# 0.6 Inverse problems

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This collection comprises Chapter 1 of the book A Wavelet Tour of Signal Processing, The Sparse Way(third edition, 2009) by Stéphane Mallat. The book's website at Academic Press ishttp://www.elsevier.com/wps/find/bookdescription.cws_home/714561/description#description The book's complementary materials are available athttp://wavelet-tour.com

Most digital measurement devices, such as cameras, microphones, or medical imaging systems, can be modeled as a linear transformation ofan incoming analog signal, plus noise due to intrinsic measurement fluctuations or to electronic noises. This linear transformationcan be decomposed into a stable analog-to-digital linear conversion followed by a discrete operator U that carries the specific transfer function of the measurement device. The resulting measured data can bewritten

$Y\left[q\right]=Uf\left[q\right]+W\left[q\right],$

where $\phantom{\rule{0.166667em}{0ex}}f\in {\mathbb{C}}^{N}$ is the high-resolution signal we want to recover, and $W\left[q\right]$ 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 imagingsystem, 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 $\phantom{\rule{0.166667em}{0ex}}f$ is an ill-posed inverse problem.

Inverse problems are among the most difficult signal-processing problems with considerable applications.When data acquisition is difficult, costly, or dangerous, or when the signal is degraded, super-resolution isimportant 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-defined inverse problems. With a sensing matrix U with random coefficients, Candès and Tao (candes-near-optimal)and Donoho (donoho-cs) proved that super-resolution becomes stable for signals having a sufficientlysparse representation in a dictionary. This remarkable result opens the door to new compression sensing devices and algorithmsthat recover high-resolution signals from a few randomized linear measurements.

## Diagonal inverse estimation

In an ill-posed inverse problem,

$Y=Uf+W$

the image space $\mathbf{ImU}=\left\{Uh\phantom{\rule{3.33333pt}{0ex}}:\phantom{\rule{3.33333pt}{0ex}}h\in {\mathbb{C}}^{N}\right\}$ of U is of dimension Q smaller than the high-resolution space N where $\phantom{\rule{0.166667em}{0ex}}f$ belongs. Inverse problems include two difficulties. In the imagespace ImU , where U is invertible, its inverse may amplify the noise W , which then needs to be reduced by an efficient denoising procedure. In the null space NullU , all signals h are set to zero $Uh=0$ and thus disappear in the measured data Y . Recovering the projection of $\phantom{\rule{0.166667em}{0ex}}f$ in NullU requires using some strong prior information. A super-resolution estimator recovers an estimation of $\phantom{\rule{0.166667em}{0ex}}f$ in a dimension space larger than Q and hopefully equal to N , but this is not alwayspossible.

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Source:  OpenStax, A wavelet tour of signal processing, the sparse way. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10711/1.3
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