Multiscale Methods for Signal Processing

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We propose a research program on multiscale methods for signal processing. My program is composed of the following two complementary components: Analysis: Adaptive Multiscale Orthogonal (AMO) bases for signal and image analysis. Synthesis: Patterns-based Simulation methods for signal and image synthesis. The success of wavelets in signal processing is based on the ability of wavelets to efficiently represent piecewise regular signals. For such signals, the discrete wavelet-transform representation is sparse in the sense that many transform coefficients are small enough to be neglected. We have been developing new AMO bases that provide exceptionally sparse representations for a signal, much sparser than wavelets for piecewise regular signals. We will develop and refine AMO bases so that they can be used effectively for a wide variety of signals and images, and revisit the main applications of wavelets with AMO bases. This work should have a major impact on a multitude of applications, e.g. medical imaging (diagnosis of disease, screening protocols), compression and denoising of medical images (MRI and ultrasound). Operational decisions made for the exploitation of natural resources (e.g, oil, water, ore) are guided by a stochastic modeling of the resource spatial distribution. In this context, the ability to characterize and model the complex spatial distribution of natural resources is essential for their assessment and exploitation. A correct description of multipoint statistics is particularly important to correctly simulate connected structures such as channels and fractures in oil reservoirs. We have been developing a simulation method that produces images that in many visual and quantitative aspects, i.e. morphology and connectivity, are superior to competing multipoint simulation approaches. Our method takes into account conditioning data, i.e. known geology at several locations, and multipoint statistics, i.e. complex patterns geometry. We propose to develop further our method to take into account constraints defined by geophysical measurements.