 Noise-whitened pressure sensing matrix

A noise-whitened pressure sensing matrix measures how a contact pressure pattern changes sensor outputs relative to their joint noise, rather than their raw numerical size. For a linear measurement modelassume the error covariance $R$ is positive definite. Choose an invertible matrix $W$ satisfying $WR{W}^{\mathsf{T}}=I$. ThenThe transformed errors have identity covariance. This is the whitening transformation documented in the primary SciPy covariance interface; its example obtains approximately identity sample covariance from simulated observations.^[ref]

A Cholesky decomposition $R=L{L}^{\mathsf{T}}$ supplies $W=L^{-1}$. Compute $Wy$ and $WH$ by solving triangular systems rather than forming an inverse unnecessarily. The least-squares residual becomesThus an output change is significant according to its noise covariance, not merely the Euclidean length of the raw readings. The original pressure basis and its units must still be stated; whitening rows does not remove arbitrary column rescalings.

For an original two-channel example, use numerical pressure values in kilopascals andThe separate standard deviations are $2$ and $1\mathrm{kPa}$ with correlation $0.9$. A residual $(2,1)\mathrm{kPa}$ whitens to $(1,0.1/\sqrt{0.19})$, giving squared length about $1.0526$. The residual $(2,-1)\mathrm{kPa}$ instead whitens to $(1,-1.9/\sqrt{0.19})$, giving squared length $20$. Treating the channels as independent would assign both residuals squared standardized length two. The positively correlated noise makes opposite-sign discrepancies much less plausible than a shared shift.

A simple repeated-measurement model illustrates the information consequence. If two sensors measure the same scalar pressure with equal standard deviation $\sigma$ and correlation $\rho$, their mean has variance ${\sigma }^2(1+\rho )/2$. At $\rho =0.9$ the standard deviation is about $0.9747\sigma$, rather than the independent-channel value $0.7071\sigma$. Adding an almost duplicate channel therefore supplies little improvement for that common pressure.

The pressure-map information matrix is $A^{\mathsf{T}}A=H^{\mathsf{T}}{R}^{-1}H$. In a full-column-rank linear Gaussian model with known $R$, the unconstrained weighted least-squares estimate has covariance $(A^{\mathsf{T}}A{)}^{-1}$. Small singular values of $A$ identify near-null directions of a pressure sensor array after accounting for noise. Alternative whiteners differ by an orthogonal row transformation and give the same information matrix; an individual whitened row need not represent one physical sensing cell.

A proposed array characterization applies repeatable known pressure patterns, estimates the channel covariance from repeated residuals, and checks the covariance on independent repeats after whitening. Drift and response-model uncertainty must be included if they belong to the intended error model. If $R$ is singular, a simple inverse whitening is unavailable: exact zero-noise constraints and redundant stochastic directions need separate treatment. A visually smooth pressure reconstruction does not establish that its weak directions were measured precisely.