Table of Contents
Advances in Electrical Engineering
Volume 2014, Article ID 276241, 23 pages
http://dx.doi.org/10.1155/2014/276241
Review Article

Fast Transforms in Image Processing: Compression, Restoration, and Resampling

Department of Physical Electronics, School of Electrical Engineering, Tel Aviv University, 69978 Tel Aviv, Israel

Received 4 March 2014; Accepted 19 May 2014; Published 6 July 2014

Academic Editor: George E. Tsekouras

Copyright © 2014 Leonid P. Yaroslavsky. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. J. W. Cooley and J. W. Tukey, “An Algorithm for the Machine Calculation of Complex Fourier Series,” Mathematics of Computation, vol. 19, no. 90, pp. 297–301, 1965. View at Google Scholar
  2. H. F. Harmuth, Transmission of Information by Orthogonal Functions, Springer, Berlin, Germany, 1970.
  3. H. C. Andrews, Computer Techniques in Image Processing, Academic Press, New York, NY, USA, 1970.
  4. H. C. Andrews, “Two-dimensional Transforms,” in Picture Processing and Digital Filtering, T. S. Huang, Ed., Springer, New, York, NY, USA, 1975. View at Google Scholar
  5. N. Ahmed and K. R. Rao, Orthogonal Transforms for Digital Signal Processing, Springer, Berlin, Germany, 1975.
  6. W. K. Pratt, W.-H. Chen, and L. R. Welch, “Slant Transform Image Coding,” IEEE Transactions on Communications, vol. 22, no. 8, pp. 1075–1093, 1974. View at Google Scholar · View at Scopus
  7. N. Ahmed, N. Natarajan, and K. R. Rao, “Discrete cosine transform,” IEEE Transactions on Computers, vol. 23, no. 1, pp. 90–93, 1974. View at Google Scholar
  8. K. R. Rao and P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications, Academic Press, Boston, Mass, USA, 1990.
  9. I. Daubechies, “Where do wavelets come from? A personal point of view,” Proceedings of the IEEE, vol. 84, no. 4, pp. 510–513, 1996. View at Publisher · View at Google Scholar · View at Scopus
  10. S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Boston, Mass, USA, 3rd edition, 2008.
  11. H. Karhunen, Über lineare Methoden in der Wahrscheinlichkeitsrechnung, Series A.I, Annales Academiæ~Scientiarum Fennicæ, 1947.
  12. M. Loeve, “Fonctions aleatoires de seconde ordre,” in Processes Stochastiques et Movement Brownien, P. Levy, Ed., Hermann, Paris, France, 1948. View at Google Scholar
  13. H. Hotelling, “Analysis of a complex of statistical variables into principal components,” Journal of Educational Psychology, vol. 24, no. 6, pp. 417–441, 1933. View at Publisher · View at Google Scholar · View at Scopus
  14. T. P. Belokova, M. A. Kronrod, P. A. Chochia, and L. P. Yaroslavakii, “Digital processing of martian surface photographs from mars 4 and mars 5,” Space Research, vol. 13, no. 6, pp. 898–906, 1975. View at Google Scholar · View at Scopus
  15. L. P. Yaroslavsky, Digital Picture Processing: An Introduction, Springer, Berlin, Germany, 1985.
  16. L. Yaroslavsky, Digital Signal Processing in Optics and Holography, Radio i Svyaz', Moscow, Russia, 1987 (Russian).
  17. L. Yaroslavsky, Digital Holography and Digital Image Processing. Principles, Methods, Algorithms, Kluwer Academic Publishers, Boston, Mass, USA, 2004.
  18. L. Yaroslavsky, Theoretical Foundations of Digital Imaging Using Matlab, Taylor & Francis, Boca Raton, Fla, USA, 2013.
