 Jacobian area correction for radiant-exposure maps

A 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 $\mathbf{x}=f(u,v)$,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.^[ref]

For an original two-dimensional example, let $0\le u,v\le 1$ 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 $H_0=2\mathrm{J}{\mathrm{m}}^{-2}$. The physical receiving area is $3ab/2=9\mathrm{m}{\mathrm{m}}^2$, andIts spatially averaged radiant exposure is $28/9\approx 3.111\mathrm{J}{\mathrm{m}}^{-2}$. Uniformly averaging the exposure values over $u,v$ without the determinant gives only $3\mathrm{J}{\mathrm{m}}^{-2}$.

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.