Abstract

The problem of digital signal and image resampling with discrete sinc interpolation is addressed. Discrete sinc interpolation is theoretically the best one among the digital convolution-based signal resampling methods because it does not distort the signal as defined by its samples and is completely reversible. However, sinc interpolation is frequently not considered in applications because it suffers from boundary effects, tends to produce signal oscillations at the image edges, and has relatively high computational complexity when irregular signal resampling is required. A solution that enables the elimination of these limitations of the discrete sinc interpolation is suggested. Two flexible and computationally efficient algorithms for boundary effects free and adaptive discrete sinc interpolation are presented: frame-wise (global) sinc interpolation in the discrete cosine transform (DCT) domain and local adaptive sinc interpolation in the DCT domain of a sliding window. The latter offers options not available with other interpolation methods: interpolation with simultaneous signal restoration/enhancement and adaptive interpolation with super resolution.

© 2003 Optical Society of America

Full Article  |  PDF Article

Errata

L. Yaroslavsky, "Boundary effect free and adaptive discrete signal sinc-interpolation algorithms for signal and image resampling: erratum," Appl. Opt. 42, 6495-6495 (2003)
https://www.osapublishing.org/ao/abstract.cfm?uri=ao-42-32-6495

References

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  1. L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  2. D. Fraser, “Interpolation by the FFT Revisited—An Experimental Investigation,” IEEE Trans. Acoust. Speech Signal Process. ASSP-37, 665–675 (1989).
    [CrossRef]
  3. T. Smith, M. S. Smith, S. T. Nichols, “Efficient sinc function interpolation technique for center padded data,” IEEE Trans. Acoust. Speech Signal Process. ASSP-38, 1512–1517 (1990).
    [CrossRef]
  4. L. Yaroslavsky, “Efficient algorithm for discrete sinc interpolation,” Appl. Opt. 36, 460–463 (1997).
    [CrossRef] [PubMed]
  5. M. Unser, P. Thevenaz, L. Yaroslavsky, “Convolution-based interpolation for fast, high-quality rotation of images,” IEEE Trans. Image Process. 4, 1371–1382 (1995).
    [CrossRef] [PubMed]
  6. L. Yaroslavsky, Digital Picture Processing: An Introduction (Springer-Verlag, Berlin, 1985).
    [CrossRef]
  7. L. Yaroslavsky, M. Eden, Fundamentals of Digital Optics (Birkhäuser, Boston, 1996).
    [CrossRef]
  8. Z. Wang, “Fast algorithms for the discrete W transform and for the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 803–816 (1984).
    [CrossRef]
  9. H. S. Hou, “A fast recursive algorithm for computing the discrete cosine transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 1455–1461 (1987).
  10. Z. Wang, “A simple structured algorithm for the DCT,” in Proc. 3rd Ann. Conf. Signal Process. Xi’an, China, Nov 1988, pp. 28–31 (in Chinese).
  11. A. Gupta, K. R. Rao, “A fast recursive algorithm for the discrete sine transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-38, 553–557 (1990).
    [CrossRef]
  12. Z. Wang, “Pruning the Fast Discrete Cosine Transform,” IEEE Trans. Commun. 39, 640–643 (1991).
    [CrossRef]
  13. Z. Cvetkovic, M. V. Popovic, “New fast recursive algorithms for the computation of discrete cosine and sine transforms,” IEEE Trans. Signal Process. 40, 2083–2086 (1992).
    [CrossRef]
  14. J. Ch. Yao, Ch-Y. Hsu, “Further results on “New fast recursive algorithms for the discrete cosine and sine transforms,” IEEE Trans. Signal Process. 42, 3254–3255 (1994).
    [CrossRef]
  15. L. Yaroslavsky, A. Happonen, Y. Katyi, “Signal discrete sinc-interpolation in DCT domain: fast algorithms,” SMMSP 2002, Second International Workshop on Spectral Methods and Multirate Signal Processing, Toulouse (France), 07.09.2002–08.09.2002.
  16. M. Unser, “Splines: A perfect fit for signal and image processing,” IEEE Trans. Signal Process. 16, 22–38 (1999).
    [CrossRef]
  17. Ph. Thevenaz, Th. Blu, M. Unser, “Interpolation Revisited,” IEEE Trans. Med. Imaging MI-19, 739–758 (2000).
    [CrossRef]
  18. L. Yaroslavsky, “Image restoration, enhancement, and target location with local adaptive filters,” in International Trends in Optics and Photonics, ICOIV, T. Asakura, ed., (Springer-Verlag, Berlin, 1999), pp. 111–127.
    [CrossRef]
  19. L. P. Yaroslavsky, K. O. Egiazarian, J. T. Astola, “Transform domain image restoration methods: review, comparison, and interpretation,” in Photonics West 2001: Electronic Imaging Nonlinear Processing and Pattern Analysis, Proc. SPIE4304, 155–170 (2001).
  20. R. Yu. Vitkus, L. P. Yaroslavsky, “Recursive Algorithms for Local Adaptive Linear Filtration,” in Mathematical Research, Computer Analysis of Images and Patterns, L. P. Yaroslavsky, A. Rosenfeld, W. Wilhelmi, eds. Band 40, (Academie Verlag, Berlin, 1987), pp. 34–39.
  21. N. Rama Murthy, N. S. Swamy, “On computation of running discrete cosine and sine transform,” IEEE Trans. Signal Process. 40, 1430–1437 (1992).
    [CrossRef]
  22. K. J. R. Liu, C. T. Chiu, R. K. Kolagotla, J. F. Jaja, “Optimal unified architectures for the real-time computation of time-recursive discrete sinusoidal transforms,” IEEE Trans. Circuits and Systems for Video Technology, Vol. 4, No. 2, April1994.
  23. J. A. R. Macias, A. Exposito, “Recursive Formulation of Short-Time Discrete Trigonometric Transforms,” IEEE Transactions Circuits Syst. II Analog and Digital Signal Processing, 45, 525–527 (1998).
    [CrossRef]
  24. V. Kober, G. Cristobal, “Fast recursive algorithms for short-time discrete cosine transform,” Electron. Lett. 35, 1236–1238 (1999).
    [CrossRef]
  25. J. Xi, J. F. Chicharo, “Computing running DCT’s and DST’s based on their second order shift properties,” IEEE Trans. Circuits Syst. - I, Fundamental Theory and Applications, 47, 779–783 (2000).
    [CrossRef]

