Abstract

We present the frequency domain analysis of the most general model of planar rigid motion, i.e., motion under the joint effects of translations and rotations with polynomial-type time dependence. We show that in the frequency domain the contribution of the object’s shape and texture is separate from that of the motion parameters in this general framework. Special attention is devoted to the case of translations and rotations with constant velocity and constant acceleration, which are the most likely candidates for modeling motion in practical contexts. This work also focuses on the distinction between the effects of the motion parameters and those of time duration, a distinction that is necessary for a clear assessment of the effects of acceleration in the frequency domain. The results presented belong to the field of signal theory and may serve as a theoretical basis in the various applications of motion models in the frequency domain, which include pattern recognition, television, image registration, image sequence analysis, and classification and neural modeling of motion perception.

© 1999 Optical Society of America

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  30. G. Cortelazzo, G. Nalesso, “A differential equation approach to the computation of the Fourier transform of images of translating objects,” IEEE Trans. Inf. Theory 40, 2049–2058 (1994).
    [CrossRef]
  31. M. Chahine, J. Konrad, “Estimation of trajectories for accelerated motion from time-varying imagery,” in Proceedings of the IEEE 1994 International Conference on Image Processing (ICIP’94) (IEEE Computer Society Press, Los Alamitos, Calif., 1994), Vol. II, pp. 800–804.
  32. A. J. Patti, M. Ibrahim, A. M. Tekalp, “Digital video standards conversion in the presence of accelerated motion,” in Proceedings of the 1994 International Conference on Acoustics Speech and Signal Processing (ICASSP’94) (IEEE Computer Society Press, Los Alamitos, Calif., 1994), Vol. 5, pp. 225–228.
  33. J. Weng, T. S. Huang, N. Ahuja, “3-D motion estimation, understanding, and prediction from noisy image sequences,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 370–389 (1987).
    [CrossRef]
  34. G.-S. J. Young, R. Chellappa, “3-D motion estimation using a sequence of noisy stereo images: models, estimation, and uniqueness results,” Trans. Pattern Anal. Mach. Intell. 12, 735–759 (1990).
    [CrossRef]
  35. G. Cortelazzo, M. Balanza, C. Monti, “Frequency domain analysis of rotational motion,” in Multidimensional Systems and Signal Processing, N. K. Bose, ed. (Kluwer Academic, Boston, Mass., 1993), Vol. 4, pp. 203–225.
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  39. L. Lucchese, G. M. Cortelazzo, M. Rizzato, “A phase correlation technique for estimating rigid planar rotations,” in Proceedings of the 5th International Workshop on Time-Varying Image Processing and Moving Object Recognition (Elsevier Science B.V., Amsterdam, 1996), pp. 244–249.
  40. L. Lucchese, G. M. Cortelazzo, C. Monti, “High resolution estimation of planar rotations based on Fourier transform and radial projections,” in Proceedings of the IEEE 1997 International Symposium on Circuits and Systems (ISCAS’97) (IEEE Computer Society Press, Los Alamitos, Calif., 1997), Vol. II, pp. 1181–1184.
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    [CrossRef] [PubMed]
  42. L. Lucchese, G. M. Cortelazzo, “Frequency domain analysis of affine motion with linear time-dependence and finite duration,” in Proceedings of the IASTED (International Association of Science and Technology for Development) International Conference on Signal Processing and Communications (IASTED/Acta Press, Anaheim, Calif., 1998), pp. 28–31.
  43. L. Lucchese, G. M. Cortelazzo, C. Monti, “Estimation of affine transformations between image pairs via Fourier transform,” in Proceedings of the IEEE 1996 International Conference on Image Processing (ICIP’96) (IEEE Computer Society Press, Los Alamitos, Calif., 1996), Vol. III, pp. 715–718.
  44. L. Lucchese, G. M. Cortelazzo, “Frequency domain estimation of 3-D rigid motion based on range and intensity data,” in Proceedings of the International Conference on Recent Advances in 3-D Digital Imaging and Modeling (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 107–112.
  45. G. M. Cortelazzo, G. Doretto, L. Lucchese, S. Totaro, “A frequency domain method for registration of range data,” in Proceedings of the IEEE 1998 International Symposium on Circuits and Systems (ISCAS’98) (IEEE Computer Society Press, Los Alamitos, Calif., 1998), Vol. 5, pp. 518–521.
  46. G. M. Cortelazzo, G. Doretto, L. Lucchese, “Free-form textured surfaces registration by a frequency domain technique,” in Proceedings of the IEEE 1998 International Conference on Image Processing (ICIP’98) (IEEE Computer Society Press, Los Alamitos, Calif., 1998), Vol. I, pp. 813–817.
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1996 (4)

W. Chen, G. B. Giannakis, N. Nandhakumar, “Spatiotemporal approach for time-varying global image motion estimation,” IEEE Trans. Image Process. 5, 1448–1461 (1996).
[CrossRef] [PubMed]

P. Burlina, R. Chellappa, “Analyzing looming motion components from their spatiotemporal spectral signature,” IEEE Trans. Pattern. Anal. Mach. Intell. 18, 1029–1033 (1996).
[CrossRef]

B. S. Reddy, B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
[CrossRef] [PubMed]

S. Alliney, G. M. Cortelazzo, G. A. Mian, “On the registration of an object translating on a static background,” Pattern Recogn. 29, 131–141 (1996).
[CrossRef]

1994 (1)

G. Cortelazzo, G. Nalesso, “A differential equation approach to the computation of the Fourier transform of images of translating objects,” IEEE Trans. Inf. Theory 40, 2049–2058 (1994).
[CrossRef]

1993 (2)

