computing the forward projection of the voxels $(\vec{y}_j^k)$ with $j \notin U^k \cup B^k$, and subtracting this from the input data $\vec{p}$ to obtain the residual sinogram $\overline{\vec{p}}^k$. The subproblem that has to be solved in this phase is: $$\begin{aligned}
\overline{\vec{W}}^k\overline{\vec{x}}^k    &= \overline{\vec{p}}^k \label{eq4}.\end{aligned}$$ In Eq. matrix $\overline{\vec{W}}^k$ is defined by $\overline{\vec{W}}^k = (w_{ij})_{ j \in U^k \cup B^k}$ and vector $\overline{\vec{x}}^k$ to be found has length $|U^k \cup B^k|$. Thus, the system of equations contains the same number of equations as Eq.  but has fewer unknowns. The system is solved using a fixed number of ARM iterations, taking the values of $(\vec{x}^{k-1}_j)_{j \in U^k \cup B^k}$ as the starting condition. The complete reconstruction $\vec{x}^k$ at the end of iteration $k$ is then formed by merging $\overline{\vec{x}}^k$ with $\vec{y}^k$.

Some DART implementations also include a smoothing step at this point. The entire loop is repeated a predefined number of times. After the loop ends, the image is segmented one more time. Note that the DART algorithm has many degrees of freedom. This includes the number of ARM iterations in the initialization phase, the number of DART iterations, the number of ARM iterations during these DART iterations, the fixing probability $\beta$, 