Abstract

A new algorithm is presented that estimates the displacement vector field from two successive image frames. In the case where the sequence is severely corrupted by additive (Gaussian or not, colored) noise of unknown covariance, then second-order statistics methods do not work well. However, we have studied this topic from a viewpoint different from the above to explore the fundamental limits in image optimal flow estimation. Our scheme is based on subpixel optimal flow estimation using the bispectrum in the parametric domain. The displacement vector of a moving object is estimated by solving linear equations involving third-order holograms and the matrix containing the Dirac delta function. To prove the feasibility of the proposed method, we compared it with a phase correlation technique and the nonparametric bispectrum method described in Res. Lett. Signal Process., ID 417915 (2008) . Our results show that our method is considerably more immune to the presence of noise.

© 2009 Optical Society of America

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References

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  1. R. M. Armitano, R. W. Schafer, F. L. Kitson, and V. Bhaskaran, “Robust block-matching motion-estimation technique for noisy sources,” in Proceedings of 1997 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1997), pp. 2685-2688.
    [CrossRef]
  2. S. Bhattacharya, N. C. Ray, and S. Sinha, “2-D signal modelling and reconstruction using third-order cumulants,” Signal Process. 62, 61-72 (1997).
    [CrossRef]
  3. E. M. Ismaili Aalaoui and E. Ibn-Elhaj, “Estimation of subpixel motion using bispectrum,” Res. Lett. Signal Process. , ID 417915 (2008).
  4. J. M. Anderson and G. B. Giannakis, “Image motion estimation algorithms using cumulants,” IEEE Trans. Image Process. 4, 346-357 (1995).
    [CrossRef] [PubMed]
  5. R. P. Kleihorst, R. L. Lagendijk, and J. Biemond, “Noise reduction of severely corrupted image sequences,” in Proceedings of 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1993), pp. 293-296.
    [CrossRef]
  6. E. Ibn-elhaj, D. Aboutajdine, S. Pateux, and L. Morin, “HOS-based method of global motion estimation for noisy image sequences,” Electron. Lett. 35, 1320-1322 (1999).
    [CrossRef]
  7. E. Sayrol, A. Gasull, and J. R. Fonollosa, “Motion estimation using higher order statistics,” IEEE Trans. Image Process. 5, 1077-1084 (1996).
    [CrossRef] [PubMed]
  8. A. N. Netravali and J. D. Robbins “Motion-compensated television coding: Part I,” Bell Syst. Tech. J. 58, 629-668 (1979).
  9. V. Murino, C. Ottonello, and S. Pagnan, “Noisy texture classification: A higher-order statistics approach,” Pattern Recogn. 31, 383-393 (1998).
    [CrossRef]
  10. M. R. Raghuveer and C. L. Nikias, “Bispectrum estimation: A parametric approach,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-33, 1213-1230 (1985).
    [CrossRef]
  11. G. B. Giannakis, “On the identifiability of non Gaussian ARMA models using cumulants,” IEEE Trans. Autom. Control 35, 18-26 (1990).
    [CrossRef]
  12. J. M. Mendel, “Tutorial on higher order statistics (spectra) in signal processing and systems theory: Theoretical results and some applications,” Proc. IEEE 79, 278-305 (1991).
    [CrossRef]
  13. C. L. Nikias and R. Pan, “Time delay estimation in unknown Gaussian spatially correlated noise,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-36, 1706-1714 (1988).
    [CrossRef]
  14. B. M. Sadler and G. B. Giannakis, “Shift- and rotation-invariant object reconstruction using the bispectrum,” J. Opt. Soc. Am. A 9, 57-69 (1992).
    [CrossRef]
  15. J. Heikkilä, “Image scale and rotation from the phase-only bispectrum,” in Proceedings of the 2004 IEEE International Conference on Image Processing (IEEE, 2004).
  16. A. P. Petropulu and H. Pozidis, “Phase reconstuction from bispectrum slices,” IEEE Trans. Image Process. 46, 527-530 (1998).
  17. C. L. Nikias and A. P. Petropulu, Higher-Order Spectra Analysis: A Nonlinear Signal Processing Framework (Prentice-Hall, 1993).
  18. Y. T. Chan, “Notes on: Time delay estimation, ARMA processes, tracking filters” (Department of Electrical Engineering, Royal Military College Canada, Kingston, Ontario, Canada K7L2W3, April 1985).
  19. G. Madec, “Half pixel accuracy in blockmatching,” in Proceedings of the Picture Coding Symposium (PCS 90), Cambridge, Massachusetts, USA, March 1990.
  20. E. M. Ismaili Aalaoui and E. Ibn-Elhaj, “Estimation of motion fields from noisy image sequences using generalized cross-correlation methods,” in Proceedings of IEEE International Conference on Signal Processing and Communications 2007 (IEEE, 2007).
  21. E. M. Ismaili Aalaoui and E. Ibn-Elhaj, “Estimation of displacement vector field from noisy data using maximum likelihood estimator,” in 14th IEEE International Conference on Electronics, Circuits and Systems (IEEE, 2007).
  22. W. K. Pratt, Digital Image Processing, PIKS Scientific Inside, 4th ed. (Wiley, 2007).
  23. S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111-119 (2007).
    [CrossRef]
  24. K. S. Lii and K. N. Helland, “Cross bispectrum computation and variance estimation,” ACM Trans. Math. Softw. 7, 284-294 (1981).
    [CrossRef]
  25. J.-M. L. Caillec and R. Garello, “Comparison of statistical indices using third order statistics for nonlinearity detection,” Signal Process. 84, 499-525 (2004).
    [CrossRef]
  26. S. A. Kruger and A. D. Calway, “A multiresolution frequency domain method for estimating affine motion parameters,” in 1996 Proceedings of International Conference on Image Processing (IEEE, 1996), Vol. 1, 113-116.
    [CrossRef]
  27. A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).

