Abstract

Imaging systems designed for point target detection and 3D reconstruction must be filtered in order to maximize signal-to-noise ratio and minimize position expectation error. An optimal whitening matched filter (WMF) is derived based on expected spatial target distribution and system colored noise. The expected noise is derived as a weighted combination of clutter aliasing, target aliasing, and detector noise. Further optimization of system performance is achieved by modification of the optical point spread function (PSF), so a sampling-balanced operation is achieved where all noise components are comparable. The improved performance of the optimized system is calculated and compared to the performance of other systems using other known linear postfilters with various optical PSF widths. It is shown that the WMF in a sampling-balanced system is a robust configuration that needs only minor modifications when scenario parameters are varied.

© 2010 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. S. K. Park and R. Hazra, “Aliasing as noise: a quantitative and qualitative assessment,” Proc. SPIE 1969, 54–65 (1993).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  7. H. Quevedo, “Modeling of aliasing effects for target detection in undersampled IR imaging systems,” Proc. SPIE 4728, 24–35 (2002).
    [CrossRef]
  8. R. E. Blahut, Theory of Remote Image Formation (Cambridge U. Press2004), pp. 51, 55, 60, 439, and 443.
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2009 (1)

2002 (1)

H. Quevedo, “Modeling of aliasing effects for target detection in undersampled IR imaging systems,” Proc. SPIE 4728, 24–35 (2002).
[CrossRef]

1999 (2)

R. D. Fiete, “Image quality and λNF/p for remote sensing systems,” Opt. Eng. 38, 1229–1240 (1999).
[CrossRef]

S. K. Park and Z. Rahman, “Fidelity analysis of sampled imaging systems,” Opt. Eng. 38, 786–800 (1999).
[CrossRef]

1993 (1)

S. K. Park and R. Hazra, “Aliasing as noise: a quantitative and qualitative assessment,” Proc. SPIE 1969, 54–65 (1993).
[CrossRef]

1987 (1)

1980 (1)

1964 (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

Blahut, R. E.

R. E. Blahut, Theory of Remote Image Formation (Cambridge U. Press2004), pp. 51, 55, 60, 439, and 443.

Burton, G. J.

Fienup, J. R.

Fiete, R. D.

R. D. Fiete, “Image quality and λNF/p for remote sensing systems,” Opt. Eng. 38, 1229–1240 (1999).
[CrossRef]

Halyo, N.

Hazra, R.

S. K. Park and R. Hazra, “Aliasing as noise: a quantitative and qualitative assessment,” Proc. SPIE 1969, 54–65 (1993).
[CrossRef]

Huck, F. O.

Moorhead, I. R.

Park, S. K.

S. K. Park and Z. Rahman, “Fidelity analysis of sampled imaging systems,” Opt. Eng. 38, 786–800 (1999).
[CrossRef]

S. K. Park and R. Hazra, “Aliasing as noise: a quantitative and qualitative assessment,” Proc. SPIE 1969, 54–65 (1993).
[CrossRef]

F. O. Huck, N. Halyo, and S. K. Park, “Aliasing and blurring in 2-D sampled imagery,” Appl. Opt. 19, 2174–2181 (1980).
[CrossRef] [PubMed]

Quevedo, H.

H. Quevedo, “Modeling of aliasing effects for target detection in undersampled IR imaging systems,” Proc. SPIE 4728, 24–35 (2002).
[CrossRef]

Rahman, Z.

S. K. Park and Z. Rahman, “Fidelity analysis of sampled imaging systems,” Opt. Eng. 38, 786–800 (1999).
[CrossRef]

Thurman, S. T.

VanderLugt, A.

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory 10, 139–145 (1964).
[CrossRef]

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

Opt. Eng. (2)

S. K. Park and Z. Rahman, “Fidelity analysis of sampled imaging systems,” Opt. Eng. 38, 786–800 (1999).
[CrossRef]

R. D. Fiete, “Image quality and λNF/p for remote sensing systems,” Opt. Eng. 38, 1229–1240 (1999).
[CrossRef]

Proc. SPIE (2)

S. K. Park and R. Hazra, “Aliasing as noise: a quantitative and qualitative assessment,” Proc. SPIE 1969, 54–65 (1993).
[CrossRef]

H. Quevedo, “Modeling of aliasing effects for target detection in undersampled IR imaging systems,” Proc. SPIE 4728, 24–35 (2002).
[CrossRef]

Other (1)

R. E. Blahut, Theory of Remote Image Formation (Cambridge U. Press2004), pp. 51, 55, 60, 439, and 443.

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Figures (13)

Fig. 1
Fig. 1

Schematic description of image acquisition process.

