Numerical methods for geometric partial differential equations: applications to freeform deformations in animation and nonrigid medical image registration

形状可以用我们熟悉的几何概念来描述，如长度、面积、体积、角度和曲率。形状的变化可以用几何偏微分方程式来描述：拉伸或弯曲，平滑粗糙的边缘，或相互映射。 这项拟议的研究将建立处理三维形状的新工具。要做到这一点，我们需要能够数值求解描述变换的方程。由于方程的复杂性和大量的数据，这项任务变得困难。到目前为止，已使用的简化方程忽略了一些几何细节。我的研究找到了这些方程的新表示法，它们克服了其中的一些困难。这项工作将允许几何细节在形状处理中得到保留。 该提案将允许在以下重要应用中使用的三维工具具有更好的几何保真度： 计算机动画中的自由变形允许艺术家通过操纵更简单的骨架或框架来移动动画图像。几何方程将运动外推到形状，同时保留精细特征。这节省了动画制作的大量时间和精力。 非刚性图像配准工具用于比较连续时间拍摄的器官图像。例如，外科医生从病人的大脑中切除肿瘤后，会在手术后立即拍摄大脑的三维图像，然后再拍摄。为了确定增长是否已经停止，通过将后面的图像映射(扭曲)到第一个图像来比较图像。

Shapes may be described by using the familiar geometric notions of length, area, volume, angle and curvature. Changing shapes may be described by Geometric Partial Differential Equations: stretching or bending, smoothing rough edges, or mapping one onto another. The proposed research will build new tools for manipulating three dimensional shapes. To do this, we need to be able to numerically solve the equations which describe the transformations. This task is made difficult by the complexity of the equations and the large amounts of data. To date, simplified equations have been used which neglect some geometric detail. My research has found new representations of these equations which have overcome some of these difficulties. This work will allow for geometric detail to be preserved in shape manipulation. The proposal will allow for better geometric fidelity in three dimensional tools used in the following important applications: Freeform deformations in computer animation allow the artist to move an animated image by manipulating a simpler skeleton or cage. The geometric equations extrapolate the motion to the shape while preserving fine features. This saves considerable time and effort in animation. Nonrigid image registration tools are used to compare images of organs taken at successive times. For example, after a surgeon removes a tumour from a patient's brain, three dimensional images are taken of the brain immediately after surgery and at later times. In order to determine if the growth has stopped, the images are compared by mapping (warping) the later images onto the first one.