Abstract

A binary phase-only filter (BPOF) bandwidth and, correspondingly, the performance with respect to stochastic noise are introduced as filter design parameters. A BPOF figure of merit is defined which references the matched filter. Analytical bounds on the BPOF signal-to-noise ratio are derived. The noise performance is illustrated with simulation results. It is demonstrated through analysis and simulation that BPOFs can be designed to perform well with respect to stochastic noise.

© 1988 Optical Society of America

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References

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  1. D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698 (1984).
    [CrossRef]
  2. J. L. Horner, J. Leger, “Pattern Recognition with Binary Phase-Only Filters,” Appl. Opt. 24, 609 (1985).
    [CrossRef] [PubMed]
  3. J. L. Horner, H. O. Bartelt “Two-Bit Correlation,” Appl. Opt. 24, 2889 (1985).
    [CrossRef] [PubMed]
  4. D. M. Cottrell, R. A. Lilly, J. A. Davis, T. Day, “Optical Correlator Performance of Binary Phase-Only Filters Using Fourier and Hartley Transforms,” Appl. Opt. 26, 3755 (1987).
    [CrossRef] [PubMed]
  5. F. M. Dickey, K. T. Stalker, “Binary Phase-Only Filters: Implications of Bandwidth and Uniqueness on Performance,” J. Opt. Soc. Am. A 4(13), P69 (1987).
  6. B. V. K. Vijaya Kumar, Z. Bahri, “Phase-Only Filters with Improved Signal-to-Noise Ratio,” Appl. Opt., to be published.
  7. F. T. S. Yu, Optical Information Processing (Wiley, New York, 1983), pp. 13–15.
  8. A. Papoulis, Probability, Random Variables, and Stochastic Processes (Publisher, New York, 1965), p. 481.
  9. J. L. Horner, P. D. Gianino, “Phase-Only Matched Filtering,” Appl. Opt. 23, 812 (1984).
    [CrossRef] [PubMed]
  10. R. N. Bracewell, The Hartley Transform (Oxford U. P., New York, 1986).

1987

F. M. Dickey, K. T. Stalker, “Binary Phase-Only Filters: Implications of Bandwidth and Uniqueness on Performance,” J. Opt. Soc. Am. A 4(13), P69 (1987).

D. M. Cottrell, R. A. Lilly, J. A. Davis, T. Day, “Optical Correlator Performance of Binary Phase-Only Filters Using Fourier and Hartley Transforms,” Appl. Opt. 26, 3755 (1987).
[CrossRef] [PubMed]

1985

1984

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

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698 (1984).
[CrossRef]

Bahri, Z.

B. V. K. Vijaya Kumar, Z. Bahri, “Phase-Only Filters with Improved Signal-to-Noise Ratio,” Appl. Opt., to be published.

Bartelt, H. O.

Bracewell, R. N.

R. N. Bracewell, The Hartley Transform (Oxford U. P., New York, 1986).

Cottrell, D. M.

Davis, J. A.

Day, T.

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), P69 (1987).

Gianino, P. D.

Horner, J. L.

Leger, J.

Lilly, R. A.

Paek, E. G.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698 (1984).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (Publisher, New York, 1965), p. 481.

Psaltis, D.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698 (1984).
[CrossRef]

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), P69 (1987).

Venkatesh, S. S.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698 (1984).
[CrossRef]

Vijaya Kumar, B. V. K.

B. V. K. Vijaya Kumar, Z. Bahri, “Phase-Only Filters with Improved Signal-to-Noise Ratio,” Appl. Opt., to be published.

Yu, F. T. S.

F. T. S. Yu, Optical Information Processing (Wiley, New York, 1983), pp. 13–15.

Appl. Opt.

J. Opt. Soc. Am. A

F. M. Dickey, K. T. Stalker, “Binary Phase-Only Filters: Implications of Bandwidth and Uniqueness on Performance,” J. Opt. Soc. Am. A 4(13), P69 (1987).

Opt. Eng.

D. Psaltis, E. G. Paek, S. S. Venkatesh, “Optical Image Correlation with a Binary Spatial Light Modulator,” Opt. Eng. 23, 698 (1984).
[CrossRef]

Other

R. N. Bracewell, The Hartley Transform (Oxford U. P., New York, 1986).

B. V. K. Vijaya Kumar, Z. Bahri, “Phase-Only Filters with Improved Signal-to-Noise Ratio,” Appl. Opt., to be published.

F. T. S. Yu, Optical Information Processing (Wiley, New York, 1983), pp. 13–15.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (Publisher, New York, 1965), p. 481.

