New Models and Algorithms in Image Processing with Partial Differential Equations

PI将与他的合作者和学生一起开发新的模型和数值算法，以解决图像处理和计算机视觉中的一些基本问题。这些模型将基于变分法和描述曲线和表面演化的偏微分方程（PDE）。该项目的主要目标是设计新的模型，将先前的形状信息整合到现有的变分图像分割技术中，如Mumford-Shah模型及其变体。新的模型将被设计成在给定的图像中找到类似于特定形状的物体，而不管物体在图像中的位置和方向。此外，他们将使用已经在相关视觉应用中证明其实用性的技术，例如水平集方法，方便地进行数值实现。第二个研究领域将是开发新的、有效的数值算法来解决在许多计算机视觉模型和其他领域（如材料科学）中出现的PDE问题。具体来说，研究者将探索新的计算技术来解决描述界面几何运动的高阶偏微分方程，如Willmore流和表面扩散运动。使用当前的技术，这些进化在计算上是非常昂贵的。新的方法是将这些运动的计算减少到交替的简单操作，而高效的算法已经可用。同样在这方面，该项目将开发新的数值算法，灵感来自于图像处理中的模型和技术，用于计算多相和结的能量驱动动力学。图像分割是计算机视觉的一个基本步骤。从数字图像中自动提取有用信息是必要的先决条件。它的目标是识别图像中属于不同对象的部分，通常不知道图像中可能存在什么对象。然而，在许多实际应用中，在图像中寻找已知形状的特定物体。例如，在航空图像中，感兴趣的对象可能是具有独特轮廓的某种交通工具。或者，在医学应用中，可能需要在脊柱的x射线图像中自动识别单个椎骨。在这种情况下，如果算法能够意识到正在寻找什么，将有助于分割过程的成功率。该项目将开发模型和数值技术，将有关感兴趣的对象的先验形状信息纳入分割过程，从而导致更好的分割方法。

The PI, together with his collaborators and students, will develop new models and numerical algorithms for the solution of a number of fundamental problems in image processing and computer vision. The models will be based on the calculus of variations and partial differential equations (PDE) that describe curve and surface evolutions. A main goal of the project will be to devise new models that incorporate prior shape information into existing variational image segmentation techniques such as the Mumford-Shah model and its variants. The new models will be designed to find in given images objects resembling a specified shape regardless of the objects' location and orientation in the image. In addition, they will have convenient numerical implementations using techniques that have already proven their utility in related vision applications, such as the level set method. A second area of research will be to develop novel, efficient numerical algorithms for the solution of PDE that arise in a number of computer vision models and in other fields such as material science. Specifically, the investigator will explore new computational techniques for the solution of high order PDE that describe geometric motion of interfaces, such as the Willmore flow and motion by surface diffusion. These evolutions are computationally very expensive using current techniques. The new approach will be to reduce the computation of these motions to alternating simple operations for which efficient algorithms are already available. Also in this vein, the project will develop new numerical algorithms inspired by models and techniques in image processing for the computation of energy driven dynamics of multiple phases and junctions.Image segmentation is a fundamental procedure of computer vision. It is a necessary preliminary step whenever useful information is to be extracted from digital images automatically. Its goal is to identify parts of the image that belong to distinct objects, often without knowing what objects might be present in the image. In many practical applications, however, a specific object of known shape is sought in the images. For example, in aerial imagery, the object of interest might be a certain vehicle that has a distinctive outline. Or, in a medical application, it might be desired to identify automatically the individual vertebrae in x-ray images of the spine. It such settings, it would help the success rate of the segmentation procedure if the algorithms could be made aware of what is being sought. This project will develop models and numerical techniques that incorporate prior shape information about objects of interest into the segmentation process, thereby leading to better segmentation methods.