Abstract

The effects of space-invariant blurring in the frequency plane of a coherent optical correlator on the system’s performance are considered. The correlator performance criteria used include loss in output correlation peak amplitude, output signal-to-noise ratio, and variation in the probability of a false alarm for a fixed probability of detection. A statistical approach is used to derive general expressions for the mean m and variance var of the output correlation. Expressions for the three correlator performance criteria are then obtained in terms of m and var. When these results are applied to several specific cases, we find that the performance of a correlator can be described and controlled by a new parameter, the space-blur bandwidth product.

© 1980 Optical Society of America

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References

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  1. D. Casasent, “Optical Pattern Recognition,” Proc. IEEE 67, 813–825 (1979).
    [Crossref]
  2. A. B. Vander Lugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [Crossref]
  3. M. Svedlow, C. D. McGillem, and P. E. Anuta, “Analytical and Experimental Design and Analysis of an Optimal Processor for Image Registration,” LARS Info. Note 090776, The Laboratory for Applications of Remote Sensing, Purdue University, West Lafayette (1976) (unpublished).
  4. A. B. Vander Lugt, “The effect of small displacements of spatial filters,” Appl. Opt. 6, 1221–1225 (1967).
    [Crossref]
  5. M. J. Lahart, “Optical Area Correlation with Magnification and Rotation,” J. Opt. Soc. Am. 60, 319–325 (1970).
    [Crossref]
  6. H. Mostafavi and F. Smith, “Image Correlation with Geometric Distortion, Part I: Acquisition Performance,” IEEE Trans. Aerosp. Electron. Syst. 14, 487–493 (1978).
    [Crossref]
  7. H. Mostafavi and F. Smith, “Image Correlation with Geometric Distortion, Part II: Effect of Local Accuracy,” IEEE Trans. Aerosp. Electron. Syst. 14, 496–501 (1978).
  8. D. Casasent and A. Furman, “Sources of correlation degradation,” Appl. Opt. 16, 1652–1661 (1977).
    [Crossref] [PubMed]
  9. K. Singh and P. C. Gupta, “Periodic non-sinusoidal vibration effects in matched filtering for detection of signals buried in noise of uniform spectral density,” Appl. Opt. 14, 2940–2943 (1975).
    [Crossref] [PubMed]
  10. D. Casasent, “Spatial Light Modulators,” Proc. IEEE 65, 143–157 (1977).
    [Crossref]
  11. A. W. Lohmann and D. P. Paris, “Variable Fresnel Zone Pattern,” Appl. Opt. 6, 1567–1570 (1967).
    [Crossref] [PubMed]
  12. E. L. O’Neill, “Spatial Filtering in Optics,” IRE Trans. Inf. Theory IT-2, 56–65 (1956).
    [Crossref]
  13. T. Luu and D. Casasent, “Effects of Wavefront Propagation Error in an Optical Correlator,” Appl. Opt. 18, 791–795 (1979).
    [Crossref] [PubMed]
  14. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  15. H. Mostafavi, “Optimal Window Functions for Image Registration in the Presence of Geometric Distortion,” IEEE Trans. Acoust. Speech Signal Process. 27, 163–169 (1979).
    [Crossref]

1979 (3)

D. Casasent, “Optical Pattern Recognition,” Proc. IEEE 67, 813–825 (1979).
[Crossref]

T. Luu and D. Casasent, “Effects of Wavefront Propagation Error in an Optical Correlator,” Appl. Opt. 18, 791–795 (1979).
[Crossref] [PubMed]

H. Mostafavi, “Optimal Window Functions for Image Registration in the Presence of Geometric Distortion,” IEEE Trans. Acoust. Speech Signal Process. 27, 163–169 (1979).
[Crossref]

1978 (2)

H. Mostafavi and F. Smith, “Image Correlation with Geometric Distortion, Part I: Acquisition Performance,” IEEE Trans. Aerosp. Electron. Syst. 14, 487–493 (1978).
[Crossref]

H. Mostafavi and F. Smith, “Image Correlation with Geometric Distortion, Part II: Effect of Local Accuracy,” IEEE Trans. Aerosp. Electron. Syst. 14, 496–501 (1978).

