Abstract

When evaluated with a spatially uniform irradiance, an imaging sensor exhibits both spatial and temporal variations, which can be described as a three-dimensional (3D) random process considered as noise. In the 1990s, NVESD engineers developed an approximation to the 3D power spectral density for noise in imaging systems known as 3D noise. The goal was to decompose the 3D noise process into spatial and temporal components identify potential sources of origin. To characterize a sensor in terms of its 3D noise values, a finite number of samples in each of the three dimensions (two spatial, one temporal) were performed. In this correspondence, we developed the full sampling corrected 3D noise measurement and the corresponding confidence bounds. The accuracy of these methods was demonstrated through Monte Carlo simulations. Both the sampling correction as well as the confidence intervals can be applied a posteriori to the classic 3D noise calculation. The Matlab functions associated with this work can be found on the Mathworks file exchange [“Finite sampling corrected 3D noise with confidence intervals,” https://www.mathworks.com/matlabcentral/fileexchange/49657-finite-sampling-corrected-3d-noise-with-confidence-intervals.].

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References

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  1. D. Haefner, “Finite sampling corrected 3D noise with confidence intervals,” https://www.mathworks.com/matlabcentral/fileexchange/49657-finite-sampling-corrected-3d-noise-with-confidence-intervals .
  2. A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 2002).
  3. J. A. D’Agostino and C. M. Webb, “Three-dimensional analysis framework and measurement methodology for imaging system noise,” Proc. SPIE 1488, 110 (1991).
    [Crossref]
  4. C. M. Webb, “Approach to three-dimensional noise spectral analysis,” Proc. SPIE 2470, 288 (1995).
    [Crossref]
  5. P. O’Shea and S. Sousk, “Practical issues with 3D noise measurements and application to modern infrared sensors,” Proc. SPIE 5784, 262 (2005).
    [Crossref]
  6. R. G. Driggers, S. J. Pruchnic, C. E. Halford, and E. E. Burroughs, “Laboratory measurement of sampled infrared imaging system performance,” Opt. Eng. 38(5), 852–861 (1999).
    [Crossref]
  7. G. C. Holst, Testing and Evaluation of Infrared Imaging Systems, 2nd Ed (JCD Publishing, 1998).
  8. A. Lundmark, “3D detector noise revisited,” Proc. SPIE 8014, 801410 (2011).
    [Crossref]
  9. Z. Bomzon, “Biases in the estimation of 3D noise in thermal imagers,” Proc. SPIE 7834, 78340 (2010).
    [Crossref]
  10. Z. Bomzon, “Removing ths statistical bias from three-dimensional noise measurements,” Proc. SPIE 8014, 801416 (2011).
    [Crossref]
  11. P. Brémaud, An Introduction to Probabilistic Modeling (Springer, 1988).

2011 (2)

A. Lundmark, “3D detector noise revisited,” Proc. SPIE 8014, 801410 (2011).
[Crossref]

Z. Bomzon, “Removing ths statistical bias from three-dimensional noise measurements,” Proc. SPIE 8014, 801416 (2011).
[Crossref]

2010 (1)

Z. Bomzon, “Biases in the estimation of 3D noise in thermal imagers,” Proc. SPIE 7834, 78340 (2010).
[Crossref]

2005 (1)

P. O’Shea and S. Sousk, “Practical issues with 3D noise measurements and application to modern infrared sensors,” Proc. SPIE 5784, 262 (2005).
[Crossref]

1999 (1)

R. G. Driggers, S. J. Pruchnic, C. E. Halford, and E. E. Burroughs, “Laboratory measurement of sampled infrared imaging system performance,” Opt. Eng. 38(5), 852–861 (1999).
[Crossref]

1995 (1)

C. M. Webb, “Approach to three-dimensional noise spectral analysis,” Proc. SPIE 2470, 288 (1995).
[Crossref]

1991 (1)

J. A. D’Agostino and C. M. Webb, “Three-dimensional analysis framework and measurement methodology for imaging system noise,” Proc. SPIE 1488, 110 (1991).
[Crossref]

Bomzon, Z.

Z. Bomzon, “Removing ths statistical bias from three-dimensional noise measurements,” Proc. SPIE 8014, 801416 (2011).
[Crossref]

Z. Bomzon, “Biases in the estimation of 3D noise in thermal imagers,” Proc. SPIE 7834, 78340 (2010).
[Crossref]

Brémaud, P.

P. Brémaud, An Introduction to Probabilistic Modeling (Springer, 1988).

Burroughs, E. E.