  19. L. Yaroslavsky, “Introduction to Digital Holography,” in Digital Signal Processing in Experimental Research, L. Yaroslavsky and J. Astola, Eds., vol. 1 of Bentham Series of e-books, Bentham, 2009, http://www.benthamscience.com/ebooks/9781608050796/index.htm. View at Google Scholar
  20. L. P. Yaroslavskii and N. S. Merzlyakov, Methods of Digital Holography, Consultance Bureau, New York, NY, USA, 1980.
  21. L. P. Jaroslavski, “Comments on FFT algorithm for both input and output pruning,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 29, no. 3, pp. 448–449, 1981. View at Google Scholar · View at Scopus
  22. J. Markel, “FFT pruning,” IEEE Trans Audio Electroacoust, vol. 19, no. 4, pp. 305–311, 1971. View at Google Scholar · View at Scopus
  23. H. V. Sorensen and C. S. Burrus, “Efficient computation of the DFT with only a subset of input or output points,” IEEE Transactions on Signal Processing, vol. 41, no. 3, pp. 1184–1200, 1993. View at Publisher · View at Google Scholar · View at Scopus
  24. R. G. Alves, P. L. Osorio, and M. N. S. Swamy, “General FFT pruning algorithm,” in Proceedings of the 43rd IEEE Midwest Symposium on Circuits and Systems, vol. 3, pp. 1192–1195, August 2000. View at Scopus
  25. L. Yaroslavsky, “Fast discrete sinc-interpolation: a gold standard for image resampling,” in Advances in Signal transforms: Theory and Applications, J. Astola and L. Yaroslavsky, Eds., vol. 7 of Eurasip Book Series on Signal Processing and Communications, pp. 337–405, 2007. View at Google Scholar · View at Scopus
  26. L. Yaroslavky, “Fast transforms in digital signal processing: theory, algorithms, applications,” in Digital Signal Processing in Experimental Research, L. Yaroslavsky and J. Astola, Eds., vol. 2 of Bentham E-book Series, 2011, http://www.benthamscience.com/ebooks/9781608052301/index.htm.
  27. R. Yu. Vitkus and L. P. Yaroslavsky, “Recursive Algorithms for Local Adaptive Linear Filtration,” in Mathematical Research. Computer Analysis of Images and Patterns, L. P. Yaroslavsky, A. Rosenfeld, and W. Wilhelmi, Eds., pp. 34–39, Academie Verlag, Berlin, Germany, 1987, Band 40. View at Google Scholar
  28. J. Xi and J. F. Chicharo, “Computing running DCT's and DST's based on their second-order shift properties,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 47, no. 5, pp. 779–783, 2000. View at Publisher · View at Google Scholar · View at Scopus
  29. E. Candès, “Compressive sampling,” in Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006.
  30. D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory, vol. 52, no. 4, pp. 1289–1306, 2006. View at Publisher · View at Google Scholar · View at Scopus
  31. E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Communications on Pure and Applied Mathematics, vol. 59, no. 8, pp. 1207–1223, 2006. View at Publisher · View at Google Scholar · View at Scopus
  32. E. J. Candes and M. B. Wakin, “An introduction to compressive sampling: A sensing/sampling paradigm that goes against the common knowledge in data acquisition,” IEEE Signal Processing Magazine, vol. 25, no. 2, pp. 21–30, 2008. View at Publisher · View at Google Scholar · View at Scopus
  33. R. G. Baraniuk, “Compressive sensing,” IEEE Signal Processing Magazine, vol. 24, no. 4, pp. 118–124, 2007. View at Publisher · View at Google Scholar · View at Scopus
  34. M. Elad, Spatse and Redundant Representations: From Theory to Applications in Signal and Image Processing, Springer, New York, NY, USA, 2010.
  35. R. M. Willet, R. F. Mareia, and J. M. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Optical Engineering, vol. 50, no. 7, Article ID 072601, 2011. View at Publisher · View at Google Scholar · View at Scopus
  36. “Compressive optical imaging: architectures and algorithms,” in Optical and Digital Image Processing. Fundamentals and Applications, G. Cristobal, P. Schelkens, and H. Thienpont, Eds., Wiley-VCH, New York, NY, USA, 2011.
  37. J. M. Whittaker, Interpolatory Function Theory, Cambridge University Press, Cambridge, UK, 1935.
  38. V. A. Kotelnikov, “on the carrying capacity of the ether and wire in telecommunications,” Material for the First All-Union Conference on Questions of Communication, Izd. Red. Upr. Svyazi RKKA, Moscow, Russia, 1933 (Russian).
