Abstract

We present an optical technique for finding the centroids of nonoverlapping objects in a scene, thus locating the objects and preserving the underlying advantage of matched filtering approaches to pattern recognition. One is then free to extract any feature desired at these centroid locations rather than restricted to the matched filter test statistic. Furthermore, this allows general feature extraction avoiding prior scene segmentation into individual objects. The technique can also be used for tracking the motion of rigid or nonrigid objects. It consists of cross-correlating the input f(x,y) with a windowed version of the function x + iy and detecting the zeros of the magnitude of the resulting correlation. At these points the x and y first moments vanish. The window is selected based on the size and separation of the objects in a scene. Experimental verification as well as restrictions are also presented.

© 1987 Optical Society of America

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References

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  1. Y. S. Fong, D. H. Brown, “A Centroid Tracking Scheme in a Weighted Coordinate System,” in IEEE 1985 Computer Vision and Pattern Recognition (1985), pp. 219–221.
  2. D. Cassasent, L. Cheatham, D. Fetterly, “Optical System to Compute Intensity Moments: Design,” Appl. Opt. 21, 3292 (1982).
    [CrossRef]
  3. M. K. Hu, “Pattern Recognition by Moment Invariants,” Proc. IRE 49, 1428 (1961).
  4. M. R. Teague, “Image Analysis via the General Theory of Moments,” J. Opt. Soc. Am. 70, 920 (1980).
    [CrossRef]
  5. Y. S. Abu-Mostafa, D. Psaltis, “Recognitive Aspects of Moment Invariants,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
    [CrossRef]
  6. Y. S. Abu-Mostafa, D. Psaltis, “Image Normalization by Complex Moments,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46 (1985).
    [CrossRef]
  7. C. H. Teh, R. T. Chin, “On Digital Approximation of Moment Invariants,” Comput. Vision Graphics Image Process. 33, 318 (1986).
    [CrossRef]
  8. A. W. Lohmann, C. Thum, Inefeaod Light Efficiency of Coherent-Optical Matched Filters,” Appl. Opt. 23, 1503 (1984).
    [CrossRef] [PubMed]
  9. D. Casasent, D. Fetterly, “Recent Advances in Optical Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 456, 105 (1984).
  10. D. Casasent, W. Rozzi, “Projection Synthetic Discriminant Function Performance,” Opt. Eng. 23, 716 (1984).
    [CrossRef]
  11. C. F. Hester, D. Casasent, “Inter-class Discrimination Using Synthetic Discriminant Functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108 (1981).
  12. D. Casasent, D. Psaltis, “New Optical Transforms for Pattern Recognition,” Proc. IEEE 65, 77 (1977).
    [CrossRef]
  13. D. Casasent, D. Psaltis, “Space–Bandwidth Product and Accuracy of the Optical Mellin Transform,” Appl. Opt. 16, 1472 (1977).
    [CrossRef] [PubMed]
  14. D. Casasent, M. Kraus, “Polar Camera for Space-Variant Pattern Recognition,” Appl. Opt. 17, 1559 (1978).
    [CrossRef] [PubMed]
  15. D. Casasent, D. Psaltis, “Multiple-Invariant Space-Variant Optical Processors,” Appl. Opt. 17, 655 (1978).
    [CrossRef] [PubMed]
  16. H. H. Arsenault, Y. N. Hsu, K. Chalasinska-Macukow, Y. Yang, “Rotation Invariant Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 359, 256 (1982).
  17. Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum Circular Symmetrical Filters and Their Uses in Optical Pattern Recognition,” Opt. Acta 29, 627 (1982).
    [CrossRef]
  18. T. Szoplik, H. H. Arsenault, “Shift and Scale-Invariant Anamorphic Fourier Correlator Using Multiple Circular Harmonic Filters,” Appl. Opt. 24, 3179 (1985).
    [CrossRef] [PubMed]
  19. H. J. Caulfield, W. T. Maloney, “Improved Discrimination in Optical Character Recognition,” Appl. Opt. 8, 2354 (1969).
    [CrossRef] [PubMed]
  20. T. A. Isberg, G. M. Morris, “Rotation-Invariant Recognition at Low Light Levels,” J. Opt. Soc. Am. A 2, P20 (1985).
  21. R. Wu, H. Stark, “Rotation-Invariant Pattern Recognition Using a Vector Reference,” Appl. Opt. 23, 838 (1984).
    [CrossRef] [PubMed]
  22. W. H. Lee, “Sampled Fourier Transform Hologram Generated by Computer,” Appl. Opt. 9, 639 (1970).
    [CrossRef] [PubMed]
  23. J. J. Burch, “A Computer Algorithm for the Synthesis of Spatial Frequency Filters,” IBM J. Res. Dev. 13, 160 (1969).