R. G. Driggers, S. J. Pruchnic, C. E. Halford, and E. E. Burroughs, “Laboratory measurement of sampled infrared imaging system performance,” Opt. Eng. 38(5), 852–861 (1999).
[Crossref]

D’Agostino, J. A.

J. A. D’Agostino and C. M. Webb, “Three-dimensional analysis framework and measurement methodology for imaging system noise,” Proc. SPIE 1488, 110 (1991).
[Crossref]

Driggers, R. G.

R. G. Driggers, S. J. Pruchnic, C. E. Halford, and E. E. Burroughs, “Laboratory measurement of sampled infrared imaging system performance,” Opt. Eng. 38(5), 852–861 (1999).
[Crossref]

Halford, C. E.

R. G. Driggers, S. J. Pruchnic, C. E. Halford, and E. E. Burroughs, “Laboratory measurement of sampled infrared imaging system performance,” Opt. Eng. 38(5), 852–861 (1999).
[Crossref]

Holst, G. C.

G. C. Holst, Testing and Evaluation of Infrared Imaging Systems, 2nd Ed (JCD Publishing, 1998).

Lundmark, A.

A. Lundmark, “3D detector noise revisited,” Proc. SPIE 8014, 801410 (2011).
[Crossref]

O’Shea, P.

P. O’Shea and S. Sousk, “Practical issues with 3D noise measurements and application to modern infrared sensors,” Proc. SPIE 5784, 262 (2005).
[Crossref]

Papoulis, A.

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 2002).

Pillai, S. U.

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 2002).

Pruchnic, S. J.

R. G. Driggers, S. J. Pruchnic, C. E. Halford, and E. E. Burroughs, “Laboratory measurement of sampled infrared imaging system performance,” Opt. Eng. 38(5), 852–861 (1999).
[Crossref]

Sousk, S.

P. O’Shea and S. Sousk, “Practical issues with 3D noise measurements and application to modern infrared sensors,” Proc. SPIE 5784, 262 (2005).
[Crossref]

Webb, C. M.

C. M. Webb, “Approach to three-dimensional noise spectral analysis,” Proc. SPIE 2470, 288 (1995).
[Crossref]

J. A. D’Agostino and C. M. Webb, “Three-dimensional analysis framework and measurement methodology for imaging system noise,” Proc. SPIE 1488, 110 (1991).
[Crossref]

Opt. Eng. (1)

R. G. Driggers, S. J. Pruchnic, C. E. Halford, and E. E. Burroughs, “Laboratory measurement of sampled infrared imaging system performance,” Opt. Eng. 38(5), 852–861 (1999).
[Crossref]

Proc. SPIE (6)

J. A. D’Agostino and C. M. Webb, “Three-dimensional analysis framework and measurement methodology for imaging system noise,” Proc. SPIE 1488, 110 (1991).
[Crossref]

C. M. Webb, “Approach to three-dimensional noise spectral analysis,” Proc. SPIE 2470, 288 (1995).
[Crossref]

P. O’Shea and S. Sousk, “Practical issues with 3D noise measurements and application to modern infrared sensors,” Proc. SPIE 5784, 262 (2005).
[Crossref]

A. Lundmark, “3D detector noise revisited,” Proc. SPIE 8014, 801410 (2011).
[Crossref]

Z. Bomzon, “Biases in the estimation of 3D noise in thermal imagers,” Proc. SPIE 7834, 78340 (2010).
[Crossref]

Z. Bomzon, “Removing ths statistical bias from three-dimensional noise measurements,” Proc. SPIE 8014, 801416 (2011).
[Crossref]

Other (4)

P. Brémaud, An Introduction to Probabilistic Modeling (Springer, 1988).

D. Haefner, “Finite sampling corrected 3D noise with confidence intervals,” https://www.mathworks.com/matlabcentral/fileexchange/49657-finite-sampling-corrected-3d-noise-with-confidence-intervals .

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 2002).

G. C. Holst, Testing and Evaluation of Infrared Imaging Systems, 2nd Ed (JCD Publishing, 1998).

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Figures (3)

Fig. 1.
Fig. 1. Example cartoon depiction of the v h random process sampled in 3D across H , V , and T samples. Note that the v h frame is independent of time. The results of the directional averaging of this process also are shown.
Fig. 2.
Fig. 2. Plot of distributions of observed variance compared to Gaussian assumption for V = 24 , H = 32 , and T = 30 ; estimated (A) temporal variance, (B) vertical variance, (C) temporal-vertical variance, (D) vertical-horizontal variance, and (E) temporal-vertical-horizontal variance.
Fig. 3.
Fig. 3. Plot of the estimated variance of the v random process in the case of limited coupling (A)  σ v = 10 , σ v h = 10 , and severe coupling (B)  σ v = 10 , σ v h = 100 . Blue circles correspond to Monte Carlo, black dashed is independent chi-squared distribution, and red solid is coupled variance derived Gaussian distribution.