G. Cortelazzo, M. Balanza, “Frequency domain analysis of translations with piecewise cubic trajectories,” IEEE Trans. Pattern. Anal. Mach. Intell. 15, 411–416 (1993).
[CrossRef]

S. Alliney, “Digital analysis of rotated images,” IEEE Trans. Pattern. Anal. Mach. Intell. 15, 499–504 (1993).
[CrossRef]

1992 (2)

L. G. Brown, “A survey of image registration techniques,” ACM Computing Surveys 24, 325–376 (1992).
[CrossRef]

R. Wilson, A. D. Calway, E. R. Pearson, “A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signals analysis,” IEEE Trans. Inf. Theory 38, 674–690 (1992).
[CrossRef]

1991 (2)

D. Adolph, R. Buschmann, “1.15 Mbit/s coding of video signals including global motion compensation,” Signal Process. Image Commun. 3, 259–274 (1991).
[CrossRef]

J. J. Koenderink, J. Van Doorn, “Affine structure from motion,” J. Opt. Soc. Am. A 8, 377–385 (1991).
[CrossRef] [PubMed]

1990 (3)

G.-S. J. Young, R. Chellappa, “3-D motion estimation using a sequence of noisy stereo images: models, estimation, and uniqueness results,” Trans. Pattern Anal. Mach. Intell. 12, 735–759 (1990).
[CrossRef]

S. F. Wu, J. Kittler, “A differential method for simultaneous estimation of rotation, change of scale and translation,” Signal Process. Image Commun. 2, 69–80 (1990).
[CrossRef]

D. J. Fleet, A. D. Jepson, “Computation of component image velocity from local phase information,” Int. J. Comput. Vis. 5, 77–104 (1990).
[CrossRef]

1989 (1)

M. Hoetter, “Differential estimation of the global motion parameters zoom and pan,” Signal Process. 16, 249–265 (1989).
[CrossRef]

1988 (1)

G. Keesman, “Motion estimation based on a motion model incorporating translation, rotation and zoom,” Signal Process. 4, 31–34 (1988).

1987 (2)

E. De Castro, C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 700–703 (1987).
[CrossRef]

J. Weng, T. S. Huang, N. Ahuja, “3-D motion estimation, understanding, and prediction from noisy image sequences,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 370–389 (1987).
[CrossRef]

1986 (2)

1985 (1)

A. Goshtasby, “Template-matching on rotated images,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-7, 338–344 (1985).
[CrossRef]

1983 (1)

P. E. Zwicke, I. Kiss, “A new implementation of the Mellin transform and its application to radar classification of ships,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-5, 191–198 (1983).
[CrossRef]

1982 (1)

D. I. Barnea, H. F. Silverman, “A class of algorithms for fast digital image registration,” IEEE Trans. Comput. C-21, 179–186 (1982).
[CrossRef]

1981 (2)

F. Kretz, J. Sabatier, “Échantillonage des images de télévision: analyse dans le domain spatio-temporel et dans le domain de Fourier,” Ann. Telecommun. 36, 231–273 (1981).

D. Casasent, “Pattern recognition: a review,” IEEE Spectr., 28–33 (March1981).

1976 (1)

D. Casasent, D. Psaltis, “Position oriented and scale-invariant optical correlation,” Appl. Opt. 15, 1793–1799 (1976).
[CrossRef]

Abramovitz, M.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Abramovitz, New York, 1968).

Adolph, D.

D. Adolph, R. Buschmann, “1.15 Mbit/s coding of video signals including global motion compensation,” Signal Process. Image Commun. 3, 259–274 (1991).
[CrossRef]

Ahuja, N.

J. Weng, T. S. Huang, N. Ahuja, “3-D motion estimation, understanding, and prediction from noisy image sequences,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 370–389 (1987).
[CrossRef]

Ahumada, A. J.

A. B. Watson, A. J. Ahumada, “A look at motion in the frequency domain,” (NASA Ames Research Center, Moffett Field, Calif., 1983).

Alliney, S.

S. Alliney, G. M. Cortelazzo, G. A. Mian, “On the registration of an object translating on a static background,” Pattern Recogn. 29, 131–141 (1996).
[CrossRef]

S. Alliney, “Digital analysis of rotated images,” IEEE Trans. Pattern. Anal. Mach. Intell. 15, 499–504 (1993).
[CrossRef]

Arsenault, H. H.

Baker, R. L.

Y. T. Tse, R. L. Baker, “Global zoom/pan estimation and compensation for video compression,” in Proceedings of the 1991 International Conference on Acoustics, Speech and Signal Processing (ICASSP’91) (IEEE Computer Society Press, Los Alamitos Calif., 1991), Vol. 4, pp. 2725–2728.

Balanza, M.

G. Cortelazzo, M. Balanza, “Frequency domain analysis of translations with piecewise cubic trajectories,” IEEE Trans. Pattern. Anal. Mach. Intell. 15, 411–416 (1993).
[CrossRef]

G. Cortelazzo, M. Balanza, C. Monti, “Frequency domain analysis of rotational motion,” in Multidimensional Systems and Signal Processing, N. K. Bose, ed. (Kluwer Academic, Boston, Mass., 1993), Vol. 4, pp. 203–225.

Barnea, D. I.

D. I. Barnea, H. F. Silverman, “A class of algorithms for fast digital image registration,” IEEE Trans. Comput. C-21, 179–186 (1982).
[CrossRef]

Bennamoun, M.

M. Bennamoun, “Application of time-frequency signal analysis to motion estimation,” in Proceedings of the IEEE 1997 International Conference on Image Processing (ICIP’97), (IEEE Computer Society Press, Los Alamitos, Calif., 1997), Vol. II, pp. 148–151.

Brown, L. G.

