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Multiresolution representations are very effective for analyzing the information content of images. We study the properties of the operator which approximates a signal at a given resolution. We show that the difference of information between the approximation of a signal at the resolutions 2 + ' and 2 can be extracted by decomposing this signal on a wavelet(More)
We introduce an algorithm, called matching pursuit , that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions. These waveforms are chosen in order to best match the signal structures. Matching pursuits are general procedures to compute adaptive signal representations. With a dictionary of(More)
A multiscale Canny edge detection is equivalent to finding the local maxima of a wavelet transform. We study the properties of multiscale edges through the wavelet theory. For pattern recognition, one often needs to discriminate different types of edges. We show that the evolution of wavelet local maxima across scales characterize the local shape of(More)
Most of a signal information is often found in irregular structures and transient phenomena. We review the mathematical characterization of singularities with Lipschitz exponents. The main theorems that estimate local Lipschitz exponents of functions, from the evolution across scales of their wavelet transform are explained. We then prove that the local(More)
In this paper we review recent multichannel models developed in psychophysiology, computer vision, and image processing. In psychophysiology, multichannel models have been particularly successful in explaining some low-level processing in the visual cortex. The expansion of a function into several frequency channels provides a representation which is(More)
A general framework for solving image inverse problems with piecewise linear estimations is introduced in this paper. The approach is based on Gaussian mixture models, which are estimated via a maximum a posteriori expectation-maximization algorithm. A dual mathematical interpretation of the proposed framework with a structured sparse estimation is(More)
This paper constructs translation invariant operators on L 2 (R d), which are Lipschitz continuous to the action of diffeomorphisms. A scattering propagator is a path ordered product of non-linear and non-commuting operators, each of which computes the modulus of a wavelet transform. A local integration defines a windowed scattering transform, which is(More)
Phase retrieval seeks to recover a signal x ∈ C p from the amplitude |Ax| of linear measurements Ax ∈ C n. We cast the phase retrieval problem as a non-convex quadratic program over a complex phase vector and formulate a tractable relaxation (called PhaseCut) similar to the classical MaxCut semidefinite program. We solve this problem using a provably(More)