2008 (1)

E. M. Ismaili Aalaoui and E. Ibn-Elhaj, “Estimation of subpixel motion using bispectrum,” Res. Lett. Signal Process. , ID 417915 (2008).

2007 (1)

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111-119 (2007).
[CrossRef]

2004 (1)

J.-M. L. Caillec and R. Garello, “Comparison of statistical indices using third order statistics for nonlinearity detection,” Signal Process. 84, 499-525 (2004).
[CrossRef]

1999 (1)

E. Ibn-elhaj, D. Aboutajdine, S. Pateux, and L. Morin, “HOS-based method of global motion estimation for noisy image sequences,” Electron. Lett. 35, 1320-1322 (1999).
[CrossRef]

1998 (2)

V. Murino, C. Ottonello, and S. Pagnan, “Noisy texture classification: A higher-order statistics approach,” Pattern Recogn. 31, 383-393 (1998).
[CrossRef]

A. P. Petropulu and H. Pozidis, “Phase reconstuction from bispectrum slices,” IEEE Trans. Image Process. 46, 527-530 (1998).

1997 (1)

S. Bhattacharya, N. C. Ray, and S. Sinha, “2-D signal modelling and reconstruction using third-order cumulants,” Signal Process. 62, 61-72 (1997).
[CrossRef]

1996 (1)

E. Sayrol, A. Gasull, and J. R. Fonollosa, “Motion estimation using higher order statistics,” IEEE Trans. Image Process. 5, 1077-1084 (1996).
[CrossRef] [PubMed]

1995 (1)

J. M. Anderson and G. B. Giannakis, “Image motion estimation algorithms using cumulants,” IEEE Trans. Image Process. 4, 346-357 (1995).
[CrossRef] [PubMed]

1992 (1)

1991 (1)

J. M. Mendel, “Tutorial on higher order statistics (spectra) in signal processing and systems theory: Theoretical results and some applications,” Proc. IEEE 79, 278-305 (1991).
[CrossRef]

1990 (1)

G. B. Giannakis, “On the identifiability of non Gaussian ARMA models using cumulants,” IEEE Trans. Autom. Control 35, 18-26 (1990).
[CrossRef]

1988 (1)

C. L. Nikias and R. Pan, “Time delay estimation in unknown Gaussian spatially correlated noise,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-36, 1706-1714 (1988).
[CrossRef]

1985 (1)

M. R. Raghuveer and C. L. Nikias, “Bispectrum estimation: A parametric approach,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-33, 1213-1230 (1985).
[CrossRef]

1981 (1)

K. S. Lii and K. N. Helland, “Cross bispectrum computation and variance estimation,” ACM Trans. Math. Softw. 7, 284-294 (1981).
[CrossRef]

1979 (1)

A. N. Netravali and J. D. Robbins “Motion-compensated television coding: Part I,” Bell Syst. Tech. J. 58, 629-668 (1979).

