Hexagonal Tight Frame Filter Banks with Idealized High-pass Filters

This paper studies the construction of hexagonal tight wavelet frame filter banks which contain three “idealized” high-pass filters. These three high-pass filters are suitable spatial shifts and frequency modulations of the associated low-pass filter, and they are used by Simoncelli and Adelson in [37] for the design of hexagonal filter banks and by Riemenschneider and Shen in [30, 31] for the construction of 2-dimensional orthogonal filter banks. For an idealized low-pass filter, these three associated highpass filters separate high frequency components of a hexagonal image in 3 different directions in the frequency domain. In this paper we show that an idealized tight frame, a frame generated by a tight frame filter bank containing the “idealized” high-pass filters, has at least 7 frame generators. We provide an approach to construct such tight frames based on the method by Lai and St¨ockler in [24] to decompose non-negative trigonometric polynomials as the summations of the absolute squares of other trigonometric polynomials. In particular, we show that if the non-negative trigonometric polynomial associated with the low-pass filter p can be written as the summation of the absolute squares of other 3 or less than 3 trigonometric polynomials, then the idealized tight frame associated with p requires exact 7 frame generators. We also discuss the symmetry of frame filters. In addition, we present in this paper several examples, including that with the scaling functions to be the Courant element B111 and the box-spline B222. The tight frames constructed in this paper will have potential applications to hexagonal image processing.