医学图像分割的新变分模型及其快速有效的最优化算法

With the rapid development of medical imaging and its more and more important roles in clinic, medical image segmentation has currently become an international popular research field. Accuracy and efficiency are two most important requirements of medical image segmentation. After a thorough and systematic research on liver segmentation, we find recently that existing segmentation models and methods are not ideal for liver tumor segmentations. This program plans to do research on more appropriate new segmentation models for liver tumor, convexation approaches to non-convex segmentation models and efficient and effective algorithms. Firstly, since variational models have the advantage of flexibility and diversity, we focus on models based on calculus of variation. We aim to use probability methods and nonlocal operators, which are very effective methods for inhomogeneous images. Meanwhile, we will do study on convexation methods for non-convex segmentation models by the functional lifting method. Moreover, due to the two demands on medical image segmentations and the huge data size of medical images, we will design fast，effective and stable numerical algorithms using calculus of variation, optimization theory, operator splitting technique, and so on. Finally, we apply our models and algorithms to practical problems to show their advantages.

随着医学影像学的快速发展及其在临床中发挥的越来越重要的作用，医学图像分割已成为当前国际上研究的热点。准确性和快速性是医学图像分割最重要的两个要求。 在对肝脏的分割进行了深入系统的研究后，最近我们发现已有的分割模型和方法对肝脏中肿瘤的分割效果和分割速度并不理想。本项目计划研究更加适合肝脏肿瘤分割的新模型、非凸分割模型的凸化方法以及快速有效的求解算法。首先，鉴于变分模型具有灵活性和多样性等优点，我们将重点研究以变分方法为基础的模型。我们计划结合概率、非局部算子等对非均匀图像非常有效的方法。同时，我们计划利用泛函提升的方法深入研究非凸分割模型的凸化方法。其次，针对医学图像分割的两个要求与医学图像数据量大的特点，我们将综合利用变分理论、优化理论、算子分裂技巧等设计快速、有效和稳定的算法。最后，我们将我们得到的模型和算法应用到实际问题中来验证其优点。

医学影像的快速发展及其在临床中的重要性给医学图像分割提出了更高的要求。本项目主要针对医学图像分割模型和算法进行研究。我们与香港浸会大学的教授合作提出了一种新的人机交互式的变分模型。文章还在投稿中。这种模型将医生的指导作用融入到变分模型中，既包含了医生给出的标识点，又包含边界信息，因此可以处理更加复杂的分割问题。在计算过程中，我们利用了流行的算子分裂，ADM等方法加速了计算速度。同时，对模型及算法的理论方面，我们也做出了详细的证明。