AIR Tools - A MATLAB Package for Algebraic Iterative ...

Chapter 2
Theory of Inverse Problems
and Regularization
Inverse problems arise in many applications in science and technology. Examples
where inverse problems are found could be in medical imaging, where it is used
e.g. in CT scanning, in geophysical prospecting or image deblurring. We will in
this chapter introduce some of the fundamental concepts of inverse problems.
We will first introduce the concept of an inverse problem and describe what
defines an ill-posed problem. Then the important tools of SVD and the discrete
Picard condition is defined followed by a few examples of spectral filtering.
Finally we will give a short description of semi-convergence for iterative methods
and define the concept of resolution limit.
2.1 Discrete Ill-Posed Problems
Inverse problems arise when we need to compute information that is either
internal or hidden. In the forward problem we have a known input and a known
system and we can then compute the output. In the inverse problem the output
is often known with errors and we then have to compute either the system or
the input, where the other one is known. For the linear problems we let the
system be represented by the matrix A ∈ R m×n , the output as the right-hand