= Near-null directions of a pressure sensor array
{title2=$Hv_j=s_ju_j,\quad s_j\approx0$}
<Near-null directions of a pressure sensor array> are changes of a modeled <pressure> field that produce small, rather than exactly zero, changes in the sensor outputs. They distinguish an ill-conditioned reconstruction from the exact ambiguity in the <null space of a spatial pressure measurement>. A formally invertible response can still contain very little usable information about some <pressure> patterns.
Choose a finite basis for the <pressure> field and a linear response $y=Hp+\eta$, where $p$ contains <pressure> coefficients, $y$ the readings, and $\eta$ measurement error. With common <pressure> units for the coefficients and readings, let
$$
Hv_j=s_ju_j,\qquad H=U\Sigma V^{\mathsf T}
$$
be a <singular value decomposition>. A change $\Delta p=a v_j$ changes the noiseless readings by $a s_j u_j$. Inverting that component divides the measurement error by $s_j$. If the errors have covariance $\sigma^2I$, the standard deviation of the reconstructed coefficient along $v_j$ is $\sigma/s_j$. An exact zero gives no inverse; a small nonzero value gives noise amplification.
An original two-region example has uniform pressures $p_1,p_2$ and sensor matrix
$$
H=\begin{pmatrix}0.5&0.5\\0.505&0.495\end{pmatrix}.
$$
Both sensors mostly measure the same average. Write $\bar p=(p_1+p_2)/2$ and $d=(p_1-p_2)/2$. Then $y_1=\bar p$ and $y_2=\bar p+0.01d$. Uniform pressures $(100,100)\,\mathrm{kPa}$ give readings $(100,100)\,\mathrm{kPa}$; the markedly different pair $(140,60)\,\mathrm{kPa}$ gives only $(100,100.4)\,\mathrm{kPa}$. The matrix is invertible, but these outputs differ by just $0.4\,\mathrm{kPa}$ in one channel.
If each channel has independent $1\,\mathrm{kPa}$ standard uncertainty, the direct contrast estimate is $\widehat d=(y_2-y_1)/0.01$, with standard deviation about $141.4\,\mathrm{kPa}$. The matrix’s smallest singular value is approximately $0.005$, consistent with roughly $200\,\mathrm{kPa}$ uncertainty in the unit-length contrast direction. Nonnegative <pressure> constraints can restrict the admissible answers, but do not turn these noisy readings into precise contrast information.
Numerical rank depends on a chosen tolerance. SciPy’s pinned null-space implementation compares singular values with a relative cutoff and returns right singular vectors beyond the retained rank.<SciPy 1.16.2 documentation>[v1.16.2 scipy/linalg/_decomp_svd.py, null_space, docstring and lines 395–403]{ref} A floating-point cutoff detects numerical scale; a physical decision about usable information must also consider sensor noise, response uncertainty and the chosen <pressure> basis. Rescaling a coefficient can change a raw singular value without changing the apparatus.
A <noise-whitened pressure sensing matrix> accounts for unequal or correlated measurement errors before comparing response strength. For positive-definite error covariance $R$, analyze $R^{-1/2}H$ together with the same coefficient units and normalization. A duplicate noisy channel does not provide the same information as an independent measurement of the weak contrast.
<Tikhonov regularization> can limit amplification by using factors $s_j/(s_j^2+\alpha^2)$ instead of $1/s_j$. Relative to the true noiseless coefficient, this introduces attenuation $s_j^2/(s_j^2+\alpha^2)$. For $s_j=0.005$ and $\alpha=0.05$, that factor is about $0.0099$: the weak component is nearly suppressed. A smoother map can therefore have less noise while omitting real contrast. A proposed validation applies independently known <pressure> patterns that excite both average and contrast directions, repeats the measurements, and reports the error covariance and response-model uncertainty alongside reconstruction results.
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