Poisson data fidelity ($\int [y\log (y/s)+s-y]$)

For positive observations $y$ and positive predicted intensities $s$, this is the generalized Kullback–Leibler divergence with observation in the first argument. Its scalar derivative in $s$ is $1-y/s$ and its second derivative is $y/s^2>0$. It differs from the divergence of a reconstructed density from a reference density by which argument is varied. A linear prediction $s=Kx$ gives derivative $K^{\ast }(1-y/(Kx))$ through the adjoint operator, on suitable admissible function spaces. This fidelity arises from the negative log likelihood of a Poisson observation model, after removal of terms depending only on observations.

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.