Abstract

In our recently published article [J. Opt. Soc. Am. A 33, 9 (2016) [CrossRef]  ], there was an error of reasoning that changes some of the results and conclusions, although the key message of the article remains unaltered. We correct these errors in the present erratum.

© 2016 Optical Society of America

We first mention a typo (error of sign in the expression of t4) in Eq. (A2). The correct version is

t1=[111]T/3t1=[111]T/3t1=[111]T/3t1=[111]T/3.
In Section 5 of Ref. [1], we claim that the solution of the minimax optimization problem expressed in Eq. (32) is given by Eq. (33). In fact, this is true only when |ΔS0|=0. If this is not the case, the expression of the minimax contrast is different. This makes Eqs. (34), (35), (38), and (39) wrong, as well as the third and fourth rows of Table 1 and, in Fig. 1, the four red curves corresponding to Cstabest1 and Cstabest1,τ0. The other results are correct, especially the experimental demonstration of Figs. 2–4, which was performed in a situation where ΔS0=0.

Tables Icon

Table 1. Summary of the Contrast Values for Noises of Type 1 and Type 2 and Different Imaging Configurations: Adaptive Imager (ada), Total Contrast of the Static Imager (sta), Worst Contrast of the Best Channel of the Static Imager for Exposure Time τ=τ0/4 (best1), and τ=τ0 (best1, τ0)a

 figure: Fig. 1.

Fig. 1. Square root of the contrast as a function Δs for the different imaging configurations in the presence of type 1 noise and type 2 noise. ΔS0=1, (ητ0/σ)2=1, and η2τ0/a=1.

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We provide in this erratum the correct versions of these wrong equations and table. A formal demonstration of these results, as well as their physical interpretation and their practical consequences, will be submitted as a separate article.

The correct expressions of Eqs. (34) and (35) of Ref. [1], representing the contrast Cstabest1 in the presence of type 1 and type 2 noise sources, are

[Cstabest1]t1=1128(ητ0σ)2F(|ΔS0|,Δs),
[Cstabest1]t2=132η2τ0aF(|ΔS0|,Δs),
where the function F(|ΔS0|,Δs) is defined as
F(|ΔS0|,Δs)={(|ΔS0|+Δs232[ΔS0]2)2if|ΔS0|Δs3(|ΔS0|+Δs3)2otherwise.
The correct expressions of Eqs. (38) and (39) of Ref. [1], representing the contrast Cstabest1,τ0 obtained when only the optimal channel is observed with the exposure time τ0, are
[Cstabest1,τ0]t1=18(ητ0σ)2F(|ΔS0|,Δs),
[Cstabest1,τ0]t2=18η2τ0aF(|ΔS0|,Δs).
In Table 1, we display the correct version of Table 1 in Ref. [1]. Only the two last rows have changed. Please note that Table 2 of Ref. [1] is correct since it deals with the case ΔS0=0.

In Fig. 1, we have plotted the correct versions of the curves in Fig. 1 of Ref. [1], which represent the square roots of the contrasts in different imaging configurations as a function of Δs. Only the red curves, corresponding to Cstabest1 and Cstabest1,τ0, have changed. Please note that since ΔS0=1 and Δs[0,3], we are always in the case |ΔS0|Δs/3, so that the second line of Eq. (4) applies.

REFERENCES

1. F. Goudail and M. Boffety, “Optimal configuration of static polarization imagers for target detection,” J. Opt. Soc. Am. A 33, 9–16 (2016). [CrossRef]  

References

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  1. F. Goudail and M. Boffety, “Optimal configuration of static polarization imagers for target detection,” J. Opt. Soc. Am. A 33, 9–16 (2016).
    [Crossref]

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Figures (1)

Fig. 1.
Fig. 1. Square root of the contrast as a function Δ s for the different imaging configurations in the presence of type 1 noise and type 2 noise. Δ S 0 = 1 , ( η τ 0 / σ ) 2 = 1 , and η 2 τ 0 / a = 1 .

Tables (1)

Tables Icon

Table 1. Summary of the Contrast Values for Noises of Type 1 and Type 2 and Different Imaging Configurations: Adaptive Imager (ada), Total Contrast of the Static Imager (sta), Worst Contrast of the Best Channel of the Static Imager for Exposure Time τ = τ 0 / 4 (best1), and τ = τ 0 (best1, τ 0 )a

Equations (6)

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t 1 = [ 1 1 1 ] T / 3 t 1 = [ 1 1 1 ] T / 3 t 1 = [ 1 1 1 ] T / 3 t 1 = [ 1 1 1 ] T / 3 .
[ C sta best 1 ] t 1 = 1 128 ( η τ 0 σ ) 2 F ( | Δ S 0 | , Δ s ) ,
[ C sta best 1 ] t 2 = 1 32 η 2 τ 0 a F ( | Δ S 0 | , Δ s ) ,
F ( | Δ S 0 | , Δ s ) = { ( | Δ S 0 | + Δ s 2 3 2 [ Δ S 0 ] 2 ) 2 if | Δ S 0 | Δ s 3 ( | Δ S 0 | + Δ s 3 ) 2 otherwise .
[ C sta best 1 , τ 0 ] t 1 = 1 8 ( η τ 0 σ ) 2 F ( | Δ S 0 | , Δ s ) ,
[ C sta best 1 , τ 0 ] t 2 = 1 8 η 2 τ 0 a F ( | Δ S 0 | , Δ s ) .

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