Abstract

A number of techniques to track rainfall patterns by use of radar observations have been developed over the years. We present a method for radar-echo tracking based on Hu invariant moments. The method has been tried on several sequences of test images, and the derived displacement fields were in good agreement with the real motions of the tested objects. For the real data obtained from the conventional meteorological radar in Legionowo the method occasionally failed when changes in the radar echo between observations were too large.

© 2003 Optical Society of America

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  1. K. A. Browning, “Meteorological application of radar, Rep. Prog. Phys. 41, 761–806 (1978).
    [CrossRef]
  2. R. E. Rinehart, E. T. Garvey, “Three-dimensional storm motion detected by conventional radar,” Nature 27, 287–289 (1978).
    [CrossRef]
  3. D. Atlas, ed., Radar in Meteorology: Battan Memorial and 40th Aniversary Radar Meteorology Conference (American Meteorological Society, New York, 1990).
  4. S. Laroche, I. Zawadzki, “Echo tracking by three variational analysis methods,” in Proceedings of the 25th Conference on Radar Meteorology (American Meteorological Society, Paris, 1991), pp. 438–440.
  5. S. Moszkowicz, “Algorithms for meteorological phenomena recognition and AP echoes suppression on automatic weather radar system AMFR,” in Proceedings of the Weather Radar Systems International Seminar (European Commission DGXII, Luxembourg, 1995), pp. 239–248.
  6. S. Laroche, I. Zawadzki, ““Retrieval of horizontal winds from single-Doppler clear-air data by methods of cross correlation and variational analysis, J. Atmos. Oceanic Technol. 12, 721–738 (1995).
    [CrossRef]
  7. M. Dixon, G. Wiener, ““TITAN: thunderstorm identification, tracking, analysis and nowcasting—a radar based methodology, J. Atmos. Oceanic Technol. 10, 785–797 (1993).
    [CrossRef]
  8. S. Moszkowicz, “Small-scale structure of rain field—preliminary results basing on a digital gauge network and on MRL-5 Legionowo radar,” Phys. Chem. Earth B 25, 933–938 (2000).
    [CrossRef]
  9. L. Li, W. Schmidt, J. Jass, “Nowcasting of motion and growth of precipitation with radar over complex orography,” J. Appl. Meteorol. 34, 1286–1300 (1995).
    [CrossRef]
  10. V. Kim, L. P. Yaroslavsky, “Rank algorithms for picture processing,” Comput. Vision Graphics Image Process. 35, 234–258 (1986).
    [CrossRef]
  11. H. H. Arsenault, T. Szoplik, B. Macukow, eds., Optical Processing and Computing (Academic, Boston, 1989), pp. 315–390, 441–467.
    [CrossRef]
  12. P. Sobey, M. V. Srinivasan, “Measurement of optical flow by a generalized gradient scheme,” J. Opt. Soc. Am. A 6, 1488–1498 (1991).
    [CrossRef]
  13. M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory IT-8, 179–187 (1962).
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    [CrossRef] [PubMed]
  15. S. Dudani, K. Breeding, R. McGhee, “Aircraft identification by moment invariants,” IEEE Trans. Comput. C-26, 39–45 (1977).
    [CrossRef]
  16. C. H. Teh, R. T. Chin, “On image analysis by the methods of moments,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 496–513 (1988).
    [CrossRef]
  17. Y. Sheng, L. Shen, “Orthogonal Fourier-Mellin moments for invariant pattern recognition,” J. Opt. Soc. Am. A 11, 1748–1757 (1994).
    [CrossRef]
  18. D. Tang, D. Fu, L. Li, S. Cai, M. Hu, ““Measuring the inner-motions in echo area with a conventional weather radar and the comparison with Doppler radar, in Proceedings of the 25th Conference on Radar Meteorology (American Meteorological Society, Paris, 1991), pp. 646–649.

2000 (1)

S. Moszkowicz, “Small-scale structure of rain field—preliminary results basing on a digital gauge network and on MRL-5 Legionowo radar,” Phys. Chem. Earth B 25, 933–938 (2000).
[CrossRef]

1995 (2)

L. Li, W. Schmidt, J. Jass, “Nowcasting of motion and growth of precipitation with radar over complex orography,” J. Appl. Meteorol. 34, 1286–1300 (1995).
[CrossRef]

S. Laroche, I. Zawadzki, ““Retrieval of horizontal winds from single-Doppler clear-air data by methods of cross correlation and variational analysis, J. Atmos. Oceanic Technol. 12, 721–738 (1995).
[CrossRef]

1994 (1)

1993 (1)

