Group theoretical methods in signal processing: learning similarities, transformations and invariants

摘要

As signal acquisition systems become increasingly interconnected, complex and diverse, new methods of data analysis and processing have become crucial. Often, the development of such methods benefits from learning inherent structure and functional equivalence present in the data due to the underlying generative mechanisms. The focus of this thesis is on developing efficient algorithms for identification and classification of such structure existing between such large data sets or within a data set. Groups are mathematical entities that allow us to study symmetries, invariance and equivalence. Group theory is therefore the natural choice for capturing structure in the data. This thesis presents novel group theoretical methods for statistical signal processing with applications to problems in computer vision, image processing, robotics and bioinformatics. Optimal filtering and transformation recovery methods are developed for isotropic signal fields in homogeneous spaces using non-commutative harmonic analysis techniques. These methods are specialized to the 3D rotation group acting on the unit sphere and applied to 3D surface smoothing, visual homing of robots and detecting embedded objects in cosmic microwave background data. Adaptive algorithms for tracking signal transformations are presented with application to beat tracking and online learning of rotations. These methods exploit Lie algebraic structure of groups. Invariant based methods are discussed with application to distributed correspondence-less curve matching in a camera network. Finally, a mapping-invariant statistical periodicity transform is presented for detecting latent repeats in genome sequences.