Proximal operator of Poisson data fidelity (${\mathrm{prox}}_{\gamma E}(z)=(z-\gamma +\sqrt{(z-\gamma {)}^2+4\gamma y})/2$)

For $y>0$ and $\gamma >0$, minimizing $|x-z{|}^2/2+\gamma [y\log (y/x)+x-y]$ on $x>0$ gives $x^2+(\gamma -z)x-\gamma y=0$. The roots have opposite signs, so only the displayed positive root belongs to the effective domain. Strict positivity of the second derivative proves uniqueness. For an integral Poisson data fidelity, the proximal operator separates pointwise under the appropriate integrability assumptions.