3D noise photon transfer curve

Bradley L. Preece; David P. Haefner

doi:10.1364/AO.452166

Applied Optics
Vol. 61,
Issue 21,
pp. 6202-6212
(2022)
•https://doi.org/10.1364/AO.452166

3D noise photon transfer curve

Bradley L. Preece and David P. Haefner

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Bradley L. Preece and David P. Haefner, "3D noise photon transfer curve," Appl. Opt. 61, 6202-6212 (2022)

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Abstract

A photon transfer curve (PTC) is used to determine fundamental detector noise parameters such as read noise, conversion gain, and fixed pattern noise. Here, the method for determining a PTC is expanded to include 3D noise parameters. 3D noise PTC provides more insight into detector noise and is treated as the next logical step to classical PTC. However, it induces several new challenges in analyzing the results, specifically the fitting of seven, or more, variance curves compared to the one (total variance) or two (temporal and fixed pattern variance) prior. Therefore, a general measurement model is created, which provides a new method to separate out all the classical terms, such as DSNU and PRNU, but can also handle high gain cameras with a noise factor. This method is then verified using Monte Carlo simulations and applied to a commercial machine vision camera. In addition, the effects of lens vignetting and non-uniformity correction (NUC) are explored, along with a comparison of the single pixel PTC.

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Supplementary Material (1)

Name	Description
Code 1	Matlab Code for Figures 3 and 4, including Ref 10 SW

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Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Type	3D $Var$ Shorthand	Full Function (Equal to Mid-Column)
$T$	$σ_{T}^{2} = Var {Y}_{T}$	$Var {E {Y}_{x}}_{t}$
$V$	$σ_{V}^{2} = Var {Y}_{V}$	$Var {E {Y}_{yt}}_{x}$
$H$	$σ_{H}^{2} = Var {Y}_{H}$	$Var {E {Y}_{xt}}_{y}$
$T V$	$σ_{TV}^{2} = Var {Y}_{TV}$	$Var {E {Y}_{x} - E {Y}_{xy} - E {Y}_{tx}}_{ty}$
$T H$	$σ_{TH}^{2} = Var {Y}_{TH}$	$Var {E {Y}_{y} - E {Y}_{xy} - E {Y}_{ty}}_{tx}$
$V H$	$σ_{VH}^{2} = Var {Y}_{VH}$	$Var {E {Y}_{t} - E {Y}_{tx} - E {Y}_{ty}}_{xy}$
$T V H$	$σ_{TVH}^{2} = Var {Y}_{TVH}$	$Var {Y - E {Y}_{x} - E {Y}_{y} - E {Y}_{t}$ $+ {E {Y}_{xy} + E {Y}_{tx} + E {Y}_{ty}}}_{xyt}$
$N A$	$σ_{t}^{2}$	$σ_{T}^{2} + σ_{TV}^{2} + σ_{TH}^{2} + σ_{TVH}^{2}$

