 Minimum-variance spectral contrast orthogonal to a reference shape

A minimum-variance spectral contrast orthogonal to a reference shape estimates a desired signal amplitude while exactly rejecting one fixed nuisance shape. Let $\mathbf{z}=\theta \mathbf{s}+\beta \mathbf{x}+\mathbf{ϵ}$, with known shapes $\mathbf{s},\mathbf{x}$ and a positive-definite noise covariance matrix $C$. Choose $\hat{\theta }={\mathbf{a}}^T\mathbf{z}$ to minimize ${\mathbf{a}}^{T}C\mathbf{a}$, subject toThe constraints retain unit signal response and reject the reference amplitude, as in projection of shared-scale uncertainty onto a spectral contrast. The quadratic variance includes the full noise covariance.^[ref]

Remove the reference component in the inverse-covariance geometry:For nonzero $\mathbf{x}$ and $D>0$, the minimizing contrast and its variance areFor a short proof, every other feasible contrast is $\mathbf{a}+\mathbf{v}$ with ${\mathbf{v}}^T\mathbf{s}={\mathbf{v}}^T\mathbf{x}=0$. Since $C\mathbf{a}$ is a linear combination of those shapes, the cross term vanishes, leaving the added nonnegative variance ${\mathbf{v}}^{T}C\mathbf{v}$. This is a constrained best linear unbiased estimator.

In an original three-channel example, take $\mathbf{s}=(1,0,0{)}^T$, $\mathbf{x}=(1,1,1{)}^T$ and $C=\mathrm{diag}(1,4,1)$. The optimum is $\mathbf{a}=(1,-1/5,-4/5{)}^T$, with variance $1.8$. Equal subtraction of the two reference channels, $(1,-1/2,-1/2{)}^T$, has variance $2.25$. The quieter third channel receives more of the subtraction.

If the desired and reference shapes are proportional, $D=0$ and no unit-response rejecting contrast exists. The noise penalty from nearly parallel signal and reference shapes already becomes large as $D$ approaches zero. This construction assumes fixed known shapes and covariance; estimating them from the same noisy spectra requires additional uncertainty terms.