Surfaces, Objects, and Their Borders in Multidimensional Images: Theory and Algorithms

本研究的目标是(i)在n维离散空间中建立物体及其边界的统一几何理论，以及（ii）开发用于检测n维图像中的物体和边界的算法。理论发展的方法使用n空间的正交镶嵌成超立方体，导致的结果是连续拓扑的一些基本结果的离散对应。因此，对象和边界可以表示为图，它们的检测转换为图遍历。算法开发的重点是设计有效的遍历算法，无论是当图像是局部可分割的还是当它们可能不可分割的。在前一种情况下，总是有一个唯一的易于遍历的生成树。在后一种情况下，没有唯一的解，因此方法是开发最优（例如，使用动态规划）和次最优遍历策略。pi对医学图像和断层扫描的分析有着悠久的贡献历史。图像可视化和分析的共同理论和算法的发展将在使用多维图像数据的许多学科中使用。

The objectives of this research are (i) to develop a unified geometric theory of objects and their boundaries in n-dimensional discrete spaces, and (ii) to develop algorithms for detecting objects and boundaries in n-dimensional images. The approach to theory development uses an orthogonal tesselation of the n-space into hypercubes, leading to results which are discrete counterparts of some essential results of continuous topology. Thus, objects and boundaries can be represented as graphs, and their detection translates to graph traversal. The algorithm development focuses on devising efficient traversal algorithms, both when the images are locally segmentable and when they may not be. In the former case there is always a unique spanning tree which is easy to traverse. In the latter case, there is no unique solutino and hence the approach is to develop optimal (e.g., using dynamic programming) and suboptimal traversal strategies. The PIs have a long history of contributions to the analysis of medical images and tomography. The development of common theory and algorithms for visualization and analysis of images will be of use in many disciplines using multidimensional image data.