The Application Of Persistent Homology To Image Classification And Recognition

For the past few years, the corresponding technology of big data was one of the most popular research directions, not only in computer science but also in some other re-lated fields, such as physics and environmental ecology. And the persistent homology is an important method to analyze various sorts of big data. The previous works of image classification and recognition have some limitation for images of nonlinear transform, rotational transformation and projectional transformation. In this paper, we will discuss how the persistent homology can be applied to make useful contributions to the qual-itative analysis of various images. we will extract the topological invariant features of images and discuss whether can we regard them as the similarity criterion of images.In our work, we used the method of persistent homology to construct a series of simplicial complexes to approach the image space, and then we got a barcode with the homology information of these complexes. From this barcode, we can not only get the topological features of the image, but also get some geometrical characteristics. Accord-ing to the experiment of simple geometric graphics and natural images, we verified that the topological invariant features can be used to get the similarities of images and the differences of images based on the geometric features. In particular, by comparing the images under different angles, we analyzed the similarities and differences of their topo-logical features. The result was that the topological features can well identify the sim-ilarity between original image and images which has been deflected, rotated, inversed, etc..In order to analyze the the similarities and differences quantitatively, we con-structed several kinds of distances of the barcode and let the distance between barcodes indicate the distance between the corresponding images. By the contrastive experiment of simple geometric images, we discussed the advantages and disadvantages of these distances. And we verified that the features have a great advantage in the classification of the deformation of images. According to the persistent homology, the information in the middle of the barcode is the most important. In order to highlight the importance of this part information, we had weighted the distance. We found such a distance was better to distinguish the images which have similar topological structure. It would lay a foundation for subsequent analyzing complex natural images.