Fig. 2
Fig. 2

Spatial distributions of randomly generated clutter before and after being filtered by the optics and the detector.

Fig. 3
Fig. 3

(a) Spatial and (b) Fourier distributions of clutter aliasing (“image-related noise”) after the unaliased clutter was subtracted. Strong white detector noise will appear as a constant addition to the Fourier distribution.

Fig. 4
Fig. 4

Point target distribution in (a) spatial and (b) Fourier domains.

Fig. 5
Fig. 5

(a) Spectrum of image noise (clutter aliasing and low detector noise) and target aliasing with the appropriate WMF. (b) Spectrum of noise and target after WMF. The nonwhite spectral distribution of aliasing noise is apparent.

Fig. 6
Fig. 6

Comparing the WMF with commonly used linear filters. Actual width of the filters can vary according to the specific scenario. However, WMF remains the widest.

Fig. 7
Fig. 7

Image noise and signal after postfiltering for (a) bilinear and (b) WMF.

Fig. 8
Fig. 8

System performance without clutter at SNR n = 10 versus optical PSF σ opt : (a) SNR and (b) PEE. WMF achieves the best performance.

Fig. 9
Fig. 9

Relative contribution of the noise components versus optical PSF for the WMF configuration described in Fig. 8.

Fig. 10
Fig. 10

(a) SNR and (b) PEE versus σ opt for aliasing limited system ( SNR n = 100 , STC n = 0.1 ).

Fig. 11
Fig. 11

Noise sources for the WMF at various optical blur. Sampling balance is at σ opt 0.75 [ pixel ] , where the noise components are comparable.

Fig. 12
Fig. 12

(a) SNR and (b) PEE of weak (solid curve) and strong (dash curve) targets versus σ opt . Sampling-balanced system (marked with circle) is optimized for maximal SNR of weak target. The appearance of a strong target improves SNR substantially. However, for an undersampled system (marked with rectangle), the improvement is less significant.

Fig. 13
Fig. 13

The WMF for different target power as described in Fig. 12 at (a) sampling-balanced system σ opt = 0.6 pixel and (b) for the undersampled system σ opt = 0.3 pixel . It is apparent that minor filter modification is needed for the sampling-balanced system, while in the unbalanced system, substantial change is needed.

Equations (21)

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i r ( x ) = i c ( x ) + i t ( x ) ,
p s ( x ) = p d ( x ) p o ( x ) ,
p d ( x ) r ( x ) r ( x ) d x = r ( x ) S d .
i p r ( x ) = p s i r ( x ) = i p c ( x ) + i p t ( x ) .
i s r ( x ) = S d · i p r ( x ) · comb ( x / X d ) + n d ( x ) · comb ( x / X d ) ,
n s ( x ) i s r ( x ) i p r ( x ) .
n s ( x ) = n u ( x ) + n t a ( x ) .
n u ( x ) S d · i p c ( x ) + n d ( x ) · comb ( x / X d ) i p c ( x ) ,
n t a ( x ) S d · i p t ( x ) · comb ( x / X d ) i p t ( x ) .
i k = k i s r i k t + n k ,
N u ( f ) = | CFT { n u } | 2 = m = m 0 | I p c ( f m · X d ) | 2 + N 0 = m = m 0 | P s ( f m · X d ) | 2 · | I c ( f m · X d ) | 2 + N 0 ,
N t a ( f ) = | CFT { n t a } | 2 = m = m 0 | I p t ( f m · X d ) | 2 = m = m 0 | P s ( f m · X d ) | 2 · | I t ( f m · X d ) | 2 ,
N s c ( f ) = N u ( f ) + C p · N t a ( f ) ,
C p image area target erea = X d 2 · M S W ( i p t ) 2 ,
S W ( i p t ) = | i p t | 2 d x max | i p t | 2 .
K ( f ) = I p t ( f ) N s c ( f ) .
k ( x ) = CIFT { K ( f ) } ,
STC n max { i p t n } STD { n p c n } ,
SNR n = max { i p t n } STD { n d } .
SNR = max { i k t } STD { n k } .
σ target = 1 2 π · SNR · B G ,

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