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。这些模型将基于描述曲线和曲面演化的变分和偏微分方程(PDE)。该项目的一个主要目标是设计新的模型，将先前的形状信息结合到现有的可变图像分割技术中，例如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与指定形状相似的对象，而不考虑这些对象在图像中的位置和方向。此外，它们将使用已经在相关视觉应用中证明其有效性的技术来进行方便的数值实现，例如水平集方法。第二个研究领域将是开发新的、高效的数值算法来求解偏微分方程组，这些算法出现在许多计算机视觉模型和其他领域，如材料科学。具体地说，研究人员将探索新的计算方法来求解描述界面几何运动的高阶偏微分方程组，如Willmore流和表面扩散运动。使用目前的技术，这些进化在计算上是非常昂贵的。新的方法将把这些运动的计算减少到交替的简单操作，对于这些操作，已经有了有效的算法。同时，该项目还将在图像处理模型和技术的启发下开发新的数值算法，用于计算多相和多结的能量驱动动力学。图像分割是计算机视觉的基本程序。当要从数字图像中自动提取有用信息时，这是必要的预备步骤。它的目标是识别属于不同对象的图像部分，通常不知道图像中可能存在哪些对象。然而，在许多实际应用中，在图像中寻找已知形状的特定对象。例如，在航空图像中，感兴趣的对象可能是具有独特轮廓的特定车辆。或者，在医学应用中，可能希望在脊柱的X射线图像中自动识别单个椎骨。在这样的设置下，如果能够使算法知道正在寻找的是什么，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，将感兴趣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.