Errata: Correcting for focal-plane-array temperature dependence in microbolometer infrared cameras lacking thermal stabilization

Paul W. Nugent; Joseph A. Shaw; Nathan J. Pust

doi:10.1117/1.OE.52.10.109801

29 October 2013 Errata: Correcting for focal-plane-array temperature dependence in microbolometer infrared cameras lacking thermal stabilization

Paul W. Nugent, Joseph A. Shaw, Nathan J. Pust

Author Affiliations +

Optical Engineering, Vol. 52, Issue 10, 109801 (October 2013). https://doi.org/10.1117/1.OE.52.10.109801

Abstract

This article [Opt. Eng.. 52, (6 ), 061304 (2013)] was originally published online on 7 January 2013 with an error in the numerator of Eq. (13) that propagated into Eqs. (14), (15), and (17). The corrected Eq. (13) and subsequent equations are given here:

This article [Opt. Eng. 52(6), 061304 (2013)] was originally published online on 7 January 2013 with an error in the numerator of Eq. (13) that propagated into Eqs. (14), (15), and (17). The corrected Eq. (13) and subsequent equations are given here:

Eq. (13)

r_{ref} = \frac{r_{2} + \frac{D_{m} G_{0} - D_{0} G_{m}}{G_{m} T_{ref} + G_{0}} (T_{ref} - T_{fpa})}{\frac{G_{m} T_{fpa} + G_{0}}{G_{m} T_{ref} + G_{0}}}

Eq. (14)

r_{ref} = \frac{r_{2} + \frac{D_{m} G_{0} - D_{0} G_{m}}{G_{m} T_{ref} + G_{0}} (T_{ref} - T_{fpa})}{\frac{G_{0} + G_{m} T_{fpa} + G_{m} T_{ref} - G_{m} T_{ref}}{G_{m} T_{ref} + G_{0}}}

Eq. (15)

r_{ref} = \frac{r_{2} + \frac{D_{m} G_{0} - D_{0} G_{m}}{G_{m} T_{ref} + G_{0}} (T_{ref} - T_{fpa})}{1 - \frac{G_{m}}{G_{m} T_{ref} + G_{0}} (T_{ref} - T_{fpa})}

Eq. (17)

b = \frac{D_{m} G_{0} - D_{0} G_{m}}{G_{m} T_{ref} + G_{0}} .

There was also a separate error that led to a singular matrix in Eqs. (24) and (25). The cause was an oversimplification of the method used to derive the correction coefficients, in which only one blackbody temperature was used instead of the two or more that are required to make Eq. (24) nonsingular so that it can be inverted. In the following, we provide a corrected version of Sec. 4 that removes this singular matrix. This discussion replaces the original Sec. 4 through Eq. (25):

4. Determining the Correction Coefficients

Rewriting the reference-temperature response function from Eq. (1) as

Eq. (19)

r_{ref} - r_{2} = r_{ref} m Δ T + b Δ T

allows us to write an equation for determining the coefficients

m

and

b

. These correction coefficients can be determined by viewing two constant-temperature blackbody scenes with radiances

L_{1}

and

L_{2}

, each with the camera at a minimum of two different temperatures,

T_{fpa 1}

and

T_{fpa 2}

. A third camera temperature

T_{fpa 3}

can be experienced while viewing the second scene, but there must be at least one common FPA temperature between the two blackbody scenes. Thus we must consider the following responses:

r_{1}

with the camera at

T_{fpa 1}

and the blackbody at radiance

L_{1}

;

r_{2}

with the camera at

T_{fpa 2}

and the blackbody at radiance

L_{1}

;

r_{3}

with the camera at

T_{fpa 1}

and the blackbody at radiance

L_{2}

, and

r_{4}

with the camera at

T_{fpa 3}

and the blackbody at radiance

L_{2}

. The camera responses

r_{1}

and

r_{3}

are at the same FPA temperature and will be used as the references. Further, note that the responses

r_{2}

and

r_{4}

can be at the same FPA temperature, but this is not required (

T_{fpa 2}

could equal

T_{fpa 3}

). Using

T_{fpa 1}

as the reference camera temperature leads to the following differences:

Eq. (20)

Δ r_{12} = r_{1} - r_{2} and Δ T_{12} = T_{fpa 1} - T_{fpa 2}

Eq. (21)

Δ r_{34} = r_{3} - r_{4} and Δ T_{34} = T_{fpa 3} - T_{fpa 4} .

These differences can be used in Eq. (19) to write the following matrix equation:

Eq. (22)

[\begin{matrix} Δ r_{12} \\ Δ r_{34} \end{matrix}] = [\begin{matrix} r_{1} Δ T_{12} & Δ T_{12} \\ r_{3} Δ T_{34} & Δ T_{34} \end{matrix}] [\begin{matrix} m \\ b \end{matrix}],

which can be inverted to obtain

m

and

b

.

Eq. (23)

[\begin{matrix} m \\ b \end{matrix}] = {[\begin{matrix} r_{1} Δ T_{12} & Δ T_{12} \\ r_{3} Δ T_{34} & Δ T_{34} \end{matrix}]}^{- 1} [\begin{matrix} Δ r_{12} \\ Δ r_{34} \end{matrix}] .

This method is the minimal approach to deriving these coefficients, but in practice we use a large range of blackbody temperatures (for example 10°C to 60°C in steps of 10°C) and a large range of camera FPA temperatures (for example from 10°C to 30°C). This can be accomplished by placing the camera in an environmental chamber and changing the ambient temperature to drive the temperature of the camera while the blackbody remains constant, then changing the blackbody temperature and repeating the ambient temperature cycle. Doing this generates multiple reference responses, one for each combination of blackbody temperature and FPA temperature, leading to an over-determined matrix as shown in Eq. (24):

Eq. (24)

[\begin{matrix} Δ r_{12} \\ ⋮ \\ Δ r_{j k} \end{matrix}] = [\begin{matrix} r_{1} Δ T_{12} & Δ T_{12} \\ ⋮ & ⋮ \\ r_{j} Δ T_{j k} & Δ T_{j k} \end{matrix}] [\begin{matrix} m \\ b \end{matrix}] .

In such a case, a pseudo-inversion is required, and in practice the Moore-Penrose pseudo-inversion is performed. This leads to a least-squares approach that reduces noise in the estimation of $m$ and $b$ through Eq. (25):

Eq. (25)

[\begin{matrix} m \\ b \end{matrix}] = {[\begin{matrix} r_{1} Δ T_{12} & Δ T_{12} \\ ⋮ & ⋮ \\ r_{j} Δ T_{j k} & Δ T_{j k} \end{matrix}]}^{- 1} [\begin{matrix} Δ r_{12} \\ ⋮ \\ Δ r_{j k} \end{matrix}] .

The paper was corrected online on 25 October 2013.

© The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

Citation Download Citation

Paul W. Nugent, Joseph A. Shaw, and Nathan J. Pust "Errata: Correcting for focal-plane-array temperature dependence in microbolometer infrared cameras lacking thermal stabilization," Optical Engineering 52(10), 109801 (29 October 2013). https://doi.org/10.1117/1.OE.52.10.109801

Published: 29 October 2013

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