Mumford-Shah models for tomography II

The general aims of the proposed project are the application, analysis and extension of recently established Mumford-Shah like methods to problems from tomography. The goal of tomography is to recover the interior structure of a body using external measurements. Tomographic imaging techniques are used in medicine or non- destructive testing and are based on several disciplines such as pure mathematics and numerical analysis as well as physics and hard- and software engineering. As existing scanning devices are enhanced (higher resolution, more computational power) and new scanning devices are developed (hybrid imaging, different physical phenomena) the same is necessary for mathematical reconstruction algorithms. Mumford-Shah like methods belong to the new field of geometrical inverse problem. These methods provide a combined output of structural and functional information. For example, in the planning of surgery one would like to have the reconstruction together with its segmentation. The standard procedure first reconstructs an image and then applies image segmentation methods to the reconstructions. This has the main disadvantage that the measured data is only used for the reconstruction and not for the segmentation. Mumford-Shah like methods for tomography have been introduced and studied in the framework of FWF project P19209-N18. They have been successfully applied to x-ray CT and SPECT/CT where SPECT is short for Single Photon Emission Computed Tomography, a nuclear medicine imaging technique providing functional information, e.g., the blood flow in the heart. In the proposed project we plan to extend Mumford-Shah like methods to tomographic imaging techniques such as PET (Positron Emission Tomography), PET/CT (combined PET and CT), electron microscopy and to apply them to real (measured) data. Furthermore, we want to study different data fit terms which are well suited to model the (physical) properties of measured data, e.g., Poisson noise, Kullback-Leibler distance, EM (expectation maximization). In order to evaluate these methods regarding well-definedness and quality of the reconstruction we plan to perform a convergence analysis and prove regularization properties if possible.

Mumford-Shah methods are a relatively new area of mathematical research. They can be applied, e.g., for tomographic imaging (medicine, non-destructive testing). The main results of the project Mumford-Shah models for tomography II are 1) a theoretical analysis of the method, 2) the development of a new and promising approach for tomography with limited data, 3) the adaptation of the method to incorporate statistical properties of data, as well as 4) the participation in a European networking program to disseminate mathematical research to the non-academic field. Tomography images the interior of an object. Data are measured outside of the object and then used to compute the image of the interior. In the same way as existing scanning device are continuously improved, mathematical algorithms for tomographic reconstruction were constructed, generalized, implemented and analyzed in the project. 1) Mathematical reconstruction methods often consists of a large number of iterative steps. Meaningful methods are designed such that each step results in a better approximation until the sought-after reconstruction (the image of the interior) is reached. One result of the project is the proof that the Mumford-Shah method has this property. The iteratively computed images get better and better until they do not change any longer and the solution is achieved. The strong challenge that Mumford-Shah methods not only compute an image but also the contours in between different materials was answered with a new notion of convergence. 2) In order to monitor the course of a localized desease while simultaneously minimizing the radiation one uses limited data tomography. In such a case standard reconstruction methods cannot be used. Within the project a new method was developed that is adapted to this setting. It was tested with very promising results. 3) Imaging methods from nuclear medicine are based on a tracer that emits radiation which is measured outside of the body. It is known that the radiation process can be described by a Poisson distribution. This information was used to generalize the Mumford-Shah method. Test calculations show that this data-driven model yields better results. 4) Within the framework of the project Austria joined a European networking program that aims at bridging the gap between the experimental X-ray tomography community and the mathematical image reconstruction community. From this, several possibilities emerged to work with real data, to apply the Mumford-Shah model to hybrid methods that use different type of radiation, as well as to disseminate mathematical results to a non-mathematical audience.