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Calculation Scheme Based on a Weighted Primitive: Application to Image Processing Transforms

Abstract

This paper presents a method to improve the calculation of functions which specially demand a great amount of computing resources. The method is based on the choice of a weighted primitive which enables the calculation of function values under the scope of a recursive operation. When tackling the design level, the method shows suitable for developing a processor which achieves a satisfying trade-off between time delay, area costs, and stability. The method is particularly suitable for the mathematical transforms used in signal processing applications. A generic calculation scheme is developed for the discrete fast Fourier transform (DFT) and then applied to other integral transforms such as the discrete Hartley transform (DHT), the discrete cosine transform (DCT), and the discrete sine transform (DST). Some comparisons with other well-known proposals are also provided.

References

  1. 1.

    Chamberlain R, Lord E, Shand DJ: Real-time 2D floating-point fast Fourier transforms for seeker simulation. In Technologies for Synthetic Environments: Hardware-in-the-Loop Testing VII, July 2002, Orlando, Fla, USA, Proceedings of SPIE Edited by: Murrer RL Jr.. 4717: 15–23.

    Google Scholar 

  2. 2.

    Yan P, Mo YL, Liu H: Image restoration based on the discrete fraction Fourier transform. In Image Matching and Analysis, September 2001, Wuhan, China, Proceedings of SPIE Edited by: Bhanu B, Shen J, Zhang T. 4552: 280–285.

    Google Scholar 

  3. 3.

    Rabadi WA, Myler HR, Weeks AR: Iterative multiresolution algorithm for image reconstruction from the magnitude of its Fourier transform. Optical Engineering 1996,35(4):1015-1024. 10.1117/1.600718

    Article  Google Scholar 

  4. 4.

    Chang C-H, Wang C-L, Chang Y-T: Efficient VLSI architectures for fast computation of the discrete Fourier transform and its inverse. IEEE Transactions on Signal Processing 2000,48(11):3206-3216. 10.1109/78.875476

    MATH  Article  Google Scholar 

  5. 5.

    Hsiao S-F, Shiue W-R: Design of low-cost and high-throughput linear arrays for DFT computations: algorithms, architectures, and implementations. IEEE Transactions on Circuits and Systems II 2000,47(11):1188-1203. 10.1109/82.885127

    Article  Google Scholar 

  6. 6.

    Cooley JW, Tukey JW: An algorithm for the machine calculation of complex Fourier series. Mathematics of Computation 1965,19(90):297-301. 10.1090/S0025-5718-1965-0178586-1

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Swarztrauber PN: Multiprocessor FFTs. Parallel Computing 1987,5(1-2):197-210. 10.1016/0167-8191(87)90018-4

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Temperton C: Self-sorting in-place fast Fourier transforms. SIAM Journal on Scientific and Statistical Computing 1991,12(4):808-823. 10.1137/0912043

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    Pease MC: An adaptation of the fast Fourier transform for parallel processing. Journal of the ACM 1968,15(2):252-264. 10.1145/321450.321457

    MATH  Article  Google Scholar 

  10. 10.

    Hope LL: A fast Gaussian method for Fourier transform evaluation. Proceedings of the IEEE 1975,63(9):1353-1354.

    Article  Google Scholar 

  11. 11.

    Wang C-L, Chang C-H: A DHT-based FFT/IFFT processor for VDSL transceivers. Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '01), May 2001, Salt Lake, Utah, USA 2: 1213–1216.

    Google Scholar 

  12. 12.

    Fang W-H, Wu M-L: An efficient unified systolic architecture for the computation of discrete trigonometric transforms. Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS '97), June 1997, Hong Kong 3: 2092–2095.

    Google Scholar 

  13. 13.

    Chan E, Panchanathan S: A VLSI architecture for DFT. Proceedings of the 36th Midwest Symposium on Circuits and Systems, August 1993, Detroit, Mich, USA 1: 292–295.

    Article  Google Scholar 

  14. 14.

    Hartley RVL: A more symmetrical Fourier analysis applied to transmission problems. Proceedings of the IRE 1942,30(3):144-150.

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Bracewell RN: Discrete Hartley transform. Journal of the Optical Society of America 1983,73(12):1832-1835. 10.1364/JOSA.73.001832

    Article  Google Scholar 

  16. 16.

    Bracewell RN: The fast Hartley transform. Proceedings of the IEEE 1984,72(8):1010-1018.