Terms from Eq. (3)	Noise Variance	3D Noise Type
$ϕ_{xy} G$	$Var {ϕ_{xy} T}_{VH} G^{2}$	$V H$
	$Var {ϕ_{xy} T}_{H} G^{2}$	$H$
	$Var {ϕ_{xy} T}_{V} G^{2}$	$V$
$ϕ_{xy} G_{vh}$	$Var {ϕ_{xy} T} Var {G_{vh}} + ϕ^{2} T^{2} Var {G_{vh}}$	$V H$ $V H^{2}$
$ϕ_{xy} G_{v}$	$Var {ϕ_{xy} T}_{VH} Var {G_{v}}$	$V H$
	$Var {ϕ_{xy} T}_{H} Var {G_{v}}$	$V H$
	$Var {ϕ_{xy} T}_{V} Var {G_{v}} + ϕ^{2} T^{2} Var {G_{v}}$	$V$ $V^{3}$
$ϕ_{xy} G_{h}$	$Var {ϕ_{xy} T}_{VH} Var {G_{h}}$	$V H$
	$Var {ϕ_{xy} T}_{H} Var {G_{h}} + ϕ^{2} T^{2} Var {G_{h}}$	$H$ $H^{4}$
	$Var {ϕ_{xy} T}_{V} Var {G_{h}}$	$V H$
$ϕ_{xy} G_{tvh}$	$Var {ϕ_{xy} T} Var {G_{tvh}} + ϕ^{2} T^{2} Var {G_{tvh}}$	$T V H$ $T V H^{1}$
$ϕ_{xy} G_{t}$	$Var {ϕ_{xy} T}_{VH} Var {G_{t}}$	$T V H$
	$Var {ϕ_{xy} T}_{H} Var {G_{t}}$	$T H$
	$Var {ϕ_{xy} T}_{V} Var {G_{t}}$	$T V$
	$ϕ^{2} T^{2} Var {G_{t}}$	$T^{7}$
$ϕ_{x y} G_{t v}$	$Var {ϕ_{x y} T}_{V H} Var {G_{t v}} Var {ϕ_{x y} T}_{H} Var {G_{t v}}$	$T V H T V H$
$ϕ_{x y} G_{t v}$	$Var {ϕ_{x y} T}_{V} Var {G_{t v}} ϕ^{2} T^{2} Var {G_{t v}}$	$T V$ $T V^{5}$
$ϕ_{x y} G_{t h}$	$Var {ϕ_{x y} T}_{V H} Var {G_{t h}}$ $Var {ϕ_{x y} T}_{V} Var {G_{t h}}$	$T V H T V H$
$ϕ_{x y} G_{t h}$	$Var {ϕ_{x y} T}_{H} Var {G_{t h}} ϕ^{2} T^{2} Var {G_{t h}}$	$T H$ $T H^{6}$
$i_{v h} T G$	$i_{D C}^{2} T^{2} Var {i_{v h}} G^{2}$	$V H$
$i_{v h} T (G_{v h} + G_{v} + G_{h})$	$Var {i_{v h} T} Var {G_{v h} + G_{v} + G_{h}} i_{D C}^{2} T^{2} Var {G_{v h}}$	$V H$ $V H^{2}$
	$i_{D C}^{2} T^{2} Var {G_{v}}$	$V^{3}$
	$i_{D C}^{2} T^{2} Var {G_{h}}$	$H^{4}$
$i_{v h} T (G_{t v h} + G_{t} + G_{t v} + G_{t h})$	$Var {i_{v h} T} Var {G_{t v h} + G_{t} + G_{t v} + G_{t h}} i_{D C}^{2} T^{2} Var {G_{t v h}}$	$T V H$ $T V H^{1}$
	$i_{D C}^{2} T^{2} Var {G_{t}}$	$T^{7}$
	$i_{D C}^{2} T^{2} Var {G_{t v}}$	$T V^{5}$
	$i_{D C}^{2} T^{2} Var {G_{t h}}$	$T H^{6}$
$S_{t v h} G$	$(ϕ T + i_{D C} T) G^{2}$	$T V H$
$S_{t v h}$ $(G_{3 D} - G)$	$(ϕ T + i_{D C} T) Var {G_{3 D}}$	$T V H$
$N_{t v h}$	$Var {N_{t v h}}$	$T V H$
$N_{t h}$	$Var {N_{t h}}$	$T H$
$N_{t v}$	$Var {N_{t v}}$	$T V$
$N_{t}$	$Var {N_{t}}$	$T$
$N_{v}$	$Var {N_{v}}$	$V$
$N_{h}$	$Var {N_{h}}$	$H$
$N_{v h}$	$Var {N_{v h}}$	$V H$

