Discrete generalized Fresnel functions and transforms in an arbitrary discrete basis

IEEE Transactions on Signal Processing

Volumn 54, Issue 11, 2006, Pages 4261-4270

  Aizenberg, Igor ^a   Astola, Jaakko T ^b  

^a TEXAS A AND M UNIVERSITY TEXARKANA   (United States)

^b TAMPERE UNIVERSITY OF TECHNOLOGY   (Finland)

Author keywords

Digital convolution Fresnel function; Fourier transform; Walsh transform

Indexed keywords

ALGEBRA; ALGORITHMS; CONVOLUTION; DISCRETE FOURIER TRANSFORMS; WALSH TRANSFORMS;

CONJUNCTIVE TRANSFORM; DIGITAL CONVOLUTION FRESNEL FUNCTION; GENERAL ALGEBRAIC APPROACH;

SIGNAL PROCESSING;

EID: 33750404022     PISSN: 1053587X     EISSN: None     Source Type: Journal    
DOI: 10.1109/TSP.2006.881189     Document Type: Article

References (36)

1. L. I. Bluestein, "A linear filtering approach to the computation of the discrete Fourier transform," in Northeast Electronics Res. Eng. Meeting Rec., 1968, vol. 10, pp. 218-219.
2. L. R. Rabiner, R. W. Schafer, and C. M. Rader, "The chirp z-transform algorithm and its application," Bell Syst. Tech. J., vol. 48, pp. 1249-1292, 1969.
3. D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications," SIAM Rev., vol. 33, pp. 389-404, 1991.
4. Bluestein's FFT algorithm Wikipedia [Online]. Available: http://en.wikipedia.org/wiki/Bluestein's_FFT_algorithm
5. J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett., vol. 11, no. 3, pp. 77-79, Aug. 1967.
6. L. P. Yaroslavsky and N. S. Merzlyakov, Methods of Digital Holography (in Russian). Moscow, Russia: Nauka, 1977.
7. L. P. Yaroslavsky and N. S. Merzlyakov, Methods of Digital Holography. New York: Consultants Bureau, 1980.
8. L. P. Yaroslavsky and N. S. Merzlyakov, Digital Holography (in Russian). Moscow, Russia: Nauka, 1982.
9. I. W. Goodman, Introduction to Fourier Optics, 2nd ed. New York: McGraw-Hill, 1996.
10. T. M. Kreis and W. P. O. Juptner, "Principles of digital holography," Hybrid Meas. Technol. Opt. Metrol., vol. 4, pp. 353-363, 1998.
11. E. Cuche, F. Bevilacqua, and C. Depeursinge, "Digital holography for quantitative phase-contrast imaging," Opt. Lett., vol. 24, no. 5, pp. 291-293, Mar. 1999.
12. M. Liebling, T. Blu, and M. Unser, "Fresnelets: New multiresolution wavelet bases for digital holography," IEEE Trans. Image Process., vol. 12, Jan. 2003.
13. C. Fournier, C. Ducottet, and T. Fournel, "Digital in-line holography: Influence of the reconstruction function on the axial profile of a reconstructed particle image," Meas. Sci. Technol., vol. 15, pp. 686-693, 2004.
14. W.-X. Cong, "Phase retrieval in the Fresnel transform system: A recursive algorithm," J. Opt. Soc. Amer. A., vol. 16, no. 7, pp. 1827-1830, 1999.
15. M. C. Parker, "Design of arrayed-waveguide gratings using hybrid Fourier-Fresnel transform techniques," IEEE J. Sel. Topics Quantum Electron., vol. 5, pp. 1379-1384, 1999.
16. A. B. Givental and T. E. Krenkel, "Number theoretical Fresnel transform and its application in multidimensional data arrays processing," in Digital Signal Processing and its Applications (in Russian), L. Yaroslavsky, Ed. Moscow, Russia: Nauka, 1981, pp. 23-32.
17. B. Hennelly and J. Sheridan, "Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms," J. Opt. Soc. Amer. A, vol. 22, no. 5, pp. 917-927, May 2005.
