## Abstract

Ben-David and Ren [Appl. Opt. **44,** 3846 (2005)] discussed methods of estimating the
concentration of chemical vapor plumes in hyperspectral images. The authors of that paper concluded that
a technique called orthogonal subspace projection (OSP) produces better concentration estimates than
background subtraction when certain stochastic noise conditions are present in the data. While that
conclusion is correct, it is worth noting that the data can be whitened to improve the performance of the
background subtraction method. In particular, if the noise is multivariate Gaussian, then whitening will
ensure that the background subtraction method is superior to OSP.

© 2007 Optical Society of America

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### Equations (9)

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(2)
$${\hat{\alpha}}_{\text{sub}}=\frac{{d}^{T}\left(r-b\right)}{{d}^{T}d}.$$
(3)
$${\hat{\alpha}}_{\text{OSP}}=\frac{{d}^{T}{{P}_{\mathbf{\text{b}}}}^{\perp}r}{{d}^{T}{{P}_{\mathbf{\text{b}}}}^{\perp}d},$$
(4)
$$r\sim \mathcal{N}\left[b+d\alpha ,C\right].$$
(5)
$$l\left(\alpha ;\text{\hspace{0.17em}}r\right)=-1/2\text{\hspace{0.17em}}\mathrm{ln}\left(\left|2\pi C\right|\right)-1/2\left[{\left(r-b-d\alpha \right)}^{T}\times {C}^{-1}\left(r-b-d\alpha \right)\right].$$
(6)
$$\frac{\partial l\left(\alpha ;\text{\hspace{0.17em}}r\right)}{\partial \alpha}=-\alpha {d}^{T}{C}^{-1}d+{r}^{T}{C}^{-1}d-{b}^{T}{C}^{-1}d.$$
(7)
$$\hat{\alpha}=\frac{{d}^{T}{C}^{-1}\left(r-b\right)}{{d}^{T}{C}^{-1}d}.$$
(8)
$$\hat{\alpha}\sim \mathcal{N}\left[\alpha ,\frac{1}{{d}^{T}{C}^{-1}d}\right].$$
(9)
$$\mathrm{Var}\left[\frac{{\mathbf{d}}^{T}{\mathbf{C}}^{-1}(\mathbf{r}-\mathbf{b})}{{\mathbf{d}}^{T}{\mathbf{C}}^{-1}\mathbf{d}}\right]\le \mathrm{Var}\left[\frac{{\mathbf{d}}^{T}{{\mathbf{P}}_{\mathbf{b}}}^{\perp}\mathbf{r}}{{\mathbf{d}}^{T}{{\mathbf{P}}_{\mathbf{b}}}^{\perp}\mathbf{d}}\right]\mathrm{.}$$