Mathematical Imaging and Data Analysis

Photoacoustic Tomography

Spherical mean values

Photoacoustic tomography is an imaging modality which is under vast progress in biology and medicine. The  related mathematical problems are multifaceted. Mathematical models for the physical processes have to be modified resp. combined in order to obtain a better understanding of the data.  Reconstruction algorithms which are fast, efficient, and stable have to be developed in order to make those imaging techniques reliable.We developed and analysed different reconstruction techniques which are related to summability kernels, spectral methods and kernel-based semi-algebraic methods. We currently work on the generalization of these methods to limited view problems and so-called quantative photoacoustic tomography. 

Approximation on Manifolds 

Scattered data on the sphere

In many fields one is confronted with increasingly detailed data in a high-dimesional ambient space. These data are typically unstructered and are located on a manifold of significantly lower dimesion.  Beyond an understanding of this manifold one needs to answer queries based on the data. These queries can be modeled mathematically as functions on the (unknown) manifold. The function is known only one some training points and we have to predict the values of items that are not yet observed. We developed approximation methods for functions on manifolds based on localized summability kernels and applied these methods to different biological data sets. It turned out that the new method perfomed better as existing kernel methods in several  cases. 

A Dimension of Complexity for Brain-Networks

NeuroImaging developed rapidly in the last years. It supports basic neuroscience and clinical analysis of the human brain. Functional or structural connectivity, caused by nerve fibers between separated Cortex regions, seems to be a biomarker for different diseases. Network or graph analysis proved to be a natural mathematical description of connectivity. We developed a new measure Ndim of complexity, quantifying the proliferation and the degree of connectivity in binary or weighted networks. By model based numerical applications and formal proofs we could show that Ndim has the properties of a Dimension. Ndim extends the scope of application of fractal box-covering Dimensions introduced recently for networks. We propose three different methods (Thresholding, Monte Carlo sampling, Transformation to functional distances) to apply Ndim in weighted graphs and explore resting-state and task-induced fMRI networks of the human brain. A characteristic difference between Ndim and a box-covering Dimension is indicated for weighted networks in the following Figure:

Ptychographical Imaging

In ptychographical far-field diffraction patterns are generated by a focused illumination of an object using X-rays. The illumination is chosen such that only a small part of the object is highlighted but then the object is shifted such that in the end the whole object is covered. The aim is now to reconstruct the phases of the object from this set of diffraction patterns, that is, we are facing a phase retrieval problem. The challenge here is that the support of the illumination is not necessarily known. The most often used reconstruction algorithm, the extended ptychographical iterative engine, basically reconstructs the illumination and the phases simultaneously. The cost of this method is that it is relatively slow. We currently work on a new non-iterative algorithm for image reconstruction.