= Jacobian area correction for radiant-exposure maps
{c}
{title2=$J_f=|\det Df|$}
A \b[Jacobian area correction for radiant-exposure maps] converts a coordinate-space integral into one over the physical receiving surface. For a one-to-one planar calibration map $\mathbf x=f(u,v)$,
$$
Q_A=\int_D H(f(u,v))|\det Df(u,v)|\,du\,dv,\qquad
A=\int_D|\det Df(u,v)|\,du\,dv.
$$
The <Jacobian determinant> supplies physical area per coordinate-area increment. This <change of variables formula> preserves the <radiant exposure> definition as incident energy per receiving area.<International Lighting Vocabulary by CIE>[Entry17-21-071]{ref}
For an original two-dimensional example, let $0\leq u,v\leq1$ and map $x=au$, $y=bv(1+u)$, with $a=2\,\mathrm{mm}$ and $b=3\,\mathrm{mm}$. Then $|\det Df|=ab(1+u)$. Stipulate $H(f(u,v))=H_0(1+u)$ with $H_0=2\,\mathrm{J\,m^{-2}}$. The physical receiving area is $3ab/2=9\,\mathrm{mm^2}$, and
$$
Q_A=abH_0\int_0^1(1+u)^2\,du=28\,\mathrm{\mu J}.
$$
Its <spatially averaged radiant exposure> is $28/9\approx3.111\,\mathrm{J\,m^{-2}}$. Uniformly averaging the exposure values over $u,v$ without the determinant gives only $3\,\mathrm{J\,m^{-2}}$.
The correction applies to exposure densities represented in distorted coordinates. If each detector sample already reports energy integrated over its physical patch, summing those calibrated energies requires no second multiplication by patch area. Otherwise the area is counted twice. An <optical camera> reduction must specify whether its values represent energy, irradiance, exposure or a further processed signal.
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