SVD-Based Minimax Design of 2-D Digital Filters

Two-dimensional (2-D) digital filters have found useful in image processing and other 2-D signal processing fields. As compared with one-dimensional (1-D) digital filter design, 2-D filter is more difficult to design because it requires more complicated mathematical formulations. Also, as the order of 2-D filter increases, the 2-D filter complexity increases drastically. One of the most efficient ways to design a 2-D filter is to decompose the difficult 2-D filter design problem into a set of 1-D filter design problems, which are relatively easy to solve. Through solving the 1-D filter design problems, a 2-D filter can be indirectly obtained. Moreover, exploiting symmetries can further reduce the complexity of the resulting 2-D filter [1]. This paper presents a special structure for designing low-complexity two-dimensional (2-D) half-band digital filter with very sharp transition band. Performing the singular-value-decomposition (SVD) of zero-phase half-band frequency response generates a set of one-dimensional (1-D) frequency responses which can be approximated using the cascade of a half-advance digital filter and a set of 1-D filters. Each such a 1-D filter only requires a single multiplication to get an output sample, so the resulting whole 2-D half-band filter has considerably reduced computational complexity as compared with the existing direct design approach. To minimize the peak errors of the magnitude response of the resulting 2-D half-band filter, we parallelize one extra section to the structure and then minimize the peak errors using a nonlinear optimization method, which reduces the design problem to a minimax design. Since the whole structure only consists of a set of simple 1-D filters, the complexity is extremely low.