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

图像处理中的数学方法在日常生活的多个领域都具有很高的意义。自放射图像采集数字化以来，在医学诊断和治疗规划中，重建、去噪、分割、量化、配准等方法得到了广泛的应用。然而，对数据的评估仍然主要受到放射科医生训练有素的眼睛的限制。图像处理的广泛领域，特别是医学图像处理，似乎没有受到科学测量、建模和误差传播文化的影响。在医学图像处理中，测量误差和不确定性不被考虑在定量评估中，例如在化疗期间测量肿瘤体积的增长或缩小。拟议项目的目标是建立一个框架，以便在分段算法中传播误差，从容易出错的输入数据到量化最终输出中的不确定性。该方法的关键是识别具有空间分布的随机变量的图像值。推广的Mumford-Shah泛函以及水平集和相场近似应类似于Chan-Vese和Ambrosio-Tortorelli的工作。因此，随机图像的分割产生随机形状，随机形状将分别由随机水平集和随机相位场来表示。对于所得到的随机偏微分方程组的离散化，我们将使用Wiener-Askey多项式混沌、随机有限元以及由随机维上的有限元和物理维上的有限差分组成的混合方法。一个特别的重点在于改进有效地求解大型方程组的技术。所开发的方法在定量图像处理的所有领域都具有很高的应用潜力。在这个项目中，我们将示范性地研究医学图像数据的分割。

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.