Abstract

We introduce the notion of optimal phase-only filters (OPOFs) that yield improved signal to noise ratios (SNRs). We illustrate the improvement in SNR resulting from the use of OPOFs with the help of several analytical examples and simulation results.

© 1989 Optical Society of America

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References

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  1. A. VanderLugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139 (1964).
    [Crossref]
  2. J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812 (1984).
    [Crossref] [PubMed]
  3. P. D. Gianino, J. L. Horner, “Additional Properties of the Phase-Only Correlation Filter,” Opt. Eng. 23, 695 (1984).
    [Crossref]
  4. J. L. Horner, P. D. Gianino, “Applying the Phase-Only Filter Concept to the Synthetic Discriminant Function Correlation Filter,” Appl. Opt. 24, 851 (1985).
    [Crossref] [PubMed]
  5. X. Su, G. Zhang, L. Guo, “Phase-Only Composite Filter,” Opt. Eng. 26, 520 (1987).
    [Crossref]
  6. J. Rosen, J. Shamir, “Distortion Invariant Pattern Recognition with Phase Filters,” Appl. Opt. 26, 2315 (1987).
    [Crossref] [PubMed]
  7. R. R. Kallman, “Direct Construction of Phase-Only Correlation Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 827, Paper 26 (Aug.1987).
  8. H. L. VanTrees, Detection, Estimation and Modulation Theory: Part I (Wiley, New York, 1968).
  9. C. S. Anderson, R. C. Anderson, “Comparison of Phase-Only and Classical Matched Filter Scale Sensitivity,” Opt. Eng. 26, 276 (1987).
    [Crossref]
  10. F. M. Dickey, K. T. Stalker, “Binary Phase-Only Filters: Implications of Bandwidth and Uniqueness on Performance,” J. Opt. Soc. Am. A 4(13), P 69 (1987).
  11. J. L. Horner, “Light Utilization in Optical Correlators,” Appl. Opt. 21, 4511 (1982).
    [Crossref] [PubMed]

1987 (5)

X. Su, G. Zhang, L. Guo, “Phase-Only Composite Filter,” Opt. Eng. 26, 520 (1987).
[Crossref]

J. Rosen, J. Shamir, “Distortion Invariant Pattern Recognition with Phase Filters,” Appl. Opt. 26, 2315 (1987).
[Crossref] [PubMed]

R. R. Kallman, “Direct Construction of Phase-Only Correlation Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 827, Paper 26 (Aug.1987).

C. S. Anderson, R. C. Anderson, “Comparison of Phase-Only and Classical Matched Filter Scale Sensitivity,” Opt. Eng. 26, 276 (1987).
[Crossref]

F. M. Dickey, K. T. Stalker, “Binary Phase-Only Filters: Implications of Bandwidth and Uniqueness on Performance,” J. Opt. Soc. Am. A 4(13), P 69 (1987).

1985 (1)

1984 (2)

J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812 (1984).
[Crossref] [PubMed]

P. D. Gianino, J. L. Horner, “Additional Properties of the Phase-Only Correlation Filter,” Opt. Eng. 23, 695 (1984).
[Crossref]

1982 (1)

1964 (1)

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

Anderson, C. S.

C. S. Anderson, R. C. Anderson, “Comparison of Phase-Only and Classical Matched Filter Scale Sensitivity,” Opt. Eng. 26, 276 (1987).
[Crossref]

Anderson, R. C.

C. S. Anderson, R. C. Anderson, “Comparison of Phase-Only and Classical Matched Filter Scale Sensitivity,” Opt. Eng. 26, 276 (1987).
[Crossref]

Dickey, F. M.

F. M. Dickey, K. T. Stalker, “Binary Phase-Only Filters: Implications of Bandwidth and Uniqueness on Performance,” J. Opt. Soc. Am. A 4(13), P 69 (1987).

Gianino, P. D.

Guo, L.

X. Su, G. Zhang, L. Guo, “Phase-Only Composite Filter,” Opt. Eng. 26, 520 (1987).
[Crossref]

Horner, J. L.

Kallman, R. R.

R. R. Kallman, “Direct Construction of Phase-Only Correlation Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 827, Paper 26 (Aug.1987).

Rosen, J.

Shamir, J.

Stalker, K. T.

F. M. Dickey, K. T. Stalker, “Binary Phase-Only Filters: Implications of Bandwidth and Uniqueness on Performance,” J. Opt. Soc. Am. A 4(13), P 69 (1987).

Su, X.

X. Su, G. Zhang, L. Guo, “Phase-Only Composite Filter,” Opt. Eng. 26, 520 (1987).
[Crossref]

VanderLugt, A.

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

VanTrees, H. L.

H. L. VanTrees, Detection, Estimation and Modulation Theory: Part I (Wiley, New York, 1968).

Zhang, G.

X. Su, G. Zhang, L. Guo, “Phase-Only Composite Filter,” Opt. Eng. 26, 520 (1987).
[Crossref]

Appl. Opt. (4)

IEEE Trans. Inf. Theory (1)

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

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

F. M. Dickey, K. T. Stalker, “Binary Phase-Only Filters: Implications of Bandwidth and Uniqueness on Performance,” J. Opt. Soc. Am. A 4(13), P 69 (1987).