1986 (1)

C. H. Teh, R. T. Chin, “On Digital Approximation of Moment Invariants,” Comput. Vision Graphics Image Process. 33, 318 (1986).
[CrossRef]

1985 (4)

Y. S. Fong, D. H. Brown, “A Centroid Tracking Scheme in a Weighted Coordinate System,” in IEEE 1985 Computer Vision and Pattern Recognition (1985), pp. 219–221.

T. A. Isberg, G. M. Morris, “Rotation-Invariant Recognition at Low Light Levels,” J. Opt. Soc. Am. A 2, P20 (1985).

T. Szoplik, H. H. Arsenault, “Shift and Scale-Invariant Anamorphic Fourier Correlator Using Multiple Circular Harmonic Filters,” Appl. Opt. 24, 3179 (1985).
[CrossRef] [PubMed]

Y. S. Abu-Mostafa, D. Psaltis, “Image Normalization by Complex Moments,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46 (1985).
[CrossRef]

1984 (5)

R. Wu, H. Stark, “Rotation-Invariant Pattern Recognition Using a Vector Reference,” Appl. Opt. 23, 838 (1984).
[CrossRef] [PubMed]

Y. S. Abu-Mostafa, D. Psaltis, “Recognitive Aspects of Moment Invariants,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
[CrossRef]

A. W. Lohmann, C. Thum, Inefeaod Light Efficiency of Coherent-Optical Matched Filters,” Appl. Opt. 23, 1503 (1984).
[CrossRef] [PubMed]

D. Casasent, D. Fetterly, “Recent Advances in Optical Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 456, 105 (1984).

D. Casasent, W. Rozzi, “Projection Synthetic Discriminant Function Performance,” Opt. Eng. 23, 716 (1984).
[CrossRef]

1982 (3)

D. Cassasent, L. Cheatham, D. Fetterly, “Optical System to Compute Intensity Moments: Design,” Appl. Opt. 21, 3292 (1982).
[CrossRef]

H. H. Arsenault, Y. N. Hsu, K. Chalasinska-Macukow, Y. Yang, “Rotation Invariant Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 359, 256 (1982).

Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum Circular Symmetrical Filters and Their Uses in Optical Pattern Recognition,” Opt. Acta 29, 627 (1982).
[CrossRef]

1981 (1)

C. F. Hester, D. Casasent, “Inter-class Discrimination Using Synthetic Discriminant Functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108 (1981).

1980 (1)

1978 (2)

1977 (2)

1970 (1)

1969 (2)

J. J. Burch, “A Computer Algorithm for the Synthesis of Spatial Frequency Filters,” IBM J. Res. Dev. 13, 160 (1969).

H. J. Caulfield, W. T. Maloney, “Improved Discrimination in Optical Character Recognition,” Appl. Opt. 8, 2354 (1969).
[CrossRef] [PubMed]

1961 (1)

M. K. Hu, “Pattern Recognition by Moment Invariants,” Proc. IRE 49, 1428 (1961).

Abu-Mostafa, Y. S.

