Inverse Problems

Inverse Problems - Connexions module: m23071 1 Inverse...

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Unformatted text preview: Connexions module: m23071 1 Inverse Problems * Stephane Mallat This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License Abstract This collection comprises Chapter 1 of the book A Wavelet Tour of Signal Processing, The Sparse Way (third edition, 2009) by Stphane Mallat. The book's website at Academic Press is http://www.elsevier.com/wps/ nd/bookdescrip The book's complementary materials are available at http://wavelet-tour.com Most digital measurement devices, such as cameras, microphones, or medical imaging systems, can be modeled as a linear transformation of an incoming analog signal, plus noise due to intrinsic measurement uctuations or to electronic noises. This linear transformation can be decomposed into a stable analog-to- digital linear conversion followed by a discrete operator U that carries the speci c transfer function of the measurement device. The resulting measured data can bewritten Y [ q ] = Uf [ q ] + W [ q ] , (1) where f C N is the high-resolution signal we want to recover, and W [ q ] is the measurement noise. For a camera with an optic that is out of focus, the operator U is a low-pass convolution producing a blur. For a magnetic resonance imaging system, U is a Radon transform integrating the signal along rays and the number Q of measurements is smaller than N . In such problems, U is not invertible and recovering an estimate of f is an ill-posed inverse problem. Inverse problems are among the most di cult signal-processing problems with considerable applications. When data acquisition is di cult, costly, or dangerous, or when the signal is degraded, super-resolution is important to recover the highest possible resolution information. This applies to satellite observations, seismic exploration, medical imaging, radar, camera phones, or degraded Internet videos displayed on high- resolution screens. Separating mixed information sources from fewer measurements is yet another super- resolution problem in telecommunication or audio recognition. Incoherence, sparsity, and geometry play a crucial role in the solution of ill-de ned inverse problems. With a sensing matrix U with random coe cients, Cands and Tao (candes-near-optimal) and Donoho (donoho- cs) proved that super-resolution becomes stable for signals having a su ciently sparse representation in a dictionary. This remarkable result opens the door to new compression sensing devices and algorithms that recover high-resolution signals from a few randomized linear measurements....
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This note was uploaded on 10/16/2010 for the course ECE 380 taught by Professor Baltazar during the Spring '10 term at Rice.

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Inverse Problems - Connexions module: m23071 1 Inverse...

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