  39. C. Shannon, “A mathematical theory of communication,” Bell System Technical Journal, vol. 27, no. 3, pp. 379–423, 623–656, 1948. View at Google Scholar
  40. T. S. Huang and O. J. Tretiak, Picture Bandwidth Compression, Gordon and Breach, New York, NY, USA, 1972.
  41. A. Habibi and P. Wintz, “Image coding by linear transformation and block quantization,” IEEE Transactions on Communications, vol. 19, no. 1, pp. 50–62, 1971. View at Google Scholar · View at Scopus
  42. T. A. Wintz, “Transform picture coding,” Proceedings of the IEEE, vol. 60, no. 7, pp. 809–823, 1972. View at Google Scholar
  43. W. B. Pennebaker and J. L. Mitchell, JPEG Still Image Data Compression Standard, Springer, Berlin, Germany, 3rd edition, 1993.
  44. I. E. G. Richardson, H.264 and MPEG-4 Video Compression, John Wiley & Sons, Chichester, UK, 2003.
  45. T. Wiegand, G. J. Sullivan, G. Bjøntegaard, and A. Luthra, “Overview of the H.264/AVC video coding standard,” IEEE Transactions on Circuits and Systems for Video Technology, vol. 13, no. 7, pp. 560–576, 2003. View at Publisher · View at Google Scholar · View at Scopus
  46. P. Schelkens, A. Skodras, and T. Ebrahimi, The JPEG, 2000 Suite, John Wiley & Sons, Chichester, UK, 2009.
  47. I. Pitas, Digital Video and Television, Ioannis Pitas, 2013.
  48. D. Vaisey and A. Gersho, “Variable block-size image coding, acoustics, speech, and signal processing,” in Proceedings of the IEEE International Conference on (ICASSP '87), vol. 12, pp. 1051–1054, 1987.
  49. W. K. Pratt, “Generalized wiener filtering computation techniques,” IEEE Transactions on Computers, vol. 21, pp. 636–641, 1972. View at Google Scholar
  50. H. C. Andrwes, “Digital computers and image processing,” Endeavour, vol. 31, no. 113, 1972. View at Google Scholar
  51. N. Wiener, The Interpolation, Extrapolation and Smoothing of Stationary Time Series, Wiley, New York, NY, USA, 1949.
  52. A. N. Kolmogorov, “Sur l'interpolation et extrapolation des suites stationnaires,” Comptes Rendus de l'Académie des Sciences, vol. 208, pp. 2043–2045, 1939. View at Google Scholar
  53. D. L. Donoho and J. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika, vol. 81, no. 3, pp. 425–455, 1994. View at Publisher · View at Google Scholar · View at Scopus
  54. D. L. Donoho, “De-noising by soft-thresholding,” IEEE Transactions on Information Theory, vol. 41, no. 3, pp. 613–627, 1995. View at Publisher · View at Google Scholar · View at Scopus
  55. L. Yaroslavsky, “Space variant and adaptive transform domain image and video restoration methods,” in Advances in Signal Transforms: Theory and Applications, J. Astola and L. Yaroslavsky, Eds., EURASIP Book Series on Signal Processing and Communications, 2007. View at Google Scholar
  56. L. Yaroslavsky, “Local criteria: a unified approach to local adaptive linear and rank filterson,” in Signal Recovery and Synthesis III, Technical Digest Series 15, 1989. View at Google Scholar
  57. M. D. Levine, Vision in Man and Machine, McGraw-Hill, New York, NY, USA, 1985.
  58. H. S. Malvar and D. H. Staelin, “LOT: transform coding without blocking effects,” IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. v, pp. 553–559, 1992. View at Google Scholar · View at Scopus
  59. R. Öktem, L. Yaroslavsky, K. Egiazarian, and J. Astola, “Transform domain approaches for image denoising,” Journal of Electronic Imaging, vol. 11, no. 2, pp. 149–156, 2002. View at Publisher · View at Google Scholar · View at Scopus
  60. L. P. Yaroslavsky, B. Fishbain, A. Shteinman, and S. Gepshtein, “Processing and fusion of thermal and video sequences for terrestrial long range observation systems,” in Proceedings of the 7th International Conference on Information Fusion (FUSION '04), pp. 848–855, Stockholm, Sweden, July 2004. View at Scopus
  61. A. Buades, B. Coll, and J. M. Morel, “A review of image denoising algorithms, with a new one,” Multiscale Modeling and Simulation, vol. 4, no. 2, pp. 490–530, 2005. View at Publisher · View at Google Scholar · View at Scopus
  62. A. Buades, B. Coll, and J.-M. Morel, “A non-local algorithm for image denoising,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR '05), pp. 60–65, June 2005. View at Publisher · View at Google Scholar · View at Scopus
