Non-linear statistical photocalibration of photodetectors without calibrated light sources

Stephen C. Cain

doi:10.1364/AO.385753

Applied Optics
Vol. 59,
Issue 9,
pp. 2767-2775
(2020)
•https://doi.org/10.1364/AO.385753

Non-linear statistical photocalibration of photodetectors without calibrated light sources

Stephen C. Cain

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Stephen C. Cain, "Non-linear statistical photocalibration of photodetectors without calibrated light sources," Appl. Opt. 59, 2767-2775 (2020)

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Table of Contents Category
- Optical Devices, Sensors, and Detectors
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About this Article
History
- Original Manuscript: December 18, 2019
- Revised Manuscript: February 15, 2020
- Manuscript Accepted: February 18, 2020
- Published: March 18, 2020

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Abstract

Calibration of CCD arrays is commonly conducted using dark frames. Non-absolute calibration techniques only measure the relative response of the detectors. For absolute calibration to be achieved, a second calibration is sometimes utilized by looking at sources with known radiances. A process like this can be used to calibrate photodetectors if a calibration source is available and sensor time can be spared to perform the operation. A previous attempt at creating a procedure for calibrating a photodetector using the underlying Poisson nature of the photodetection statistics relied on a linear model. This effort produced the statistically applied non-uniformity calibration algorithm, which demonstrated an ability to relate the measured signal with the true radiance of the source. Reliance on a completely linear model does not allow for non-linear behaviors to be described, thus potentially producing poor photocalibration over large dynamic ranges. In this paper, a photocalibration procedure is defined that requires only first and second moments of the measurements and allows the response to be modeled using a non-linear function over the dynamic range of the detector. The technique is applied to image data containing a light source measured with different integration times showing that the non-linear technique achieves significant improvement over the linear model over a large dynamic range.

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Step#	Variance Method and SANUC	NLSNUC
Compute offset, $O$ 1 linear algorithms 1 NLSNUC	$O = C a m e r a$ black level set by user (Can also be determined by dark frame measurements)	$O = C a m e r a$ black level set by user (Can also be determined by dark frame measurements)
Compute mean of datasets $D_{1}$ , $D_{2}$ 2 linear algorithms 2 NLSNUC	Average the data in the frame dimension after subtracting the offset $O$	Same as SANUC
Compute variances of datasets $D_{1}$ , $D_{2}$ 3 linear algorithms N/A NLSNUC	For each pixel in the array, compute the variance	Not done with NLSNUC
Compute detector gain 4 linear algorithms 7 NLSNUC	$G (x, y) = \frac{σ_{D_{2}}^{2} (x, y) - σ_{D_{1}}^{2} (x, y)}{{\bar{D}}_{2} (x, y) - {\bar{D}}_{1} (x, y)}$	$α (x, y) = \frac{σ_{H_{2}}^{2} (x, y) - σ_{H_{1}}^{2} (x, y)}{{\bar{H}}_{2} (x, y) - {\bar{H}}_{1} (x, y)}$
Compute $\bar{K}$ 5 linear algorithms 8 NLSNUC	$\bar{K} (x, y) = \frac{{\bar{D}}_{2} (x, y) - {\bar{D}}_{1} (x, y)}{(N - 1) G (x, y)}$	$\bar{K} (x, y) = \frac{{\bar{H}}_{2} (x, y) - {\bar{H}}_{1} (x, y)}{(N - 1) α (x, y)}$
Compute model bias 6 linear algorithms N/A NLSNUC	Not done for the variance method. For the SANUC algorithm: $B (x, y) = {\bar{D}}_{1} (x, y) - G (x, y) \bar{K} (x, y)$	Not necessary for NLSNUC
Compute the parameter $\hat{g}$ N/A linear algorithms 3 NLSNUC	Not necessary for the variance method or SANUC	Numerical procedure of finding the value of $\hat{g}$ $\hat{g} (x, y) = \underset{g (x, y)}{\arg min} (\frac{{\bar{D}}_{2} (x, y)}{{\bar{D}}_{1} (x, y)} - \frac{1 - e^{N_{2} g (x, y) / N_{1}}}{1 - e^{g (x, y)}})^{2}$
Compute the saturation, $C (x, y)$ N/A SANUC 4 NLSNUC	Not necessary for the variance method or SANUC	$C (x, y) = \sum_{f = 1}^{M} {\bar{D}}_{1} (x, y) / (1 - e^{\hat{g} (x, y)})$
Non-linear correction N/A linear algorithms 5 NLSNUC	Not necessary for the variance method or SANUC	$H_{1} (x, y) = - \log (1 - (D_{1} (x, y) - O) / C (x, y)) H_{2} (x, y) = - \log (1 - (D_{2} (x, y) - O) / C (x, y))$
Compute means and variances of data with non-linear correction N/A linear algorithms 6 NLSNUC	Not necessary for the variance method or SANUC	$\begin{array}{l} {\bar{H}}_{1} (x, y) = E [H_{1} (x, y)] \\ {\bar{H}}_{2} (x, y) = E [H_{2} (x, y)] \\ σ_{H_{1}}^{2} (x, y) = E [(H_{1} (x, y) - {\bar{H}}_{1} (x, y))^{2}] \\ σ_{H_{2}}^{2} (x, y) = E [(H_{2} (x, y) - {\bar{H}}_{2} (x, y))^{2}] \end{array}$
Compute estimated value of K from data sample 7 linear algorithms 9 NLSNUC	$\hat{K}$ is an estimate of the number of electrons. For the variance method: $\overset{⌢}{K} (x, y, f) = (D (x, y, f) - O) / G (x, y)$ For SANUC: $\overset{⌢}{K} (x, y, f) = (D (x, y, f) - B (x, y) - O) / G (x, y)$	$\hat{K}$ is an estimate of the number of electrons $\hat{K} (x, y, f) = - \log (1 - (D (x, y, f) - O) / C (x, y)) / \hat{α} (x, y)$