Aboutajdine, D.

E. Ibn-elhaj, D. Aboutajdine, S. Pateux, and L. Morin, “HOS-based method of global motion estimation for noisy image sequences,” Electron. Lett. 35, 1320-1322 (1999).
[CrossRef]

Anderson, J. M.

J. M. Anderson and G. B. Giannakis, “Image motion estimation algorithms using cumulants,” IEEE Trans. Image Process. 4, 346-357 (1995).
[CrossRef] [PubMed]

Armitano, R. M.

R. M. Armitano, R. W. Schafer, F. L. Kitson, and V. Bhaskaran, “Robust block-matching motion-estimation technique for noisy sources,” in Proceedings of 1997 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1997), pp. 2685-2688.
[CrossRef]

Bhaskaran, V.

R. M. Armitano, R. W. Schafer, F. L. Kitson, and V. Bhaskaran, “Robust block-matching motion-estimation technique for noisy sources,” in Proceedings of 1997 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1997), pp. 2685-2688.
[CrossRef]

Bhattacharya, S.

S. Bhattacharya, N. C. Ray, and S. Sinha, “2-D signal modelling and reconstruction using third-order cumulants,” Signal Process. 62, 61-72 (1997).
[CrossRef]

Biemond, J.

R. P. Kleihorst, R. L. Lagendijk, and J. Biemond, “Noise reduction of severely corrupted image sequences,” in Proceedings of 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1993), pp. 293-296.
[CrossRef]

Caillec, J.-M. L.

J.-M. L. Caillec and R. Garello, “Comparison of statistical indices using third order statistics for nonlinearity detection,” Signal Process. 84, 499-525 (2004).
[CrossRef]

Calway, A. D.

S. A. Kruger and A. D. Calway, “A multiresolution frequency domain method for estimating affine motion parameters,” in 1996 Proceedings of International Conference on Image Processing (IEEE, 1996), Vol. 1, 113-116.
[CrossRef]

Chan, Y. T.

Y. T. Chan, “Notes on: Time delay estimation, ARMA processes, tracking filters” (Department of Electrical Engineering, Royal Military College Canada, Kingston, Ontario, Canada K7L2W3, April 1985).

Fonollosa, J. R.

E. Sayrol, A. Gasull, and J. R. Fonollosa, “Motion estimation using higher order statistics,” IEEE Trans. Image Process. 5, 1077-1084 (1996).
[CrossRef] [PubMed]

Frigo, M.

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111-119 (2007).
[CrossRef]

Garello, R.

J.-M. L. Caillec and R. Garello, “Comparison of statistical indices using third order statistics for nonlinearity detection,” Signal Process. 84, 499-525 (2004).
[CrossRef]

Gasull, A.

E. Sayrol, A. Gasull, and J. R. Fonollosa, “Motion estimation using higher order statistics,” IEEE Trans. Image Process. 5, 1077-1084 (1996).
[CrossRef] [PubMed]

Giannakis, G. B.

J. M. Anderson and G. B. Giannakis, “Image motion estimation algorithms using cumulants,” IEEE Trans. Image Process. 4, 346-357 (1995).
[CrossRef] [PubMed]

B. M. Sadler and G. B. Giannakis, “Shift- and rotation-invariant object reconstruction using the bispectrum,” J. Opt. Soc. Am. A 9, 57-69 (1992).
[CrossRef]

G. B. Giannakis, “On the identifiability of non Gaussian ARMA models using cumulants,” IEEE Trans. Autom. Control 35, 18-26 (1990).
[CrossRef]

Heikkilä, J.

