= Null space of a spatial pressure measurement
{title2=$\ker\mathcal M=\{\delta p:\mathcal M\delta p=0\}$}
The <null space of a spatial pressure measurement> consists of changes to a contact <pressure> field that leave every measured channel unchanged. For a linear response model on a contact region $A$, write
$$
(\mathcal Mp)_j=\int_A K_j(x,y)p(x,y)\,dA.
$$
Then $\ker\mathcal M=\{\delta p:\mathcal M\delta p=0\}$. Two fields have identical outputs precisely when their difference lies in this <kernel of a linear map>. The sensitivity kernels $K_j$, their support and the admissible class of <pressure> fields are part of the model.
For an ideal area-average sensor, $K_j=1/|A_j|$ inside cell $A_j$ and zero elsewhere. A positive and negative perturbation within one cell can cancel in its integral while remaining invisible to the other cells. This is the mechanism behind <spatial pressure-map resolution below a sensor cell>. It is a nonuniqueness of the observations, even if their numerical values are exact.
An original finite model divides each of two equal-area sensor cells into two equal halves with constant pressures $p_1,\ldots,p_4$. The observed averages are
$$
y=Hp,\qquad
H=\begin{pmatrix}1/2&1/2&0&0\\0&0&1/2&1/2\end{pmatrix}.
$$
This matrix has rank two and a two-dimensional null space spanned by $(1,-1,0,0)^{\mathsf T}$ and $(0,0,1,-1)^{\mathsf T}$. Thus the fields
$$
p^{(1)}=(100,100,100,100)^{\mathsf T}\,\mathrm{kPa},\qquad
p^{(2)}=(80,120,140,60)^{\mathsf T}\,\mathrm{kPa}
$$
both produce $y=(100,100)^{\mathsf T}\,\mathrm{kPa}$. Their peak pressures differ by $40\,\mathrm{kPa}$. Adding an independent measurement of $p_1$ determines the first pair, but still leaves the second pair’s difference undetermined. An extra channel helps only if it adds an independent constraint on the missing directions.
The linear null space is not itself a set of physical <pressure> fields: its perturbations can have either sign. For a nonadhesive contact, require $p+\delta p\ge0$ as well. With the uniform $100\,\mathrm{kPa}$ field above, perturbations $t(1,-1,0,0)$ are admissible for $|t|\le100\,\mathrm{kPa}$. Positivity restricts the ambiguity without removing it. In contrast, a zero area-average reading together with a nonnegative <pressure> assumption <forces> <pressure> to vanish almost everywhere in that cell under the ideal model.
For a chosen finite basis of <pressure> fields, a <singular value decomposition> identifies the matrix null space. SciPy’s pinned implementation constructs its basis from right singular vectors and uses a relative singular-value cutoff to determine effective numerical rank.<SciPy 1.16.2 documentation>[scipy.linalg.null_space, pinned v1.16.2 scipy/linalg/_decomp_svd.py, docstring and implementation lines 333–403]{ref} A numerical cutoff and an exact zero singular value are different notions. <Near-null directions of a pressure sensor array> have small nonzero responses and can amplify measurement error: a dimensionless singular value of $0.01$ turns a $1\,\mathrm{kPa}$ channel-error component into a $100\,\mathrm{kPa}$ coefficient error when inverted without regularization.
A proposed apparatus check applies controlled local loads to estimate the sensitivity kernels, then compares fields with similar predicted channel outputs using independent spatial measurements. Contact area, load spreading and cross-coupling must enter that comparison. A reconstruction algorithm may select one smooth or nonnegative solution, but the selected structure is additional modeling information; it is not automatically an independently observed subcell <pressure> map.
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