Shifted Poisson data fidelity ($D_g(u)=\int [Tu-g\log (1+Tu)]$)

For $u\ge 0$ and a positivity-preserving operator, this is the negative log-likelihood for independent Poisson observations with intensity $1+Tu$, up to constants depending only on the data and the unit background. For bounded $g\ge 0$, its scalar derivatives are $1-g/(1+s)$ and $g/(1+s{)}^2$ on $s=Tu\ge 0$. It is convex, and strictly convex in the predicted intensity when $g>0$ almost everywhere. On a unit-area domain, the Jensen inequality gives $D_g(u)\ge \Vert Tu{\Vert }_1-\Vert g{\Vert }_{\infty }\log (1+\Vert Tu{\Vert }_1)$. The shift makes $\log (1+Tu)$ integrable whenever $Tu\in L^1$; no integrability of $\log u$ is needed.

Let $\Omega$ be a bounded connected Lipschitz domain, $\alpha >0$, $g\ge 0$ bounded, and $T:L^1\to L^1$ bounded and positivity preserving with $T1\ne 0$. Minimize $\alpha \mathrm{TV}(u)+D_g(u)$ over nonnegative BV space functions. The logarithmic Jensen inequality bounds $\Vert Tu{\Vert }_1$ and variation on sublevels; mean control for positive imaging operators then bounds the full $BV$ norm. Bounded-variation compactness gives strong $L^1$ convergence, including nonnegativity. The scalar fidelity has bounded derivative on $s\ge 0$, so continuity of $T$ makes its integral continuous in that limit. Lower semicontinuity of variation proves existence by the direct method in the calculus of variations. If $g>0$ almost everywhere and $T$ is injective, strict convexity of the fidelity gives uniqueness. For $0<g<1$, zero is the unique minimizer even without injectivity, since zero energy forces both $Tu=0$ and a constant nonnegative $u$.

On a bounded domain, suppose $0<g<1$, $\alpha >0$, and a bounded positivity-preserving operator satisfies $T1\ne 0$. Every strictly positive $u$ has $Tu\ne 0$: otherwise $T{\chi }_{\{u\ge 1/n\}}=0$ for all $n$, and continuity gives $T1=0$. Since $s-g\log (1+s)>0$ whenever $s>0$, every such $u$ has positive energy. The constants $u=\epsilon$ have zero variation and energy at most $\epsilon \Vert T1{\Vert }_1\to 0$. Their excluded zero limit proves nonattainment. Coercivity and bounded-variation compactness cannot by themselves preserve a nonclosed strict-positivity constraint.