J. Heikkilä, “Image scale and rotation from the phase-only bispectrum,” in Proceedings of the 2004 IEEE International Conference on Image Processing (IEEE, 2004).

Helland, K. N.

K. S. Lii and K. N. Helland, “Cross bispectrum computation and variance estimation,” ACM Trans. Math. Softw. 7, 284-294 (1981).
[CrossRef]

Ibn-Elhaj, E.

E. M. Ismaili Aalaoui and E. Ibn-Elhaj, “Estimation of subpixel motion using bispectrum,” Res. Lett. Signal Process. , ID 417915 (2008).

E. Ibn-elhaj, D. Aboutajdine, S. Pateux, and L. Morin, “HOS-based method of global motion estimation for noisy image sequences,” Electron. Lett. 35, 1320-1322 (1999).
[CrossRef]

E. M. Ismaili Aalaoui and E. Ibn-Elhaj, “Estimation of motion fields from noisy image sequences using generalized cross-correlation methods,” in Proceedings of IEEE International Conference on Signal Processing and Communications 2007 (IEEE, 2007).

E. M. Ismaili Aalaoui and E. Ibn-Elhaj, “Estimation of displacement vector field from noisy data using maximum likelihood estimator,” in 14th IEEE International Conference on Electronics, Circuits and Systems (IEEE, 2007).

Ismaili Aalaoui, E. M.

E. M. Ismaili Aalaoui and E. Ibn-Elhaj, “Estimation of subpixel motion using bispectrum,” Res. Lett. Signal Process. , ID 417915 (2008).

E. M. Ismaili Aalaoui and E. Ibn-Elhaj, “Estimation of motion fields from noisy image sequences using generalized cross-correlation methods,” in Proceedings of IEEE International Conference on Signal Processing and Communications 2007 (IEEE, 2007).

E. M. Ismaili Aalaoui and E. Ibn-Elhaj, “Estimation of displacement vector field from noisy data using maximum likelihood estimator,” in 14th IEEE International Conference on Electronics, Circuits and Systems (IEEE, 2007).

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).

Johnson, S. G.

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111-119 (2007).
[CrossRef]

Kitson, F. L.

R. M. Armitano, R. W. Schafer, F. L. Kitson, and V. Bhaskaran, “Robust block-matching motion-estimation technique for noisy sources,” in Proceedings of 1997 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1997), pp. 2685-2688.
[CrossRef]

Kleihorst, R. P.

R. P. Kleihorst, R. L. Lagendijk, and J. Biemond, “Noise reduction of severely corrupted image sequences,” in Proceedings of 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1993), pp. 293-296.
[CrossRef]

Kruger, S. A.

S. A. Kruger and A. D. Calway, “A multiresolution frequency domain method for estimating affine motion parameters,” in 1996 Proceedings of International Conference on Image Processing (IEEE, 1996), Vol. 1, 113-116.
[CrossRef]

Lagendijk, R. L.

R. P. Kleihorst, R. L. Lagendijk, and J. Biemond, “Noise reduction of severely corrupted image sequences,” in Proceedings of 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1993), pp. 293-296.
[CrossRef]

Lii, K. S.

K. S. Lii and K. N. Helland, “Cross bispectrum computation and variance estimation,” ACM Trans. Math. Softw. 7, 284-294 (1981).
[CrossRef]

Madec, G.

G. Madec, “Half pixel accuracy in blockmatching,” in Proceedings of the Picture Coding Symposium (PCS 90), Cambridge, Massachusetts, USA, March 1990.

Mendel, J. M.

J. M. Mendel, “Tutorial on higher order statistics (spectra) in signal processing and systems theory: Theoretical results and some applications,” Proc. IEEE 79, 278-305 (1991).
[CrossRef]

Morin, L.

E. Ibn-elhaj, D. Aboutajdine, S. Pateux, and L. Morin, “HOS-based method of global motion estimation for noisy image sequences,” Electron. Lett. 35, 1320-1322 (1999).
[CrossRef]

Murino, V.

V. Murino, C. Ottonello, and S. Pagnan, “Noisy texture classification: A higher-order statistics approach,” Pattern Recogn. 31, 383-393 (1998).
[CrossRef]

Netravali, A. N.