Noise Type	Output 3D Noise Variance: [ ${D N}^{2}]$
$T V H$	$Var {Y}_{TVH} = (T^{2} Var {ϕ_{xy}} + {(ϕ T + i_{D C})}^{2}) Var {G_{tvh}} + i_{D C}^{2} T^{2} {D S N U}^{2} Var {G_{tvh} + G_{t} + G_{tv} + G_{th}} + T^{2} Var {ϕ_{xy}}_{VH} Var {G_{tv} + G_{th} + G_{t}} + T^{2} Var {ϕ_{xy}}_{H} Var {G_{tv}} + T^{2} Var {ϕ_{xy}}_{V} Var {G_{th}} + (ϕ T + i_{D C} T) (G^{2} + Var {G_{3}}) + Var {N_{tvh}}$
$V H$	$Var {Y}_{VH} =$ $({(ϕ T + i_{D C} T)}^{2} + T^{2} Var {ϕ_{xy}} + i_{D C}^{2} T^{2} {D S N U}^{2}) Var {G_{vh}} + (T^{2} Var {ϕ_{xy}}_{VH} + i_{D C}^{2} T^{2} {D S N U}^{2}) (G^{2} + Var {G_{v} + G_{h}}) + T^{2} Var {ϕ_{xy}}_{H} Var {G_{v}} + T^{2} Var {ϕ_{xy}}_{V} Var {G_{h}} + Var {N_{vh}}$
$T V$	$Var {Y}_{TV} = T^{2} Var {ϕ_{xy}}_{V} Var {G_{t}} + (T^{2} Var {ϕ_{xy}}_{V} + {(ϕ T + i_{D C} T)}^{2}) Var {G_{v}} + Var {N_{tv}}$
$T H$	$Var {Y}_{TH} = T^{2} Var {ϕ_{xy}}_{H} Var {G_{t}} + (T^{2} Var {ϕ_{xy}}_{H} + {(ϕ T + i_{D C} T)}^{2}) Var {G_{th}} + Var {N_{th}}$
$V$	$Var {Y}_{V} = T^{2} Var {ϕ_{xy}}_{V} G^{2} + (T^{2} Var {ϕ_{xy}}_{V} + {(ϕ T + i_{D C} T)}^{2}) Var {G_{v}} + Var {N_{v}}$
$H$	$Var {Y}_{H} = T^{2} Var {ϕ_{xy}}_{H} G^{2} + (T^{2} Var {ϕ_{xy}}_{H} + {(ϕ T + i_{D C} T)}^{2}) Var {G_{h}} + Var {N_{h}}$
$T$	$Var {Y}_{T} = {(ϕ T + i_{D C} T)}^{2} Var {G_{t}} + Var {N_{t}}$

$S i z e (Y) = [64, 64, 128]$	$T = 1 / 60 [s e c]$
$c = A_{d} τ Ω η = 1.309 E - 11$	$A_{d} = 1.0 E - 10 [m^{2}]$
$i_{D C} = 200 [e^{-} / sec/pix]$	$η = 60 %$
$τ = 90 %$	$Ω = 0.2424 [s r]$
$D S N U = 10 %$	$P R N U = 5 %$
$Var {Ω_{xy}} / Ω^{2} = {(23.5 %)}^{2}$	$Var {Ω_{xy}}_{VH} / Ω^{2} = {(2.7 %)}^{2}$
$Var {Ω_{xy}}_{V} / Ω^{2} = {(16.5 %)}^{2}$	$Var {Ω_{xy}}_{H} / Ω^{2} = {(16.5 %)}^{2}$
$G = 1$	$L = 80$
$Var {G_{t}} = {0.05}^{2}$	$Var {N_{t}} = {0.1}^{2}$
$Var {G_{v}} = {0.02}^{2}$	$Var {N_{v}} = {0.07}^{2}$
$Var {G_{h}} = {0.02}^{2}$	$Var {N_{h}} = {0.03}^{2}$
$Var {G_{tv}} = {0.02}^{2}$	$Var {N_{tv}} = {0.5}^{2}$
$Var {G_{th}} = {0.01}^{2}$	$Var {N_{th}} = {0.5}^{2}$
$Var {G_{vh}} = {0.1}^{2}$	$Var {N_{vh}} = 20^{2}$
$Var {G_{tvh}} = {0.005}^{2}$	$Var {N_{tvh}} = 3^{2}$

