 Pressure-map information matrix

A pressure-map information matrix quantifies which coefficients of a modeled contact pressure field are constrained by a sensor array, relative to its measurement noise. For $y=Hp+\eta$ with fixed positive-definite error covariance $R$, defineHere $A$ is the noise-whitened pressure sensing matrix. Whitening removes the output-noise covariance from the residual metric; the primary SciPy covariance interface documents this transformation and demonstrates near-identity covariance on simulated observations.^[ref]

The matrix is positive semidefinite because $v^{\mathsf{T}}\mathcal{I}v=\Vert Av{\Vert }^2\ge 0$. Its null space is exactly the null space of $H$: an unobserved pressure change remains unobserved after invertible row whitening. With independent Gaussian errors of known covariance, the log likelihood is a constant minus $(y-Hp{)}^{\mathsf{T}}{R}^{-1}(y-Hp)/2$. Differentiating twice with respect to $p$ gives negative Hessian $\mathcal{I}$, identifying it as the Fisher information for this stated model.

If $H$ has full column rank, weighted least squares givesThis covariance follows by substituting $y=Hp+\eta$ and multiplying the error matrices. It concerns the unconstrained linear estimator with known response and covariance, not every reconstruction algorithm. Nonnegativity constraints, regularization and uncertain sensitivity kernels require additional treatment.

An original two-cell example begins with one average-pressure measurement, $H=(1/2,1/2)$, with $1\mathrm{kPa}$ standard error. In numerical kilopascal coordinates,Its eigenvalues are $1/2$ and zero: the mean is observed, but the pressure difference is not. A second independent copy doubles the information matrix while leaving the difference unobserved. More repeats of the same mixture improve precision only in the already measured direction.

Instead add a direct measurement of the first cell with the same independent error. NowThe inferred first-cell pressure has standard deviation $1\mathrm{kPa}$ and the second $\sqrt{5}\approx 2.236\mathrm{kPa}$. Their errors are negatively correlated because the second pressure is reconstructed by subtracting the first from twice the measured mean.

A rank-one update of pressure-map information follows when an independent new channel has sensitivity row $a^{\mathsf{T}}$ and error variance ${\sigma }_a^2$: its added information is $a{a}^{\mathsf{T}}/{\sigma }_a^2$. A channel aligned with a weak pressure direction can therefore be more useful than another precise copy of a strong direction. Correlated new errors cannot be added by that independent-channel formula.

A proposed characterization uses repeatable known loading patterns to estimate $H$, independent repeats to estimate $R$, and withheld patterns to test predicted reconstruction covariance. State the pressure basis and units: rescaling a coefficient changes the numerical information entries. A sharp reconstructed image or a large matrix determinant alone does not establish that the underlying response model or noise assumptions are correct.