M. Dixon, G. Wiener, ““TITAN: thunderstorm identification, tracking, analysis and nowcasting—a radar based methodology, J. Atmos. Oceanic Technol. 10, 785–797 (1993).
[CrossRef]

1991 (1)

P. Sobey, M. V. Srinivasan, “Measurement of optical flow by a generalized gradient scheme,” J. Opt. Soc. Am. A 6, 1488–1498 (1991).
[CrossRef]

1988 (1)

C. H. Teh, R. T. Chin, “On image analysis by the methods of moments,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 496–513 (1988).
[CrossRef]

1987 (1)

1986 (1)

V. Kim, L. P. Yaroslavsky, “Rank algorithms for picture processing,” Comput. Vision Graphics Image Process. 35, 234–258 (1986).
[CrossRef]

1978 (2)

K. A. Browning, “Meteorological application of radar, Rep. Prog. Phys. 41, 761–806 (1978).
[CrossRef]

R. E. Rinehart, E. T. Garvey, “Three-dimensional storm motion detected by conventional radar,” Nature 27, 287–289 (1978).
[CrossRef]

1977 (1)

S. Dudani, K. Breeding, R. McGhee, “Aircraft identification by moment invariants,” IEEE Trans. Comput. C-26, 39–45 (1977).
[CrossRef]

1962 (1)

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory IT-8, 179–187 (1962).

Breeding, K.

S. Dudani, K. Breeding, R. McGhee, “Aircraft identification by moment invariants,” IEEE Trans. Comput. C-26, 39–45 (1977).
[CrossRef]

Browning, K. A.

K. A. Browning, “Meteorological application of radar, Rep. Prog. Phys. 41, 761–806 (1978).
[CrossRef]

Cai, S.

D. Tang, D. Fu, L. Li, S. Cai, M. Hu, ““Measuring the inner-motions in echo area with a conventional weather radar and the comparison with Doppler radar, in Proceedings of the 25th Conference on Radar Meteorology (American Meteorological Society, Paris, 1991), pp. 646–649.

Chin, R. T.

C. H. Teh, R. T. Chin, “On image analysis by the methods of moments,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 496–513 (1988).
[CrossRef]

Dixon, M.

M. Dixon, G. Wiener, ““TITAN: thunderstorm identification, tracking, analysis and nowcasting—a radar based methodology, J. Atmos. Oceanic Technol. 10, 785–797 (1993).
[CrossRef]

Dudani, S.

S. Dudani, K. Breeding, R. McGhee, “Aircraft identification by moment invariants,” IEEE Trans. Comput. C-26, 39–45 (1977).
[CrossRef]

Duvernoy, J.

Fu, D.

D. Tang, D. Fu, L. Li, S. Cai, M. Hu, ““Measuring the inner-motions in echo area with a conventional weather radar and the comparison with Doppler radar, in Proceedings of the 25th Conference on Radar Meteorology (American Meteorological Society, Paris, 1991), pp. 646–649.

Garvey, E. T.

R. E. Rinehart, E. T. Garvey, “Three-dimensional storm motion detected by conventional radar,” Nature 27, 287–289 (1978).
[CrossRef]

Hu, M.

D. Tang, D. Fu, L. Li, S. Cai, M. Hu, ““Measuring the inner-motions in echo area with a conventional weather radar and the comparison with Doppler radar, in Proceedings of the 25th Conference on Radar Meteorology (American Meteorological Society, Paris, 1991), pp. 646–649.

Hu, M. K.

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory IT-8, 179–187 (1962).

Jass, J.

L. Li, W. Schmidt, J. Jass, “Nowcasting of motion and growth of precipitation with radar over complex orography,” J. Appl. Meteorol. 34, 1286–1300 (1995).
[CrossRef]

Kim, V.

V. Kim, L. P. Yaroslavsky, “Rank algorithms for picture processing,” Comput. Vision Graphics Image Process. 35, 234–258 (1986).
[CrossRef]

Laroche, S.

S. Laroche, I. Zawadzki, ““Retrieval of horizontal winds from single-Doppler clear-air data by methods of cross correlation and variational analysis, J. Atmos. Oceanic Technol. 12, 721–738 (1995).
[CrossRef]

S. Laroche, I. Zawadzki, “Echo tracking by three variational analysis methods,” in Proceedings of the 25th Conference on Radar Meteorology (American Meteorological Society, Paris, 1991), pp. 438–440.

Li, L.

