0.6 Inverse problems

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

where $f\in {\mathbb{C}}^N$ 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 $f$ 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,

the image space $\mathbf{ImU}=\{Uh:h\in {\mathbb{C}}^N\}$ of U is of dimension Q smaller than the high-resolution space N where $f$ 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 $f$ in NullU requires using some strong prior information. A super-resolution estimator recovers an estimation of $f$ in a dimension space larger than Q and hopefully equal to N , but this is not alwayspossible.