Simplified weighting function for high dynamic range video frame formation

David J. Griffiths; Alfred Wicks

doi:10.1364/AO.55.0000C9

Applied Optics
Vol. 55,
Issue 31,
pp. C9-C17
(2016)
•https://doi.org/10.1364/AO.55.0000C9

Simplified weighting function for high dynamic range video frame formation

David J. Griffiths and Alfred Wicks

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David J. Griffiths and Alfred Wicks, "Simplified weighting function for high dynamic range video frame formation," Appl. Opt. 55, C9-C17 (2016)

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About this Article
History
- Original Manuscript: June 1, 2016
- Revised Manuscript: August 30, 2016
- Manuscript Accepted: August 30, 2016
- Published: October 3, 2016

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Abstract

High dynamic range imagery has matured from a laboratory demonstration to an integrated capability in consumer-level cameras. However, high dynamic range video has lagged in development despite the introduction of several different frame formation methodologies and system architectures. The most common formation methodology involves the calculation of a high dynamic range image from a series of rapidly bracketed frames at varying exposure levels, then through the use of a radiometric calibration, recombined utilizing a weighted maximum likelihood estimator. The ideal weighting scheme to minimize the error on final image formation has been investigated by several authors. A new nonrecursive, simplified weighting scheme is proposed that utilized only knowledge of the optical attenuation, integration time, and sensor gain. A method is developed to test the various weighting schemes for their resilience to error with limited available exposures over a wide range of exposure value offsets, a common scenario for multi-imager-based high dynamic range video systems.

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Author	Weighting Function
Mann and Picard	$w (z) = \frac{1}{f^{'} [Log (z)]}$
Debevec and Malik	$w (z) = {\begin{matrix} z - z_{\min}, & for z \leq \frac{1}{2} (z_{\min} + Z_{\max}) \\ z_{max} - z & for z \leq \frac{1}{2} (z_{\min} + Z_{\max}) \end{matrix}$
Robertson et al.	$w (z) = e^{- 16 (\frac{z^{2}}{q_{s}^{2}} + \frac{1}{q_{s}} + \frac{1}{4})}$
Mitsunaga and Nayar	$w (z) = \frac{f^{- 1} (z)}{f^{'} (z)}$
Tsin et al.	$w (z) = \frac{Δ t}{\sqrt{g Φ σ_{s}^{2} + σ_{d c}^{2} + f^{'} {(z)}^{2} σ_{q}^{2}}}$
Ward et al.	$w (z) = f^{- 1} (z) f^{'} (z) [1 - {(\frac{2 z}{q_{s}} - 1)}^{12}]$
Kirk and Anderson	$w (z) = \frac{Δ t^{2}}{f^{'} {(z)}^{2} {(σ_{r} + σ_{q})}^{2}}$
Akyüz and Reinhold	$w (z) = f^{- 1} (z) f^{'} (z) [1 - {(\frac{2 Φ}{q_{s}} - 1)}^{12}]$
Hasinoff et al.	$w (z) = \frac{Δ t_{j}^{2}}{Φ Δ t_{j} + σ_{r}^{2} + g_{j}^{2} σ_{q}^{2}}$
Granados et al.	$w (z) = \frac{Δ t_{j}^{2} g^{2} a_{i}^{2}}{g^{2} Δ t_{j} (a_{i} Φ_{i} + 2 \bar{μ_{D}}) + 2 σ_{r}^{2}}$

Term	Range of Values	Units
$g$	$1 \to 10$	$e_{out}^{-} / e_{in}^{-}$
$q$	$1.602 * 10^{- 19}$	$coulomb / e^{-}$
$C$	$10^{- 13} - 10^{- 15}$	Farads
$q_{s}$	$2^{8}, 2^{10}, 2^{12}, 2^{14}, 2^{16}$	ADU
$Z$	$0 \to q_{s}$	ADU
$η$	$0.1 \to 0.6$	$e^{-} / s / γ / s$
$λ$	$300 * 10^{- 9} \to 700 * 10^{- 9}$	m
$A$	$4 * 10^{- 12} - 9 * 10^{- 10}$	$m^{2}$
$N_{well}$	$10,000 \to 150,000$	$e^{-}$
$h c$	$1.9864 * 10^{- 25}$	$J * m$
$k$	$1.3806 * 10^{- 23}$	J/K
$t$	$273 \to 330$	K
$A_{phy}$	$1 * 10^{- 8} \to 1$	–
$μ_{d c}$	$0 \to 0.0008 * q_{s}$	ADU
$μ_{r d}$	$0 \to 0.0012 * q_{s}$	ADU

Author	Weighting Function
Mann and Picard	$w (z) = \frac{1}{f^{'} [Log (z)]}$
Debevec and Malik	$w (z) = {\begin{matrix} z - z_{\min}, & for z \leq \frac{1}{2} (z_{\min} + Z_{\max}) \\ z_{max} - z & for z \leq \frac{1}{2} (z_{\min} + Z_{\max}) \end{matrix}$
Robertson et al.	$w (z) = e^{- 16 (\frac{z^{2}}{q_{s}^{2}} + \frac{1}{q_{s}} + \frac{1}{4})}$
Mitsunaga and Nayar	$w (z) = \frac{f^{- 1} (z)}{f^{'} (z)}$
Tsin et al.	$w (z) = \frac{Δ t}{\sqrt{g Φ σ_{s}^{2} + σ_{d c}^{2} + f^{'} {(z)}^{2} σ_{q}^{2}}}$
Ward et al.	$w (z) = f^{- 1} (z) f^{'} (z) [1 - {(\frac{2 z}{q_{s}} - 1)}^{12}]$
Kirk and Anderson	$w (z) = \frac{Δ t^{2}}{f^{'} {(z)}^{2} {(σ_{r} + σ_{q})}^{2}}$
Akyüz and Reinhold	$w (z) = f^{- 1} (z) f^{'} (z) [1 - {(\frac{2 Φ}{q_{s}} - 1)}^{12}]$
Hasinoff et al.	$w (z) = \frac{Δ t_{j}^{2}}{Φ Δ t_{j} + σ_{r}^{2} + g_{j}^{2} σ_{q}^{2}}$
Granados et al.	$w (z) = \frac{Δ t_{j}^{2} g^{2} a_{i}^{2}}{g^{2} Δ t_{j} (a_{i} Φ_{i} + 2 \bar{μ_{D}}) + 2 σ_{r}^{2}}$

Term	Range of Values	Units
$g$	$1 \to 10$	$e_{out}^{-} / e_{in}^{-}$
$q$	$1.602 * 10^{- 19}$	$coulomb / e^{-}$
$C$	$10^{- 13} - 10^{- 15}$	Farads
$q_{s}$	$2^{8}, 2^{10}, 2^{12}, 2^{14}, 2^{16}$	ADU
$Z$	$0 \to q_{s}$	ADU
$η$	$0.1 \to 0.6$	$e^{-} / s / γ / s$
$λ$	$300 * 10^{- 9} \to 700 * 10^{- 9}$	m
$A$	$4 * 10^{- 12} - 9 * 10^{- 10}$	$m^{2}$
$N_{well}$	$10,000 \to 150,000$	$e^{-}$
$h c$	$1.9864 * 10^{- 25}$	$J * m$
$k$	$1.3806 * 10^{- 23}$	J/K
$t$	$273 \to 330$	K
$A_{phy}$	$1 * 10^{- 8} \to 1$	–
$μ_{d c}$	$0 \to 0.0008 * q_{s}$	ADU
$μ_{r d}$	$0 \to 0.0012 * q_{s}$	ADU

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Applied Optics