L. Li, W. Schmidt, J. Jass, “Nowcasting of motion and growth of precipitation with radar over complex orography,” J. Appl. Meteorol. 34, 1286–1300 (1995).
[CrossRef]

D. Tang, D. Fu, L. Li, S. Cai, M. Hu, ““Measuring the inner-motions in echo area with a conventional weather radar and the comparison with Doppler radar, in Proceedings of the 25th Conference on Radar Meteorology (American Meteorological Society, Paris, 1991), pp. 646–649.

McGhee, R.

S. Dudani, K. Breeding, R. McGhee, “Aircraft identification by moment invariants,” IEEE Trans. Comput. C-26, 39–45 (1977).
[CrossRef]

Moszkowicz, S.

S. Moszkowicz, “Small-scale structure of rain field—preliminary results basing on a digital gauge network and on MRL-5 Legionowo radar,” Phys. Chem. Earth B 25, 933–938 (2000).
[CrossRef]

S. Moszkowicz, “Algorithms for meteorological phenomena recognition and AP echoes suppression on automatic weather radar system AMFR,” in Proceedings of the Weather Radar Systems International Seminar (European Commission DGXII, Luxembourg, 1995), pp. 239–248.

Rinehart, R. E.

R. E. Rinehart, E. T. Garvey, “Three-dimensional storm motion detected by conventional radar,” Nature 27, 287–289 (1978).
[CrossRef]

Schmidt, W.

L. Li, W. Schmidt, J. Jass, “Nowcasting of motion and growth of precipitation with radar over complex orography,” J. Appl. Meteorol. 34, 1286–1300 (1995).
[CrossRef]

Shen, L.

Sheng, Y.

Sobey, P.

P. Sobey, M. V. Srinivasan, “Measurement of optical flow by a generalized gradient scheme,” J. Opt. Soc. Am. A 6, 1488–1498 (1991).
[CrossRef]

Srinivasan, M. V.

P. Sobey, M. V. Srinivasan, “Measurement of optical flow by a generalized gradient scheme,” J. Opt. Soc. Am. A 6, 1488–1498 (1991).
[CrossRef]

Tang, D.

D. Tang, D. Fu, L. Li, S. Cai, M. Hu, ““Measuring the inner-motions in echo area with a conventional weather radar and the comparison with Doppler radar, in Proceedings of the 25th Conference on Radar Meteorology (American Meteorological Society, Paris, 1991), pp. 646–649.

Teh, C. H.

C. H. Teh, R. T. Chin, “On image analysis by the methods of moments,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 496–513 (1988).
[CrossRef]

Wiener, G.

M. Dixon, G. Wiener, ““TITAN: thunderstorm identification, tracking, analysis and nowcasting—a radar based methodology, J. Atmos. Oceanic Technol. 10, 785–797 (1993).
[CrossRef]

Yaroslavsky, L. P.

V. Kim, L. P. Yaroslavsky, “Rank algorithms for picture processing,” Comput. Vision Graphics Image Process. 35, 234–258 (1986).
[CrossRef]

Zawadzki, I.

S. Laroche, I. Zawadzki, ““Retrieval of horizontal winds from single-Doppler clear-air data by methods of cross correlation and variational analysis, J. Atmos. Oceanic Technol. 12, 721–738 (1995).
[CrossRef]

S. Laroche, I. Zawadzki, “Echo tracking by three variational analysis methods,” in Proceedings of the 25th Conference on Radar Meteorology (American Meteorological Society, Paris, 1991), pp. 438–440.

Appl. Opt. (1)

Comput. Vision Graphics Image Process. (1)

V. Kim, L. P. Yaroslavsky, “Rank algorithms for picture processing,” Comput. Vision Graphics Image Process. 35, 234–258 (1986).
[CrossRef]

IEEE Trans. Comput. (1)

S. Dudani, K. Breeding, R. McGhee, “Aircraft identification by moment invariants,” IEEE Trans. Comput. C-26, 39–45 (1977).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

C. H. Teh, R. T. Chin, “On image analysis by the methods of moments,” IEEE Trans. Pattern Anal. Mach. Intell. 10, 496–513 (1988).
[CrossRef]

IRE Trans. Inf. Theory (1)

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Inf. Theory IT-8, 179–187 (1962).

