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

数学科学：曲线演化和图像分割的通用框架

• 批准号: 9531293
• 金额: $ 6.7万
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9531293&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Common; Framework; Curve

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.

小行星9531293 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，并已被应用于各种各样的问题，如形状的平滑，形状分析和形状恢复。 其基本原理是一个简单的封闭曲线的发展，其点以规定的速度在法线方向上移动。 该方法的一个基本限制是它不能处理重要的图像特征，如三重点。 该方法还需要选择在图像域上定义的“边缘强度”函数，指示对象边界存在于图像域中的任何点处的可能性。 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先计算近似边界。 初始曲线也必须预先选定。 研究人员证明，在计算机视觉中使用的不同版本的曲线演化与预处理步骤和初始曲线的选择可以集成在一个新的分割功能，统一和扩展曲线演化模型的形式。 此外，获得的数值解保留了尖锐的不连续性或“冲击”，从而提供了对象边界的尖锐分界。 新的功能描述提供了一个统一的方法分割问题和形状分析。 建议的研究的主要目标是进行数学分析的新功能，将其应用到真实的图像，并研究其可能的扩展。 该项目属于机器视觉领域，特别是在医学成像方面的应用。 这里提出的基本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。 计算机通过照相机接收每帧图像中数以百万计的数字形式的视觉信息，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对象之前的第一个处理任务是找到对象的轮廓。 这就是所谓的分割问题。 这里的基本困难是如何将对象的轮廓从杂乱的场景中分离出来。 即使当计算机已经成功地定位对象轮廓时，也很难识别对象，因为随着对象和相机的移动，对象的外观会根据其方向而变化。 更糟糕的是，物体可能会被部分遮挡。 因此，必须定义和计算将一个对象与另一个对象区分开并且在对象的外观改变时保持不变的特征。 这就是形状分析的问题。 该项目制定了一个新的统一的方法来解决这两个问题的基础数学问题。 它基于这种方法开发了新的数学工具，并将其应用于特定类别的实际问题，特别是生物医学图像的分析。