Multidimensional Nonseparable Oversampled Lapped Transforms: Theory and Design
Abstract: This paper extends the theory of the one one-dimensional dimensional oversampled linear-phase linear perfect reconstruction filter banks (OLPPRFBs) developed by Gan et al. to multidimensional (MD) cases and proposes MD nonseparable oversampled lapped transforms (NSOLTs). NSOLTs allow us to achieve an overcomplete analysis-synthesis synthesis system with nonseparable, symmetric, real-valued, real overlapping, and compact-supported supported ffilters. ilters. The proposed systems are based on lattice structures and the redundancy is flexibly controlled by the number of channels and downsampling ratio. The proposed structure is shown to be capable of constructing Parseval tight frames in any number of di dimensions. mensions. The number of design parameters are examined under the Parseval tight frame constraint. In order to design NSOLTs specified for sparse approximation of image or volume data, an example-based based design procedure is introduced. The effectiveness of this th method is verified by examining design samples and evaluating their sparse approximation performance using the iterative hard thresholding algorithm for a natural image and MRI volume data patch.