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Local contact pressure from additional spatial measurements requires more information than total normal force and two first moments. One explicit reconstruction model partitions the contact into cells of known areas $A_i$ and centroids $(x_i,y_i)$, assuming uniform pressure $p_i$ within each cell. Writing $F_i=A_i{p}_i$ givesThese are three linear measurements of the cell loads. They need not determine every $F_i$, as the nonuniqueness argument in pressure distributions with identical resultant force and center of pressure shows. The primary centroid definition specifies precisely which force/moment information is retained.^[ref]

For an original four-cell example, use equal $0.01{\mathrm{m}}^2$ cells centered at successive positions $(-a,-a),(a,-a),(a,a),(-a,a)$ with $a=0.05\mathrm{m}$. Loads $(10,20,30,40)\mathrm{N}$ give $N=100\mathrm{N}$ and centroid $(0,0.020)\mathrm{m}$. Adding $s(1,-1,1,-1)$ to these loads changes neither the sum nor either first moment. Nonnegative loads permit $-10\le s\le 20\mathrm{N}$. The alternative $(15,15,35,35)\mathrm{N}$ therefore gives the same global measurements.

An independently calibrated measurement of the first cell load resolves this one missing degree of freedom under the four-cell model. If it reads $10\mathrm{N}$, the global moment equations recover the original other three loads. Their cell pressures are $(1,2,3,4)\mathrm{kPa}$. Measuring all four cells instead provides redundant data for checking total load and the assumed geometry.

The local sensor output must actually represent a known cell-integrated force, or have a validated response model; a point sample cannot automatically be treated as an area average. Calibration errors, cell cross-coupling and imperfect coverage become additional measurement inputs. Spatial pressure-map resolution below a sensor cell remains limited even when every cell average is known: unmeasured within-cell peaks are not recovered by simply drawing a finer image. Constitutive or smoothness assumptions can add information, but must be identified as assumptions rather than extra measurements.

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 $A$, writeThen $\ker \mathcal{M}=\{\delta p:\mathcal{M}\delta p=0\}$. 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 $p_1,\dots ,p_4$. The observed averages areThis 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 fieldsboth 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 $p+\delta p\ge 0$ as well. With the uniform $100\mathrm{kPa}$ field above, perturbations $t(1,-1,0,0)$ are admissible for $|t|\le 100\mathrm{kPa}$. 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.^[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.

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,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 $y$: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 byThe 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 mostEvery 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 $L_p=2kPa/mm$, 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.^[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.