A. N. Netravali and J. D. Robbins “Motion-compensated television coding: Part I,” Bell Syst. Tech. J. 58, 629-668 (1979).

Nikias, C. L.

C. L. Nikias and R. Pan, “Time delay estimation in unknown Gaussian spatially correlated noise,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-36, 1706-1714 (1988).
[CrossRef]

M. R. Raghuveer and C. L. Nikias, “Bispectrum estimation: A parametric approach,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-33, 1213-1230 (1985).
[CrossRef]

C. L. Nikias and A. P. Petropulu, Higher-Order Spectra Analysis: A Nonlinear Signal Processing Framework (Prentice-Hall, 1993).

Ottonello, C.

V. Murino, C. Ottonello, and S. Pagnan, “Noisy texture classification: A higher-order statistics approach,” Pattern Recogn. 31, 383-393 (1998).
[CrossRef]

Pagnan, S.

V. Murino, C. Ottonello, and S. Pagnan, “Noisy texture classification: A higher-order statistics approach,” Pattern Recogn. 31, 383-393 (1998).
[CrossRef]

Pan, R.

C. L. Nikias and R. Pan, “Time delay estimation in unknown Gaussian spatially correlated noise,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-36, 1706-1714 (1988).
[CrossRef]

Pateux, S.

E. Ibn-elhaj, D. Aboutajdine, S. Pateux, and L. Morin, “HOS-based method of global motion estimation for noisy image sequences,” Electron. Lett. 35, 1320-1322 (1999).
[CrossRef]

Petropulu, A. P.

A. P. Petropulu and H. Pozidis, “Phase reconstuction from bispectrum slices,” IEEE Trans. Image Process. 46, 527-530 (1998).

C. L. Nikias and A. P. Petropulu, Higher-Order Spectra Analysis: A Nonlinear Signal Processing Framework (Prentice-Hall, 1993).

Pozidis, H.

A. P. Petropulu and H. Pozidis, “Phase reconstuction from bispectrum slices,” IEEE Trans. Image Process. 46, 527-530 (1998).

Pratt, W. K.

W. K. Pratt, Digital Image Processing, PIKS Scientific Inside, 4th ed. (Wiley, 2007).

Raghuveer, M. R.

M. R. Raghuveer and C. L. Nikias, “Bispectrum estimation: A parametric approach,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-33, 1213-1230 (1985).
[CrossRef]

Ray, N. C.

S. Bhattacharya, N. C. Ray, and S. Sinha, “2-D signal modelling and reconstruction using third-order cumulants,” Signal Process. 62, 61-72 (1997).
[CrossRef]

Robbins, J. D.

A. N. Netravali and J. D. Robbins “Motion-compensated television coding: Part I,” Bell Syst. Tech. J. 58, 629-668 (1979).

Sadler, B. M.

Sayrol, E.

E. Sayrol, A. Gasull, and J. R. Fonollosa, “Motion estimation using higher order statistics,” IEEE Trans. Image Process. 5, 1077-1084 (1996).
[CrossRef] [PubMed]

Schafer, R. W.

R. M. Armitano, R. W. Schafer, F. L. Kitson, and V. Bhaskaran, “Robust block-matching motion-estimation technique for noisy sources,” in Proceedings of 1997 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1997), pp. 2685-2688.
[CrossRef]

Sinha, S.

S. Bhattacharya, N. C. Ray, and S. Sinha, “2-D signal modelling and reconstruction using third-order cumulants,” Signal Process. 62, 61-72 (1997).
[CrossRef]

ACM Trans. Math. Softw. (1)

K. S. Lii and K. N. Helland, “Cross bispectrum computation and variance estimation,” ACM Trans. Math. Softw. 7, 284-294 (1981).
[CrossRef]

Bell Syst. Tech. J. (1)

A. N. Netravali and J. D. Robbins “Motion-compensated television coding: Part I,” Bell Syst. Tech. J. 58, 629-668 (1979).

