Applications of Algebraic Topology to Video Compression
In this study, we applied Algebraic Topology techniques to extract information about the shape of data and applied these insights to the research problem of video compression. Specifically, we applied a computational tool known as Persistent Homology to point-cloud data sets and extracted insights on the data from the induced barcodes. We generalized results from a study of the local behavior of spaces of natural images by Carlsson et al  to the study of videos. To this end, we considered an ambient space of 81-dimensional points containing arrangements of 3 x 3 patches of pixels extracted from the frames within a video. We developed a computational model for the high-contrast dense sub-manifolds of this point cloud and found that these sub-manifolds have the topological properties of a connected bouquet of spheres. The reduction of dimension to a bouquet of spheres could have potential applications for video compression.