Y. S. Abu-Mostafa, D. Psaltis, “Image Normalization by Complex Moments,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46 (1985).
[CrossRef]

Y. S. Abu-Mostafa, D. Psaltis, “Recognitive Aspects of Moment Invariants,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
[CrossRef]

Arsenault, H. H.

T. Szoplik, H. H. Arsenault, “Shift and Scale-Invariant Anamorphic Fourier Correlator Using Multiple Circular Harmonic Filters,” Appl. Opt. 24, 3179 (1985).
[CrossRef] [PubMed]

Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum Circular Symmetrical Filters and Their Uses in Optical Pattern Recognition,” Opt. Acta 29, 627 (1982).
[CrossRef]

H. H. Arsenault, Y. N. Hsu, K. Chalasinska-Macukow, Y. Yang, “Rotation Invariant Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 359, 256 (1982).

Brown, D. H.

Y. S. Fong, D. H. Brown, “A Centroid Tracking Scheme in a Weighted Coordinate System,” in IEEE 1985 Computer Vision and Pattern Recognition (1985), pp. 219–221.

Burch, J. J.

J. J. Burch, “A Computer Algorithm for the Synthesis of Spatial Frequency Filters,” IBM J. Res. Dev. 13, 160 (1969).

Casasent, D.

D. Casasent, D. Fetterly, “Recent Advances in Optical Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 456, 105 (1984).

D. Casasent, W. Rozzi, “Projection Synthetic Discriminant Function Performance,” Opt. Eng. 23, 716 (1984).
[CrossRef]

C. F. Hester, D. Casasent, “Inter-class Discrimination Using Synthetic Discriminant Functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108 (1981).

D. Casasent, M. Kraus, “Polar Camera for Space-Variant Pattern Recognition,” Appl. Opt. 17, 1559 (1978).
[CrossRef] [PubMed]

D. Casasent, D. Psaltis, “Multiple-Invariant Space-Variant Optical Processors,” Appl. Opt. 17, 655 (1978).
[CrossRef] [PubMed]

D. Casasent, D. Psaltis, “New Optical Transforms for Pattern Recognition,” Proc. IEEE 65, 77 (1977).
[CrossRef]

D. Casasent, D. Psaltis, “Space–Bandwidth Product and Accuracy of the Optical Mellin Transform,” Appl. Opt. 16, 1472 (1977).
[CrossRef] [PubMed]

Cassasent, D.

Caulfield, H. J.

Chalasinska-Macukow, K.

H. H. Arsenault, Y. N. Hsu, K. Chalasinska-Macukow, Y. Yang, “Rotation Invariant Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 359, 256 (1982).

Cheatham, L.

Chin, R. T.

C. H. Teh, R. T. Chin, “On Digital Approximation of Moment Invariants,” Comput. Vision Graphics Image Process. 33, 318 (1986).
[CrossRef]

Fetterly, D.

D. Casasent, D. Fetterly, “Recent Advances in Optical Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 456, 105 (1984).

D. Cassasent, L. Cheatham, D. Fetterly, “Optical System to Compute Intensity Moments: Design,” Appl. Opt. 21, 3292 (1982).
[CrossRef]

Fong, Y. S.

Y. S. Fong, D. H. Brown, “A Centroid Tracking Scheme in a Weighted Coordinate System,” in IEEE 1985 Computer Vision and Pattern Recognition (1985), pp. 219–221.

Hester, C. F.

C. F. Hester, D. Casasent, “Inter-class Discrimination Using Synthetic Discriminant Functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108 (1981).

Hsu, Y. N.

H. H. Arsenault, Y. N. Hsu, K. Chalasinska-Macukow, Y. Yang, “Rotation Invariant Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 359, 256 (1982).

Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum Circular Symmetrical Filters and Their Uses in Optical Pattern Recognition,” Opt. Acta 29, 627 (1982).
[CrossRef]

Hu, M. K.

