Stochastic level-sets and random sets in image processing with partial differential equations

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Mathematical methods in image processing have reached a high significance in multiple areas of daily life. In medical diagnosis and treatment planning a multitude of applications for reconstruction, denoising, segmentation, quantification, registration and many more methods have been established since the digitalization of the radiological image acquisition. The evaluation of the data, however, is still mostly limited by the trained eye of the radiologist. Wide areas of image processing and in particular medical image processing seem to be untouched by the culture of scientific measurements, modeling and error propagation. In medical image processing measurement errors and uncertainties are not considered in quantitative evaluations like the measurement of the growth or shrinkage of the tumor volume during chemotherapy. The goal of the proposed project is to establish a framework that allows for the propagation of errors in segmentation algorithms, from error prone input data to a quantification of the uncertainty in the final output. The key ingredient of the approach is the identification of image values with spatially distributed random variables. A generalized Mumford- Shah functional as well as level set and phase field approximations shall be developed in analogy to the works of Chan-Vese and Ambrosio-Tortorelli. Thus, the segmentation of the stochastic images leads to stochastic shapes, which will be represented by stochastic level sets and stochastic phase fields, respectively. For the discretization of the resulting stochastic partial differential equations we will use the Wiener-Askey polynomial chaos, stochastic finite elements and hybrid approaches made of finite elements in the stochastic dimension and finite differences in the physical dimension. A particular emphasis lies on the improvement of techniques for the efficient solution of the large systems of equations. The methods developed have a high potential for application in all areas of quantitative image processing. In this project we will exemplarily investigate the segmentation of medical image data.