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Let $s=Kx$ and assume $s>0$. The scalar Poisson data fidelity integrand has derivative $1-y/s$. Applying the permitted interchange of differentiation and integration in a direction $z$ givesThus the ambient-space Fréchet derivative, or its Hilbert space gradient when the pairing is an inner product, isThe quotient is pointwise, and the adjoint operator transports it back from the data domain to the source domain. The chosen spaces must ensure the integrals and derivative pairing are finite; an output bounded away from zero is a useful sufficient condition.

The printed set of probability density functions and positive cone are not vector spaces, so the notation for a bounded linear map between them is understood as a positive bounded linear operator on ambient function spaces, restricted to admissible inputs. If unit mass is a constraint, admissible directions satisfy ${\int }_{\Omega }z=0$. The displayed gradient is then a representative modulo an additive constant; optimization with that constraint additionally includes its normal cone.

A Poisson observation is an integer-valued count modeled by a Poisson distribution of specified intensity, possibly conditionally on a latent field. Independent Poisson observations motivate Poisson data fidelity through their log-likelihood.

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.