M. K. Hu, “Pattern Recognition by Moment Invariants,” Proc. IRE 49, 1428 (1961).

Isberg, T. A.

T. A. Isberg, G. M. Morris, “Rotation-Invariant Recognition at Low Light Levels,” J. Opt. Soc. Am. A 2, P20 (1985).

Kraus, M.

Lee, W. H.

Lohmann, A. W.

Maloney, W. T.

Morris, G. M.

T. A. Isberg, G. M. Morris, “Rotation-Invariant Recognition at Low Light Levels,” J. Opt. Soc. Am. A 2, P20 (1985).

Psaltis, D.

Y. S. Abu-Mostafa, D. Psaltis, “Image Normalization by Complex Moments,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46 (1985).
[CrossRef]

Y. S. Abu-Mostafa, D. Psaltis, “Recognitive Aspects of Moment Invariants,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
[CrossRef]

D. Casasent, D. Psaltis, “Multiple-Invariant Space-Variant Optical Processors,” Appl. Opt. 17, 655 (1978).
[CrossRef] [PubMed]

D. Casasent, D. Psaltis, “Space–Bandwidth Product and Accuracy of the Optical Mellin Transform,” Appl. Opt. 16, 1472 (1977).
[CrossRef] [PubMed]

D. Casasent, D. Psaltis, “New Optical Transforms for Pattern Recognition,” Proc. IEEE 65, 77 (1977).
[CrossRef]

Rozzi, W.

D. Casasent, W. Rozzi, “Projection Synthetic Discriminant Function Performance,” Opt. Eng. 23, 716 (1984).
[CrossRef]

Stark, H.

Szoplik, T.

Teague, M. R.

Teh, C. H.

C. H. Teh, R. T. Chin, “On Digital Approximation of Moment Invariants,” Comput. Vision Graphics Image Process. 33, 318 (1986).
[CrossRef]

Thum, C.

Wu, R.

Yang, Y.

H. H. Arsenault, Y. N. Hsu, K. Chalasinska-Macukow, Y. Yang, “Rotation Invariant Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 359, 256 (1982).

Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum Circular Symmetrical Filters and Their Uses in Optical Pattern Recognition,” Opt. Acta 29, 627 (1982).
[CrossRef]

Appl. Opt. (9)

Comput. Vision Graphics Image Process. (1)

C. H. Teh, R. T. Chin, “On Digital Approximation of Moment Invariants,” Comput. Vision Graphics Image Process. 33, 318 (1986).
[CrossRef]

IBM J. Res. Dev. (1)

J. J. Burch, “A Computer Algorithm for the Synthesis of Spatial Frequency Filters,” IBM J. Res. Dev. 13, 160 (1969).

IEEE 1985 Computer Vision and Pattern Recognition (1)

Y. S. Fong, D. H. Brown, “A Centroid Tracking Scheme in a Weighted Coordinate System,” in IEEE 1985 Computer Vision and Pattern Recognition (1985), pp. 219–221.

IEEE Trans. Pattern Anal. Mach. Intell. (2)

Y. S. Abu-Mostafa, D. Psaltis, “Recognitive Aspects of Moment Invariants,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 698–706 (1984).
[CrossRef]

Y. S. Abu-Mostafa, D. Psaltis, “Image Normalization by Complex Moments,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-7, 46 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

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

T. A. Isberg, G. M. Morris, “Rotation-Invariant Recognition at Low Light Levels,” J. Opt. Soc. Am. A 2, P20 (1985).

Opt. Acta (1)

Y. Yang, Y. N. Hsu, H. H. Arsenault, “Optimum Circular Symmetrical Filters and Their Uses in Optical Pattern Recognition,” Opt. Acta 29, 627 (1982).
[CrossRef]

Opt. Eng. (1)

D. Casasent, W. Rozzi, “Projection Synthetic Discriminant Function Performance,” Opt. Eng. 23, 716 (1984).
[CrossRef]

Proc. IEEE (1)

D. Casasent, D. Psaltis, “New Optical Transforms for Pattern Recognition,” Proc. IEEE 65, 77 (1977).
[CrossRef]

Proc. IRE (1)

M. K. Hu, “Pattern Recognition by Moment Invariants,” Proc. IRE 49, 1428 (1961).

Proc. Soc. Photo-Opt. Instrum. Eng. (3)

D. Casasent, D. Fetterly, “Recent Advances in Optical Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 456, 105 (1984).