Tables (2)

Tables Icon

Table 1. Percent Error in Average Variance from the Monte Carlo Simulation. Class Refers to Mixing with Eq. (7), UB0 Refers to Eq. (9), and the New Refers to the Mixing Derived of Eq. (6)

Tables Icon

Table 2. Observed Percentage of Confidence Intervals Capturing the Population Statistic

Equations (49)

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Noise ( h , v , t ) = noise t ( t ) + noise v ( v ) + noise h ( h ) + noise t v ( v , t ) + noise t h ( h , t ) + noise v h ( h , v ) + noise tvh ( h , v , t ) .
σ t 2 = 1 T 1 ( = 1 T noise t ( ) 2 ) , σ t v 2 = 1 T V 1 ( = 1 T k = 1 V noise t v ( k , ) 2 ) , σ v 2 = 1 V 1 ( k = 1 V noise v ( k ) 2 ) , σ t h 2 = 1 T H 1 ( = 1 T j = 1 H noise t h ( j , ) 2 ) , σ h 2 = 1 H 1 ( j = 1 H noise h ( j ) 2 ) , σ v h 2 = 1 V H 1 ( j = 1 H k = 1 V noise t v ( j , k ) 2 ) , σ tvh 2 = 1 T V H 1 ( = 1 T j = 1 H k = 1 V noise tvh ( j , k , ) 2 ) .
x ¯ = 1 n j = 1 n x j .
σ 2 = 1 n 1 j = 1 n ( x j x ¯ ) 2 .
σ D t 2 = σ D t , t 2 + σ D t , v 2 + σ D t , h 2 + σ D t , t v 2 + σ D t , t h 2 + σ D t , v h 2 + σ D t , tvh 2 .
[ σ D v h 2 σ D t h 2 σ D t v 2 σ D h 2 σ D v 2 σ D t 2 σ D tvh 2 ] = M [ σ t 2 σ v 2 σ h 2 σ t v 2 σ t h 2 σ v h 2 σ tvh 2 ] ,
M = [ 1 0 0 1 V 1 H 0 1 V H 0 1 0 1 T 0 1 H 1 T H 0 0 1 0 1 T 1 V 1 T V V ( T 1 ) T V 1 T ( V 1 ) T V 1 0 1 V T V 1 T 1 H T T V 1 V 1 H 1 H H ( T 1 ) T H 1 0 T ( H 1 ) T H 1 H T H 1 T 1 V 1 T T H 1 H 1 V 1 V 0 H ( V 1 ) V H 1 V ( H 1 ) V H 1 H V H 1 V 1 T V V H 1 H 1 T 1 1 T V H ( T 1 ) T V H 1 T H ( V 1 ) T V H 1 T V ( H 1 ) T V H 1 H ( T V 1 ) T V H 1 V ( T H 1 ) T V H 1 T ( V H 1 ) T V H 1 1 ] ,
[ σ D v h 2 σ D t h 2 σ D t v 2 σ D h 2 σ D v 2 σ D t 2 σ D tvh 2 ] = [ 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 1 0 0 0 1 0 1 0 1 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 1 ] [ σ t 2 σ v 2 σ h 2 σ t v 2 σ t h 2 σ v h 2 σ tvh 2 ] .
[ σ ¯ t 2 σ ¯ v 2 σ ¯ h 2 σ ¯ t v 2 σ ¯ t h 2 σ ¯ v h 2 σ ¯ tvh 2 ] = [ 1 0 0 1 V 1 H 0 1 V H 0 1 0 1 T 0 1 H 1 T H 0 0 1 0 1 T 1 V 1 T V 0 0 0 1 0 0 1 H 0 0 0 0 1 0 1 V 0 0 0 0 0 1 1 T 0 0 0 0 0 0 1 ] [ σ t 2 σ v 2 σ h 2 σ t v 2 σ t h 2 σ v h 2 σ tvh 2 ] .
[ σ D v h 2 σ D t h 2 σ D t v 2 σ D h 2 σ D v 2 σ D t 2 σ D tvh 2 ] = M U B 0 [ σ t 2 σ v 2 σ h 2 σ t v 2 σ t h 2 σ v h 2 σ tvh 2 ] ,
