Given a set of intensity measurements ${\left\{{n}_{i}\right\}}_{i=1}^{M}$, the negative log-likelihood function for transmission tomography for statistical image reconstruction is defined as

$$\begin{array}{rl}l(f)=& \sum _{i=1}^{M}\left(-{n}_{i}\mathrm{ln}({n}_{0})+{n}_{i}\sum _{j=1}^{n}{a}_{ij}{f}_{j}\right.\\ & \left.+\mathrm{ln}({n}_{i}!)+{n}_{0}\mathrm{exp}(-\sum _{j=1}^{N}{a}_{ij}{f}_{j})\right)\end{array}$$(1)where *f* ∈ ℝ^{N} is a vector that consists of the expected attenuation coefficients [2]. The number of photons that are detected in the absence of absorption is denoted by *n*_{0}, the total number of projections is denoted by *M*, and the number of pixels within the image is denoted by *N*. The forward projection step is given by

$${p}_{i}={\displaystyle \sum _{j=1}^{N}{a}_{ij}{f}_{j}},$$(2)where *a*_{ij} is an element of the system matrix *A* ∈ ℝ^{M×N}. Based on these formulations, let

$$f*{=}_{f}^{\mathrm{arg}\mathrm{min}}\left(l(f))+\delta R(f,\Gamma (g,y))\right)$$(3)be the optimization problem that is to be solved in order to reconstruct an image of the tomographed object, where *R*(*f*, *Γ*(*g*, *y*)) is a regularization term that is added to the objective function and a regularization parameter that controls the influence of the regularization. Here a non-local regularization is proposed, which is defined as

$$R(f,\Gamma (g,y))=\sqrt{{\displaystyle \sum _{x=0}^{N}{\lambda}_{x}{\left({f}_{x}-\frac{1}{{\omega}_{x}}{\Psi}_{x}(f,\Gamma (g,y))\right)}^{2},}}$$(4)with

$$\begin{array}{l}{\Psi}_{x}(f,\Gamma (g,y))=\\ {\displaystyle \sum _{y\in {N}_{x}}\Gamma ({g}_{x},y)\mathrm{exp}\left(\frac{-\left|\right|f{\eta}_{x}-\Gamma (g,y){\eta}_{y}|{|}_{p}}{{h}^{2}}\right)}\end{array}$$(5)where *Γ*(*g*, *y*) is the transformed prior image *g* with the transformation parameter. Within (5) _{x} denotes a patch window around pixel *x*, 𝒩_{x} denotes a search window around pixel *x*, and ||·||_{p} denotes the Minkowski distance of order *p*. Furthermore is *λ* ∈ {0, 1}^{N} a mask with

$${\lambda}_{x}=\left\{\begin{array}{ll}0& \text{if}\phantom{\rule{thickmathspace}{0ex}}{g}_{x}=0\\ 0& \text{if}\phantom{\rule{thickmathspace}{0ex}}{g}_{x}\ne 0\end{array}\right.,$$(6)which forces the regularization to ignore all pixels where the prior image holds no information.

The optimization problem (3) is solved by the l-BFGS-b algorithm [3]. In order save computation time while preserving accuracy, the registration problem

$$\mathcal{D}({f}^{(k)},\mathrm{\Gamma}(g,y))\stackrel{!}{=}min$$(7)where 𝒟 : ℝ^{2N} → ℝ denotes a distance measure, is solved only in specific iteration steps. These iteration steps are specified by a set of convergence tolerances *ω*^{(k + 1)} > *ω*^{*} with *k* ∈ ℕ such that *ω*^{(k+1)} < *ω*^{(k)}, where *ω*^{*} denotes the final convergence tolerance at which point the algorithm stops and returns the reconstructed image. While for the distance measure D mutual information Γ is chosen, the calculated transformation is based on an affine transformation.

In order to further reduce metal artifacts, an initial reconstruction based on a filtered backprojection is generated. The resulting image is thresholded in order to gain a mask *μ* ∈ {0, 1}^{N} that describes the position of the metal object. Based on *μ* the set 𝓜_{1} of projection indices is introduced, which corresponds to projection values that are not affected by metal and 𝓜_{2}, a set of projection indices which corresponds to projection values that are affected by metal, such that {1, ..., *M*} = 𝓜 = 𝓜_{1} ∪ 𝓜_{2} and 𝓜_{1} ∩ 𝓜_{2} = ø. Before the algorithm is started, the projection values *p*_{j} with *j* ∈ 𝓜_{2} are removed from the measurement in order to discard all projection values that are associated with the metal object. However, in the course of the algorithm, likewise the solving of the registration problem (7), the projection values *p*_{j} with *j* ∈ 𝓜_{2} are replaced by a forward projection of a filtered version of the current image *f*^{*}(*k*), which is represented by the intermediate result of (3) in iteration *k*. For the filtering step a bilateral filter has shown to be appropriate [4]. A bilateral filter is a non-linear local filter operator that takes into account a local intensity context and as such preserves edges. Let ${\widehat{f}}^{(k)}$ denote the filtered image in iteration *k* such that the new projection values can be defined as

$${\hat{p}}_{i}^{(k+1)}=\sum _{j=1}^{N}{a}_{ij}{\hat{f}}_{j}^{(k)}\phantom{\rule{1em}{0ex}}\mathrm{\forall}i\in {\mathcal{M}}_{\mathcal{2}}$$(8)and the intensity values $\{{\hat{n}}_{i}{\}}_{i\in {\mathcal{M}}_{\mathcal{2}}}$ can be defined as

$${\hat{n}}_{i}^{(k)}={n}_{0}\mathrm{exp}(-{p}_{i}^{(k)})\phantom{\rule{1em}{0ex}}\mathrm{\forall}i\in {\mathcal{M}}_{\mathcal{2}}$$(9)such that the negative log-likelihood function for iteration *k* > 1 is defined as

$$\begin{array}{rl}{l}^{(k)}(f)=& (\sum _{i\in {\mathcal{M}}_{\mathcal{1}}}-n\mathrm{ln}({n}_{0})+{n}_{i}\sum _{j=1}^{N}{a}_{ij}{f}_{j}+\mathrm{ln}({n}_{i}!)\\ & +\sum _{i\in {\mathcal{M}}_{\mathcal{2}}}-{\hat{n}}_{i}^{(k)}\mathrm{ln}({n}_{0})+{\hat{n}}_{i}^{(k)}\sum _{j=1}^{N}{a}_{ij}{f}_{j}+\mathrm{ln}({\hat{n}}_{i}^{(k)}!)\\ & +\sum _{i=1}^{M}{n}_{0}\mathrm{exp}(-\sum _{j=1}^{N}{a}_{ij}{f}_{j})).\end{array}$$(10)In algorithm 1 the complete algorithm is given by a pseudo-code representation.

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