= Spatial pressure-map resolution below a sensor cell
<Spatial pressure-map resolution below a sensor cell> concerns which variations of a contact <pressure> field can actually be distinguished by spatial measurements. A smaller display pixel does not imply a smaller independently measured contact region. The response of a <pressure sensor>, its sampling positions and any imposed reconstruction assumptions must be specified.
For an original ideal array model, cell $A_j$ of known area reports an area-average <pressure>,
$$
y_j=\frac{1}{|A_j|}\int_{A_j}p(x,y)\,dA.
$$
It determines that cell’s load $F_j=|A_j|y_j$, but not every value inside it. For a more general linear response, $y_j=\int K_j(x,y)p(x,y)\,dA$ with known sensitivity $K_j$. The <null space of a spatial pressure measurement> consists of perturbations $\delta p$ whose weighted integrals vanish for all measured channels. Nonnegative fields that differ by such a perturbation remain indistinguishable in that model.
Consider square cells of side $a$ aligned with $x=ja$, and compare fields independent of $y$:
$$
p_1(x)=p_0,\qquad p_2(x)=p_0\left[1+\frac12\cos(2\pi x/a)\right].
$$
Every cell contains one full cosine period, so both fields give every channel the identical value $p_0$. Yet $p_2$ varies between $0.5p_0$ and $1.5p_0$. With $a=10\,\mathrm{mm}$ and $p_0=100\,\mathrm{kPa}$, a perfectly uniform field and a field spanning $50$–$150\,\mathrm{kPa}$ produce the same ideal map. On an aligned finite rectangular array, the added cosine also has zero first moment within each cell, so even the global resultant and <center of pressure> agree. This is an inverse-problem construction, not a claim that both profiles arise from the same elastic contact geometry.
Cell averaging has a calculable spatial response. For a one-dimensional centered interval of width $a$, averaging a complex wave $e^{2\pi i\nu x}$ multiplies its value at the interval center by
$$
\frac1a\int_{-a/2}^{a/2}e^{2\pi i\nu u}\,du
=\operatorname{sinc}(a\nu).
$$
The first zero at $\nu=1/a$ explains the erased cosine above. Sampling introduces further ambiguity; even translating this ideal array cannot recover a component erased by its aperture at every position.
A regularity assumption can supply a conditional bound. If $p$ has a validated spatial <Lipschitz constant> $L_p$ over a square cell, its center value differs from its area average by at most
$$
\left|p(x_c,y_c)-y_j\right|\le L_p a/\sqrt2.
$$
Every point in the cell is at most half the diagonal from its center, so integrating the Lipschitz inequality proves this bound. For $a=10\,\mathrm{mm}$ and $L_p=2\,\mathrm{kPa/mm}$, it gives about $14.14\,\mathrm{kPa}$. The array readings alone do not establish that slope bound.
Interpolation chooses a finer representation from the measured values. SciPy explicitly describes its array zoom operation as spline interpolation; it does not describe collecting new <pressure> data.<SciPy 1.16.2 documentation>[scipy.ndimage.zoom, description and order parameter]{ref} A proposed apparatus test scans a known small indenter across a sensor array at controlled total load, comparing independent load measurements with channel responses. The indenter’s actual contact area, load spreading, sensor cross-coupling and repeatability must be measured before inferring a spatial response. Additional smaller-aperture measurements or justified mechanical constraints can improve identifiability; a smoother image alone supplies neither.
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