Dataset	Integration Time (milliseconds)	Median Digital Count
Dark	1, 2, 20, 40, and 100	52
$D_{1}$	1	188
$D_{2}$	2	309
$D_{20}$	20	2400
$D_{40}$	40	4704
$D_{100}$	100	11215

Parameter	Value	Unit
${\bar{K}}_{1}$	443.42	Electrons
Gain (G)	0.1687	Digital counts/electron
Bias (B)	Not computed for the variance method 201.975 for SANUC	Digital counts

Parameter	Value	Unit
$K_{1}$	391.84	Electrons
Nonlinear gain (α)	$2.366 x 10^{- 6}$	${E l e c t r o n s}^{- 1}$
Saturation parameter (C)	$1.0277 x 10^{5}$	Digital counts

Algorithm	$E_{1}$	$E_{2}$	$E_{20}$	$E_{40}$	$E_{100}$	$σ_{1}$	$σ_{2}$	$σ_{20}$	$σ_{40}$	$σ_{100}$
Variance method	25.97	37.03	152.63	260.89	260.90	0.02	0.03	0.19	0.38	0.43
SANUC	184.17	173.10	59.73	57.67	63.24	0.37	0.36	0.19	0.04	0.05
NLSNUC	17.14	19.53	27.79	37.84	45.70	0.01	0.02	0.02	0.03	0.08
Min Error Difference	8.83	17.50	31.94	19.83	17.54	N/A	N/A	N/A	N/A	N/A
$\frac{min E r r D i f f}{\sqrt{2 σ_{N}^{2}}}$	16.8	33.5	118.9	70.4	28.8	N/A	N/A	N/A	N/A	N/A
Confidence	$> 99.99 %$	$> 99.99 %$	$> 99.99 %$	$> 99.99 %$	$> 99.99 %$	N/A	N/A	N/A	N/A	N/A

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Applied Optics