Abstract

The principle of representing continuous complex-valued functions by their decomposition into three positive-valued ones is proposed for the generation of complex reference functions for a joint transform correlator. Three basic approaches involving coherent and incoherent superposition of the component functions are analyzed. The potentials and limitations of the techniques are discussed.

© 1994 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

1991 (2)

1990 (1)

1989 (1)

1982 (1)

1977 (1)

1976 (1)

1966 (1)

Arsenault, H. H.

Chaveland, P.

Goodman, J. W.

Gregory, D. A.

Hsu, Y.-N.

Hugonin, J.-P.

Jutamulia, S.

Konforti, N.

Li, X.

Mahlab, U.

Marom, E.

Mendlovic, D.

Rosen, J.

Shamir, J.

Tam, E. C.

Weaver, C. S.

Woody, L. M.

Yu, F. T. S.

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Figures (6)

Fig. 1
Fig. 1

Representation of a complex function by real positive functions: (a) addition of a bias to the real and imaginary components with reduction of the available dynamic range, (b) decomposition into three functions along constant phasors.

Fig. 2
Fig. 2

Interferometric arrangements to perform correlation with complex references through (a) coherent superposition, (b) incoherent superposition. PT, piezoelectric transducer; M, mirror; BS, beam splitter; L, lens; f, focal length of L; PW, plane wave. h1, h2, h3, and f are positive transparencies.

Fig. 3
Fig. 3

Reconstructed images for a univalued rectangle: (a) conventional joint correlation with a delta function, (b) joint correlation with a bipolar filter for edge enhancement.

Fig. 4
Fig. 4

Edge enhancement with nonlinearities in the JTH recording: (a) experimental result, (b) simulation with the same degree of saturation.

Fig. 5
Fig. 5

Superposition at the Fourier plane correlation of an input pattern with a CHC of the letter E: (a)–(c) three components of the filter by the input, (d)–(f) three JTH’s, (g) the JTH obtained after superposition (approximate solution), (h) the correlation output after Fourier transformation.

Fig. 6
Fig. 6

Incoherent superposition at the correlation plane. Correlation of an input pattern with a CHC of the letter E: (a)–(c) partial correlations of the three inputs from Figs. 5(a)–(c), (d) the final exact result after superposition.

Equations (12)

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h ( x , y ) = h 1 ( x , y ) + h 2 ( x , y ) exp ( i 2 π 3 ) + h 3 ( x , y ) exp ( i 4 π 3 ) .
F { h 1 ( x , y ) + f ( x , y + b ) } + F { h 2 ( x , y ) exp ( i ϕ 2 ) } + F { h 3 ( x , y ) exp ( i ϕ 3 ) } = H ( u , υ ) + F ( u , υ ) exp ( i 2 π b υ ) ,
| F + H 2 | 2 = | F | 2 + | H 2 | 2 + | F | | H 2 | × exp { i [ 2 π b υ + ϕ F ( u , υ ) ϕ 2 ( u , υ ) 2 π 3 ] } + | F | | H 2 | exp { i [ 2 π b υ + ϕ F ( u , υ ) ϕ 2 ( u , υ ) 2 π 3 ] } ,
i | F + H i | 2 = 3 | F | 2 + | H 1 | 2 + | H 2 | 2 + | H 3 | 2 + | F | | H 1 | exp [ i ( 2 π b υ + ϕ F ϕ 1 ) ] + | F | | H 1 | exp [ i ( 2 π b υ + ϕ F ϕ 1 ) ] + | F | | H 2 | exp [ i ( 2 π b υ + ϕ F ϕ 2 2 π 3 ) ] + | F | | H 2 | exp [ i ( 2 π b υ + ϕ F ϕ 2 2 π 3 ) ] + | F | | H 3 | exp [ i ( 2 π b υ + ϕ F ϕ 3 4 π 3 ) ] + | F | | H 3 | exp [ i ( 2 π b υ + ϕ F ϕ 3 4 π 3 ) ] .
c ( x , y ) = 3 f f + h 1 h 1 + h 2 h 2 + h 3 h 3 + f ( x , y ) [ h 1 ( x , y ) + h 2 ( x , y ) exp ( i 2 π 3 ) + h 3 ( x , y ) exp ( i 4 π 3 ) ] * δ ( x , y b ) + f ( x , y ) [ h 1 ( x , y ) + h 2 ( x , y ) exp ( i 2 π 3 ) + h 3 ( x , y ) exp ( i 4 π 3 ) ] * δ ( x , y + b ) ,
c ( x , y ) = 3 f f + h 1 h 1 + h 2 h 2 + h 3 h 3 + f ( x , y ) h ( x , y ) * δ ( x , y b ) + f ( x , y ) h * ( x , y ) * δ ( x , y + b ) .
c k = f h k , k = 1 , 2 , 3 .
c = f h = c 1 ( x , y ) + c 2 ( x , y ) exp ( i 2 π 3 ) + c 3 ( x , y ) exp ( i 4 π 3 ) .
| c | 2 = c 1 2 + c 2 2 + c 3 2 c 1 c 2 c 1 c 3 c 2 c 3 .
JTH 2 = | F + H 2 | 2 ( u , υ 1 3 b ) = | F ( u , υ 1 3 b ) | 2 + | H 2 ( u , υ 1 3 b ) | 2 + | F ( u , υ 1 3 b ) | | H 2 ( u , υ 1 3 b ) | × exp { i [ 2 π b ( υ 1 3 b ) + ϕ F ( u , υ 1 3 b ) ϕ 2 ( u , υ 1 3 b ) ] } + | F ( u , υ 1 3 b ) | | H 2 ( u , υ 1 3 b ) | × exp { i [ 2 π b ( υ 1 3 b ) + ϕ F ( u , υ 1 3 b ) ϕ 2 ( u , υ 1 3 b ) ] } .
F { JTH 2 } = f ( x , y ) exp [ i ( 2 π 3 b ) y ] f ( x , y ) exp [ i ( 2 π 3 b ) y ] + h ( x , y ) exp [ i ( 2 π 3 b ) y ] h ( x , y ) exp [ i ( 2 π 3 b ) y ] + f ( x , y ) exp [ i ( 2 π 3 b ) y ] h ( x , y ) × exp [ i ( 2 π 3 b ) y ] δ ( x , y + b ) exp ( i 2 π 3 ) + f ( x , y ) exp [ i ( 2 π 3 b ) y ] h ( x , y ) × exp [ i ( 2 π 3 b ) y ] δ ( x , y b ) exp ( i 2 π 3 )
c a p r = f ( x , y b ) h 1 ( x , y b ) + f ( x , y b ) exp [ i ( 2 π 3 b ) ( y b ) ] h 2 ( x , y b ) × exp [ i ( 2 π 3 b ) ( y b ) ] exp ( i 2 π 3 ) + f ( x , y b ) exp [ i ( 4 π 3 b ) ( y b ) ] h 3 ( x , y b ) × exp [ i ( 4 π 3 b ) ( y b ) ] exp ( i 4 π 3 ) .

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