Opt. Eng. (3)

C. S. Anderson, R. C. Anderson, “Comparison of Phase-Only and Classical Matched Filter Scale Sensitivity,” Opt. Eng. 26, 276 (1987).
[Crossref]

X. Su, G. Zhang, L. Guo, “Phase-Only Composite Filter,” Opt. Eng. 26, 520 (1987).
[Crossref]

P. D. Gianino, J. L. Horner, “Additional Properties of the Phase-Only Correlation Filter,” Opt. Eng. 23, 695 (1984).
[Crossref]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. R. Kallman, “Direct Construction of Phase-Only Correlation Filters,” Proc. Soc. Photo-Opt. Instrum. Eng. 827, Paper 26 (Aug.1987).

Other (1)

H. L. VanTrees, Detection, Estimation and Modulation Theory: Part I (Wiley, New York, 1968).

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Figures (8)

Fig. 1
Fig. 1

A 32 × 32 tank image used in our simulations.

Fig. 2
Fig. 2

Sample images when noise of different variances was added to the image in Fig. 1: (a) σ2 = 1; (b) σ2 = 2; (c) σ2 = 5; (d) σ2 = 10; (e) σ2 = 50; (f) σ2 = 100; (g) σ2 = 200; (h) σ2 = 500; and (i) σ2 = 1000.

Fig. 3
Fig. 3

Result of a 2-D FFT of size 64 × 64 performed on the zero-padded image in Fig. 1.

Fig. 4
Fig. 4

Phase function of the conventional phase-only filter obtained from the 2-D FFT in Fig. 3.

Fig. 5
Fig. 5

Passband of the optimal phase-only filter for the image in Fig. 1.

Fig. 6
Fig. 6

Output SNR plotted as a function of input SNR for the three different filters: SMF represents the classical matched filter, POF represents the conventional phase-only filter, and OPO denotes the optimal phase-only filter.

Fig. 7
Fig. 7

Output cross-correlations of the three filters for an input noise variance of 100: (a) classical matched filter; (b) conventional phase-only filter; and (c) optimal phase-only filter.

Fig. 8
Fig. 8

Output cross-correlations of the three filters for an input noise variance of 1000: (a) classical matched filter; (b) conventional phase-only filter; and (c) optimal phase-only filter.

Equations (28)

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SNR = S ( f ) H ( f ) d f 2 P n ( f ) H ( f ) 2 d f ,
H ( f ) = S * ( f ) P n ( f ) = S ( f ) P n ( f ) exp [ - j Φ s ( f ) ] ,
SNR MSF = S ( f ) 2 P n ( f ) d f .
H MSF ( f ) = S ( f ) exp [ - j Φ s ( f ) ] .
H POF = exp [ - j Φ s ( f ) ] .
H ( f ) = { exp [ j Φ ( f ) ] for f W h , 0 otherwise ,
SNR = | - W h W h S ( f ) exp [ j Φ ( f ) ] d f | 2 - W h W h P n ( f ) d f .
S ( f ) = S ( f ) exp [ j Φ s ( f ) ] ,
SNR POF = | - W h W h S ( f ) exp { j [ Φ s ( f ) + Φ ( f ) ] } d f | 2 - W h W h P n ( f ) d f ( - W h W h S ( f ) exp { j [ Φ s ( f ) + Φ ( f ) ] } d f ) 2 - W h W h P n ( f ) d f = [ - W h W h S ( f ) d f ] 2 - W h W h P n ( f ) d f ,
SNR OPOF = 2 [ 0 W h S ( f ) d f ] 2 0 W h P n ( f ) d f ,
Φ ( f ) = - Φ s ( f ) + θ ,
[ 2 0 W h P n ( f ) d f ] S ( W h ) = [ 0 W h S ( f ) d f ] P n ( W h ) .
S ( f ) = { ( 1 - f W s ) for f W s 0 otherwise .
SNR OPOF ( W h ) = 2 [ 0 W h ( 1 - f W s ) d f ] 2 N 0 W h d f = 2 W h N 0 ( 1 - W h 2 W s ) 2 .
W h = W s .
SNR OPOF = 16 27 W s N 0 .
SNR POF = 2 W s N 0 ( 1 - W s 2 W s ) 2 = W s 2 N 0 .
SNR MSF = 2 N 0 0 W s ( 1 - f W s ) 2 d f = 2 W s 3 N 0 .
SNR POF SNR OPOF SNR MSF .
S ( f ) = { cos ( π f 2 W s ) for f W s , 0 otherwise .
W h W s = 1 π tan ( π 2 W h W s ) .
W h = 0.742 W s .
S ( f ) = sinc ( f T ) ,
SNR MSF = 1 N 0 - sinc 2 ( f T ) d f = 1 N 0 T .
SNR POF = [ - sinc ( f T ) d f ] 2 - N 0 d f = 0.
SNR OPOF = 0.8245 N 0 T .
S ( 1 ) S ( 2 ) S ( d ) ,
SNR ( K ) = 1 K [ k = 1 K S ( k ) ] 2 .

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