Noise Type	Dark Current Noise Variance [ ${D N}^{2}]$
$T V H$	${(i_{D C} T)}^{2} Var {G_{tvh}} + i_{D C}^{2} T^{2} {D S N U}^{2} Var {G_{tvh} + G_{t} + G_{tv} + G_{th}} + (i_{D C} T) (G^{2} + Var {G_{3 D}}) + Var {N_{tvh}}$
$V H$	${(i_{D C} T)}^{2} (1 + {D S N U}^{2}) Var {G_{vh}} + (i_{D C}^{2} T^{2} {D S N U}^{2}) (G^{2} + Var {G_{v} + G_{h}}) + Var {N_{vh}}$
$T V$	${(i_{D C} T)}^{2} Var {G_{tv}} + Var {N_{tv}}$
$T H$	${(i_{D C} T)}^{2} Var {G_{th}} + Var {N_{th}}$
$V$	${(i_{D C} T)}^{2} Var {G_{v}} + Var {N_{v}}$
$H$	${(i_{D C} T)}^{2} Var {G_{h}} + Var {N_{h}}$
$T$	${(i_{D C} T)}^{2} Var {G_{t}} + Var {N_{t}}$

3D Noise Type	Output 3D Noise Variance: [ ${D N}^{2}]$
$V H^{*}$	$Var {F_{HP} {Y}}_{VH} \approx$ $({(ϕ T + i_{D C} T)}^{2} + T^{2} Var {ϕ_{xy}} + i_{D C}^{2} T^{2} {D S N U}^{2}) Var {G_{vh}} + (ϕ^{2} T^{2} [{P R N U}^{2} (1 + \frac{Var {Ω_{x y}}}{Ω^{2}})] + i_{D C}^{2} T^{2} {D S N U}^{2}) G^{2} + (T^{2} Var {ϕ_{xy}}_{VH} + i_{D C}^{2} T^{2} {D S N U}^{2}) Var {G_{v} + G_{h}} + T^{2} Var {ϕ_{xy}}_{H} Var {G_{v}} + T^{2} Var {ϕ_{xy}}_{V} Var {G_{h}} + Var {N_{vh}^{*}}$
V*	$Var {F_{HP} {Y}}_{V} \approx (T^{2} Var {ϕ_{xy}}_{V} + {(ϕ T + i_{D C} T)}^{2}) Var {G_{v}} + Var {N_{v}^{*}}$
$H^{*}$	$Var {F_{HP} {Y}}_{H} \approx (T^{2} Var {ϕ_{xy}}_{H} + {(ϕ T + i_{D C} T)}^{2}) Var {G_{h}} + Var {N_{h}^{*}}$

