]. Our approach emphasizes the idea of average-interpolation { synthesizing a smooth function on the line having prescribed boxcar averages. This particular approach leads to methods with subpixel resolution and to wavelet transforms with the advantage that, for a signal of length n, all n pixel-level segmented wavelet transforms can be computed simultaneously in a total time and space which are both O(n log(n)). We consider the search for a segmented wavelet basis which, among all such segmented bases, minimizes the \entropy" of the resulting coe cients. Fast access to all segmentations enables fast search for a best segmentation. When the \entropy" is Stein's Unbiased Risk Estimate, one obtains a new method of edge-preserving de-noising. When the \entropy" is the `-energy, one obtains a new multi-resolution edge detector, which works not only for step discontinuities but also for cusp and higherorder discontinuities, and in a near-optimal fashion in the presence of noise. We describe an iterative approach, Segmentation Pursuit, for identifying edges by the fast segmentation algorithm and removing them from the data.