Multidimensional Nonsubsampled Hourglass Filter Banks: Geometry of Passband Support and Filter Design

Recently, the classical two-dimensional directional filter banks have been extended to higher dimensions. In this paper, we study one of the key components in this new construction, namely the multidimensional nonsubsampled hourglass filter banks. Starting with a rigorous analysis on the geometry of multidimensional hourglass-shaped passband supports, we propose a novel design for these filter banks in arbitrary dimensions, featuring perfect reconstruction and finite impulse response (FIR) filters. We analyze necessary and sufficient conditions for the resulting filters to achieve good frequency responses, and provide an optimal solution that satisfies these conditions using simplest filters. The proposed filter design technique is verified by a design example in 3-D.