2000 (2)

Ph. Thevenaz, Th. Blu, M. Unser, “Interpolation Revisited,” IEEE Trans. Med. Imaging MI-19, 739–758 (2000).
[CrossRef]

J. Xi, J. F. Chicharo, “Computing running DCT’s and DST’s based on their second order shift properties,” IEEE Trans. Circuits Syst. - I, Fundamental Theory and Applications, 47, 779–783 (2000).
[CrossRef]

1999 (2)

V. Kober, G. Cristobal, “Fast recursive algorithms for short-time discrete cosine transform,” Electron. Lett. 35, 1236–1238 (1999).
[CrossRef]

M. Unser, “Splines: A perfect fit for signal and image processing,” IEEE Trans. Signal Process. 16, 22–38 (1999).
[CrossRef]

1998 (1)

J. A. R. Macias, A. Exposito, “Recursive Formulation of Short-Time Discrete Trigonometric Transforms,” IEEE Transactions Circuits Syst. II Analog and Digital Signal Processing, 45, 525–527 (1998).
[CrossRef]

1997 (1)

1995 (1)

M. Unser, P. Thevenaz, L. Yaroslavsky, “Convolution-based interpolation for fast, high-quality rotation of images,” IEEE Trans. Image Process. 4, 1371–1382 (1995).
[CrossRef] [PubMed]

1994 (1)

J. Ch. Yao, Ch-Y. Hsu, “Further results on “New fast recursive algorithms for the discrete cosine and sine transforms,” IEEE Trans. Signal Process. 42, 3254–3255 (1994).
[CrossRef]

1992 (2)

N. Rama Murthy, N. S. Swamy, “On computation of running discrete cosine and sine transform,” IEEE Trans. Signal Process. 40, 1430–1437 (1992).
[CrossRef]

Z. Cvetkovic, M. V. Popovic, “New fast recursive algorithms for the computation of discrete cosine and sine transforms,” IEEE Trans. Signal Process. 40, 2083–2086 (1992).
[CrossRef]

1991 (1)

Z. Wang, “Pruning the Fast Discrete Cosine Transform,” IEEE Trans. Commun. 39, 640–643 (1991).
[CrossRef]

1990 (2)

A. Gupta, K. R. Rao, “A fast recursive algorithm for the discrete sine transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-38, 553–557 (1990).
[CrossRef]

T. Smith, M. S. Smith, S. T. Nichols, “Efficient sinc function interpolation technique for center padded data,” IEEE Trans. Acoust. Speech Signal Process. ASSP-38, 1512–1517 (1990).
[CrossRef]

1989 (1)

D. Fraser, “Interpolation by the FFT Revisited—An Experimental Investigation,” IEEE Trans. Acoust. Speech Signal Process. ASSP-37, 665–675 (1989).
[CrossRef]

1987 (1)

H. S. Hou, “A fast recursive algorithm for computing the discrete cosine transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 1455–1461 (1987).

