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Conversion and Trade-offs between Scaled Gaussian Parallel and Hierarchic Analysis Multirate Filter Banks

Abstract Scaled Gaussian analysis multirate filter banks (MRBs) are analysis MRBs whose filters are Gaussians scaled differently in width and height. They are frequently employed in image processing and visual systems modeling. We define generalized scaled Gaussian analysis MRBs for both parallel and hierarchic architectures and derive closed-form equations for conversions between them. The MRBs have arbitrary rational number changes in sampling rate between successive outputs, and arbitrary vertical and horizontal scaling of the Gaussian filters for each output. We calculate the number of multiplications, number of additions, and total number of filter coefficients to compare the parallel and hierarchic architectures as implemented with direct form and time-varying filters. In all cases, the parallel MRB requires considerably more multiplications, additions, and filter coefficients than the equivalent hierarchic MRB. However, the relative differences are far less severe in the time-varying case. We also derive a useful approximation for the discrete time Fourier transform of a Gaussian.