    Article  Google Scholar 

  17. 17.

    Bracewell RN: The Hartley Transform. Oxford University Press, New York, NY, USA; 1986.

    Google Scholar 

  18. 18.

    Bracewell RN: Computing with the Hartley transform. Computers in Physics 1995,9(4):373-379. 10.1063/1.168534

    Article  Google Scholar 

  19. 19.

    Sorensen HV, Jones DL, Heideman MT, Burrus CS: Real-valued fast Fourier transfer algorithms. IEEE Transactions on Acoustics, Speech, and Signal Processing 1987,35(6):849-863. 10.1109/TASSP.1987.1165220

    Article  Google Scholar 

  20. 20.

    Duhamel P, Vetterli M: Improved Fourier and Hartley transform algorithms: application to cyclic convolution of real data. IEEE Transactions on Acoustics, Speech, and Signal Processing 1987,35(6):818-824. 10.1109/TASSP.1987.1165218

    Article  Google Scholar 

  21. 21.

    Popović M, Šević D: A new look at the comparison of the fast Hartley and Fourier transforms. IEEE Transactions on Signal Processing 1994,42(8):2178-2182. 10.1109/78.301854

    Article  Google Scholar 

  22. 22.

    Frigo M, Johnson SG: The design and implementation of FFTW3. Proceedings of the IEEE 2005,93(2):216-231.

    Article  Google Scholar 

  23. 23.

    Arico A, Serra-Capizzano S, Tasche M: Fast and numerically stable algorithms for discrete Hartley transforms and applications to preconditioning. Communications in Information Systems 2005,5(1):21-68.

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Rao KR, Yip P: Discrete Cosine Transform: Algorithms, Advantages, Applications. Academic Press, Boston, Mass, USA; 1990.

    Google Scholar 

  25. 25.

    Martucci SA: Symmetric convolution and the discrete sine and cosine transforms. IEEE Transactions on Signal Processing 1994,42(5):1038-1051. 10.1109/78.295213

    Article  Google Scholar 

  26. 26.

    Pennebaker WB, Mitchell JL: JPEG Still Image Data Compression Standard. Van Nostrand Reinhold, New York, NY, USA; 1993.

    Google Scholar 

  27. 27.

    Shi YQ, Sun H: Image and Video Compression for Multimedia Engineering. CRC Press, Boca Raton, Fla, USA; 2000.

    Google Scholar 

  28. 28.

    Duhamel P, Guillemot C: Polynomial transform computation of the 2-D DCT. Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '90), April 1990, Albuquerque, NM, USA 3: 1515–1518.

    Google Scholar 

  29. 29.

    Feig E, Winograd S: Fast algorithms for the discrete cosine transform. IEEE Transactions on Signal Processing 1992,40(9):2174-2193. 10.1109/78.157218

    MATH  Article  Google Scholar 

  30. 30.

    Hung AC, Meng TH-Y: A comparison of fast inverse discrete cosine transform algorithms. Multimedia Systems 1994,2(5):204-217. 10.1007/BF01215398

    Article  Google Scholar 

  31. 31.

    Duhamel P, Vetterli M: Fast Fourier transforms: a tutorial review and a state of the art. Signal Processing 1990,19(4):259-299. 10.1016/0165-1684(90)90158-U

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Serra-Capizzano S: A note on antireflective boundary conditions and fast deblurring models. SIAM Journal on Scientific Computing 2003,25(4):1307-1325.

    MathSciNet  MATH  Article  Google Scholar 

  33. 33.

    Ercegovac M, Lang T: Division and Square Root: Digit-Recurrence, Algorithms and Implementations. Klüwer Academic Publishers, Boston, Mass, USA; 1994.

    Google Scholar 

  34. 34.

    Piñeiro J-A, Ercegovac MD, Bruguera JD: High-radix logarithm with selection by rounding. Proceedings of the 13th IEEE International Conference on Application-Specific Systems, Architectures and Processors (ASAP '02), July 2002, San Jose, Calif, USA 101–110.

    Google Scholar 

  35. 35.

    García Chamizo JM, Signes Pont MT, Mora Mora H, de Miguel Casado G: Parametrizable architecture for function recursive evaluation. Proceedings of the 18th Conference on Design of Circuits and Integrated Systems (DCIS '03), November 2003, Ciudad Real, Spain

    Google Scholar 

  36. 36.