1984 (1)

Z. Wang, “Fast algorithms for the discrete W transform and for the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 803–816 (1984).
[CrossRef]

Astola, J. T.

L. P. Yaroslavsky, K. O. Egiazarian, J. T. Astola, “Transform domain image restoration methods: review, comparison, and interpretation,” in Photonics West 2001: Electronic Imaging Nonlinear Processing and Pattern Analysis, Proc. SPIE4304, 155–170 (2001).

Blu, Th.

Ph. Thevenaz, Th. Blu, M. Unser, “Interpolation Revisited,” IEEE Trans. Med. Imaging MI-19, 739–758 (2000).
[CrossRef]

Chicharo, J. F.

J. Xi, J. F. Chicharo, “Computing running DCT’s and DST’s based on their second order shift properties,” IEEE Trans. Circuits Syst. - I, Fundamental Theory and Applications, 47, 779–783 (2000).
[CrossRef]

Chiu, C. T.

K. J. R. Liu, C. T. Chiu, R. K. Kolagotla, J. F. Jaja, “Optimal unified architectures for the real-time computation of time-recursive discrete sinusoidal transforms,” IEEE Trans. Circuits and Systems for Video Technology, Vol. 4, No. 2, April1994.

Cristobal, G.

V. Kober, G. Cristobal, “Fast recursive algorithms for short-time discrete cosine transform,” Electron. Lett. 35, 1236–1238 (1999).
[CrossRef]

Cvetkovic, Z.

Z. Cvetkovic, M. V. Popovic, “New fast recursive algorithms for the computation of discrete cosine and sine transforms,” IEEE Trans. Signal Process. 40, 2083–2086 (1992).
[CrossRef]

Eden, M.

L. Yaroslavsky, M. Eden, Fundamentals of Digital Optics (Birkhäuser, Boston, 1996).
[CrossRef]

Egiazarian, K. O.

L. P. Yaroslavsky, K. O. Egiazarian, J. T. Astola, “Transform domain image restoration methods: review, comparison, and interpretation,” in Photonics West 2001: Electronic Imaging Nonlinear Processing and Pattern Analysis, Proc. SPIE4304, 155–170 (2001).

Exposito, A.

J. A. R. Macias, A. Exposito, “Recursive Formulation of Short-Time Discrete Trigonometric Transforms,” IEEE Transactions Circuits Syst. II Analog and Digital Signal Processing, 45, 525–527 (1998).
[CrossRef]

Fraser, D.

D. Fraser, “Interpolation by the FFT Revisited—An Experimental Investigation,” IEEE Trans. Acoust. Speech Signal Process. ASSP-37, 665–675 (1989).
[CrossRef]

Gold, B.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Gupta, A.

A. Gupta, K. R. Rao, “A fast recursive algorithm for the discrete sine transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-38, 553–557 (1990).
[CrossRef]

Hou, H. S.

H. S. Hou, “A fast recursive algorithm for computing the discrete cosine transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 1455–1461 (1987).

Hsu, Ch-Y.

J. Ch. Yao, Ch-Y. Hsu, “Further results on “New fast recursive algorithms for the discrete cosine and sine transforms,” IEEE Trans. Signal Process. 42, 3254–3255 (1994).
[CrossRef]

Jaja, J. F.

K. J. R. Liu, C. T. Chiu, R. K. Kolagotla, J. F. Jaja, “Optimal unified architectures for the real-time computation of time-recursive discrete sinusoidal transforms,” IEEE Trans. Circuits and Systems for Video Technology, Vol. 4, No. 2, April1994.

Kober, V.

V. Kober, G. Cristobal, “Fast recursive algorithms for short-time discrete cosine transform,” Electron. Lett. 35, 1236–1238 (1999).
[CrossRef]

Kolagotla, R. K.

