Exploiting Data Topology and Manifolds in Medical Image Analysis

该研究将探讨拓扑学（数学科学）和医学图像分析（计算机科学）之间的接口。拓扑学和欧几里得几何是研究不同类型空间的数学的一部分。欧几里得几何是我们日常世界的一部分，其中物体保持不变，无论它们是否移动或旋转，长度，面积，体积，角度等概念都很重要，可以测量。拓扑是不同的，因为主要的重点是孔的数量，连通性，以及对象可能具有的不同部分的数量（从拓扑角度来看，茶杯和甜甜圈是相同的，因为一个可以变形为另一个而不引入额外的孔或部分）。一些拓扑不变测度已经被发展出来，它们倾向于集中在连通性和同调性测度上。流形是拓扑空间，它是局部欧几里得的（即球面是二维流形），并且可能配备有距离测度。流形和拓扑都可以用来描述一维数据，便于从二维数据中直观地表达和推断结论。然而，在医学成像领域中的许多数据具有较高的维度。当添加时间分量时，这对于3D体积（例如CT和MR身体扫描）或3D+T（即4D）数据是清楚的。与医学图像分析中使用的维度相比，这些仍然是低维情况，其中数据集中的区域可能必须由n个特征（即nD）表示，其中n>>4。在这种情况下，它是预期的数据分析的拓扑方法将提供真实的beneficies.To能够应用的拓扑原则，医学图像数据，我们需要能够表示为nD流形的数据。拓扑不变的措施将被开发和用于医学图像数据的分类和分割。此外，图像分辨率（规模）对这些拓扑方面的影响将进行研究。从应用的角度来看，这将集中在乳腺癌和前列腺癌的检测和分类。

The proposed research will investigate the interface between topology (mathematical sciences) and medical image analysis (computer science).Topological and Euclidean geometry are parts of mathematics that study different types of spaces. Euclidean geometry is part of our everyday world where objects stay the same regardless if they are moved or rotated and notions like length, area, volume, angle, etc. are important and can be measured. Topology is different as the main emphasis is on the number of holes, the connectedness, and the number of distinct parts an object might have (from a topological point a teacup and a doughnut are identical as one can be deformed into the other without introducing additional holes or parts). A number of topological invariant measures have been developed, which tend to concentrate on the connectivity and homology measures. Manifolds are topological space, which are locally Euclidean (i.e. the surface of a sphere is a 2D manifold) and possibly equipped with a measure of distance. Both manifolds and topology can be used to describe nD data.It is easy to visualize and infer conclusions from 2D data. However, a lot of the data in the medical imaging domain has a higher dimension. This is clear for 3D volumetric (e.g. CT and MR body scans) or for 3D+T (i.e. 4D) data when a time component is added. These are still low dimensional cases compared to the dimensionality used in medical image analysis, where a region in a dataset might have to be represented by n features (i.e. nD) where n>>4. In this case it is expected that a topological approach to data analysis will provide real benefits.To be able to apply the topological principles to medical image data we need to be able to represent the data as nD manifolds. Topological invariant measures will be developed and used for the classification and segmentation of medical image data. In addition, the effects of image resolution (scale) on these topological aspects will be investigated. From an applications point of view this will concentrate on the detection and classification of breast and prostate cancer.