    Chien-Chang L, Chih-Da Ch, Guo JI: A parameterized hardware design for the variable length discrete Fourier transform. Proceedings of the 15th International Conference on VLSI Design (VLSID '02), August 2002, Taiwan, China

    Google Scholar 

  37. 37.

    Chang LW, Chen MY: A new systolic array for discrete Fourier transform. IEEE Transactions on Acoustics, Speech, and Signal Processing 1988,36(10):1665-1666. 10.1109/29.7554

    MATH  Article  Google Scholar 

  38. 38.

    Fang W-H, Wu M-L: An efficient unified systolic architecture for the computation of discrete trigonometric transforms. Proceedings of IEEE International Symposium on Circuits and Systems (ISCAS '97), June 1997, Hong Kong 3: 2092–2095.

    Google Scholar 

  39. 39.

    Murthy NR, Swamy MNS: On the real-time computation of DFT and DCT through systolic architectures. IEEE Transactions on Signal Processing 1994,42(4):988-991. 10.1109/78.285671

    Article  Google Scholar 

  40. 40.

    Chang T-S, Guo J-I, Jen C-W: Hardware-efficient DFT designs with cyclic convolution and subexpression sharing. IEEE Transactions on Circuits and Systems II 2000,47(9):886-892. 10.1109/82.868456

    Article  Google Scholar 

  41. 41.

    Chen C-H, Liu B-D, Yang J-F:Direct recursive structures for computing radix- two-dimensional DCT/IDCT/DST/IDST. IEEE Transactions on Circuits and Systems I: Regular Papers 2004,51(10):2017-2030. 10.1109/TCSI.2004.835685

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Cho NI, Lee SU:A fast DCT algorithm for the recursive 2-D DCT. IEEE Transactions on Signal Processing 1992,40(9):2166-2173. 10.1109/78.157217

    MATH  Google Scholar 

  43. 43.

    Wang JL, Wu CB, Liu D-B, Yang J-F: Recursive architecture for realizing modified discrete cosine transform and its inverse. Proceedings of IEEE Workshop on Signal Processing Systems (SIPS '99), October 1999, Taipei, Taiwan 120–130.

    Google Scholar 

  44. 44.

    Chau L-P, Siu W-C: Recursive algorithm for the discrete cosine transform with general lengths. Electronics Letters 1994,30(3):197-198. 10.1049/el:19940182

    Article  Google Scholar 

  45. 45.

    Wang Z, Jullien GA, Miller WC: Recursive algorithms for the forward and inverse discrete cosine transform with arbitrary length. IEEE Signal Processing Letters 1994,1(7):101-102. 10.1109/97.311803

    Article  Google Scholar 

  46. 46.

    Chan Y-H, Chau L-P, Siu W-C: Efficient implementation of discrete cosine transform using recursive filter structure. IEEE Transactions on Circuits and Systems for Video Technology 1994,4(6):550-552. 10.1109/76.340198

    Article  Google Scholar 

  47. 47.

    Yang J-F, Fan C-P: Compact recursive structures for discrete cosine transform. IEEE Transactions on Circuits and Systems II 2000,47(4):314-321. 10.1109/82.839667

    MATH  Article  Google Scholar 

  48. 48.

    Kober V, Cristobal G: Fast recursive algorithms for short-time discrete cosine transform. Electronics Letters 1999,35(15):1236-1238. 10.1049/el:19990877

    Article  Google Scholar 

  49. 49.

    Aburdene MF, Zheng J, Kosick RJ: Computation of discrete cosine transform using Clenshaw's recurrence formula. IEEE Signal Processing Letters 1995,2(8):155-156. 10.1109/97.404131

    Article  Google Scholar 

  50. 50.

    Cho NI, Lee SU: Fast algorithm and implementation of 2-D discrete cosine transform. IEEE Transactions on Circuits and Systems 1991,38(3):297-305. 10.1109/31.101322

    Article  Google Scholar 

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Correspondence to María Teresa Signes Pont.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://doi.org/creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Signes Pont, M., García Chamizo, J., Mora Mora, H. et al. Calculation Scheme Based on a Weighted Primitive: Application to Image Processing Transforms. EURASIP J. Adv. Signal Process. 2007, 045321 (2007). https://doi.org/10.1155/2007/45321

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Keywords

  • Fourier Transform
  • Time Delay
  • Information Technology
  • Signal Processing
  • Sine