Bias of pooled flat-field ratio gain ($g_{\mathrm{naive}}/g=1/(⟨N⟩⟨1/N⟩)$)

For stable positive detector pixel means $N_i$ and uniform detector conversion gain $g$, high-count independent Poisson distribution noise gives $\mathrm{Var}(A_i/B_i)\simeq 2/(g{N}_i)$. Thus substituting a global mean in the equal-signal formula yields $g_{\mathrm{naive}}=g/[⟨N_i⟩⟨1/N_i⟩]\le g$, by the Jensen inequality. Equality holds for uniform means. The flat-field correction pattern cancels from the ratio mean but not from the signal-dependent noise. Local signal bins, or a variance fit after normalization of differences by the square root of their local mean, avoid this leading bias.