Symmetry in two-dimensional rectangularly-sampled digital filters

It is well known that exploiting the quadrantal, diagonal or octagonal symmetry existing in the 2-D filter frequency response results in a reduction in the design and implementation complexities. This paper introduces new symmetries that could possibly exist in the 2-D filter frequency response. The symmetries are presented in a general framework that encompasses all previously known and newly discovered symmetries. The commonly found cases such as displacement, rotation and reflection symmetries are discussed in some detail to present their geometric properties in the frequency-plane. An example is solved at the end to illustrate the significant savings resulting from exploiting those previously unknown symmetries.