  63. J. M. Morel and A. Buades Capo, Non Local Image Processing, http://www.farman.ens-cachan.fr/6_JeanMichel_Morel.pdf.
  64. S. E. Hecht and J. J. Vidal, “generation of ECG prototype waveforms by piecewise correlational averaging,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 2, no. 5, pp. 415–420, 1980. View at Google Scholar · View at Scopus
  65. K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Transactions on Image Processing, vol. 16, no. 8, pp. 2080–2095, 2007. View at Publisher · View at Google Scholar · View at Scopus
  66. L. Yaroslavsky and M. Eden, “Correlational accumulation as a method for signal restoration,” Signal Processing, vol. 39, no. 1-2, pp. 89–106, 1994. View at Google Scholar · View at Scopus
  67. V. Katkovnik, K. Egiazarian, and J. Astola, “Adaptive varying window methods in signal and image processing,” in Advances in Signal Transforms. Theory and Applications, J. Astola and L. Yaroslavsky, Eds., vol. 7 of Eurasip Book Series on Signal Processing and Communications, pp. 241–284, 2007. View at Google Scholar · View at Scopus
  68. D. Gabor, “Theory of Communication,” Journal of IEEE, vol. 93, pp. 429–457, 1946. View at Google Scholar
  69. E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Physical Review, vol. 40, no. 5, pp. 749–759, 1932. View at Publisher · View at Google Scholar · View at Scopus
  70. R. K. Potter, G. Koppand, and H. C. Green, Visible Speech, Van Nostrand, New York, NY, USA, 1947.
  71. D. L. Donoho, “Nonlinear wavelet methods for recovery of signals, densities, and spectra from indirect and noisy data,” Proceedings of Symposia in Applied Mathematics, American Mathematical Society, vol. 47, pp. 173–205, 1993. View at Google Scholar
  72. R. R. Coifman and D. L. Donoho, “Translation- Invariant De-Noising,” in Wavelets and Statistics, A. Antoniadis, Ed., vol. 103 of Lecture Notes in Statistics, pp. 125–150, Springer, Berlin, Germany, 1995. View at Google Scholar
  73. ftp://ftp.cis.upenn.edu/pub/eero/matlabPyrTools.tar.gz.
  74. B. Z. Shaick, L. Ridel, and L. Yaroslavsky, “A hybrid transform method for image denoising,” in Proceedings of the 10th European Signal Processing Conference (EUSIPCO '00), M. Gabbouj and P. Kuosmanen, Eds., Tampere, Finland, September 2000.
  75. M. Vetterli and J. Kovacevic, Wavelets and Sub-Band Coding, Prentice Hall, Englewood Cliffs, NJ, USA, 1995.
  76. J. Backus, The Acoustical Foundations of Music, Norton and Co., New York, NY, USA, 1969.
  77. M. Unser, A. Aldroubi, and M. Eden, “Polynomial spline signal approximations: Filter design and asymptotic equivalence with Shannon's sampling theorem,” IEEE Transactions on Information Theory, vol. 38, no. 1, pp. 95–103, 1992. View at Publisher · View at Google Scholar · View at Scopus
  78. A. Gotchev, “Image interpolation by optimized spline-based kernels,” in Advances in Signal transforms: Theory and Applications, J. Astola and L. Yaroslavsky, Eds., vol. 7 of Eurasip Book Series on Signal Processing and Communications, pp. 285–335, 2007. View at Google Scholar · View at Scopus
  79. L. P. Yaroslavsky, “Efficient algorithm for discrete sine interpolation,” Applied Optics, vol. 36, no. 2, pp. 460–463, 1997. View at Google Scholar · View at Scopus
  80. L. P. Yaroslavsky, “Shifted discrete fourier transforms,” in Digital Signal Processing, V. Cappellini and A. G. Constantinides, Eds., pp. 69–74, Avademic Press, London, UK, 1980. View at Google Scholar
  81. L. Yaroslavsky, “Boundary effect free and adaptive discrete signal sinc-interpolation algorithms for signal and image resampling,” Applied Optics, vol. 42, no. 20, pp. 4166–4175, 2003. View at Google Scholar · View at Scopus
  82. L. Bilevich and L. Yaroslavsky, “Fast DCT-based image convolution algorithms and application to image resampling and hologram reconstruction,” in Real-Time Image and Video Processing, Proceedings of SPIE, Brussels, Belgium, April 2010.
