13 - Application III: image restoration and inverse problems

Summary

An inverse problem assumes a direct problem that is a well-posed problem of mathematical physics. In other words, if we know completely a “physical device,” we have a classical mathematical description of this device including uniqueness, stability and existence of a solution of the corresponding mathematical problem.

Victor Isakov. Inverse Problems for Partial Differential Equations. Springer-Verlag (1998)

Image restoration is historically one of the oldest concerns in image processing and is still a necessary preprocessing step for many applications.

Gilles Aubert and Pierre Kornprobst. Mathematical Problems in Image Processing. Springer-Verlag (2002)

However, for the time being it is worthwhile recalling the remark of Lanczos: “A lack of information cannot be remedied by any mathematical trickery.” Hence in order to determine what we mean by a solution it will be necessary to introduce “nonstandard” information that reflects the physical situation we are trying to model.

David Colton and Rainer Kress. Integral Equation Methods in Scattering Theory. Wiley (1983)

The research of inverse problems has become increasingly popular for two reasons:

1. (i) there is an urgent need to understand these problems and find adequate solution methods; and

2. (ii) the underlying mathematics is intriguingly nonlinear and is naturally posed as a challenge to mathematicians and engineers alike.

It is customary for an introduction to inverse problems of boundary value problems to discuss the somewhat unhelpful terms of ‘ill-posed problems’ or ‘improperly-posed problems’.