L. G. Brown, “A survey of image registration techniques,” ACM Computing Surveys 24, 325–376 (1992).
[CrossRef]

Burlina, P.

P. Burlina, R. Chellappa, “Analyzing looming motion components from their spatiotemporal spectral signature,” IEEE Trans. Pattern. Anal. Mach. Intell. 18, 1029–1033 (1996).
[CrossRef]

Buschmann, R.

D. Adolph, R. Buschmann, “1.15 Mbit/s coding of video signals including global motion compensation,” Signal Process. Image Commun. 3, 259–274 (1991).
[CrossRef]

Calway, A. D.

R. Wilson, A. D. Calway, E. R. Pearson, “A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signals analysis,” IEEE Trans. Inf. Theory 38, 674–690 (1992).
[CrossRef]

Casasent, D.

D. Casasent, “Pattern recognition: a review,” IEEE Spectr., 28–33 (March1981).

D. Casasent, D. Psaltis, “Position oriented and scale-invariant optical correlation,” Appl. Opt. 15, 1793–1799 (1976).
[CrossRef]

Chahine, M.

M. Chahine, J. Konrad, “Estimation of trajectories for accelerated motion from time-varying imagery,” in Proceedings of the IEEE 1994 International Conference on Image Processing (ICIP’94) (IEEE Computer Society Press, Los Alamitos, Calif., 1994), Vol. II, pp. 800–804.

Chatterji, B. N.

B. S. Reddy, B. N. Chatterji, “An FFT-based technique for translation, rotation, and scale-invariant image registration,” IEEE Trans. Image Process. 5, 1266–1271 (1996).
[CrossRef] [PubMed]

Chellappa, R.

P. Burlina, R. Chellappa, “Analyzing looming motion components from their spatiotemporal spectral signature,” IEEE Trans. Pattern. Anal. Mach. Intell. 18, 1029–1033 (1996).
[CrossRef]

G.-S. J. Young, R. Chellappa, “3-D motion estimation using a sequence of noisy stereo images: models, estimation, and uniqueness results,” Trans. Pattern Anal. Mach. Intell. 12, 735–759 (1990).
[CrossRef]

Chen, W.

W. Chen, G. B. Giannakis, N. Nandhakumar, “Spatiotemporal approach for time-varying global image motion estimation,” IEEE Trans. Image Process. 5, 1448–1461 (1996).
[CrossRef] [PubMed]

Cortelazzo, G.

G. Cortelazzo, G. Nalesso, “A differential equation approach to the computation of the Fourier transform of images of translating objects,” IEEE Trans. Inf. Theory 40, 2049–2058 (1994).
[CrossRef]

G. Cortelazzo, M. Balanza, “Frequency domain analysis of translations with piecewise cubic trajectories,” IEEE Trans. Pattern. Anal. Mach. Intell. 15, 411–416 (1993).
[CrossRef]

G. Cortelazzo, M. Balanza, C. Monti, “Frequency domain analysis of rotational motion,” in Multidimensional Systems and Signal Processing, N. K. Bose, ed. (Kluwer Academic, Boston, Mass., 1993), Vol. 4, pp. 203–225.

Cortelazzo, G. M.

S. Alliney, G. M. Cortelazzo, G. A. Mian, “On the registration of an object translating on a static background,” Pattern Recogn. 29, 131–141 (1996).
[CrossRef]

L. Lucchese, G. M. Cortelazzo, C. Monti, “A frequency domain technique for estimating rigid planar rotations,” in Proceedings of the IEEE 1996 International Symposium on Circuits and Systems (ISCAS’96) (IEEE Computer Society Press, Los Alamitos, Calif., 1996), Vol. 2, pp. 774–777.

L. Lucchese, G. M. Cortelazzo, M. Rizzato, “A phase correlation technique for estimating rigid planar rotations,” in Proceedings of the 5th International Workshop on Time-Varying Image Processing and Moving Object Recognition (Elsevier Science B.V., Amsterdam, 1996), pp. 244–249.

L. Lucchese, G. M. Cortelazzo, “Frequency domain analysis of affine motion with linear time-dependence and finite duration,” in Proceedings of the IASTED (International Association of Science and Technology for Development) International Conference on Signal Processing and Communications (IASTED/Acta Press, Anaheim, Calif., 1998), pp. 28–31.

G. M. Cortelazzo, G. Doretto, L. Lucchese, S. Totaro, “A frequency domain method for registration of range data,” in Proceedings of the IEEE 1998 International Symposium on Circuits and Systems (ISCAS’98) (IEEE Computer Society Press, Los Alamitos, Calif., 1998), Vol. 5, pp. 518–521.

L. Lucchese, G. M. Cortelazzo, C. Monti, “Estimation of affine transformations between image pairs via Fourier transform,” in Proceedings of the IEEE 1996 International Conference on Image Processing (ICIP’96) (IEEE Computer Society Press, Los Alamitos, Calif., 1996), Vol. III, pp. 715–718.

G. M. Cortelazzo, G. Doretto, L. Lucchese, “Free-form textured surfaces registration by a frequency domain technique,” in Proceedings of the IEEE 1998 International Conference on Image Processing (ICIP’98) (IEEE Computer Society Press, Los Alamitos, Calif., 1998), Vol. I, pp. 813–817.

L. Lucchese, G. M. Cortelazzo, “Frequency domain estimation of 3-D rigid motion based on range and intensity data,” in Proceedings of the International Conference on Recent Advances in 3-D Digital Imaging and Modeling (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 107–112.

L. Lucchese, G. M. Cortelazzo, C. Monti, “High resolution estimation of planar rotations based on Fourier transform and radial projections,” in Proceedings of the IEEE 1997 International Symposium on Circuits and Systems (ISCAS’97) (IEEE Computer Society Press, Los Alamitos, Calif., 1997), Vol. II, pp. 1181–1184.

