= Solution
A stable multiplicative sensitivity factor $s_i$ at <detector pixel> $i$ changes both means to $N_i=s_iN_0$. The noise-free ratio is still one, so <flat-field correction> is not required merely to remove that fixed pattern from the mean ratio. This is the principal advantage of using two matched flats rather than the <variance> of a single image.
However, the <photon shot noise> is not cancelled. At high counts the conditional ratio <variance> is $2/(gN_i)$. Pooling <detector pixels> gives $E^2\simeq(2/g)\langle1/N_i\rangle$, where brackets denote a spatial average. If the formula uses $\overline N=\langle N_i\rangle$, it returns
$$
\boxed{g_{\rm naive}\simeq\frac{g}{\langle N_i\rangle\langle1/N_i\rangle}\le g.}
$$
The inequality follows from the <Jensen inequality> for $1/x$ on positive signals. For small fractional response variation, the relative bias is approximately minus its squared <coefficient of variation>. A locally nearly uniform region or a fit using <detector pixel>-dependent <variances> avoids this bias from mixing the <harmonic mean> with the <arithmetic mean>. Stable response cancellation also presumes the conversion gain itself is uniform; gain variations, offset errors or changing sensitivity do not obey that simple cancellation. \b[The fixed response pattern cancels in the ratio mean; unequal shot-noise levels can still bias a globally pooled gain estimate.]
Back to article page
