Representation of natural signals such as sounds and images is critically important in a broad range of fields such as multimedia, data communication and storage, biomedical imaging, robotics, and computational neuroscience. Often it is crucial that the representation be efficient, i.e., the signals of interest are encoded economically. It is also desirable that the representation be robust to various types of noise. In this thesis, we advocate several ways to expand current signal encoding approaches via the framework of adaptive representations. In recent decades, the multiresolution paradigm has provided powerful mathematical and algorithmic tools to signal encoding. In spite of widely proven effectiveness, such methods ignore statistical structure of the class of signals they should represent. On the other hand, high computational costs artificially confine standar
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