1977 (2)

1975 (1)

1970 (1)

1967 (2)

1964 (1)

A. B. Vander Lugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[Crossref]

1956 (1)

E. L. O’Neill, “Spatial Filtering in Optics,” IRE Trans. Inf. Theory IT-2, 56–65 (1956).
[Crossref]

Anuta, P. E.

M. Svedlow, C. D. McGillem, and P. E. Anuta, “Analytical and Experimental Design and Analysis of an Optimal Processor for Image Registration,” LARS Info. Note 090776, The Laboratory for Applications of Remote Sensing, Purdue University, West Lafayette (1976) (unpublished).

Casasent, D.

Furman, A.

Gupta, P. C.

Lahart, M. J.

Lohmann, A. W.

Luu, T.

McGillem, C. D.

M. Svedlow, C. D. McGillem, and P. E. Anuta, “Analytical and Experimental Design and Analysis of an Optimal Processor for Image Registration,” LARS Info. Note 090776, The Laboratory for Applications of Remote Sensing, Purdue University, West Lafayette (1976) (unpublished).

Mostafavi, H.

H. Mostafavi, “Optimal Window Functions for Image Registration in the Presence of Geometric Distortion,” IEEE Trans. Acoust. Speech Signal Process. 27, 163–169 (1979).
[Crossref]

H. Mostafavi and F. Smith, “Image Correlation with Geometric Distortion, Part I: Acquisition Performance,” IEEE Trans. Aerosp. Electron. Syst. 14, 487–493 (1978).
[Crossref]

H. Mostafavi and F. Smith, “Image Correlation with Geometric Distortion, Part II: Effect of Local Accuracy,” IEEE Trans. Aerosp. Electron. Syst. 14, 496–501 (1978).

O’Neill, E. L.

E. L. O’Neill, “Spatial Filtering in Optics,” IRE Trans. Inf. Theory IT-2, 56–65 (1956).
[Crossref]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Paris, D. P.

Singh, K.

Smith, F.

H. Mostafavi and F. Smith, “Image Correlation with Geometric Distortion, Part II: Effect of Local Accuracy,” IEEE Trans. Aerosp. Electron. Syst. 14, 496–501 (1978).

H. Mostafavi and F. Smith, “Image Correlation with Geometric Distortion, Part I: Acquisition Performance,” IEEE Trans. Aerosp. Electron. Syst. 14, 487–493 (1978).
[Crossref]

Svedlow, M.

M. Svedlow, C. D. McGillem, and P. E. Anuta, “Analytical and Experimental Design and Analysis of an Optimal Processor for Image Registration,” LARS Info. Note 090776, The Laboratory for Applications of Remote Sensing, Purdue University, West Lafayette (1976) (unpublished).

Vander Lugt, A. B.

A. B. Vander Lugt, “The effect of small displacements of spatial filters,” Appl. Opt. 6, 1221–1225 (1967).
[Crossref]

A. B. Vander Lugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[Crossref]

Appl. Opt. (5)

IEEE Trans. Acoust. Speech Signal Process. (1)

H. Mostafavi, “Optimal Window Functions for Image Registration in the Presence of Geometric Distortion,” IEEE Trans. Acoust. Speech Signal Process. 27, 163–169 (1979).
[Crossref]

IEEE Trans. Aerosp. Electron. Syst. (2)

H. Mostafavi and F. Smith, “Image Correlation with Geometric Distortion, Part I: Acquisition Performance,” IEEE Trans. Aerosp. Electron. Syst. 14, 487–493 (1978).
[Crossref]

H. Mostafavi and F. Smith, “Image Correlation with Geometric Distortion, Part II: Effect of Local Accuracy,” IEEE Trans. Aerosp. Electron. Syst. 14, 496–501 (1978).

