Texture recognition by the methods of topological data analysis

High spatial resolution satellite images are different from Gaussian statistics of counts. Therefore, texture recognition methods based on variances become ineffective. The aim of this paper is to study possibilities of completely different, topological approach to problems of structures classification. Persistent Betti numbers are signs of texture recognition. They are not associated with metrics and received directly fromdata in form of so-called persistence diagram (PD). The different structures built on PD are used to get convenient numerical statistics. At the present time, three of such objects are known: topological landscapes, persistent images and rank functions. They have been introduced recently and appeared as an attempt to vectorize PD. Typically, each of the proposed structures was illustrated by the authors with simple examples.However, the practical application of these approaches to large data sets requires to evaluate their efficiency within the frame of the selected task at the same standard database. In our case, such a task is to recognize different textures of the Remote Sensing Data (RSD). We check efficiency of structure, called persistent images in this work. We calculate PD for base containing 800 images of high resolution representing 20 texture classes. We have found out that average efficiency of separate image recognition in the classes is 84%, and in 11 classes, it is not less than 90%. By comparison topological landscapes provide 68% for average efficiency, and only 3 classes of not less than 90%. Reached conclusions are of interest for new methods of texture recognition in RSD.

Estimating Multidimensional Persistent Homology Through a Finite Sampling

An exact computation of the persistent Betti numbers of a submanifold [Formula: see text] of a Euclidean space is possible only in a theoretical setting. In practical situations, only a finite sample of [Formula: see text] is available. We show that, under suitable density conditions, it is possible to estimate the multidimensional persistent Betti numbers of [Formula: see text] from the ones of a union of balls centered on the sample points; this even yields the exact value in restricted areas of the domain. Using these inequalities we improve a previous lower bound for the natural pseudodistance to assess dissimilarity between the shapes of two objects from a sampling of them. Similar inequalities are proved for the multidimensional persistent Betti numbers of the ball union and the one of a combinatorial description of it.