K. J. R. Liu, C. T. Chiu, R. K. Kolagotla, J. F. Jaja, “Optimal unified architectures for the real-time computation of time-recursive discrete sinusoidal transforms,” IEEE Trans. Circuits and Systems for Video Technology, Vol. 4, No. 2, April1994.

Liu, K. J. R.

K. J. R. Liu, C. T. Chiu, R. K. Kolagotla, J. F. Jaja, “Optimal unified architectures for the real-time computation of time-recursive discrete sinusoidal transforms,” IEEE Trans. Circuits and Systems for Video Technology, Vol. 4, No. 2, April1994.

Macias, J. A. R.

J. A. R. Macias, A. Exposito, “Recursive Formulation of Short-Time Discrete Trigonometric Transforms,” IEEE Transactions Circuits Syst. II Analog and Digital Signal Processing, 45, 525–527 (1998).
[CrossRef]

Nichols, S. T.

T. Smith, M. S. Smith, S. T. Nichols, “Efficient sinc function interpolation technique for center padded data,” IEEE Trans. Acoust. Speech Signal Process. ASSP-38, 1512–1517 (1990).
[CrossRef]

Popovic, M. V.

Z. Cvetkovic, M. V. Popovic, “New fast recursive algorithms for the computation of discrete cosine and sine transforms,” IEEE Trans. Signal Process. 40, 2083–2086 (1992).
[CrossRef]

Rabiner, L. R.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Rama Murthy, N.

N. Rama Murthy, N. S. Swamy, “On computation of running discrete cosine and sine transform,” IEEE Trans. Signal Process. 40, 1430–1437 (1992).
[CrossRef]

Rao, K. R.

A. Gupta, K. R. Rao, “A fast recursive algorithm for the discrete sine transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-38, 553–557 (1990).
[CrossRef]

Smith, M. S.

T. Smith, M. S. Smith, S. T. Nichols, “Efficient sinc function interpolation technique for center padded data,” IEEE Trans. Acoust. Speech Signal Process. ASSP-38, 1512–1517 (1990).
[CrossRef]

Smith, T.

T. Smith, M. S. Smith, S. T. Nichols, “Efficient sinc function interpolation technique for center padded data,” IEEE Trans. Acoust. Speech Signal Process. ASSP-38, 1512–1517 (1990).
[CrossRef]

Swamy, N. S.

N. Rama Murthy, N. S. Swamy, “On computation of running discrete cosine and sine transform,” IEEE Trans. Signal Process. 40, 1430–1437 (1992).
[CrossRef]

Thevenaz, P.

M. Unser, P. Thevenaz, L. Yaroslavsky, “Convolution-based interpolation for fast, high-quality rotation of images,” IEEE Trans. Image Process. 4, 1371–1382 (1995).
[CrossRef] [PubMed]

Thevenaz, Ph.

Ph. Thevenaz, Th. Blu, M. Unser, “Interpolation Revisited,” IEEE Trans. Med. Imaging MI-19, 739–758 (2000).
[CrossRef]

Unser, M.

Ph. Thevenaz, Th. Blu, M. Unser, “Interpolation Revisited,” IEEE Trans. Med. Imaging MI-19, 739–758 (2000).
[CrossRef]

M. Unser, “Splines: A perfect fit for signal and image processing,” IEEE Trans. Signal Process. 16, 22–38 (1999).
[CrossRef]

M. Unser, P. Thevenaz, L. Yaroslavsky, “Convolution-based interpolation for fast, high-quality rotation of images,” IEEE Trans. Image Process. 4, 1371–1382 (1995).
[CrossRef] [PubMed]

Vitkus, R. Yu.

R. Yu. Vitkus, L. P. Yaroslavsky, “Recursive Algorithms for Local Adaptive Linear Filtration,” in Mathematical Research, Computer Analysis of Images and Patterns, L. P. Yaroslavsky, A. Rosenfeld, W. Wilhelmi, eds. Band 40, (Academie Verlag, Berlin, 1987), pp. 34–39.

Wang, Z.

Z. Wang, “Pruning the Fast Discrete Cosine Transform,” IEEE Trans. Commun. 39, 640–643 (1991).
[CrossRef]

Z. Wang, “Fast algorithms for the discrete W transform and for the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 803–816 (1984).
[CrossRef]

Z. Wang, “A simple structured algorithm for the DCT,” in Proc. 3rd Ann. Conf. Signal Process. Xi’an, China, Nov 1988, pp. 28–31 (in Chinese).