IEEE Trans. Inf. Theory (1)

A. B. Vander Lugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[Crossref]

IRE Trans. Inf. Theory (1)

E. L. O’Neill, “Spatial Filtering in Optics,” IRE Trans. Inf. Theory IT-2, 56–65 (1956).
[Crossref]

J. Opt. Soc. Am. (1)

Proc. IEEE (2)

D. Casasent, “Optical Pattern Recognition,” Proc. IEEE 67, 813–825 (1979).
[Crossref]

D. Casasent, “Spatial Light Modulators,” Proc. IEEE 65, 143–157 (1977).
[Crossref]

Other (2)

M. Svedlow, C. D. McGillem, and P. E. Anuta, “Analytical and Experimental Design and Analysis of an Optimal Processor for Image Registration,” LARS Info. Note 090776, The Laboratory for Applications of Remote Sensing, Purdue University, West Lafayette (1976) (unpublished).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

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Figures (5)

FIG. 1
FIG. 1

Schematic of the optical frequency plane correlator

FIG. 2
FIG. 2

Variation of IpNB with SBBP = LD for cases (i)–(iv).

FIG. 3
FIG. 3

Variation of SNRout with SBBP = LD for SBWP = 10, 30, 100.

FIG. 4
FIG. 4

Enlargement of Fig. 3 near SBBP = LD = 1 with vertical normalization of all SBWP = LW curves at low SBBP ≪ 1.

FIG. 5
FIG. 5

Variation of PFA with SBBP = LD for PD = 0.999 and SBWP = 10 for the cases of no noise (SNRin = ∞) and for SNRin = 0 dB.

Equations (51)

