Algebraic signal processing theory: Cooley–Tukey type algorithms on the 2-D hexagonal spatial lattice - Püschel, Markus; Rtteler, Martin (Applicable algebra in engineering, communication and computing, 2008)
Recently, we introduced the framework for signal processing on a nonseparable 2-D hexagonal spatial lattice including the associated notion of Fourier transform called discrete triangle transform (DTT). Spatial means that the lattice is undirected in contrast to earlier work by Mersereau introducing hexagonal discrete Fourier transforms. In this paper we derive a general-radix algorithm for the DTT of an n × n 2-D signal, focusing on the radix-2 × 2 case. The runtime of the algorithm is O ( n 2 log( n )), which is the same as for commonly used separable 2-D transforms. The DTT algorithm derivation is based on the algebraic signal processing theory. This means that instead of manipulating transform coefficients, the algorithm is derived through a stepwise decomposition of its underlying polynomial algebra based on a general theorem that we introduce. The theorem shows that the obtained DTT algorithm is the precise equivalent of the well-known Cooley–Tukey fast Fourier transform, which motivates the title of this paper. It is with great sadness that the authors contribute this paper to this special issue in memory of their former PhD advisor Thomas Beth. Beth was an extremely versatile researcher with contributions in a wide range of disciplines. However, one pervading theme can be identified in all of his work: the belief that mathematics, and in particular abstract algebra, was the language and key to uncovering the structure in many real world problems. One testament to this vision is his seminal habilitation thesis on the theory of Fourier transform algorithms, which ingeniously connects one of the principal tools in signal processing with group theory to open up an entirely new field of research. The authors deeply regret that Beth’s untimely death prevented him from seeing the Algebraic Signal Processing Theory, a body of work, including the present paper, that develops an axiomatic approach to and generalization of signal processing based on the representation theory of algebras. The theory is a logical continuation of Beth’s ideas and would not exist without him and his influence as PhD advisor. The authors like to think that he would have approved of this work and wish to dedicate this paper to him and his memory.