Xi, J.

J. Xi, J. F. Chicharo, “Computing running DCT’s and DST’s based on their second order shift properties,” IEEE Trans. Circuits Syst. - I, Fundamental Theory and Applications, 47, 779–783 (2000).
[CrossRef]

Yao, J. Ch.

J. Ch. Yao, Ch-Y. Hsu, “Further results on “New fast recursive algorithms for the discrete cosine and sine transforms,” IEEE Trans. Signal Process. 42, 3254–3255 (1994).
[CrossRef]

Yaroslavsky, L.

L. Yaroslavsky, “Efficient algorithm for discrete sinc interpolation,” Appl. Opt. 36, 460–463 (1997).
[CrossRef] [PubMed]

M. Unser, P. Thevenaz, L. Yaroslavsky, “Convolution-based interpolation for fast, high-quality rotation of images,” IEEE Trans. Image Process. 4, 1371–1382 (1995).
[CrossRef] [PubMed]

L. Yaroslavsky, Digital Picture Processing: An Introduction (Springer-Verlag, Berlin, 1985).
[CrossRef]

L. Yaroslavsky, M. Eden, Fundamentals of Digital Optics (Birkhäuser, Boston, 1996).
[CrossRef]

L. Yaroslavsky, “Image restoration, enhancement, and target location with local adaptive filters,” in International Trends in Optics and Photonics, ICOIV, T. Asakura, ed., (Springer-Verlag, Berlin, 1999), pp. 111–127.
[CrossRef]

Yaroslavsky, L. P.

L. P. Yaroslavsky, K. O. Egiazarian, J. T. Astola, “Transform domain image restoration methods: review, comparison, and interpretation,” in Photonics West 2001: Electronic Imaging Nonlinear Processing and Pattern Analysis, Proc. SPIE4304, 155–170 (2001).

R. Yu. Vitkus, L. P. Yaroslavsky, “Recursive Algorithms for Local Adaptive Linear Filtration,” in Mathematical Research, Computer Analysis of Images and Patterns, L. P. Yaroslavsky, A. Rosenfeld, W. Wilhelmi, eds. Band 40, (Academie Verlag, Berlin, 1987), pp. 34–39.

Appl. Opt. (1)

Electron. Lett. (1)

V. Kober, G. Cristobal, “Fast recursive algorithms for short-time discrete cosine transform,” Electron. Lett. 35, 1236–1238 (1999).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (5)

D. Fraser, “Interpolation by the FFT Revisited—An Experimental Investigation,” IEEE Trans. Acoust. Speech Signal Process. ASSP-37, 665–675 (1989).
[CrossRef]

T. Smith, M. S. Smith, S. T. Nichols, “Efficient sinc function interpolation technique for center padded data,” IEEE Trans. Acoust. Speech Signal Process. ASSP-38, 1512–1517 (1990).
[CrossRef]

Z. Wang, “Fast algorithms for the discrete W transform and for the discrete Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 803–816 (1984).
[CrossRef]

H. S. Hou, “A fast recursive algorithm for computing the discrete cosine transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-35, 1455–1461 (1987).

A. Gupta, K. R. Rao, “A fast recursive algorithm for the discrete sine transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-38, 553–557 (1990).
[CrossRef]

IEEE Trans. Circuits Syst. - I, Fundamental Theory and Applications (1)

J. Xi, J. F. Chicharo, “Computing running DCT’s and DST’s based on their second order shift properties,” IEEE Trans. Circuits Syst. - I, Fundamental Theory and Applications, 47, 779–783 (2000).
[CrossRef]

IEEE Trans. Commun. (1)

Z. Wang, “Pruning the Fast Discrete Cosine Transform,” IEEE Trans. Commun. 39, 640–643 (1991).
[CrossRef]

IEEE Trans. Image Process. (1)

M. Unser, P. Thevenaz, L. Yaroslavsky, “Convolution-based interpolation for fast, high-quality rotation of images,” IEEE Trans. Image Process. 4, 1371–1382 (1995).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging (1)

Ph. Thevenaz, Th. Blu, M. Unser, “Interpolation Revisited,” IEEE Trans. Med. Imaging MI-19, 739–758 (2000).
[CrossRef]

IEEE Trans. Signal Process. (4)

Z. Cvetkovic, M. V. Popovic, “New fast recursive algorithms for the computation of discrete cosine and sine transforms,” IEEE Trans. Signal Process. 40, 2083–2086 (1992).
[CrossRef]

