The homology groups of a topological space provide us with information about its connectivity and the number and type of holes in it. This type of information can find practical applications in describing the intrinsic structure of an image, as well as in identifying equivalence classes in collections of images. When computing homological characteristics, the existence and strength of the relationships between each pair of points in the topological space are studied. The practical use of this approach begins by building a topological space from the image, in which the computation of the homology groups can be carried out in a feasible time.

Once the homological properties are obtained, what follows is the task of translating such information into operations such as image segmentation. This work presents a technique for denoising persistent diagrams and reconstructing the shape of segmented objects using the remaining classes on the diagram. A case study for the segmentation of cell nuclei in histological images is used for demonstration purposes. With this approach: a) topological denoising is achieved by aggregating trivial classes on the persistence diagram, and b) a growing seed algorithm uses the information obtained during the construction of the persistence diagram for the reconstruction of the segmented cell structures.

拓扑空间的同调群为我们提供了有关其连通性以及其中孔的数量和类型的信息。这种类型的信息可以在描述图像的内在结构以及识别图像集合中的等价类方面找到实际应用。在计算同调特征时，研究拓扑空间中每对点之间关系的存在性和强度。这种方法的实际应用开始于从图像构建拓扑空间，其中可以在可行的时间内进行同源群的计算。

一旦获得了同源性，接下来就是将这些信息转化为图像分割等操作的任务。这项工作提出了一种技术，用于对持久图进行去噪并使用图中剩余的类重建分段对象的形状。组织学图像中细胞核分割的案例研究用于演示目的。使用这种方法：a) 拓扑去噪是通过在持久图上聚合琐碎的类来实现的，并且 b) 不断增长的种子算法使用在构建持久图期间获得的信息来重建分段的单元结构。