Inverse problems and paramter choice strategies

Ill-posed (or inverse) problems are occuring in numerous real situations. Almost everytime when we measure data by a kind of averaging process (which is an easy forward problem) the inversion, namely recovering the original information out of the measured data is extremely unstable. Examples for such problems are

- Numerical differentiation
- Downward-continuation of satellite data, i.e. recovering the gravitational field out of satellite data measured outside the Earth's atmosphere.
- Learning Theory, in particular kernel based learning methods.
- Financial mathematics
- Image processing, in particular denoising, debluring etc.
- Parallell Magnetic Resonance Imaging, a medical imaging method which is up to current knowledge completely harmless.

If we used a naive inversion procedure the error in the high frequency compentents would get extremely amplified, the solution has nothing to do with the original information. Therefore one needs to damp this high frequency part, despite that one also looses the high frequency components in the solution, i.e. makes by purpose some small error in order to avoid a large one. This procedure is called regularization, how much one damps down these high frequency components is associated with the regularization parameter. Choosing this parameter in an optimal way is crucial.

Image Processing

Image processing is a part of computer science and mathematics which is getting increasingly important in technical applications and industry for automatization and quality control. Particular examples are

- Denoising of images

Some measurement procecures, such as time-of-flight cameras (an active 3D-measurement technique) generate extremely noisy measurements. Before one can automatically process these images one needs to remove the noise, e.g. using wavelet shrinkage or anisotropic diffusion. - Tracking, Image Registration and Quality Control

A common task in Computer Vision is searching a prototype image in a bigger one (respectively if there are outliers which do not match the prototype). For these tasks one needs to measure the similarity of two images (respectively patches of images). One possibility is using the discrepancy norm which exhibits very interesting features like monotonicity and stability in the presence of noise. - Deformation

In some medical and technical applications one needs to determine the deformation between two images, sometimes even with sub-pixel accuracy.