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H ( f ) = k S * ( f ) ,
C ( τ ) = ( 1 / A ) A s ( x + τ ) [ s ( x ) + n ( x ) ] d x .
H ( f ) = H ( u ) B ( f u ) d u ,
( i ) B ( f ) is real , ( ii ) B ( 0 ) B ( f ) for all f , ( iii ) B ( u ) d u = 1 ( normalization ) ,
( iv ) Equivalent blur bandwidth D = [ B ( 0 ) ] 1 / 2 ( v ) Equivalent amplititude B ( 0 ) .
SBBP = D L .
H ( f ) = H ( u ) B ( f u ) d u = H ( f ) * B ( f )
h ( x ) = h ( x ) · b ( x ) ,
C B ( τ ) = ( 1 / A ) A s B ( x + τ ) [ s ( x ) + n ( x ) ] d x ,
s B ( x ) = s ( x ) b ( x ) .
E { C B ( τ ) } = E { ( 1 / A ) A s B ( x + τ ) [ s ( x ) + n ( x ) ] d x } .
E { C B ( τ ) } = ( 1 / A ) A E { s B ( x + τ ) s ( x ) } d x .
E { C B ( τ ) } = R s ( τ ) [ ( 1 / A ) A b ( x + τ ) d x ] .
C N B ( τ ) E { C B ( τ ) } / R s ( τ ) ( 1 / A ) A b ( x + τ ) dx 1.
I p N B = C N B ( 0 ) = ( 1 / A ) A b ( x ) d x .
I p N B = C N B ( 0 ) = 1.
I p N B = C N B ( 0 ) = B ( 0 ) / A = ( L D ) 2 ,
b ( x ) d x = B ( 0 ) = 1 / D 2
B ( f ) = B ( f 1 , f 2 ) = { 1 / Δ 2 for | f 1 | Δ / 2 and | f 2 | Δ / 2 0 elsewhere .
b ( x ) = b ( x 1 , x 2 ) = ( sinc x 1 Δ ) ( sinc x 2 Δ ) ,
I p N B = ( 2 S i ( π L D / 2 ) π L D ) 2
S i ( y ) = 0 x [ ( sin t ) / t ] d t .
B ( f ) = B ( f 1 , f 2 ) = ( 1 / 2 π Δ 2 ) exp [ ( f 1 2 + f 2 2 ) / 2 Δ 2 ] ,
b ( x ) = b ( x 1 , x 2 ) = exp [ 2 π 2 Δ 2 ( x 1 2 + x 2 2 ) ] .
I p N B = { ( 2 / L D ) [ ϕ ( π 1 / 2 L D / 2 1 / 2 ) 0.5 ] } 2 ,
ϕ ( y ) = ( 2 π ) 1 / 2 y exp ( u 2 / 2 ) d u .
B ( f ) = B ( f 1 , f 2 ) = { ( 1 / π Δ 2 ) for ( f 1 2 + f 2 2 ) 1 / 2 Δ 0 elsewhere
b ( x ) = b ( x 1 , x 2 ) = [ J 1 ( 2 π ρ Δ ) ] / ( π ρ Δ ) ,
I p N B = ( 1 / L D ) 2 [ 1 J 0 ( 2 L D ) ] ,
I p N B = ( 1 / L D ) 2 [ 1 exp ( L 2 D 2 ) ] .
( Case i ) : I p N B 1 ( π / 6 ) 2 ( L D ) 2 1 ( 1 4 ) ( L D ) 2 ,
( Case iii ) : I p N B 1 ( 1 4 ) ( L D ) 2 ,
( Case ii ) : I p N B 1 ( π / 6 ) ( L D ) 2 1 ( 1 2 ) ( L D ) 2 ,
( Case iv ) : I p N B 1 ( 1 2 ) ( L D ) 2 .
SNR out = E 2 { C ( 0 ) } Var { C ( τ ) } | τ 0 ,
Var [ C B ( τ ) ] = ( 1 / A 2 ) A A b ( x + τ ) b ( z + τ ) [ R s 2 ( z x ) + R s ( z x τ ) R s ( z x + τ ) + R s ( z x ) R n ( z x ) ] d x d z .
Var N [ C B ( τ ) ] = Var [ C ( τ ) ] | τ 0 ( 1 / A 2 ) A A b ( x + τ ) b ( z + τ ) × [ R s 2 ( z x ) + R s ( z x ) R n ( z x ) ] d x d z
Var p [ C B ( 0 ) ] = Var [ C ( τ ) ] | τ = 0 = ( 1 / A 2 ) A A b ( x ) b ( z ) × [ 2 R s 2 ( z x ) + R s ( z x ) R n ( z x ) ] d x d z .
SNR out = [ A b ( x ) d x ] 2 A A b ( x + τ ) b ( z + τ ) R s 2 ( z x ) d x d z × 1 1 + 1 / SNR in ,
SNR in = R s ( 0 ) / R n ( 0 ) = A A b ( x + τ ) b ( z + τ ) R s 2 ( z x ) d x d z A A b ( x + τ ) b ( z + τ ) R s ( z x ) R n ( z x ) d x d z .
R s ( τ ) = R s ( τ 1 , τ 2 ) = exp [ ( τ 1 2 + τ 2 2 ) / 2 a 2 ] ,
SNR out = { 2 S i [ π ( L D ) / 2 ] } 4 π 4 ( L D ) 4 V ( L D , L W ) ,
V ( L D , L W ) = ( 1 + 1 SNR in ) { ( 1 4 ) 1 + 1 sin c [ ( L D ) ( u + 0.5 ) / 2 ] × sinc ( L D ) ( υ + 0.5 ) / 2 ] exp [ π ( L W ) 2 ( u υ ) 2 / 2 ] d u d υ } 2 ,
E [ C B ( 0 ) ] = ( 1 / A ) A b ( x ) d x ,
E [ C B ( τ ) ] = 0 ,
var [ C B ( 0 ) ] = ( 1 / A 2 ) ( 2 + 1 / SNR in ) × A A b ( x ) b ( z ) R s 2 ( z x ) d x d z ,
var [ C B ( τ ) ] = ( 1 / A 2 ) ( 1 + 1 / SNR in ) × A A b ( x + τ ) b ( z + τ ) R s 2 ( z x ) d x d z .
E [ C B ( τ ) ] = R s ( τ ) [ ( 1 / A ) A b ( x + τ ) d x ] .
E [ | C B ( τ ) | 2 ] = E { ( 1 / A ) 2 A A s ( x + τ ) s ( z + τ ) b ( x + τ ) b ( z + τ ) × [ s ( x ) + n ( x ) ] [ s ( z ) + n ( z ) d x d z ] } ,
E [ | C B ( τ ) | 2 ] = ( 1 / A ) 2 A A b ( x + τ ) b ( z + τ ) × { [ s ( x + τ ) s ( z + τ ) s ( x ) s ( z ) ] av + [ s ( x + τ ) s ( z + τ ) ] av · [ n ( x ) n ( z ) ] } av d x d z .
E [ | C B ( τ ) | 2 ] = ( 1 / A 2 ) A A b ( x + τ ) b ( z + τ ) × [ R s 2 ( z x ) + R s 2 ( τ ) + R s ( z x τ ) R s ( z x + τ ) + R s ( z x ) R n ( z x ) ] d x d z .