  83. L. Bilevich and L. Yaroslavsky, “Fast DCT-based algorithm for signal and image accurate scaling,” in Image Processing: Algorithms and Systems XI, vol. 8655 of Proceedings of SPIE, February 2013. View at Publisher · View at Google Scholar · View at Scopus
  84. L. P. Yaroslavsky, “Linking analog and digital image processing,” in Optical and Digital Image Processing. Fundamentals and Applications, G. Cristobal, P. Schelkens, and H. Thienpont, Eds., pp. 397–418, Wiley-VCH, New York, NY, USA, 2011. View at Google Scholar
  85. M. Unser, P. Thevenaz, and L. Yaroslavsky, “Convolution-based interpolation for fast, high-quality rotation of images,” IEEE Transactions on Image Processing, vol. 4, no. 10, pp. 1371–1381, 1995. View at Publisher · View at Google Scholar · View at Scopus
  86. P. Thévenaz, EPFL/STI/IOA/LIB, Bldg. +BM-Ecublens 4.137, Station 17, CH-1015 Lausanne VD, Switzerland, http://bigwww.epfl.ch/thevenaz/.
  87. A. Gotchev, “Spline and wavelet based techniques for signal and image processing,” Tampere University of Technology, Publication 429, Tampere, Finland, TTY-Paino 2003.
  88. http://www.eng.tau.ac.il/~yaro/Etudes/PDF/MonaLisa320x256CurvMirr.avi.
  89. L. Yaroslavsky, B. Fishbain, G. Shabat, and I. Ideses, “Superresolution in turbulent videos: making profit from damage,” Optics Letters, vol. 32, no. 21, pp. 3038–3040, 2007. View at Publisher · View at Google Scholar · View at Scopus
  90. B. Fishbain, I. A. Ideses, G. Shabat, B. G. Salomon, and L. P. Yaroslavsky, “Superresolution in color videos acquired through turbulent media,” Optics Letters, vol. 34, no. 5, pp. 587–589, 2009. View at Publisher · View at Google Scholar · View at Scopus
  91. B. Fishbain, L. P. Yaroslavsky, and I. A. Ideses, “Real-time stabilization of long range observation system turbulent video,” Journal of Real-Time Image Processing, vol. 2, no. 1, pp. 11–22, 2007. View at Publisher · View at Google Scholar · View at Scopus
  92. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes. the Art of Scientific Computing, Cambridge University Press, Cambridge, UK, 1987.
  93. J. H. Mathew and K. D. Fink, Numerical Methods Using MATLAB, Prentice-Hall, Englewood Cliffs, NJ, USA, 1999.
  94. L. P. Yaroslavsky, A. Moreno, and J. Campos, “Frequency responses and resolving power of numerical integration of sampled data,” Optics Express, vol. 13, no. 8, pp. 2892–2905, 2005. View at Publisher · View at Google Scholar · View at Scopus
  95. C. F. Gauss, “Nachclass: theoria interpolationis methodo nova tractata,” IEEE ASSP Magazine, vol. 1, no. 4, pp. 14–81, 1984. View at Google Scholar
  96. L. F. Yaroslavsky, G. Shabat, B. G. Salomon, I. A. Ideses, and B. Fishbain, “Nonuniform sampling, image recovery from sparse data and the discrete sampling theorem,” Journal of the Optical Society of America, vol. 26, no. 3, pp. 566–575, 2009. View at Publisher · View at Google Scholar · View at Scopus
  97. L. P. Yaroslavsky, “Image recovery from sparse samples, discrete sampling theorem, and sharply bounded band-limited discrete signals,” in Advances in Imaging and Electron Physics, P. Hawkes, Ed., vol. 167, chapter 5, pp. 295–331, Academic Press, 2011. View at Publisher · View at Google Scholar · View at Scopus
  98. R. W. Gerchberg and W. O. Saxton, “Practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik, vol. 35, no. 2, pp. 237–250, 1972. View at Google Scholar · View at Scopus
  99. A. Papoulis, “New algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans Circuits Syst, vol. 22, no. 9, pp. 735–742, 1975. View at Google Scholar · View at Scopus
  100. D. Gabor, “Theory of communication,” Journal of IEEE, vol. 93, pp. 429–457, 1946. View at Google Scholar