Ky Fan norm application for video segmentation

Abstract

This article presents results of applying the Ky Fan norm in the context of solving the problem of video segmentation. Since the task of video analysis can be considered as analysis of the sequence of images, it was decided to find a way to formalize the description of the video frame using the mathematical apparatus of non-square matrices. When choosing a method, particular attention was paid precisely to universality with respect to the dimension of the initial data due to the technical characteristics and nature of the video data - video frames are matrices of arbitrary dimension. The ability to skip the step of matrix transformation to square dimension, or vectorization using some descriptor allows you to reduce computational costs required for this transformation. It was decided to use the value of the Ky Fan norm as an image descriptor, since it is built on top of matrix singular values. As it is known, singular values are calculated during the singular decomposition of the matrix and can be used, among other features, to reduce the dimension of the source data. A singular decomposition does not impose restrictions on either the dimension or the character of the elements of the original matrix. In addition, it can be used to derive other matrix decompositions with required characteristics. A comparative analysis of the effectiveness of the obtained descriptor was carried out in the case of using the k-norm and 1-norm, which showed that the 1-norm allows us to identify the most significant changes in the scene, while k -norm is able to detect minor. In other words, depending on the context of the source video data and the scope of the developed application, it is possible to configure the sensitivity of the application to a change in the scene by varying the number of singular values involved. The decision about the presence of changes in the context of video scene is made based on a comparison of descriptors of two consecutive images, that is, the values of the Ky Fan norm.