Algebraic signal processing theory: Cooley-Tukey type algorithms on the 2-D hexagonal spatial lattice. (English) Zbl 1151.65104

Summary: 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 R. M. Mersereau [The processing of hexagonally, Proc. IEEE 67, No. 6, 930–949 (1979)] introducing hexagonal discrete Fourier transforms.
In this paper we derive a general-radix algorithm for the DTT of an \(n \times n\) 2-D signal, focusing on the radix-\(2 \times 2\) 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 [cf. J. W. Cooley and J. W. Tukey, Math. Comput. 19, 297–301 (1965; Zbl 0127.09002)], which motivates the title of this paper.


65T50 Numerical methods for discrete and fast Fourier transforms
94A08 Image processing (compression, reconstruction, etc.) in information and communication theory


Zbl 0127.09002


Full Text: DOI


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