De Castro, E.

E. De Castro, C. Morandi, “Registration of translated and rotated images using finite Fourier transforms,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 700–703 (1987).
[CrossRef]

Doretto, G.

G. M. Cortelazzo, G. Doretto, L. Lucchese, “Free-form textured surfaces registration by a frequency domain technique,” in Proceedings of the IEEE 1998 International Conference on Image Processing (ICIP’98) (IEEE Computer Society Press, Los Alamitos, Calif., 1998), Vol. I, pp. 813–817.

G. M. Cortelazzo, G. Doretto, L. Lucchese, S. Totaro, “A frequency domain method for registration of range data,” in Proceedings of the IEEE 1998 International Symposium on Circuits and Systems (ISCAS’98) (IEEE Computer Society Press, Los Alamitos, Calif., 1998), Vol. 5, pp. 518–521.

Drewery, J. O.

J. O. Drewery, “The filtering of luminance and chrominance signals to avoid cross-colour in a PAL colour system,” (BBC, London, 1975).

Duvernoy, J.

Erdélyi, A.

A. Erdélyi, W. Magnus, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

Fleet, D. J.

D. J. Fleet, A. D. Jepson, “Computation of component image velocity from local phase information,” Int. J. Comput. Vis. 5, 77–104 (1990).
[CrossRef]

Giannakis, G. B.

W. Chen, G. B. Giannakis, N. Nandhakumar, “Spatiotemporal approach for time-varying global image motion estimation,” IEEE Trans. Image Process. 5, 1448–1461 (1996).
[CrossRef] [PubMed]

Goshtasby, A.

A. Goshtasby, “Template-matching on rotated images,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-7, 338–344 (1985).
[CrossRef]

Hoetter, M.

M. Hoetter, “Differential estimation of the global motion parameters zoom and pan,” Signal Process. 16, 249–265 (1989).
[CrossRef]

Huang, T. S.

J. Weng, T. S. Huang, N. Ahuja, “3-D motion estimation, understanding, and prediction from noisy image sequences,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-9, 370–389 (1987).
[CrossRef]

T. S. Huang, Image Sequence Analysis (Springer-Verlag, Berlin, 1983).

Ibrahim, M.

A. J. Patti, M. Ibrahim, A. M. Tekalp, “Digital video standards conversion in the presence of accelerated motion,” in Proceedings of the 1994 International Conference on Acoustics Speech and Signal Processing (ICASSP’94) (IEEE Computer Society Press, Los Alamitos, Calif., 1994), Vol. 5, pp. 225–228.

Jepson, A. D.

D. J. Fleet, A. D. Jepson, “Computation of component image velocity from local phase information,” Int. J. Comput. Vis. 5, 77–104 (1990).
[CrossRef]

Keesman, G.

G. Keesman, “Motion estimation based on a motion model incorporating translation, rotation and zoom,” Signal Process. 4, 31–34 (1988).

Kiss, I.

P. E. Zwicke, I. Kiss, “A new implementation of the Mellin transform and its application to radar classification of ships,” IEEE Trans. Pattern. Anal. Mach. Intell. PAMI-5, 191–198 (1983).
[CrossRef]

Kittler, J.

S. F. Wu, J. Kittler, “A differential method for simultaneous estimation of rotation, change of scale and translation,” Signal Process. Image Commun. 2, 69–80 (1990).
[CrossRef]

Koenderink, J. J.

Konrad, J.

M. Chahine, J. Konrad, “Estimation of trajectories for accelerated motion from time-varying imagery,” in Proceedings of the IEEE 1994 International Conference on Image Processing (ICIP’94) (IEEE Computer Society Press, Los Alamitos, Calif., 1994), Vol. II, pp. 800–804.

Kretz, F.

F. Kretz, J. Sabatier, “Échantillonage des images de télévision: analyse dans le domain spatio-temporel et dans le domain de Fourier,” Ann. Telecommun. 36, 231–273 (1981).

Lucchese, L.

L. Lucchese, G. M. Cortelazzo, C. Monti, “Estimation of affine transformations between image pairs via Fourier transform,” in Proceedings of the IEEE 1996 International Conference on Image Processing (ICIP’96) (IEEE Computer Society Press, Los Alamitos, Calif., 1996), Vol. III, pp. 715–718.

G. M. Cortelazzo, G. Doretto, L. Lucchese, S. Totaro, “A frequency domain method for registration of range data,” in Proceedings of the IEEE 1998 International Symposium on Circuits and Systems (ISCAS’98) (IEEE Computer Society Press, Los Alamitos, Calif., 1998), Vol. 5, pp. 518–521.

L. Lucchese, G. M. Cortelazzo, “Frequency domain analysis of affine motion with linear time-dependence and finite duration,” in Proceedings of the IASTED (International Association of Science and Technology for Development) International Conference on Signal Processing and Communications (IASTED/Acta Press, Anaheim, Calif., 1998), pp. 28–31.

L. Lucchese, G. M. Cortelazzo, C. Monti, “A frequency domain technique for estimating rigid planar rotations,” in Proceedings of the IEEE 1996 International Symposium on Circuits and Systems (ISCAS’96) (IEEE Computer Society Press, Los Alamitos, Calif., 1996), Vol. 2, pp. 774–777.

L. Lucchese, G. M. Cortelazzo, M. Rizzato, “A phase correlation technique for estimating rigid planar rotations,” in Proceedings of the 5th International Workshop on Time-Varying Image Processing and Moving Object Recognition (Elsevier Science B.V., Amsterdam, 1996), pp. 244–249.

