Analysis of Integral Equations with Parameter-Dependent Kernels

(German version)

Physical measuring instruments such as mass spectroscopes, heat sensors or optical cameras often experience the problem that the quantities to be measured are falsified. Two effects play an important role here: degradation due to faults in the measuring device or in the measuring process, and noise. This type of process can be modelled as an integral equation in which the degradation is represented by the integral kernel and the noise by an additive contribution. If the cause of the degradation, i.e. the integral kernel, is known, then this amounts to an ill-posed inverse problem. Problems of this type have been thoroughly investigated and regularisation methods lead here to well-posed linear or non-linear problems, with very good results in some cases. In image processing, in particular, the bounded variation regularisation is very popular. It removes the noise while at the same time focusing the outlines in the image. In many fields of application, however, the cause of the degradation is not fully known or understood. Examples are motional distortion when the speed of the object is unknown or very complicated physical systems for which there is no satisfactory model. In these cases, the kernel function of the integral equation is completely unknown, or at best it depends on one or several parameters. Both the measurand and the kernel itself have to be estimated. Such problems are more difficult to deal with and as yet they have hardly been analysed mathematically. Initial methods presented in the literature estimated kernel parameters using zeros of the Fourier transformed measurand. Later on, regularised functionals were minimised by alternately holding the kernel and the image. Both methods reduce the problem essentially to the case of the known kernel.

Original image	Degraded image (without noise),     here with depth of field blur	Image reconstructed without knowing the integral kernel

With this "blind desmoothing" it is primarily the non-uniqueness of the solution which causes problems. The first subject of Lutz Justen's doctoral project is therefore the search for suitable means of limiting the number of solutions. Minimum norm criteria, which can be applied to various Hilbert scales, play a significant role here. Since this approach is fundamentally different to the previous approach, the connections to familiar methods are explained. The existence and uniqueness of minimum norm solutions is demonstrated and the dependency of the input data investigated. If this data is noisy, additional regularisation is required. Various methods are conceivable here, for example Tikhonov regularisation or a pre-smoothing of the data. Including convex constraints such as the non-negativity of the kernel function or restricting the support generally leads to better results and is being investigated.
One field of application is provided by the collaboration with Bruker Daltonik GmbH: to provide an early diagnosis of ovarian or prostate cancer, protein spectra from patients are generated in MALDI-TOF mass spectrometers in order that the protein distribution can be used to draw conclusions about the clinical picture. These spectra can be modelled by means of an integral equation with parameter-dependent kernels so that the results of the dissertation can be applied to them.

Back to: Projects