J. Appl. Meteorol. (1)

L. Li, W. Schmidt, J. Jass, “Nowcasting of motion and growth of precipitation with radar over complex orography,” J. Appl. Meteorol. 34, 1286–1300 (1995).
[CrossRef]

J. Atmos. Oceanic Technol. (2)

S. Laroche, I. Zawadzki, ““Retrieval of horizontal winds from single-Doppler clear-air data by methods of cross correlation and variational analysis, J. Atmos. Oceanic Technol. 12, 721–738 (1995).
[CrossRef]

M. Dixon, G. Wiener, ““TITAN: thunderstorm identification, tracking, analysis and nowcasting—a radar based methodology, J. Atmos. Oceanic Technol. 10, 785–797 (1993).
[CrossRef]

J. Opt. Soc. Am. A (2)

P. Sobey, M. V. Srinivasan, “Measurement of optical flow by a generalized gradient scheme,” J. Opt. Soc. Am. A 6, 1488–1498 (1991).
[CrossRef]

Y. Sheng, L. Shen, “Orthogonal Fourier-Mellin moments for invariant pattern recognition,” J. Opt. Soc. Am. A 11, 1748–1757 (1994).
[CrossRef]

Nature (1)

R. E. Rinehart, E. T. Garvey, “Three-dimensional storm motion detected by conventional radar,” Nature 27, 287–289 (1978).
[CrossRef]

Phys. Chem. Earth B (1)

S. Moszkowicz, “Small-scale structure of rain field—preliminary results basing on a digital gauge network and on MRL-5 Legionowo radar,” Phys. Chem. Earth B 25, 933–938 (2000).
[CrossRef]

Rep. Prog. Phys. (1)

K. A. Browning, “Meteorological application of radar, Rep. Prog. Phys. 41, 761–806 (1978).
[CrossRef]

Other (5)

D. Atlas, ed., Radar in Meteorology: Battan Memorial and 40th Aniversary Radar Meteorology Conference (American Meteorological Society, New York, 1990).

S. Laroche, I. Zawadzki, “Echo tracking by three variational analysis methods,” in Proceedings of the 25th Conference on Radar Meteorology (American Meteorological Society, Paris, 1991), pp. 438–440.

S. Moszkowicz, “Algorithms for meteorological phenomena recognition and AP echoes suppression on automatic weather radar system AMFR,” in Proceedings of the Weather Radar Systems International Seminar (European Commission DGXII, Luxembourg, 1995), pp. 239–248.

D. Tang, D. Fu, L. Li, S. Cai, M. Hu, ““Measuring the inner-motions in echo area with a conventional weather radar and the comparison with Doppler radar, in Proceedings of the 25th Conference on Radar Meteorology (American Meteorological Society, Paris, 1991), pp. 646–649.

H. H. Arsenault, T. Szoplik, B. Macukow, eds., Optical Processing and Computing (Academic, Boston, 1989), pp. 315–390, 441–467.
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Radar images: (a) recorded at t 1 = 0, vectors show the displacement field of (b) recorded at t 2 = 20 min. For better visibility vectors are shown at double lengths.

Fig. 2
Fig. 2

The second sequence of radar images taken 1 h after that of Fig. 1(a). Image recorded at t 1 = 0, vectors show the displacement field of (b) recorded at t 2 = 20 min. For better visibility vectors are shown at double lengths.

Equations (24)

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

Rp, q=covp, qσt1σt2,
Ju, v=u-u02+v-v02dxdy,
ux+vy=0.
ηt+uηx+vηy=Sη,
dηdt=ηt+uηx+vηy=0.
η2r-η1r-vΔt=0.
J12=i=1Lη2ri-η1ri-vΔt2.
Xk+1=Xk+αkdk,
J12u=2Δti=1Lη2ri-η1ri-vΔtη1ri-vΔtx,
J12v=2Δti=1Lη2ri-η1ri-vΔtη1ri-vΔty.
mp,q=--xpyqfx, ydxdy,
x0=m1,0m0,0, y0=m0,1m0,0.
μp,q=--x-x0py-y0qfx, ydxdy.
H1=m2,0+m0,2,
H2=m2,0+m0,22+4m1,12,
H3=m3,0-3m1,22+3m2,1-m0,32,
H4=m3,0+m1,22+m2,1+m0,32,
H5=m3,0-3m1,2m3,0+m1,2m3,0+m1,22-3m2,1+m0,32+3m2,1-m0,3m2,1+m0,3×3m3,0+m1,22-m2,1+m0,32,
H6=m2,0-m0,2m3,0+m1,22-m2,1+m0,32+4m1,1m3,0+m1,2m2,1+m0,3,
H7=3m2,1-m0,3m3,0+m1,2m3,0+m1,22-3m3,0+m1,22-m3,0-3m1,2m3,0+m1,2×3m3,0+m1,22-m3,0-3m1,22.
mp,q=i=0Mj=0Nipjqηi, j,
μp,q=i=0Mj=0Ni-i0pj-j0qηi, j,
Φk=Hkj=16Hj2, k=1, , 6.
di2, j2=k=16Φki2, j2-Φki1, j12.

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