Multiscale Signal and Image Processing using Singularity Grammars

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Many real-world signals and images are naturally processed in a multiscale fashion, just as we move away from an object to view its general appearance and zoom in to see its fine details. The multiscale data representation of the wavelet transform naturally matches important signal and image singularity structures such as edges, ridges, bursts, textures, and other transients. This project focuses on characterizing these key structures using wavelets. The resulting new signal and image processing algorithms will impact adiverse range of important problems, from the management of computer network traffic in the next-generation Internet to seamless handling of complicated data such as documents containing both text and images.Since wavelets form a basis, they can reproduce arbitrary functions, from highly structured real-world signals and images to completely unstructured noise. In linguistic terms, the wavelet vocabulary canbe too expressive. To perform useful modeling of real-world signals and images, the vocabulary's scope must be narrowed by imposing a set of constraints -- a grammar -- that capture the salient structures ofsingularities. While much research has concentrated on developing new wavelet vocabularies, the most important current challenges to wavelet-based processing lie in grammatical modeling. This projectstudies a new statistical approach to multiscale modeling designed to match singularity-rich data. Grammars under consideration include efficient and trainable probabilistic graphs (hidden Markov tree models) and multiplicative pyramidal cascades. New grammar design is being guided by a detailed study of the fine structure of singularities using Besov spaces and multifractal analysis. From these new models follow new algorithms for signal and image estimation, detection/classification, segmentation, fusion, and synthesis.