Abstract
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.
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References
Püschel M. and Rötteler M. (2007). Algebraic signal processing theory: 2-D hexagonal spatial lattice. IEEE Trans. Image Process. 16(6): 1506–1521
Püschel, M., Moura, J.M.F.: Algebraic signal processing theory: foundation and 1-D time. Part of [3]; IEEE Trans. Signal Process. (2008, to appear)
Püschel, M., Moura, J.M.F.: Algebraic signal processing theory. Available at http://arxiv.org/abs/cs.IT/0612077, parts of this manuscript are submitted as [2,4]
Püschel, M., Moura, J.M.F.: Algebraic signal processing theory: 1-D space. Part of [3]; IEEE Trans. Signal Process. (2008, to appear)
Rivlin T.J. (1974). The Chebyshev Polynomials. Wiley Interscience, New York
Koornwinder T. (1974). Orthogonal polynomials in two variables which are eigenfunctions of two algebraically independent partial differential operators (part III). Indag. Math. 36: 357–369
Püschel, M., Moura, J.M.F.: Algebraic signal processing theory: Cooley–Tukey type algorithms for DCTs and DSTs. IEEE Trans. Signal Process. (2008, to appear); a longer version is available at http://arxiv.org/abs/cs.IT/0702025
Püschel M. and Moura J.M.F. (2003). The algebraic approach to the discrete cosine and sine transforms and their fast algorithms. SIAM J. Comput. 32(5): 1280–1316
Püschel, M., Rötteler, M.: Cooley–Tukey FFT like algorithm for the discrete triangle transform. In: Proceedings of the 11th IEEE DSP Workshop, pp. 158–162 (2004)
Mersereau R.M. (1979). The processing of hexagonally sampled two-dimensional signals. Proc. IEEE 67(6): 930–949
Dudgeon D.E. and Mersereau R.M. (1984). Multidimensional Digital Signal Processing. Prentice-Hall, Englewood Cliffs
Middleton L. and Sivaswamy J. (2005). Hexagonal Image Processing. Springer, Heidelberg
Grigoryan A.M. (2002). Hexagonal discrete cosine transform for image coding. IEEE Trans. Signal Process. 50(6): 1438–1448
Nussbaumer H.J. (1982). Fast Fourier Transformation and Convolution Algorithms, 2nd edn. Springer, Heidelberg
Steidl G. and Tasche M. (1991). A polynomial approach to fast algorithms for discrete Fourier-cosine and Fourier-sine transforms. Math. Comput. 56(193): 281–296
Driscoll J.R., Healy D.M. Jr. and Rockmore D. (1997). Fast discrete polynomial transforms with applications to data analysis for distance transitive graphs. SIAM J. Comput. 26: 1066–1099
Potts D., Steidl G. and Tasche M. (1998). Fast algorithms for discrete polynomial transforms. Math. Comput. 67(224): 1577–1590
Curtis W.C. and Reiner I. (1962). Representation Theory of Finite Groups. Interscience, New York
Beth, Th.: Verfahren der Schnellen Fouriertransformation [Fast Fourier Transform Methods]. Teubner (1984)
Clausen, M.: Beiträge zum Entwurf schneller Spektraltransformationen (Habilitationsschrift), University of Karlsruhe (1988)
Clausen M. (1989). Fast generalized Fourier transforms. Theor. Comput. Sci. 67: 55–63
Clausen, M., Baum, U.: Fast Fourier Transforms. BI-Wiss.-Verl. (1993)
Maslen, D., Rockmore, D.: Generalized FFTs—a survey of some recent results. In: Proceedings of IMACS Workshop in Groups and Computation, vol. 28, pp. 182–238 (1995)
Rockmore D. (1990). Fast Fourier analysis for abelian group extensions. Adv. Appl. Math. 11: 164–204
Maslen, D., Rockmore, D.: Double coset decompositions and computational harmonic analysis on groups. J. Fourier Anal. Appl. 6(4), (2000)
Jacobson, N.: Basic Algebra I. W.H. Freeman and Co., San Francisco (1974)
Cox D., Little J. and O’Shea D. (1997). Ideals, Varieties, and Algorithms. Springer, Heidelberg
Becker Th. and Weispfenning V. (1993). Gröbner Bases. Springer, Heidelberg
Fuhrman P.A. (1996). A Polynomial Approach to Linear Algebra. Springer, New York
Ahmed N., Natarajan T. and Rao K.R. (1974). Discrete cosine transform. IEEE Trans. Comput. C-23: 90–93
Püschel, M., Rötteler, M.: The discrete triangle transform. Proc. Int. Conf. Acoust. Speech Process., vol. 3, pp. 45–48 (2004)
Voronenko, Y., Püschel, M.: Algebraic derivation of general radix Cooley–Tukey algorithms for the real discrete Fourier transform. Proc. Int. Conf. Acoust. Speech Signal Process., vol. 3, pp. 876–879 (2006)
Auslander L., Feig E. and Winograd S. (1984). Abelian semi-simple algebras and algorithms for the discrete Fourier transform. Adv. Appl. Math. 5: 31–55
Püschel M. (2002). Decomposing monomial representations of solvable groups. J. Symbolic Comput. 34(6): 561–596
Egner S. and Püschel M. (2001). Automatic generation of fast discrete signal transforms. IEEE Trans. Signal Process. 49(9): 1992–2002
Egner S. and Püschel M. (2004). Symmetry-based matrix factorization. J. Symbolic Comput. 37(2): 157–186
Chihara T.S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York
Eier R. and Lidl R. (1982). A class of orthogonal polynomials in k variables. Math. Ann. 260: 93–99
Ricci P.E. (1986). An iterative property of Chebyshev polynomials of the first kind in several variables. Rendiconti di Matematica e delle sue Applicazioni 6(4): 555–563
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This work was supported by NSF through awards 0310941 and 0634967.
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Püschel, M., Rtteler, M. Algebraic signal processing theory: Cooley–Tukey type algorithms on the 2-D hexagonal spatial lattice. AAECC 19, 259–292 (2008). https://doi.org/10.1007/s00200-008-0077-x
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DOI: https://doi.org/10.1007/s00200-008-0077-x