## 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)
$$\mathrm{Var}\left[\frac{{\mathbf{d}}^{T}{\mathbf{C}}^{-1}(\mathbf{r}-\mathbf{b})}{{\mathbf{d}}^{T}{\mathbf{C}}^{-1}\mathbf{d}}\right]\le \mathrm{Var}\left[\frac{{\mathbf{d}}^{T}{{\mathbf{P}}_{\mathbf{b}}}^{\perp}\mathbf{r}}{{\mathbf{d}}^{T}{{\mathbf{P}}_{\mathbf{b}}}^{\perp}\mathbf{d}}\right]\mathrm{.}$$