= Area weighting of radiant exposure
{title2=$\overline H=\sum_jA_jH_j/\sum_jA_j$}
\b[Area weighting of radiant exposure] combines patch-average exposures in proportion to the receiving areas they represent. For disjoint patches with areas $A_j$ and mean <radiant exposures> $H_j$, the coefficients $\alpha_j=A_j/\sum_kA_k$ satisfy $\sum_j\alpha_j=1$, and the combined <spatially averaged radiant exposure> is $\sum_j\alpha_jH_j$. This follows by adding the received energies $A_jH_j$, rather than averaging a list of values without its geometry.<International Lighting Vocabulary by CIE>[Entry17-21-071, incident energy per receiving area]{ref}
An original coordinate example explains a calibration issue in image-based measurements. Let a rectangular receiving strip have length $\ell$ and constant width. Stipulate $H(x)=H_0x/\ell$ for $0\leq x\leq\ell$. Suppose a calibrated image coordinate $u\in[0,1]$ maps to physical position $x=\ell u^2$. Uniform sampling in $u$ is therefore nonuniform in physical area.
The physical area average is
$$
\overline H=\int_0^1H_0u^2(2u)\,du=\frac{H_0}{2}.
$$
An unweighted average over increasingly dense, uniformly spaced $u$ samples would instead converge to $\int_0^1H_0u^2\,du=H_0/3$. The factor $2u$ is the <change of variables formula>'s area correction for the chosen mapping; the constant strip width and $\ell$ cancel in the normalized average.
Thus a <Jacobian area correction for radiant-exposure maps> can matter even if every local exposure value is correct. An <optical camera> measurement needs its spatial calibration and the receiving-surface map, not just pixel values. Overlapping patches must not be counted as disjoint areas, and local point samples need an explicit quadrature model before they can stand for patch averages.
Back to article page