M U B 0 = [ 1 0 0 1 V 1 H 0 1 V H 0 1 0 1 T 0 1 H 1 T H 0 0 1 0 1 T 1 V 1 T V 1 1 0 V + T + T V T V 1 H 1 H T + V + T V T V H 1 0 1 1 V H + T + T H T H 1 V H + T + H T T V H 0 1 1 1 T 1 T H + V + H V V H H + V + H V T V H 1 1 1 V + T + T V T V H + T + T H T H H + V + H V V H C T V H ] ,
cov ( [ σ t 2 σ v 2 σ h 2 σ t v 2 σ t h 2 σ v h 2 σ tvh 2 ] , [ σ t 2 σ v 2 σ h 2 σ t v 2 σ t h 2 σ v h 2 σ tvh 2 ] ) = M 1 ( cov ( [ σ D v h 2 σ D t h 2 σ D t v 2 σ D h 2 σ D v 2 σ D t 2 σ D tvh 2 ] , [ σ D v h 2 σ D t h 2 σ D t v 2 σ D h 2 σ D v 2 σ D t 2 σ D tvh 2 ] ) ) ( M 1 ) T ,
cov ( x , y ) = E [ ( x μ x ) ( y μ y ) ] = σ x σ y corr ( x , y ) .
X 2 = ( n 1 ) S 2 σ 2
f scaled χ 2 ( x ; v , a ) = { ( x / a ) ( v / 2 1 ) e x / 2 a 2 v / 2 Γ ( v / 2 ) a x 0 0 x < 0 ,
E [ S 2 ] = f scaled χ 2 ( x ; v , a ) x d x = a v = σ 2 .
V a r S 2 = E [ ( S 2 ) 2 ] E [ S 2 ] = f scaled χ 2 ( x ; v , a ) x 2 d x a v = 2 a 2 ν .
a ¯ t = [ a D v h , t a D t h , t a D t v , t a D h , t a D v , t a D t , t a D tvh , t ] = [ 1 T 1 0 0 V T V 1 H T V 1 0 V H T V H 1 ] [ σ t 2 ] , ν ¯ t = [ ν D v h , t ν D t h , t ν D t v , t ν D h , t ν D v , t ν D t , t ν D tvh , t ] = [ T 0 0 T T 0 T ] .
cor r t ( [ σ D v h , t 2 σ D t h , t 2 σ D t v , t 2 σ D h , t 2 σ D v , t 2 σ D t , t 2 σ D tvh , t 2 ] , [ σ D v h , t 2 σ D t h , t 2 σ D t v , t 2 σ D h , t 2 σ D v , t 2 σ D t , t 2 σ D tvh , t 2 ] ) = [ 1 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 1 1 0 0 1 1 0 1 0 0 0 0 0 0 0 1 0 0 1 1 0 1 ] .
cov ( [ σ D v h , t 2 σ D t h , t 2 σ D t v , t 2 σ D h , t 2 σ D v , t 2 σ D t , t 2 σ D tvh , t 2 ] , [ σ D v h , t 2 σ D t h , t 2 σ D t v , t 2 σ D h , t 2 σ D v , t 2 σ D t , t 2 σ D tvh , t 2 ] ) = corr t · ( ( 2 a ¯ t 2 ν ¯ t ) ( 2 a ¯ t 2 ν ¯ t ) T ) ,
C I L PDF ( x ) d x = α 2 , C I H PDF ( x ) d x = 1 α 2 .
[ σ ^ t 2 σ ^ v 2 σ ^ h 2 σ ^ t v 2 σ ^ t h 2 σ ^ v h 2 σ ^ tvh 2 ] = [ 100 100 100 100 100 100 100 ] , CI σ 2 = [ 43.3 48.6 41.8 9.0 7.8 8.7 1.6 ] , [ σ ^ t 2 σ ^ v 2 σ ^ h 2 σ ^ t v 2 σ ^ t h 2 σ ^ v h 2 σ ^ tvh 2 ] = [ 100 100 100 100 100 10000 100 ] , CI σ 2 = [ 43.3 161.4 182.5 9.0 7.8 869.9 1.6 ] .
σ ^ = sign ( σ ^ 2 ) | σ ^ 2 | ,
CI σ = arg max ( | σ ^ sign ( σ ^ 2 ± CI σ 2 ) | σ ^ 2 ± CI σ 2 | | ) .
[ σ ^ t σ ^ v σ ^ h σ ^ t v σ ^ t h σ ^ v h σ ^ tvh ] = [ 10 10 10 10 10 100 10 ] , CI σ , α = . 1 = [ 2.5 17.8 19.1 0.5 0.4 4.4 0.1 ] .
D t , t ( h , v ) = 1 T j = 1 T noise t ( t j ) = constant ,
