29 October 2013 Errata: Correcting for focal-plane-array temperature dependence in microbolometer infrared cameras lacking thermal stabilization
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Optical Engineering, 52(10), 109801 (2013). doi: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:
Nugent, Shaw, and Pust: Errata: Correcting for focal-plane-array temperature dependence in microbolometer infrared cameras lacking thermal stabilization

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:

(13)

rref=r2+DmG0D0GmGmTref+G0(TrefTfpa)GmTfpa+G0GmTref+G0

(14)

rref=r2+DmG0D0GmGmTref+G0(TrefTfpa)G0+GmTfpa+GmTrefGmTrefGmTref+G0

(15)

rref=r2+DmG0D0GmGmTref+G0(TrefTfpa)1GmGmTref+G0(TrefTfpa)

(17)

b=DmG0D0GmGmTref+G0.

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

(19)

rrefr2=rrefmΔT+bΔT
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 L1 and L2, each with the camera at a minimum of two different temperatures, Tfpa1 and Tfpa2. A third camera temperature Tfpa3 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: r1 with the camera at Tfpa1 and the blackbody at radiance L1; r2 with the camera at Tfpa2 and the blackbody at radiance L1; r3 with the camera at Tfpa1 and the blackbody at radiance L2, and r4 with the camera at Tfpa3 and the blackbody at radiance L2. The camera responses r1 and r3 are at the same FPA temperature and will be used as the references. Further, note that the responses r2 and r4 can be at the same FPA temperature, but this is not required (Tfpa2 could equal Tfpa3). Using Tfpa1 as the reference camera temperature leads to the following differences:

(20)

Δr12=r1r2andΔT12=Tfpa1Tfpa2

(21)

Δr34=r3r4andΔT34=Tfpa3Tfpa4.

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

(22)

[Δr12Δr34]=[r1ΔT12ΔT12r3ΔT34ΔT34][mb],
which can be inverted to obtain m and b.

(23)

[mb]=[r1ΔT12ΔT12r3ΔT34ΔT34]1[Δr12Δr34].

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):

(24)

[Δr12Δrjk]=[r1ΔT12ΔT12rjΔTjkΔTjk][mb].

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):

(25)

[mb]=[r1ΔT12ΔT12rjΔTjkΔTjk]1[Δr12Δrjk].
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.
Paul W. Nugent, Joseph A. Shaw, 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). http://dx.doi.org/10.1117/1.OE.52.10.109801
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KEYWORDS
Black bodies

Cameras

Microbolometers

Infrared cameras

Staring arrays

Magnesium

Optical engineering

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