Electron. Lett. (1)

E. Ibn-elhaj, D. Aboutajdine, S. Pateux, and L. Morin, “HOS-based method of global motion estimation for noisy image sequences,” Electron. Lett. 35, 1320-1322 (1999).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process. (2)

M. R. Raghuveer and C. L. Nikias, “Bispectrum estimation: A parametric approach,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-33, 1213-1230 (1985).
[CrossRef]

C. L. Nikias and R. Pan, “Time delay estimation in unknown Gaussian spatially correlated noise,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-36, 1706-1714 (1988).
[CrossRef]

IEEE Trans. Autom. Control (1)

G. B. Giannakis, “On the identifiability of non Gaussian ARMA models using cumulants,” IEEE Trans. Autom. Control 35, 18-26 (1990).
[CrossRef]

IEEE Trans. Image Process. (3)

A. P. Petropulu and H. Pozidis, “Phase reconstuction from bispectrum slices,” IEEE Trans. Image Process. 46, 527-530 (1998).

E. Sayrol, A. Gasull, and J. R. Fonollosa, “Motion estimation using higher order statistics,” IEEE Trans. Image Process. 5, 1077-1084 (1996).
[CrossRef] [PubMed]

J. M. Anderson and G. B. Giannakis, “Image motion estimation algorithms using cumulants,” IEEE Trans. Image Process. 4, 346-357 (1995).
[CrossRef] [PubMed]

IEEE Trans. Signal Process. (1)

S. G. Johnson and M. Frigo, “A modified split-radix FFT with fewer arithmetic operations,” IEEE Trans. Signal Process. 55, 111-119 (2007).
[CrossRef]

J. Opt. Soc. Am. A (1)

Pattern Recogn. (1)

V. Murino, C. Ottonello, and S. Pagnan, “Noisy texture classification: A higher-order statistics approach,” Pattern Recogn. 31, 383-393 (1998).
[CrossRef]

Proc. IEEE (1)

J. M. Mendel, “Tutorial on higher order statistics (spectra) in signal processing and systems theory: Theoretical results and some applications,” Proc. IEEE 79, 278-305 (1991).
[CrossRef]

Res. Lett. Signal Process. (1)

E. M. Ismaili Aalaoui and E. Ibn-Elhaj, “Estimation of subpixel motion using bispectrum,” Res. Lett. Signal Process. , ID 417915 (2008).

Signal Process. (2)

J.-M. L. Caillec and R. Garello, “Comparison of statistical indices using third order statistics for nonlinearity detection,” Signal Process. 84, 499-525 (2004).
[CrossRef]

S. Bhattacharya, N. C. Ray, and S. Sinha, “2-D signal modelling and reconstruction using third-order cumulants,” Signal Process. 62, 61-72 (1997).
[CrossRef]

Other (11)

R. M. Armitano, R. W. Schafer, F. L. Kitson, and V. Bhaskaran, “Robust block-matching motion-estimation technique for noisy sources,” in Proceedings of 1997 IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 1997), pp. 2685-2688.
[CrossRef]

R. P. Kleihorst, R. L. Lagendijk, and J. Biemond, “Noise reduction of severely corrupted image sequences,” in Proceedings of 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1993), pp. 293-296.
[CrossRef]

J. Heikkilä, “Image scale and rotation from the phase-only bispectrum,” in Proceedings of the 2004 IEEE International Conference on Image Processing (IEEE, 2004).

C. L. Nikias and A. P. Petropulu, Higher-Order Spectra Analysis: A Nonlinear Signal Processing Framework (Prentice-Hall, 1993).

Y. T. Chan, “Notes on: Time delay estimation, ARMA processes, tracking filters” (Department of Electrical Engineering, Royal Military College Canada, Kingston, Ontario, Canada K7L2W3, April 1985).

G. Madec, “Half pixel accuracy in blockmatching,” in Proceedings of the Picture Coding Symposium (PCS 90), Cambridge, Massachusetts, USA, March 1990.

E. M. Ismaili Aalaoui and E. Ibn-Elhaj, “Estimation of motion fields from noisy image sequences using generalized cross-correlation methods,” in Proceedings of IEEE International Conference on Signal Processing and Communications 2007 (IEEE, 2007).

