Abstract

A coherent optical processor capable of calculating generalized moments of a two-dimensional pattern is described. A spatial-frequency multiplexing scheme is used to provide for parallel computation of multiple moments. The use of a computer-generated holographic mask permits complete flexibility in choosing moment-generating functions; e.g., the functions could be complex or have a predetermined weighting function. Experimentally, calculation of five geometric moments (corresponding to x, y, xy, x2, and y2) is demonstrated for simple objects. The special features of the proposed coherent optical processor and its space–bandwidth requirements are also discussed.

© 1982 Optical Society of America

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References

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  1. M. K. Hu, IEEE Trans. Inf. Theory IT-8, 179 (1962).
  2. S. Dudani, K. Breeding, R. McGhee, IEEE Trans. Comput. C-26, 39 (1977).
    [CrossRef]
  3. S. S. Reddi, IEEE Trans. Pattern Anal. Machine Intell. PAMI-3, 240 (1981).
    [CrossRef]
  4. M. R. Teague, Appl. Opt. 19, 1353 (1980).
    [CrossRef] [PubMed]
  5. D. Casasent, D. Psaltis, Opt. Lett. 5, 395 (1980).
    [CrossRef] [PubMed]
  6. W. H. Lee, Computer-Generated Holograms: Technique and Applications, Vol. XVI in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978).
    [CrossRef]
  7. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).

1981 (1)

S. S. Reddi, IEEE Trans. Pattern Anal. Machine Intell. PAMI-3, 240 (1981).
[CrossRef]

1980 (2)

1977 (1)

S. Dudani, K. Breeding, R. McGhee, IEEE Trans. Comput. C-26, 39 (1977).
[CrossRef]

1962 (1)

M. K. Hu, IEEE Trans. Inf. Theory IT-8, 179 (1962).

Breeding, K.

S. Dudani, K. Breeding, R. McGhee, IEEE Trans. Comput. C-26, 39 (1977).
[CrossRef]

Casasent, D.

Dudani, S.

S. Dudani, K. Breeding, R. McGhee, IEEE Trans. Comput. C-26, 39 (1977).
[CrossRef]

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).

Hu, M. K.

M. K. Hu, IEEE Trans. Inf. Theory IT-8, 179 (1962).

Lee, W. H.

W. H. Lee, Computer-Generated Holograms: Technique and Applications, Vol. XVI in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978).
[CrossRef]

McGhee, R.

S. Dudani, K. Breeding, R. McGhee, IEEE Trans. Comput. C-26, 39 (1977).
[CrossRef]

Psaltis, D.

Reddi, S. S.

S. S. Reddi, IEEE Trans. Pattern Anal. Machine Intell. PAMI-3, 240 (1981).
[CrossRef]

Teague, M. R.

Appl. Opt. (1)

IEEE Trans. Comput. (1)

S. Dudani, K. Breeding, R. McGhee, IEEE Trans. Comput. C-26, 39 (1977).
[CrossRef]

IEEE Trans. Inf. Theory (1)

M. K. Hu, IEEE Trans. Inf. Theory IT-8, 179 (1962).

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

S. S. Reddi, IEEE Trans. Pattern Anal. Machine Intell. PAMI-3, 240 (1981).
[CrossRef]

Opt. Lett. (1)

Other (2)

W. H. Lee, Computer-Generated Holograms: Technique and Applications, Vol. XVI in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1978).
[CrossRef]

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of a multiplexed coherent optical processor for calculating generalized moments.

Fig. 2
Fig. 2

Plot of the generating functions x and x2 with x multiplied by a weight of 0.5.

Fig. 3
Fig. 3

Arrangement of the five geometric moments in the output plane.

Fig. 4
Fig. 4

Photograph of the output of the optical processor. The input was a square binary object with x and y symmetry.

Fig. 5
Fig. 5

(a) Line scan through the m10 moment of a binary square with x symmetry. (b) Line scan through the m10 moment of a binary square shifted in the x direction with respect to its position in (a).

Fig. 6
Fig. 6

Line scans through two geometric moments of a rectangle with 2:1 aspect ratio: (a) m20, (b) m02.

Fig. 7
Fig. 7

Distribution in the output plane when u0 = Bi + Bm, leading to zero cross talk between adjacent moments.

Equations (5)

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G m n = f ( x , y ) g m n ( x , y ) d x d y ,
G m n = f ( x , y ) g m n ( x , y ) × exp [ j ( u x + υ y ) ] d x d y | u = 0 v = 0 .
G m n = f ( x , y ) g m n ( x , y ) exp { j [ ( u m u 0 ) x + ( υ n υ 0 ) y ] } d x d y | u = mu 0 v = nv 0 .
g ( x , y ) = m = M M n = N M g m n ( x , y ) exp [ j ( m u 0 x + n υ 0 y ) ] ,
N = [ 2 ( B s B i + B m ) 1 ] 2 .

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