\begin{cases}
                \rho_1, & \vec{x}_j^{k-1} < \frac{1}{2}(\rho_1 + \rho_2) \\
                \rho_2, & \frac{1}{2}(\rho_1 + \rho_2) \leq \vec{x}_j^{k-1} < \frac{1}{2}(\rho_2 + \rho_3) \\
                \ \ \ \ \vdots \\
                \rho_m, & \frac{1}{2}(\rho_{m-1} + \rho_{m}) \leq \vec{x}_j^{k-1}
                \end{cases} \\
            &= \operatorname*{arg\,min}_{\rho \in \mathcal{R}} ||\vec{x}^{k-1}_j - \rho||_2.\end{aligned}$$ The second expression is easier to generalize to a higher-dimensional setting, which will we done in Section \[sec:MCDART\].

### Boundary detection and masking

A set of voxels in the figure is then selected for a new reconstruction to refine the resulting image. First, the set $B^k \subset \{1, \ldots, n\}$ of boundary voxel indices is determined based on the segmentation. Various schemes can be applied for boundary detection. Additionally, a set $U^k \subset \{1, \ldots, n\}$ of free voxel indices is determined, where each voxel is included with a certain probability $1-\beta$, with $0 \leq \beta \leq 1$. The process of selecting the voxels $U^k \cup B^k$ to be reconstructed and the voxels to be left out is called masking. Note that in the initialization phase all voxels are included in the mask.

### Masked ARM Reconstructions

The set of free voxel indices $U^k \cup B^k$ are subjected to a new ARM reconstruction. This is done by 