G. M. Cortelazzo, G. Doretto, L. Lucchese, “Free-form textured surfaces registration by a frequency domain technique,” in Proceedings of the IEEE 1998 International Conference on Image Processing (ICIP’98) (IEEE Computer Society Press, Los Alamitos, Calif., 1998), Vol. I, pp. 813–817.

L. Lucchese, G. M. Cortelazzo, “Frequency domain estimation of 3-D rigid motion based on range and intensity data,” in Proceedings of the International Conference on Recent Advances in 3-D Digital Imaging and Modeling (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 107–112.

L. Lucchese, G. M. Cortelazzo, C. Monti, “High resolution estimation of planar rotations based on Fourier transform and radial projections,” in Proceedings of the IEEE 1997 International Symposium on Circuits and Systems (ISCAS’97) (IEEE Computer Society Press, Los Alamitos, Calif., 1997), Vol. II, pp. 1181–1184.

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S. Alliney, G. M. Cortelazzo, G. A. Mian, “On the registration of an object translating on a static background,” Pattern Recogn. 29, 131–141 (1996).
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L. Lucchese, G. M. Cortelazzo, C. Monti, “A frequency domain technique for estimating rigid planar rotations,” in Proceedings of the IEEE 1996 International Symposium on Circuits and Systems (ISCAS’96) (IEEE Computer Society Press, Los Alamitos, Calif., 1996), Vol. 2, pp. 774–777.

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L. Lucchese, G. M. Cortelazzo, C. Monti, “Estimation of affine transformations between image pairs via Fourier transform,” in Proceedings of the IEEE 1996 International Conference on Image Processing (ICIP’96) (IEEE Computer Society Press, Los Alamitos, Calif., 1996), Vol. III, pp. 715–718.

L. Lucchese, G. M. Cortelazzo, C. Monti, “High resolution estimation of planar rotations based on Fourier transform and radial projections,” in Proceedings of the IEEE 1997 International Symposium on Circuits and Systems (ISCAS’97) (IEEE Computer Society Press, Los Alamitos, Calif., 1997), Vol. II, pp. 1181–1184.

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T. R. Reed, “The analysis of motion in natural scenes using a spatiotemporal/spatiotemporal-frequency representation,” in Proceedings of IEEE 1997 International Conference on Image Processing (ICIP’97) (IEEE Computer Society Press, Los Alamitos, Calif., 1997), Vol. I, pp. 93–96.

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L. Lucchese, G. M. Cortelazzo, M. Rizzato, “A phase correlation technique for estimating rigid planar rotations,” in Proceedings of the 5th International Workshop on Time-Varying Image Processing and Moving Object Recognition (Elsevier Science B.V., Amsterdam, 1996), pp. 244–249.

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A. Erdélyi, W. Magnus, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

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Y. T. Tse, R. L. Baker, “Global zoom/pan estimation and compensation for video compression,” in Proceedings of the 1991 International Conference on Acoustics, Speech and Signal Processing (ICASSP’91) (IEEE Computer Society Press, Los Alamitos Calif., 1991), Vol. 4, pp. 2725–2728.

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W. Chen, G. B. Giannakis, N. Nandhakumar, “Spatiotemporal approach for time-varying global image motion estimation,” IEEE Trans. Image Process. 5, 1448–1461 (1996).
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R. Wilson, A. D. Calway, E. R. Pearson, “A generalized wavelet transform for Fourier analysis: the multiresolution Fourier transform and its application to image and audio signals analysis,” IEEE Trans. Inf. Theory 38, 674–690 (1992).
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S. Alliney, “Digital analysis of rotated images,” IEEE Trans. Pattern. Anal. Mach. Intell. 15, 499–504 (1993).
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G. Cortelazzo, M. Balanza, “Frequency domain analysis of translations with piecewise cubic trajectories,” IEEE Trans. Pattern. Anal. Mach. Intell. 15, 411–416 (1993).
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Other (21)

G. Cortelazzo, M. Balanza, C. Monti, “Frequency domain analysis of rotational motion,” in Multidimensional Systems and Signal Processing, N. K. Bose, ed. (Kluwer Academic, Boston, Mass., 1993), Vol. 4, pp. 203–225.

A. Papoulis, Systems and Transforms with Applications in OpticsMcGraw-Hill, New York, 1968.

M. Abramovitz, I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Abramovitz, New York, 1968).

L. Lucchese, G. M. Cortelazzo, C. Monti, “A frequency domain technique for estimating rigid planar rotations,” in Proceedings of the IEEE 1996 International Symposium on Circuits and Systems (ISCAS’96) (IEEE Computer Society Press, Los Alamitos, Calif., 1996), Vol. 2, pp. 774–777.

L. Lucchese, G. M. Cortelazzo, M. Rizzato, “A phase correlation technique for estimating rigid planar rotations,” in Proceedings of the 5th International Workshop on Time-Varying Image Processing and Moving Object Recognition (Elsevier Science B.V., Amsterdam, 1996), pp. 244–249.

L. Lucchese, G. M. Cortelazzo, C. Monti, “High resolution estimation of planar rotations based on Fourier transform and radial projections,” in Proceedings of the IEEE 1997 International Symposium on Circuits and Systems (ISCAS’97) (IEEE Computer Society Press, Los Alamitos, Calif., 1997), Vol. II, pp. 1181–1184.

J. O. Drewery, “The filtering of luminance and chrominance signals to avoid cross-colour in a PAL colour system,” (BBC, London, 1975).

Y. T. Tse, R. L. Baker, “Global zoom/pan estimation and compensation for video compression,” in Proceedings of the 1991 International Conference on Acoustics, Speech and Signal Processing (ICASSP’91) (IEEE Computer Society Press, Los Alamitos Calif., 1991), Vol. 4, pp. 2725–2728.