J. Ch. Yao, Ch-Y. Hsu, “Further results on “New fast recursive algorithms for the discrete cosine and sine transforms,” IEEE Trans. Signal Process. 42, 3254–3255 (1994).
[CrossRef]

M. Unser, “Splines: A perfect fit for signal and image processing,” IEEE Trans. Signal Process. 16, 22–38 (1999).
[CrossRef]

N. Rama Murthy, N. S. Swamy, “On computation of running discrete cosine and sine transform,” IEEE Trans. Signal Process. 40, 1430–1437 (1992).
[CrossRef]

IEEE Transactions Circuits Syst. II Analog and Digital Signal Processing (1)

J. A. R. Macias, A. Exposito, “Recursive Formulation of Short-Time Discrete Trigonometric Transforms,” IEEE Transactions Circuits Syst. II Analog and Digital Signal Processing, 45, 525–527 (1998).
[CrossRef]

Other (9)

K. J. R. Liu, C. T. Chiu, R. K. Kolagotla, J. F. Jaja, “Optimal unified architectures for the real-time computation of time-recursive discrete sinusoidal transforms,” IEEE Trans. Circuits and Systems for Video Technology, Vol. 4, No. 2, April1994.

L. R. Rabiner, B. Gold, Theory and Application of Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

L. Yaroslavsky, A. Happonen, Y. Katyi, “Signal discrete sinc-interpolation in DCT domain: fast algorithms,” SMMSP 2002, Second International Workshop on Spectral Methods and Multirate Signal Processing, Toulouse (France), 07.09.2002–08.09.2002.

L. Yaroslavsky, “Image restoration, enhancement, and target location with local adaptive filters,” in International Trends in Optics and Photonics, ICOIV, T. Asakura, ed., (Springer-Verlag, Berlin, 1999), pp. 111–127.
[CrossRef]

L. P. Yaroslavsky, K. O. Egiazarian, J. T. Astola, “Transform domain image restoration methods: review, comparison, and interpretation,” in Photonics West 2001: Electronic Imaging Nonlinear Processing and Pattern Analysis, Proc. SPIE4304, 155–170 (2001).

R. Yu. Vitkus, L. P. Yaroslavsky, “Recursive Algorithms for Local Adaptive Linear Filtration,” in Mathematical Research, Computer Analysis of Images and Patterns, L. P. Yaroslavsky, A. Rosenfeld, W. Wilhelmi, eds. Band 40, (Academie Verlag, Berlin, 1987), pp. 34–39.

L. Yaroslavsky, Digital Picture Processing: An Introduction (Springer-Verlag, Berlin, 1985).
[CrossRef]

L. Yaroslavsky, M. Eden, Fundamentals of Digital Optics (Birkhäuser, Boston, 1996).
[CrossRef]

Z. Wang, “A simple structured algorithm for the DCT,” in Proc. 3rd Ann. Conf. Signal Process. Xi’an, China, Nov 1988, pp. 28–31 (in Chinese).

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Figures (11)

Fig. 1
Fig. 1

Illustration of the discrete sampling theorem: (a) initial signal, (b) initial signal with zeros placed between its samples, (c) spectrum of signal (a), (d) spectrum of signal (b): periodical replication of the initial signal spectrum; (e) removing spectrum replicas that may cause aliasing by low pass filter; (f) sinc-interpolated signal between samples of signal (b).

Fig. 2
Fig. 2

Principle of signal convolution in the DCT domain with signal extension by its mirror reflection.

Fig. 3
Fig. 3

Flow diagram of the discrete sinc interpolation in the DCT domain for generating a p-shifted copy of a signal.

Fig. 4
Fig. 4

Enlarging a fragment of an image (left) by sinc interpolation in the DFT domain (upper-right image) and in the DCT domain (bottom-right image). Oscillations due to boundary effects that are clearly seen in a DFT-interpolated image completely disappear in the DCT-interpolated image.

Fig. 5
Fig. 5

Windowed discrete sinc functions with a window size of 11 and 15 samples (left) and their DFT spectra for 3× signal enlarging (right).

Fig. 6
Fig. 6

Flow diagram of a simultaneous signal sliding-window sinc interpolation and restoration/enhancement in the DCT domain.

Fig. 7
Fig. 7

Image rectification and denoising by resampling with sinc interpolation in the sliding window in the DCT domain.

Fig. 8
Fig. 8

Principle of local adaptive interpolation.

