4 December 2000 Curvelets and reconstruction of images from noisy radon data
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The problem of recovering an input signal form noisy and linearly distorted data arises in many different areas of scientific investigation; e.g., noisy Radon inversion is a problem of special interest and considerable practical relevance in medical imaging. We will argue that traditional methods for solving inverse problems - damping of the singular value decomposition or cognate methods - behave poorly when the object to recover has edges. We apply a new system of representation, namely the curvelets in this setting. Curvelets provide near-optimal representations of otherwise smooth objects with discontinuities along smooth C2 edges. Inspired by some recent work on nonlinear estimation, we construct a curvelet-based biorthogonal decomposition of the Radon operator and build a reconstruction based on the shrinkage of the noisy curvelet coefficients. This novel approach is shown to give a new theoretical understanding of the problem of edges in the Radon inversion problem.
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Emmanuel J. Candes, David L. Donoho, "Curvelets and reconstruction of images from noisy radon data", Proc. SPIE 4119, Wavelet Applications in Signal and Image Processing VIII, (4 December 2000); doi: 10.1117/12.408569; https://doi.org/10.1117/12.408569

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