A. B. Watson, A. J. Ahumada, “A look at motion in the frequency domain,” (NASA Ames Research Center, Moffett Field, Calif., 1983).

M. Chahine, J. Konrad, “Estimation of trajectories for accelerated motion from time-varying imagery,” in Proceedings of the IEEE 1994 International Conference on Image Processing (ICIP’94) (IEEE Computer Society Press, Los Alamitos, Calif., 1994), Vol. II, pp. 800–804.

A. J. Patti, M. Ibrahim, A. M. Tekalp, “Digital video standards conversion in the presence of accelerated motion,” in Proceedings of the 1994 International Conference on Acoustics Speech and Signal Processing (ICASSP’94) (IEEE Computer Society Press, Los Alamitos, Calif., 1994), Vol. 5, pp. 225–228.

A. M. Tekalp, Digital Video Processing (Prentice-Hall, Englewood Cliffs, N.J., 1995).

T. R. Reed, “The analysis of motion in natural scenes using a spatiotemporal/spatiotemporal-frequency representation,” in Proceedings of IEEE 1997 International Conference on Image Processing (ICIP’97) (IEEE Computer Society Press, Los Alamitos, Calif., 1997), Vol. I, pp. 93–96.

M. Bennamoun, “Application of time-frequency signal analysis to motion estimation,” in Proceedings of the IEEE 1997 International Conference on Image Processing (ICIP’97), (IEEE Computer Society Press, Los Alamitos, Calif., 1997), Vol. II, pp. 148–151.

T. S. Huang, Image Sequence Analysis (Springer-Verlag, Berlin, 1983).

L. Lucchese, G. M. Cortelazzo, “Frequency domain analysis of affine motion with linear time-dependence and finite duration,” in Proceedings of the IASTED (International Association of Science and Technology for Development) International Conference on Signal Processing and Communications (IASTED/Acta Press, Anaheim, Calif., 1998), pp. 28–31.

L. Lucchese, G. M. Cortelazzo, C. Monti, “Estimation of affine transformations between image pairs via Fourier transform,” in Proceedings of the IEEE 1996 International Conference on Image Processing (ICIP’96) (IEEE Computer Society Press, Los Alamitos, Calif., 1996), Vol. III, pp. 715–718.

L. Lucchese, G. M. Cortelazzo, “Frequency domain estimation of 3-D rigid motion based on range and intensity data,” in Proceedings of the International Conference on Recent Advances in 3-D Digital Imaging and Modeling (IEEE Computer Society Press, Los Alamitos, Calif., 1997), pp. 107–112.

G. M. Cortelazzo, G. Doretto, L. Lucchese, S. Totaro, “A frequency domain method for registration of range data,” in Proceedings of the IEEE 1998 International Symposium on Circuits and Systems (ISCAS’98) (IEEE Computer Society Press, Los Alamitos, Calif., 1998), Vol. 5, pp. 518–521.

G. M. Cortelazzo, G. Doretto, L. Lucchese, “Free-form textured surfaces registration by a frequency domain technique,” in Proceedings of the IEEE 1998 International Conference on Image Processing (ICIP’98) (IEEE Computer Society Press, Los Alamitos, Calif., 1998), Vol. I, pp. 813–817.

A. Erdélyi, W. Magnus, F. G. Tricomi, Tables of Integral Transforms (McGraw-Hill, New York, 1954).

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Figures (12)

Fig. 1
Fig. 1

Example of rototranslational motion: signal (a) at time t=t0 and (b) at a generic time t>t0.

Fig. 2
Fig. 2

Circular sector.

Fig. 3
Fig. 3

(a) Approximation of the magnitude of Ls(k) relative to the circular sector corresponding only to the terms from N=-5 to N=+5 in Eq. (24); (b) detail at low frequencies.

Fig. 4
Fig. 4

Magnitude of Lsn(k) for n=0, 1, 2, 3 (from top-left to bottom-right).

Fig. 5
Fig. 5

Planes described by Eq. (31).

Fig. 6
Fig. 6

Directional vectors of the plane set (h), h=0, 1,, H-1.

Fig. 7
Fig. 7

Qualitative illustration of the contribution to Ψˆ(k, f) from the sinc terms of Eq. (56) on the plane ky=0 (notation vx refers to the component of the vector v along the x axis).

Fig. 8
Fig. 8

Region V={(k, f)3|(0,-TkTv2)f(0,-TkTv2)}.

Fig. 9
Fig. 9

Magnitude of Ψ(k, f) at frequencies f=0, 50, 200, 500 for v2=[1;-2]T and T=1.

Fig. 10
Fig. 10

Qualitative illustration of the contribution of the functions sinc[Δ(f+kT(v2th+v1))], h=0,1,, H-1, to Eq. (56).

Fig. 11
Fig. 11

Trajectory of the circular sector.

Fig. 12
Fig. 12

Magnitude of L(k, f) relative to the circular sector at the frequencies f reported.

