Mathematical Sciences: A Common Framework for Curve Evolution and Image Segmentation

近年来，曲线演化已发展成为计算机视觉中的一个重要工具，并广泛应用于形状平滑、形状分析和形状恢复等问题。其基本原理是一个简单的闭合曲线的演化，其点以规定的速度沿法线方向移动。该方法的一个基本限制是，它不能处理重要的图像特征，如三点。该方法还需要选择在图像域上定义的“边缘强度”函数，指示物体边界在图像域中任意点存在的可能性。这意味着一个单独的预处理步骤，本质上是在存在噪声的情况下预计算近似边界。初始曲线也必须预先选定。研究人员证明，计算机视觉中使用的不同版本的曲线演化以及预处理步骤和初始曲线的选择可以以统一和扩展曲线演化模型的新分割函数的形式集成。此外，获得的数值解保留了明显的不连续或“冲击”，从而提供了物体边界的明确划分。所描述的新功能为分割问题和形状分析提供了统一的方法。本研究的主要目的是对新函数进行数学分析，将其应用于真实图像，并研究其可能的扩展。这个项目是在机器视觉领域，特别是在医学成像方面的应用。这里提出的基础数学理论也可能适用于看似无关的领域，如液晶技术、冶金、火焰传播和燃烧。计算机通过摄像头接收视觉信息，每帧图像有数百万个数字，每个数字描述场景中某一点的光的强度和颜色。视觉信息的绝对数量使得计算机对其进行解释的任务极其困难。计算机在开始识别场景中的物体之前的第一个处理任务是找到物体的轮廓。这就是所谓的分割问题。这里的基本困难是如何将物体的轮廓与混乱场景的其余部分分开。即使计算机成功地定位了物体的轮廓，也很难识别出物体，因为随着物体和摄像机的移动，物体的外观会随着其方向而变化。更糟糕的是，物体可能被部分遮挡。因此，必须定义和计算区分一个物体与另一个物体的特征特征，并随着物体外观的变化而保持不变。这就是形状分析的问题。这个项目为这两个问题背后的基本数学问题制定了一种新的统一方法。它基于这种方法开发了新的数学工具，并将它们应用于特定类别的实际问题，特别是生物医学图像的分析。

9531293 Shah In recent years, curve evolution has developed into an important tool in Computer Vision and has been applied to a wide variety of problems such as smoothing of shapes, shape analysis and shape recovery. The underlying principle is the evolution of a simple closed curve whose points move in the direction of the normal with a prescribed velocity. A fundamental limitation of the method as it stands is that it cannot deal with important image features such as triple points. The method also requires a choice of an "edge-strength" function, defined over the image domain, indicating the likelihood of an object boundary being present at any point in the image domain. This implies a separate preprocessing step that is in essence precomputing approximate boundaries in the presence of noise. The initial curve also has to be preselected. The investigator demonstrates that the different versions of curve evolution used in Computer Vision together with the preprocessing step and the choice of the initial curve can be integrated in the form of a new segmentation functional that unifies and extends curve evolution models. Moreover, the numerical solutions obtained retain sharp discontinuities or "shocks," thus providing sharp demarcation of object boundaries. The new functional described provides a unified approach to the segmentation problem and shape analysis. The principal objective of the proposed research is to carry out a mathematical analysis of the new functional, apply it to real images, and study its possible extensions. This project is in the field of machine vision, with applications especially in medical imaging. The underlying mathematical theory proposed here is potentially applicable also to seemingly unrelated fields such as liquid crystal technology, metallurgy, flame propagation and combustion. A computer receives visual information through a camera in the form of millions of numbers per picture frame, each number describing th e intensity and the color of light at a point in the scene. The sheer volume of the visual information makes the task of its interpretation by the computer extremely difficult. The first processing task for the computer before it can begin to recognize an object in a scene is to find the object's outline. This is what is called the segmentation problem. The fundamental difficulty here is how to separate the object's outline from the rest of a cluttered scene. Even when the computer has successfully located the object outline, it is difficult to identify the object because its appearance changes depending on its orientation as the object and the camera move. Worse, the object might be partially obscured. Therefore, characteristic features that distinguish one object from another and that remain unchanged as the appearance of the object changes must be defined and calculated. This is the problem of shape analysis. This project formulates a new unified approach to the fundamental mathematical questions underlying the two problems. It develops new mathematical tools based upon this approach and applies them to specific classes of practical problems, in particular to analysis of biomedical images.