E. M. Ismaili Aalaoui and E. Ibn-Elhaj, “Estimation of displacement vector field from noisy data using maximum likelihood estimator,” in 14th IEEE International Conference on Electronics, Circuits and Systems (IEEE, 2007).

W. K. Pratt, Digital Image Processing, PIKS Scientific Inside, 4th ed. (Wiley, 2007).

S. A. Kruger and A. D. Calway, “A multiresolution frequency domain method for estimating affine motion parameters,” in 1996 Proceedings of International Conference on Image Processing (IEEE, 1996), Vol. 1, 113-116.
[CrossRef]

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).

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Figures (8)

Fig. 1
Fig. 1

Physical representation of cumulants [2].

Fig. 2
Fig. 2

Motion field for the Hall Monitor sequence in the presence of AGN with SNR = 14 dB .

Fig. 3
Fig. 3

Motion field for the Silent sequence in the presence of NGN with SNR = 14 dB .

Fig. 4
Fig. 4

Motion field for the Table Tennis sequence in the presence of CGN with SNR = 10 dB .

Fig. 5
Fig. 5

Phase correlation surfaces between two blocks.

Fig. 6
Fig. 6

PSNR versus frame number for motion-compensated prediction.

Fig. 7
Fig. 7

Prediction for frame 36 of the Hall Monitor sequence in the presence of AGN with SNR = 18 .

Fig. 8
Fig. 8

Enlarged portions of the motion-compensated pictures of the Hall Monitor sequence.

Tables (2)

Tables Icon

Table 1 Comparison of Three Methods for Computation Time

Tables Icon

Table 2 Average PSNR (dB) of Motion-Compensated Images for Three Techniques with Hall Monitor Sequence

Equations (26)