C. F. Hester, D. Casasent, “Inter-class Discrimination Using Synthetic Discriminant Functions (SDFs),” Proc. Soc. Photo-Opt. Instrum. Eng. 302, 108 (1981).

H. H. Arsenault, Y. N. Hsu, K. Chalasinska-Macukow, Y. Yang, “Rotation Invariant Pattern Recognition,” Proc. Soc. Photo-Opt. Instrum. Eng. 359, 256 (1982).

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Figures (8)

Fig. 1
Fig. 1

Finding the centroid of a function by cross-correlating with a windowed version of the first moment generating kernel: (a) input function; (b) windowed first moment generating function; (c) correlation output. The zero-crossing corresponds to the centroid of f(x).

Fig. 2
Fig. 2

Finding the centroids of multiple objects in an input using the correlation technique: (a) input; (b) correlation output resulting from cross-correlating the input with the first moment generating kernel truncated by a window of length M. The isolated zero-crossings correspond to individual object centroids.

Fig. 3
Fig. 3

Schematic of the complete hybrid system.

Fig. 4
Fig. 4

Illustrating the three possible background-level conditions: (a) background level falls below the object gray levels; (b) background level falls within the object gray levels; (c) background level falls above the object gray levels. The actual centroid is shown by ×, while the centroid located with this technique is shown by ○.

Fig. 5
Fig. 5

Using the centroid locater to extract pattern recognition features without prior scene segmentation.

Fig. 6
Fig. 6

Comparison of the magnitude of the desired point spread function (a) vs the achieved point spread function used in our experimental verification (b).

Fig. 7
Fig. 7

Experimentally tracking the motion of the letter e across the optical field of view. The successive correlation outputs shown on the left were masked by the thresholded objects themselves to locate the centroids. The path of the centroids is shown on the right.

Fig. 8
Fig. 8

Input scene containing multiple objects; (b) correlation output when the window size used was matched to the size of a single letter. Notice that two objects are located for the composite letter ae, even though the object spacing restrictions are violated.

Equations (15)

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

[ M ( 1 , 0 ) M ( 0 , 0 ) , M ( 0 , 1 ) M ( 0 , 0 ) ] ,
M ( p , q ) = - - ( x - y 0 ) p ( y - y 0 ) q f ( x , y ) d x d y ,
M ( x 0 , y 0 , p , q ) = - - ( x - x 0 ) p ( y - y 0 ) q f ( x , y ) d x d y ,
g ( x 0 , y 0 ) = M ( x 0 , y 0 1 , 0 ) + i M ( x 0 , y 0 , 0 , 1 )
= - - [ ( x - x 0 ) + i ( y - y 0 ) ] f ( x , y ) d x d y ,
= ( x 0 + i y 0 ) f ( x 0 , y 0 ) ,
m ( x , y ) = ( x + i y ) circ ( 2 r W )
= r exp ( i θ ) circ ( 2 r W ) ,
[ 1 1 1 1 - 1 1 1 1 1 ] .
L j = - - f ( x , y ) g j ( x , y ) d x d y ,
= - - f ( x - a , y - b ) g j ( x , y ) d x d y a = b = 0 ,
1 Δ 1 λ f ( l f + W ) ,
M ( ρ , ϕ ) = FT [ m ( x , y ) ]
= FT [ r exp ( i θ ) circ ( r W ) ]
= i W 2 4 ρ J 2 ( W π ρ ) exp ( i θ ) ,

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