18. B. Hennelly and J. Sheridan, "Fast numerical algorithm for the linear canonical transform," J. Opt. Soc. Amer. A, vol. 22, no. 5, pp. 928-937, May 2005.
19. N. N. Aizenberg, I. N. Aizenberg, J. Astola, and K. Egiazarian, J. Astola, Ed., "An introduction to algebraic theory of discrete signals," in Proc. Int. Workshop Transforms Filter Banks, Tampere, Finland, Feb. 1998, pp. 70-94, TICSP Series 1.
20. N. N. Aizenberg, "Spectrum of the convolution of the discrete signals in an arbitrary basis," (in Russian) Rep. Acad. Sci. USSR (Doklady Akdemii Nauk SSSR), vol. 241, no. 3, pp. 551-554, 1978.
21. N. N. Aizenberg and O. T. Trofimluk, "Shift, convolution and correlation function of the discrete signals in an arbitrary basis", (in Russian) Rep. Acad. Sci. USSR (Doklady Akdemii Nauk SSSR), vol. 250, no. 1, pp. 47-51, 1980.
22. N. Ahmed and K. R. Rao, Orthogonal Transforms for Digital Signal Processing. Berlin, Germany: Springer-Verlag, 1975.
23. N. N. Aizenberg, V. D. Butakov, T. E. Krenkel, and Y. G. Harbash, "Fresnel functions and transforms for the linear nonsingular transforms," (in Russian) J. Commun. Technol. Electron. (Radiotechnika i Elektronika), vol. 29, no. 4, pp. 697-704, 1984.
24. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light. Amsterdam, The Netherlands: North-Holland, 1977.
25. I. J. Good, "The interaction algorithm and practical Fourier analysis," J. Roy. Stat. Soc. Ser. B, vol. 20, no. 2, pp. 372-375, 1958.
26. B. L. van der Warden, Algebra I. Auflage der Modernen Algebra. Berlin, Germany: Springer-Verlag, 1971.
27. P. J. Nicholson, "Algebraic theory of finite Fourier transforms," J. Comput. Syst. Sci., vol. 5, pp. 524-547, 1971.
28. A. Terras, Fourier Analysis on Finite Groups and Applications. Cambridge, U.K.: London Mathematical Society, 1999, vol. 43, Student Texts.
29. F. F. Bonsall and J. Duncan, Complete Normed Algebra. New York: Springer-Verlag, 1973.
30. D. E. Muller, "Application of Boolean algebra to switching circuit design and error detection," IRE Trans., vol. 1, pp. 6-12, 1954.
31. I. S. Reed, "Aclass of multiple-error-correcting codesand their decoding scheme," IRE Trans. Inf. Theory, vol. IT-3, pp. 6-12, 1954.
32. B. J. Falkowski, "A note on the polynomial form of Boolean functions and related topics," IEEE Trans. Comput., vol. 48, no. 8, pp. 860-864, Aug. 1999.
33. K. D. Heidtmann, "Arithmetic spectrum applied to fault detection for combinational networks," IEEE Trans. Comput., vol. 40, no. 3, pp. 320-324, 1991.
34. M. A. Thornton, D. M. Miller, and R. Drechsler, "Transformations amongst the Walsh, Haar, arithmetic and Reed-Muller spectral domains," in Int. Workshop Applicat. Reed-Muller Expansion Circuit Design (RMW), Aug. 10-11, 2001, pp. 215-225.
35. N. N. Aizenberg and O. T. Trofimluk, "The conjunctive transform of the discrete signals and its application for finding the tests and recognition of monotonicity of the Boolean functions," (in Russian) Kibernetika, no. 5, pp. 138-139, 1981.
36. N. N. Aizenberg, "Fast methods of solving the systems of Boolean equations and system of equations over the field of real numbers with the Boolean constraints," Kibernetika, no. 3, pp. 126-127, 1991.