σ D t , t 2 = 1 V H 1 k = 1 V j = 1 H ( D t , t ( j , k ) D ¯ t , t ) 2 = 0 .
D t , v ( h , v ) = 1 T = 1 T noise v ( h , v , t ) = noise v ( v ) .
σ D t , v 2 = 1 V H 1 k = 1 V j = 1 H ( D t , v ( j , k ) D ¯ t , v ) 2 = H V H 1 k = 1 V · noise v ( k ) 2 = H ( V 1 ) σ v 2 V H 1 .
D t , v h ( h , v ) = 1 T = 1 T noise v h ( h , v , t ) = noise v h ( h , v ) .
σ D t , v h 2 = 1 V H 1 k = 1 V j = 1 H ( D t , v h ( j , k ) D ¯ t , v h ) 2 σ D t , v h 2 = 1 V H 1 k = 1 V j = 1 H ( noise v h ( j , k ) ) 2 = σ v h 2 .
D t , t v ( h , v ) = 1 T = 1 T noise t v ( h , v , t ) = noise t , t v ( v ) .
D t , t v ( h , v ) = N ( 0 , σ t , t v 2 ) σ t , t v 2 = 1 V 1 ( k = 1 V ( noise t , t v ( k ) n ¯ ois e t , t v ( k ) ) 2 ) = σ t v 2 T .
σ D t , t v 2 = 1 V H 1 k = 1 V j = 1 H ( D t , t v ( j , k ) D ¯ t , t v ) 2 = H V H 1 ( k = 1 V ( noise t , t v ( k ) n ¯ ois e t , t v ( k ) ) 2 ) = H V H 1 ( V 1 ) σ t v 2 T .
D t , tvh ( h , v ) = 1 T = 1 T noise tvh ( h , v , t ) = noise t , tvh ( 0 , σ tvh 2 T ) .
σ D t , tvh 2 = 1 V H 1 k = 1 V j = 1 H ( D t , tvh ( j , k ) D ¯ t , tvh ) 2 = σ tvh 2 T .
D t v , t ( h ) = 1 V 1 T k = 1 V = 1 T noise t ( h , v k , t ) = constant .
σ D t v , t 2 = 1 H 1 j = 1 H ( D t v , t ( j ) D ¯ t v , t ) 2 = 0 .
D t v , h ( h ) = 1 V 1 T k = 1 V = 1 T noise h ( h , v k , t ) = noise h ( h ) .
σ D t v , h 2 = 1 H 1 j = 1 H ( D t v , t ( j ) D ¯ t v , t ) 2 = σ h 2 .
D t v , t v ( h ) = 1 V 1 T k = 1 V = 1 T noise t v ( h , v k , t ) = constant .
σ D t v , t v 2 = 1 H 1 j = 1 H ( D t v , t v ( j ) D ¯ t v , t v ) 2 = 0 .
σ D t v , t v 2 = 0 , σ D t h , t h 2 = 0 , σ D v h , v h 2 = 0 .
D t v , t h ( h ) = 1 V 1 T k = 1 V = 1 T noise t v ( h , v k , t ) D t v , t h ( h ) = noise t v , t h ( 0 , σ t h 2 T ) .
σ D t v , t h 2 = 1 H 1 j = 1 H ( D t v , t v ( j ) D ¯ t v , t v ) 2 = σ t h 2 T .
D t v , tvh ( h ) = 1 V 1 T k = 1 V = 1 T noise tvh ( h , v k , t ) D t v , tvh ( h ) = noise t v , t h h ( 0 , σ tvh 2 T V ) ,
σ D t v , tvh 2 = 1 H 1 j = 1 H ( D t v , t v ( j ) D ¯ t v , t v ) 2 = σ tvh 2 T V .
σ D t v , tvh 2 = σ tvh 2 T V , σ D t h , tvh 2 = σ tvh 2 T H , σ D v h , tvh 2 = σ tvh 2 V H .
σ N t , tvh 2 = 1 T V H 1 m = 1 H k = 1 V = 1 T noise t ( h m , v k , t ) σ N t , tvh 2 = V H T V H 1 = 1 T noise t ( h m , v k , t ) σ N t , tvh 2 = V H ( T 1 ) T V H 1 σ t 2 .

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