Equations (120)

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

x=R(t)x0+xC(t),
R(t)=cos θ(t)-sin θ(t)sin θ(t)cos θ(t)SO(2),
θ(t)=l=1Lωltll!,ωl,-<t<+,
xC(t)=m=1Mvmtmm!,vm2,-<t<+.
l(x, t)=ls(R-1(t)[x-xC(t)]).
x0=r cos φ,r0,
y0=r sin φ,0φ<2π,
kx=kr cos kφ,kr0,
ky=kr sin kφ,0kφ<2π.
L(k, f)F[l(x, t)|k, f]
=-+l(x, t)exp[-j2π(kTx+ft)]dxdt=-+ls(R-1(t)[x-xC(t)])×exp[-j2π(kTx+ft)]dxdt.
ls(R-1(t)[x-xC(t)])
=ls(RT(t)[x-xC(t)])=ls([x-xC(t)] cos θ(t)+[y-yC(t)] sin θ(t), [x-xC(t)] sin θ(t)+[y-yC(t)] cos θ(t))=ls(r cos φ cos θ(t)+r sin φ sin θ(t), -r cos φ sin θ(t)+r sin φ cos θ(t))=ls(r cos[φ-θ(t)], r sin[φ-θ(t)])
l¯s(r, φ-θ(t)).
L¯(kr, kφ, f)L(kr cos kφ, kr sin kφ, f)
=-+-ππ0rl¯s(r, φ-θ(t))×exp[-j2πrkr(cos kφ cos φ+sin kφ sin φ)]×exp{-j2πkr[xC(t)cos kφ+yC(t)sin kφ]}×exp(-j2πft)drdφdt
=-+-ππ0rl¯s(r, φ-θ(t))×exp[-j2πrkr cos(φ-kφ)]×exp{-j2πkr[xC(t)×coskφ+yC(t)sin kφ]}exp(-j2πft)drdφdt.
l¯s(r, φ-θ(t))=n=-+an(r)exp{jn[φ-θ(t)]},
an(r)=12π-ππl¯s(r, φ)exp(-jnφ)dφ.
L¯(kr, kφ, f)
=-+-ππ0rn=-+an(r)×exp{jn[φ-θ(t)]} exp[-j2πrkr cos(φ-kφ)]×exp{-j2πkr[xC(t)cos kφ+yC(t)sin kφ]}×exp(-j2πft)drdφdt=n=-+-ππ0ran(r)exp(jnφ)×exp[-j2πrkr cos(φ-kφ)]drdφ×-+ exp[-jnθ(t)]exp{-j2πkr×[xC(t)cos kφ+yC(t)sin kφ]}exp(-j2πft)dt=n=-+L¯sn(kr, kφ)m=1MT¯m(kr, kφ,.)l=1LRl(., n)(f),
L¯sn(kr, kφ)2π exp[jn(kφ-π/2)]S¯n(2πkr),
S¯n(2πkr)=0ran(r)Jn(2πrkr)dr,
Jn(2πrkr)=12π-ππ exp[j(nφ-2πrkr sin φ)]dφ,
T¯m(kr, kφ, f)Fexp-j2πkr[cos kφsin kφ]vmtmm!|f,
Rl(f, n)Fexp-jnωltll!|f,
L(k, f)=n=-+Lsn(k)P(M,L)(k, f, n),
Lsn(k)L¯sn(kx2+ky2, arctan(ky/kx)),
P(M,L)(k, f, n)m=1MTm(k,.)l=1LRl(., n)(f),
Tm(k, f)T¯m(kx2+ky2, arctan(ky/kx), f).
l¯s(r, φ)
=1if0rR,γ-αφγ+α0elsewhere,
Ls(k)=n=-+Lsn(k),
Lsn(k)=2π exp[jn(arctan(ky/kx)-π/2)]Sn(2πk),
Sn(2πk)S¯n(2πkr)
=απexp(-jnγ)sincnαπ×14π2(kx2+ky2)02πRkx2+ky2τJn(τ)dτ.
T1(k, f)=δ(f+kTv1),
R1(f, n)=δ(f+nω1/2π),
P(1)(k, f, n)[T1(k,.)R1(., n)](f)
=δ(f+kTv1+nω1/2π),
L(k, f)=n=-+Lsn(k)δ(f+kTv1+nω1/2π).
f+kTv1+nω1/2π=0,n.
d(1)=|ω1|2π[v12+1]1/2
T2(k, f)=δ(f)ifkTv2=01|kTv2|expjπf2kTv2-14sign(kTv2)  ifkTv20,
R2(f, n)=δ(f)ifnω2=02π|nω2|expjπ2πf2nω2  -14sign(nω2)ifnω20.
P(2)(k, f, n)
[P(1)(k,., n)T2(k,.)R2(., n)](f)
=δ(f+kTv1+nω1/2π)ifkTv2+nω2/2π=0Q(k, f, n)ifkTv2+nω2/2π0,
Q(k, f, n)1[|kTv2+nω2/2π|]1/2
×expjπ(f+kTv1+nω1/2π)2kTv2+nω2/2π-14sign(kTv2+nω2/2π)
L(k, f)=n=-+Lsn(k)P(2)(k, f, n).
|Q(k, f, n)|=1[|kTv2+nω2/2π|]1/2
arg[Q(k, f, n)]=π(f+kTv1+nω1/2π)2kTv2+nω2/2π-π4sign(kTv2+nω2/2π).
d(2)=|ω2|2πv2.
w(t)rectt-T/2T=1if0tT0elsewhere.
W(f)F[w(t)|f]=T exp(-jπfT)sinc(fT),
L(k, f)F[l(x, t)w(t)|k, f]
=n=+Lsn(k)PW(M,L)(k, f, n),
PW(M,L)(k, f, n)(P(M,L)(k,., n)W(.))(f).
PW(1)(k, f, n)=(P(1)(k,., n)W(.))(f)=[δ(.+kTv1+nω1/2π)W(.)](f)=W(f+kTv1+nω1/2π).
