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，用于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变的变分法和偏微分方程（PDE）。该项目的一个主要目标将是设计新的模型，将先验形状信息纳入现有的变分图像分割技术，如Mumford-Shah模型及其变体。新的模型将被设计为在给定的图像中找到类似于指定形状的对象，而不管对象在图像中的位置和方向。此外，他们将有方便的数值实现使用的技术，已经证明了他们的效用在相关的视觉应用，如水平集方法。研究的第二个领域将是开发新的，有效的数值算法的PDE的解决方案，出现在一些计算机视觉模型和其他领域，如材料科学。具体来说，研究人员将探索新的计算技术的解决方案，高阶偏微分方程描述的几何运动的接口，如Willmore流和运动的表面扩散。使用当前技术，这些演进在计算上非常昂贵。新的方法将减少这些运动的计算，以交替的简单操作，有效的算法已经可用。此外，该项目还将开发新的数值算法，这些算法的灵感来自图像处理中的模型和技术，用于计算多个相位和连接点的能量驱动动力学。图像分割是计算机视觉的基本程序。这是一个必要的初步步骤，每当有用的信息是从数字图像中自动提取。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，可能期望在脊柱的X射线图像中自动识别各个椎骨。在这样的设置中，如果算法能够知道正在寻找什么，则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.