$S i z e (Y)$ $= [1280, 720, 100]$	With Lens	Without Lens	Accuracy/Precision
$c = A_{d} τ Ω η$	Unknown	Unknown	NA
$G [D N / e^{-}]$	64.1	62.9	0.0/0.0%
$L [D N]$	3480	3483	0.0/0.0%
$i_{D C} [e^{-} / (s e c p i x)]$	18.8	23.8	2.8/5.7%
$D S N U [u l]$	64.1%	82.5%	5.2/3.6%*
$P R N U [u l]$	0.47%	0.54%	0.0/0.0%
$Var {Ω_{xy}} / Ω^{2} [u l]$	$(13.0 %)^{2}$	$(0 %)^{2}$	NA
$Var {Ω_{xy}}_{VH} / Ω^{2} [u l]$	$(0.6 %)^{2}$	$(0 %)^{2}$	NA
$Var {Ω_{xy}}_{V} / Ω^{2} [u l]$	$(4.0 %)^{2}$	$(0 %)^{2}$	NA
$Var {Ω_{xy}}_{H} / Ω^{2} [u l]$	$(12.4 %)^{2}$	$(0 %)^{2}$	NA
$Var {N_{t}} / G^{2} {[e^{-}]}^{2}$	( $0.06 [e^{-}])^{2}$	( $0.03 [e^{-}])^{2}$	32/5.0%
$Var {N_{v}} / G^{2} {[e^{-}]}^{2}$	( $0.03 [e^{-}])^{2}$	( $0.05 [e^{-}])^{2}$	17/6.4%
$Var {N_{h}} / G^{2} {[e^{-}]}^{2}$	( $0.03 [e^{-}])^{2}$	( $0.03 [e^{-}])^{2}$	6.7/3.2%
$Var {N_{tv}} / G^{2} {[e^{-}]}^{2}$	( $0.84 [e^{-}])^{2}$	( $0.86 [e^{-}])^{2}$	0.0/0.1%
$Var {N_{th}} / G^{2} {[e^{-}]}^{2}$	( $0.02 [e^{-}])^{2}$	( $0.01 [e^{-}])^{2}$	70/13%
$Var {N_{vh}} / G^{2} {[e^{-}]}^{2}$	( $0.67 [e^{-}])^{2}$	( $0.68 [e^{-}])^{2}$	0.1/0.2%
$Var {N_{tvh}} / G^{2} {[e^{-}]}^{2}$	( $3.74 [e^{-}])^{2}$	( $3.85 [e^{-}])^{2}$	0.0/0.0%
$Var {G_{t}} / G^{2} [u l]$	( $0.01 %)^{2}$	( $0.01 %)^{2}$	8.7/41%
$Var {G_{v}} / G^{2} [u l]$	( $0.17 %)^{2}$	( $0.13 %)^{2}$	3.0/4.4%
$Var {G_{h}} / G^{2} [u l]$	( $0.16 %)^{2}$	( $0.20 %)^{2}$	0.8/1.5%
$Var {G_{tv}} / G^{2} [u l]$	( $0.00 %)^{2}$	( $0.02 %)^{2}$	0.0/0.0%
$Var {G_{th}} / G^{2} [u l]$	( $0.01 %)^{2}$	( $0.02 %)^{2}$	60/28%
$Var {G_{vh}} / G^{2} [u l]$	( $0.00 %)^{2}$	( $0.00 %)^{2}$	0.0/0.0%
$Var {G_{tvh}} / G^{2} [u l]$	( $0.08 %)^{2}$	( $0.47 %)^{2}$	1.3/1.6%
Noise Factor( $N F$ )	$1.00$	$1.00$	NA

Noise Type	Extra Correlated Noise Variance Terms
$T V H$	$2 ρ_{tvh} (ϕ T + i_{D C} T) \sqrt{Var {G_{th}} Var {L_{tvh}}} + 2 ρ_{i η} ϕ i_{D C} T^{2} Var {G_{tvh} + G_{tv} + G_{th} +_{t}} D S N U P R N U$
$V H$	$2 ρ_{vh} (ϕ T + i_{D C} T) \sqrt{Var {G_{vh}} Var {L_{vh}}} + 2 ρ_{i η} ϕ i_{D C} T^{2} (G^{2} + Var {G_{v} + G_{h} + G_{vh}}) D S N U P R N U$
$T V$	$2 ρ_{tv} (ϕ T + i_{D C} T) \sqrt{Var {G_{tv}} Var {L_{tv}}}$
$T H$	$2 ρ_{th} (ϕ T + i_{D C} T) \sqrt{Var {G_{th}} Var {L_{th}}}$
$V$	$2 ρ_{v} (ϕ T + i_{D C} T) \sqrt{Var {G_{v}} Var {L_{v}}}$
$H$	$2 ρ_{h} (ϕ T + i_{D C} T) \sqrt{Var {G_{h}} Var {L_{h}}}$
$T$	$2 ρ_{t} (ϕ T + i_{D C} T) \sqrt{Var {G_{t}} Var {L_{t}}}$

Abstract

Supplementary Material (1)

Data availability

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Tables (8)

Equations (48)

Applied Optics