Preservation of 2-D signal symmetries in Quincunx filter-banks

An important problem in the analysis of symmetric extension methods is to determine the conditions under which signal symmetries are preserved by the filtering (convolution) and downsampling operations. In this letter, we provide a complete characterization of four-fold two-dimensional signal symmetries viz. quadrantal symmetry, diagonal symmetry, and 90 degrees rotational symmetry. We then consider Quincunx filter-banks and determine the conditions under which the four-fold signal symmetries are preserved by the filtering and downsampling operations.