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Figures (6)

Fig. 1
Fig. 1

Similar functions (a) and (b) with BPOFs determined by dissimilar functions (a) and (c), respectively. Function (c) is the even part of function (b).

Fig. 2
Fig. 2

Figure of merit for a 2-D square pulse.

Fig. 3
Fig. 3

Figure of merit for a 2-D sinc2 function.

Fig. 4
Fig. 4

Autocorrelations using Hartley BPOF made from an even object function: (a) object function centered at the origin; (b) object function center offset from the origin; (c) object function located completely in one quadrant.

Fig. 5
Fig. 5

Not purely even function adapted from a Hadamard matrix (a) and the resulting Hartley BPOF autocorrelation (b) where the filter was made from the function after being centered well away from the origin.

Fig. 6
Fig. 6

Autocorrelation with stochastic input noise present: Autocorrelation response for filters given by (a) F*, (b) F*/P n . (c) a Hartley BPOF with large bandwidth and (d) Hartley BPOF with bandwidth reduced to improve noise performance.

Equations (40)

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( x ) = ( x , y ) , ( ω ) = ( ω x , ω y ) = ( 2 π f x , 2 π f y ) , d 2 x = d x d y , d 2 ω = ( 1 2 π d ω x , 1 2 π d ω x ) ,
H ( ω ) = F * ( ω ) P n ( ω ) ,
SNR = - F ( ω ) 2 d 2 ω N 0 .
P n ( ω ) = K δ ( ω ) + 1 π S ( ω ) S ( ω ) ,
H ( ω ) = F ( ω ) P n ( ω ) exp [ - i φ ( ω ) ]
H POF = exp [ - i φ ( ω ) ]
H OPOF = exp [ - i φ ( ω ) ] rect ( ω 2 π f 0 ) .
H o ( ω ) = { 1 , Im [ F ( ω ) ] 0 , - 1 , otherwise , = sgn [ - F o ( ω ) ] ,
H e ( ω ) = { 1 , Re [ F ( ω ) ] 0 , - 1 , otherwise , = sgn [ F e ( ω ) ] ,
H H ( ω ) = { 1 , H ( ω ) 0 , - 1 , otherwise , = sgn [ H ( ω ) ] ,
H ( ω ) = - f ( x ) [ cos ( ω · x ) + sin ( ω · x ) ] d 2 x = Re [ F ( ω ) ] - Im [ F ( ω ) ] = F e ( ω ) + F o ( ω ) ,
f e ( x ) = ½ [ f ( x ) + f ( - x ) ] , f o ( x ) = ½ [ f ( x ) - f ( - x ) ] .
- f e 2 ( x ) d 2 x - f 2 ( x ) d 2 x = 1 2 - f 2 ( x ) d 2 x + 1 2 - f ( x ) f ( - x ) d 2 x - f 2 ( x ) d 2 x .
1 2 - f e 2 ( x ) d 2 x - f 2 ( x ) d 2 x 1.
H H ( ω ) = sgn [ F ( ω ) ( cos ω · x 0 + sin ω · x 0 ) ] ,
C ( x ) = - F ( ω ) exp ( - i α ω · x 0 ) H H ( ω ) exp ( i ω · x ) d 2 ω ,
exp [ i ω · ( x - α x 0 ) ] = cos ω · x 0 ± i sin ω · x 0 .