L(k, f)=n=-+Lsn(k)W(f+kTv1+nω1/2π).
PW(2)(k, f, n)=(P(2)(k,., n)W(.))(f)=PW(1)(k, f, n)ifkTv2+nω2/2π=0QW(k, f, n)ifkTv2+nω2/2π0,
QW(k, f, n)Q(k, f, n)Ψ(k, f, n),
Ψ(k, f, n)12erfT+v(k, f, n)u(k, n)c(k, n)
-erfv(k, f, n)u(k, n)c(k, n),
u(k, n)kTv2+nω2/2π,
v(k, f, n)f+kTv1+nω1/2π,
c(k, n)π|u(k, n)| expjπ21-12sign[u(k, n)].
L(k, f)=n=-+Lsn(k)PW(2)(k, f, n),
L(k, f)=n=-+Lsn(k)QW(k, f, n)=n=-+Lsn(k)Q(k, f, n)Ψ(k, f, n).
l(x, t)=lsx-v2t22lsx-h=0H-1[v1(h)(t-h Δ)]×rectt-Δ(h+1/2)Δ,
L(k, f)Ls(k)Δh=0H-1sinc[Δ(f+kTv1(h))]×exp-j2πΔfh+12+kTv1(h)2(h+1)
Ls(k)Ψˆ(k, f),
Ψˆ(k, f)Δh=0H-1sinc[Δ(f+kTv1(h))]
×exp-j2πΔfh+12+kTv1(h)2(h+1).
L(k, f)=Ls(k)Q(k, f,0)Ψ˜(k, f).
(h){kTv1(h)+f=0}h=0,1,,H-1.
V={(k, f)3|min(0,-TkTv2)fmax(0,-TkTv2)}
Ψ˜(k, f)=12erfT+fkTv2c˜(k)-erffkTv2c˜(k),
c˜(k)c(k, 0)
=π|kTv2| expjπ21-12sign(kTv2).
Ψ˜(k, 0)=12erf[Tc˜(k)],
Ψ˜(k, f+kTv1)=12erfT+f+kTv1kTv2c˜(k)-erff+kTv1kTv2c˜(k),
V={(k, f)3|min[-kTv1,-kT(v1+Tv2)]fmax[-kTv1,-kT(v1+Tv2)]}.
Vn=(k, f)3|min-kTv1+nω12π, -kT(v1+Tv2)+n2π(ω1+Tω2)fmax-kTv1+nω12π, -kT(v1+Tv2)+n2π(ω1+Tω2).
ϕ=arccosv12+Tv1·v2+1v12+2Tv1·v2+T2v22+1v12+1,
an(r)=12π-ππl¯s(r, φ) exp(-jnφ)dφ=12πγ-αγ+α exp(-jnφ)dφrectr-R/2R=exp(-jnγ)×exp(jnα)-exp(-jnα)2πjnrectr-R/2R=απexp(-jnγ)sincnαπrectr-R/2R.
S¯n(2πkr)=0ran(r)Jn(2πrkr)dr=απexp(-jnγ)sincnαπ0RrJn(2πrkr)dr=απexp(-jnγ)sincnαπ14π2kr2×02πRkrτJn(τ)dτ.
τnJn-1(τ)dτ=τnJn(τ).
S¯0(2πkr)=R2απJ1(2πRkr)2πRkr.
Sn(2πkx, 2πky)S¯n(2πkr)
=απexp(-jnγ)sincnαπ14π2(kx2+ky2)×02πRkx2+ky2τJn(τ)dτ.
W(f)F[w(t)|f]=-+w(t)exp(-j2πft)dt=0T exp(-j2πft)dt.
a1|kTv2|exp-jπ4sign(kTv2),bkTv2.
QW(k, f)=[T2(k,.)W(.)](f)=-+a expjπλ2bW(f-λ)dλ=a-+ expjπλ2b×0T exp[-j2π(f-λ)t]dtdλ=a0T exp(-j2πft)×-+ expjπλ2b+2λtdλdt=a0T exp-j2πft+bt22×-+ expjπb(λ+bt)2dλdt.
QW(k, f)=a0T exp-j2πft+bt22dt×-+ expjπν2bdν.
F[exp(jσt2)|ω]=-+ exp(jσt2)exp(-jωt)dt=π/σ expjπ4exp-jω24σ,
-+ expjπν2bdν=Fexpjπν2b|ωω=0=|b| expjπ21-12sign(b).
0T exp-j2πft+bt22dt
=expjπf2b0T exp-jπbt+fb2dt=expjπf2bf/bT+f/b exp(-jπbτ2)dτ.
erfx expjπ4=2b expjπ40x/πb exp(-jπbξ2)dξ,
f/bT+f/b exp(-jπbτ2)dτ
=12bexp-jπ4erfT+fbπb expjπ4-erffbπb expjπ4.
QW(k, f)=12a expjπf2berfT+fbπ|b|×expjπ21-12sign(b)-erffbπ|b| expjπ21-12sign(b),
QW(k, f)=T2(k, f)Ψ˜(k, f),
Ψ˜(k, f)12erfT+fkTv2c˜(k)
-erffkTv2c˜(k),
c˜(k)π|kTv2| expjπ21-12sign(kTv2).
erf(z)=2 expiπ4Cz2π exp-iπR-iSz2π exp-iπ4,
C(z)0z cosπτ22dτ,S(z)0z sinπτ22dτ.
C(z)=12+f(z)sinπ2|z|2-g(z)cosπ2|z|2,
S(z)=12-f(z)cosπ2|z|2-g(z)sinπ2|z|2,
f(z)=1+0.926|z|2+1.792|z|+3.104|z|2,
g(z)=12+4.142|z|+3.492|z|2+6.670|z|3.
kT(v1+Tv2)+f+n2π(ω1+Tω2)=0,
kTv1+f+n2πω1=0,
kTv2+n2πω2=0,
kTv1+f+n2πω1=0.
p=e1e2e3v2xv2y0v1xv1y1=v2ye1-v2xe2+(v2xv1y-v1xv2y)e3,
dn=|nω2|2πv2

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