Analytical Design of Two-Dimensional Filters and Applications in Biomedical Image Processing

The field of two-dimensional filters and their design methods has known a large development due to its importance in image processing (Lim, 1990; Lu & Antoniou, 1992). There are methods based on numerical optimization and also analytical methods relying on 1D prototypes. A commonly-used design technique for 2D filters is to start from a specified 1D prototype filter and transform its transfer function using various frequency mappings in order to obtain a 2D filter with a desired frequency response. These are essentially spectral transformations from s to z plane, followed by z to (z1, z2) mappings, approached in early papers (Chakrabarti & Mitra, 1977; Hirano & Aggarwal, 1978; Harn & Shenoi, 1986; Nie & Unbehauen, 1989). Generally these transformations conserve stability, so from 1D prototypes various stable recursive 2D filters can be obtained. The most common types are directional, fan-shaped, diamond-shaped and circular filters. Diamond filters are commonly used as anti-aliasing filters in the conversion between signals sampled on the rectangular sampling grid and the quincunx sampling grid. Various design methods for diamond-shaped filters were studied in (Tosic, 1997; Lim & Low, 1997; Low & Lim, 1998; Ito, 2010; Matei, 2010).