Fig. 9
Fig. 9

Signal (upper plot) shift by nonadaptive (middle plot) and adaptive (bottom plot) sliding-window DCT sinc interpolation. One can notice the disappearance of oscillations at the edges of the rectangular impulses when interpolation is adaptive.

Fig. 10
Fig. 10

Comparison of nearest neighbor, linear, bicubic spline, and adaptive sliding-window sinc-interpolation methods for enlarging a digital signal. (From left to right, from top to bottom: Continuous signal, initial sampled signal, nearest-neighbor 8×-interpolated signal; linearly 8×-interpolated signal, cubic spline 8×-interpolated signal, sliding-window 8× sinc-interpolated signal).

Fig. 11
Fig. 11

Image (upper) rotation with sliding window nonadaptive (left) and adaptive DCT sinc interpolation (right). Note disappearance of oscillations at sharp edges thanks to switching between sinc interpolation and nearest-neighbor interpolation at the boundaries of black and white squares.

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

ax=- ak sincπx-kΔx/Δx,
ak=1Δx- axsincπx-kΔxdx,
ax=k=0N-1 ak sincdN; N; x/Δx-k,
sincdN; N; x=sinπxN sinπx/N
ak=k1=0N-1k2=0L-1 ak1δk2hintk-k1L+k2, k=0, 1,, LN-1,
α˜r=1LNk=0LN-1 ãk expi2π krLN=1LNk1=0N-1k2=0L-1 ak1δk2expi2π k1L+k2LN r=1LNk1=0N-1 ak1 expi2π k1N r=1L αrmod N,
DFTak=ar=1L αrmod NDFThintk.
DFThintk=1-rectr-N+1/2LN-N-1,
r=0, 1,, LN-1; rectx=1,0<x<10,otherwise.
ak=IDFT1-rectr-N+1/2LN-N-1αrmod NL =1Ln=0N-1 an1 sincdN; N; k-n1L,
sincdK; N; x=sinπKx/NN sinπx/N
ak=IDFT1-rectr-N/2LN-Nαrmod NL =1Ln1=0N-1 an1 sincdN-1; N; k-n1L;
ak=IDFT1-rectr-N/2-1LN-N-2αrmod NL =1Ln1=0N-1 an1 sincdN+1; N; k-n1L.
ak=1Ln1=0N-1 an1 sincd±1; N; k-n1L,
sincd±1; N; k=sincdN+1; N; k+sincdN-1; N; k/2.
akp=IDFTαrφrp,
φrp=1Nk=0N-1 sincdK; N; k-pexpi2π krN
φrp=1Nexpi2πpr/N;r=0, 1,, N-1/2-1φN-r*;r=N-1/2+1,, N-1
φrp=1Nexpi2πpr/N;r=0, 1,, N/2-11Ncos2πpr/N;r=N/2φN-r*;r=N/2+1,, N-1
αru,v=1Nk=0N-1 ak expi2π k+uN rexpi2π kvN
hintpk=sincdK; N; k-p;k=0, 1,N-10;k=N, N+1,2N-1,
akp=ISDFT1/2,0αrDCT · ηrp,
ηrp=ηrrep+iηrimp=12Nk=02N-1 hintpkexpi2π kr2N.
αrDCT=αrDCT=DCTak,r=0, 1,, N-1;0,r=N-α2N-1-rDCT,r=N+1, N+2,, 2N-1,
akp=12Nr=02N-1 αrDCTηrpexp-i2π k+1/2r2N =12Nα0DCTη0+r=1N-1 αrDCT×ηr exp-iπ k+1/2N r+ηr* expiπ k+1/2N r =12Nα0DCTη0+2 r=1N-1 αrDCTηrre cosπ k+1/2N r-2 r=1N-1 αrDCTηrim sinπ k+1/2N r,
η2rp=1Nexpi2π prN.
η2r+1p=12Nk=0N-1sincdK; N; k-pexpi2π k2r+12N,
âk=Tk-1αˆrβr.
αˆrβr=max0, |βr|2-Thr|βr|2βr
αˆr=βr,|βr|2>Thr0,otherwise
akp=ISDFT1/2,0αˆrDCTβrDCTηrp.
akp=ISDFT1/2,0αˆrDCTβrDCTηrp, βrDCT.
bk=12Nα0DCT+r=1N-1/2-1rα2rDCTη2rrep-α2r-1DCTη2r-1rep.

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