F m e = | Δ f [ F e ( ω ) - i F o ( ω ) ] sgn [ F e ( ω ) ] exp ( i ω · x ) d 2 ω | max 2 Δ f P n ( ω ) d 2 ω - F ( ω ) 2 P n ( ω ) d 2 ω ,
F m H = | Δ f [ F e ( ω ) - i F o ( ω ) ] sgn [ F e ( ω ) + F o ( ω ) ] exp ( i ω · x ) d 2 ω | max 2 Δ f P n ( ω ) d 2 ω - F ( ω ) 2 P n ( ω ) d 2 ω .
C e ( x ) = Δ f [ F e ( ω ) - i F o ( ω ) ] sgn [ F e ( ω ) ] exp ( i ω · x d 2 ω ) 2 .
C e ( x ) = Δ f [ F e ( ω ) cos ω · x + F o ( ω ) sin ω · x ] sgn [ F e ( ω ) ] d 2 ω 2 .
C H ( x ) = Δ f [ F e ( ω ) - i F o ( ω ) ] sgn [ F e ( ω ) + F o ( ω ) ] exp ( i ω · x ) d 2 ω 2 = Δ f [ F e ( ω ) cos ω · x + F o ( ω ) sin ω · x - i F o ( ω ) cos ω · x + i F e ( ω ) sin ω · x ] sgn [ F e ( ω ) + F o ( ω ) ] d 2 ω 2 .
S ( ω ) = { 1 , F e ( ω ) F o ( ω ) , 0 , otherwise ,
1 - S ( ω ) = { 1 , F o ( ω ) F e ( ω ) , 0 , otherwise ,
C H ( x ) = [ Δ f F e ( ω ) S ( ω ) cos ω · x + F o ( ω ) S ( ω ) sgn [ F e ( ω ) · sin ω · x d 2 w ] 2 + [ Δ f F o ( ω ) [ 1 - S ( ω ) ] cos ω · x - F e ( ω ) [ 1 - S ( ω ) ] sgn [ F o ( ω ) ] sin ω · x d 2 ω ] 2 .
F m e = | Δ f [ F e ( ω ) - i F o ( ω ) ] sgn [ F e ( ω ) ] exp ( i ω · x ) d 2 ω | max 2 Δ f - F ( ω ) 2 d 2 ω ,
F m H = | Δ f [ F e ( ω ) - i F o ( ω ) ] sgn [ F e ( ω ) + F o ( ω ) ] exp ( i ω · x ) d 2 ω | max 2 Δ f - F ( ω ) 2 d 2 ω ,
0 < F m 1 , F m 0 as Δ f 0.
C H ( 0 ) = [ Δ f F e ( ω ) S ( ω ) d 2 ω ] 2 + { Δ f F o ( ω ) [ 1 - S ( ω ) ] d 2 ω } 2 .
H ¯ ( ω ) = max [ F e ( ω ) , F o ( ω ) ] .
C H ( 0 ) = [ Δ f H ¯ ( ω ) S ( ω ) d 2 ω ] 2 + [ Δ f H ¯ ( ω ) [ 1 - S ( ω ) ] d 2 ω ] 2 = [ Δ f H ¯ ( ω ) d 2 ω ] 2 - 2 [ Δ f H ¯ ( ω ) d 2 f ] [ Δ f H ¯ ( ω ) S ( ω ) d 2 ω ] + 2 [ Δ f H ¯ ( ω ) S ( ω ) d 2 ω ] 2 .
C H ( 0 ) [ Δ f H ¯ ( ω ) d 2 ω ] 2 = 1 - 2 Δ f H ¯ ( ω ) S ( ω ) d 2 ω Δ f H ¯ ( ω ) d 2 ω + 2 [ Δ f H ¯ ( ω ) S ( ω ) d 2 ω ] 2 [ Δ f H ¯ ( ω ) d 2 ω ] 2 = 1 - 2 R + 2 R 2 ,             0 < R 1 ,
1 2 C H ( 0 ) [ Δ f H ¯ ( ω ) d 2 ω ] 1.
[ Δ f H ( ω ) d 2 ω ] 2 = [ Δ f [ F e 2 ( ω ) + F o 2 ( ω ) ] 1 / 2 d 2 ω ] 2 ,
[ Δ f H ¯ ( ω ) d 2 ω ] 2 [ Δ f H ( ω ) d 2 ω ] 2 2 [ Δ f H ¯ ( ω ) d 2 ω ] 2 .
1 4 C H ( 0 ) [ Δ f H ( ω ) d 2 ω ] 2 1 ,
Δ f F e ( ω ) d 2 ω Δ f F e ( ω ) S ( ω ) d 2 ω = Δ f H ¯ ( ω ) S ( ω ) d 2 ω
R 2 2 C e ( 0 ) [ Δ H ( ω ) d 2 ω ] 2 1.
Δ f F e ( ω ) d 2 ω Δ f F ( ω ) d 2 ω
F e ( ω ) 2 d 2 ω F ( ω ) 2 d 2 ω 1 2 .

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