Geometric and statistical methods for image analysis and control

该提案分为三个部分：（i）图像配准和分析;（ii）时空建模;和（iii）旋转体（例如卫星）的控制。前两个主题是密切相关的，我的重点是适用于医学成像的方法。第三个主题可能看起来不相关，但实际上借鉴了几何和力学的共同数学基础。 该提案的第一个主题是图像配准和分析的几何和统计方法。我们将开发新的方法来配准，即对齐，图像，并描述形状变形。这些任务是医学图像分析和许多计算机视觉应用（如跟踪和变化检测）的核心。在医学图像领域，配准允许我们比较来自不同人的图像，或来自同一个人在不同时间的图像，并对图像组进行统计。目前的配准方法对于来自健康成年人的大多数图像都很有效，但对于不寻常的图像，包括患病或非常年轻或年老的受试者的图像，这些图像往往正是最感兴趣的图像。我们将开发更强大，更准确或更快的通用方法，以及同时注册图像和检测异常的方法。 第二对象是来自多个对象（或对象）的纵向数据的时空模型，例如医学成像或临床疾病进展数据。时空模型考虑到变化的模式和变化的速率。例如，要找到某种疾病进展的统计模式，考虑到疾病在不同个体中进展或快或慢的事实是有用的，即使他们遵循相同的进展路径。我们将开发改进的模型和统计技术，例如，纳入不同空间区域的变化率。 本建议的第三个主题是通过质量分布的变化来控制旋转体的姿态。就像一个旋转的溜冰者可以通过拉她的手臂来加速一样，一般旋转的“刚体”（即没有控制力的刚体）的角速度可以通过改变其形状来改变，即它的质量分布。一般的情况比滑冰者的例子更有趣：角速度可以改变方向，而不仅仅是幅度。这种形状变化永远不能用来减慢旋转物体的速度，使其静止，或从静止开始运动，因此它不适合大多数控制任务。然而，在某些应用中，例如在成像卫星中，它可以提供有效的解决方案，其中相机可以瞬间指向期望的方向，同时总是以非零速度移动。我将研究这个问题的运动规划和最优控制，并研究一个相关的问题，其中一个刚性物体通过移动内部质量移动通过流体。

This proposal has three parts: (i) image registration and analysis; (ii) spatiotemporal modeling; and (iii) control of rotating bodies (e.g. satellites). The first two topics are closely related, and my focus is on methods applicable to medical imaging. The third topic may seem unrelated, but in fact draws on a common mathematical base of geometry and mechanics. The first subject of this proposal is geometric and statistical methods for image registration and analysis. We will develop new methods for registering, i.e. aligning, images, and describing shape deformations. These tasks are central to medical image analysis and many computer vision applications such as tracking and change detection. In the medical image domain, registration allows us to compare images from different people, or from the same person at different times, and do statistics on groups of images. Current registration methods work well for most images from healthy adults, but struggle with unusual images, including those of diseased or very young or old subjects, which are often precisely the ones of most interest. We will develop general methods that are more robust, more accurate or faster, and methods to simultaneously register images and detect abnormalities. The second subject is spatiotemporal models of longitudinal data from multiple subjects (or objects), for example medical imaging or clinical disease progression data. Spatiotemporal models take into account both the patterns of change and the rates of changes. For example, to find statistical patterns in the progression of a certain disease, it is useful to take into account the fact that the disease may progress more or less quickly in different individuals, even if they are following the same path of progression. We will develop improved models and statistical techniques, for example incorporating rates of change that vary in different spatial regions. A third subject of this proposal is the attitude control of rotating bodies via changes in the mass distribution. Just as a spinning skater can speed up by pulling her arms in, angular velocity in a general rotating “rigid” body (i.e. rigid in the absence of control forces) can be changed by changing its shape, i.e. its mass distribution. The general case is much more interesting than the skater example suggests: the angular velocity can change direction, not just amplitude. This kind of shape change can never be used to slow a rotating body down to rest, or initiate motion from rest, so it is not suitable for most control tasks. However, it may provide efficient solutions in some applications, for example in an imaging satellite, where it may suffice for a camera to point in the desired directions instantaneously, while always moving with nonzero velocity. I will study motion planning and optimal control for this problem, and also investigate a related problem in which a rigid object moves through a fluid by moving internal masses.