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

数学方法在图像处理中已经在日常生活的多个领域中发挥了重要的作用。在医学诊断和治疗规划中，自放射图像采集数字化以来，已经建立了重建、去噪、分割、量化、配准等多种方法。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，测量误差和不确定度在定量评估中不被考虑，比如在化疗期间测量肿瘤体积的生长或缩小。拟议项目的目标是建立一个框架，允许在分割算法中传播错误，从容易出错的输入数据到最终输出中的不确定性的量化。该方法的关键是识别具有空间分布的随机变量的图像值。将根据Chan-Vese和Ambrosio-Tortorelli的工作，提出广义的Mumford- Shah泛函以及水平集和相场近似。因此，对随机图像进行分割得到随机形状，分别用随机水平集和随机相场表示。对于所得到的随机偏微分方程的离散化，我们将使用Wiener-Askey多项式混沌、随机有限元和由随机维度的有限元和物理维度的有限差分组成的混合方法。特别强调的是技术的改进，以有效地解决大方程组。所开发的方法在定量图像处理的各个领域具有很高的应用潜力。在这个项目中，我们将举例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.