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

g k 1 ( x , y ) = f k 1 ( x , y ) + n k 1 ( x , y ) ,
g k ( x , y ) = f k 1 ( x d x , y d y ) + n k ( x , y ) ,
C 3 g k g k g k ( r 1 , r 2 ; s 1 , s 2 ) = E [ g k ( x , y ) g k ( x + r 1 , y + r 2 ) g k ( x + s 1 , y + s 2 ) ] ,
C 3 g k g k 1 g k ( r 1 , r 2 ; s 1 , s 2 ) = E [ g k ( x , y ) g k 1 ( x + r 1 , y + r 2 ) g k ( x + s 1 , y + s 2 ) ] ,
B 3 g k g k g k ( u 1 , u 2 ; v 1 , v 2 ) = F 4 [ C 3 g k g k g k ( r 1 , r 2 ; s 1 , s 2 ) ] ,
B 3 g k g k g k ( u 1 , u 2 ; v 1 , v 2 ) = G g k ( u 1 , u 2 ) G g k ( v 1 , v 2 ) G g k * ( u 1 + v 1 , u 2 + v 2 ) ,
B 3 g k g k 1 g k ( u 1 , u 2 ; v 1 , v 2 ) = F 4 [ C 3 g k g k 1 g k ( r 1 , r 2 ; s 1 , s 2 ) ] = G g k ( u 1 , u 2 ) G g k 1 ( v 1 , v 2 ) G g k * ( u 1 + v 1 , u 2 + v 2 ) .
C 3 g k g k g k ( r 1 , r 2 ; s 1 , s 2 ) = C 3 g k g k g k ( s 1 , s 2 ; r 1 s 1 , r 2 s 2 ) = C 3 g k g k g k ( s 1 , s 2 ; r 1 , r 2 ) = C 3 g k g k g k ( s 1 r 1 , s 2 r 2 ; r 1 , r 2 ) = C 3 g k g k g k ( r 1 s 1 , r 2 s 2 ; s 1 , s 2 ) = C 3 g k g k g k ( r 1 , r 2 ; s 1 r 1 , s 2 r 2 ) .
B 3 g k g k g k ( u 1 , u 2 ; v 1 , v 2 ) = B 3 g k g k g k ( v 1 , v 2 ; u 1 , u 2 ) = B 3 * g k g k g k ( v 1 , v 2 ; u 1 , u 2 ) = B 3 * g k g k g k ( u 1 , u 2 ; v 1 , v 2 ) = B 3 g k g k g k ( u 1 v 1 , u 2 v 2 ; v 1 , v 2 ) = B 3 g k g k g k ( u 1 , u 2 ; u 1 v 1 , u 2 v 2 ) = B 3 g k g k g k ( u 1 v 1 , u 2 v 2 ; u 1 , u 2 ) = B 3 g k g k g k ( v 1 , v 2 ; u 1 v 1 , u 2 v 2 )
g k ( x , y ) = i R j R α ( i , j ) g k 1 ( x i , y j ) + n k ( x , y ) + n k 1 ( x d x , y d y ) ,
E [ g k 1 ( x , y ) g k ( x + r 1 , y + r 2 ) g k 1 ( x + s 1 , y + s 2 ) ] = i R j R α ( i , j ) E [ g k 1 ( x , y ) g k 1 ( x i + r 1 , y j + r 2 ) × g k 1 ( x + s 1 , y + s 2 ) ] + E [ g k 1 ( x , y ) n k ( x + r 1 , y + r 2 ) g k 1 ( x + s 1 , y + s 2 ) ] + E [ g k 1 ( x , y ) n k 1 ( x + r 1 d x , y + r 2 d y ) g k 1 ( x + s 1 , y + s 2 ) ] .
C 3 g k 1 g k g k 1 ( r 1 , r 2 ; s 1 , s 2 ) = i R j R α ( i , j ) C 3 g k 1 g k 1 g k 1 ( r 1 i , r 2 j ; s 1 , s 2 ) .
B 3 g k 1 g k g k 1 ( u 1 , u 2 ; v 1 , v 2 ) = B 3 g k 1 g k 1 g k 1 ( u 1 , u 2 ; v 1 , v 2 ) i R j R α ( i , j ) e j 2 π ( u 1 i + u 2 j ) .
B 3 , l g k 1 g k 1 g k 1 ( u 1 , u 2 ) = B 3 , l g k 1 g k 1 g k 1 ( u 1 , u 2 ; l u 1 , l u 2 ) ,
B 3 , l g k 1 g k g k 1 ( u 1 , u 2 ) = B 3 , l g k 1 g k g k 1 ( u 1 , u 2 ; l u 1 , l u 2 ) .
h l ( r 1 , r 2 ) = F 2 [ B 3 , l g k 1 g k g k 1 ( u 1 , u 2 ) B 3 , l g k 1 g k 1 g k 1 ( u 1 , u 2 ) ] = i R j R α ( i , j ) δ ( r 1 i , r 2 j ) ,
h ̂ l = α δ ̂ ,
α l s = [ δ ̂ T δ ̂ ] 1 δ ̂ T h ̂ l .
α = m = 1 L t m 1 q m ( p m * ) T h ̂ l ,
h ̂ l ( r 1 , r 2 ) = α ̂ ( d ̂ x , d ̂ y ) δ ̂ ( r 1 d ̂ x , r 2 d ̂ y ) = δ ̂ ( r 1 d ̂ x , r 2 d ̂ y ) .
( r 1 m , r 2 m ) = arg max [ h ̂ l ( r 1 , r 2 ) ] .
{ h ̂ l ( r 1 m 1 , r 2 m ) , h ̂ l ( r 1 m , r 2 m ) , h ̂ l ( r 1 m + 1 , r 2 m ) } ,
{ h ̂ l ( r 1 m , r 2 m 1 ) , h ̂ l ( r 1 m , r 2 m ) , h ̂ l ( r 1 m , r 2 m + 1 ) } ,
d ̂ ̂ x = h ̂ l ( r 1 m 1 , r 2 m ) h ̂ l ( r 1 m + 1 , r 2 m ) 2 [ h ̂ ( r 1 m + 1 , r 2 m ) 2 h ̂ ( r 1 m , r 2 m ) + h ̂ ( r 1 m 1 , r 2 m ) ] .
E [ n k ( x , y ) n k 1 ( x d x , y d y ) ] = σ n 2 a x d x a y d y a k ,
PSNR avg = 1 F i = 1 F PSNR i .

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