Multiplicative Algebraic Reconstruction Technique (MART)#

Algebraic reconstruction techniques (ARTs) are a classic approach to solving the computed tomography problem [Gordon et al., 1970]. There are two possible types of this technique: additive and multiplicative. For limited-angle tomography problems (such as reconstructing a scene using a CTIS), the multiplicative method is generally preferred due to its positivity-preserving properties. The multiplicative algebraic reconstruction technique (MART) has become the de-facto standard algorithm for reconstructing the solar transition region using the Multi-Order Solar EUV Spectrograph (MOSES) [Fox et al., 2010] and the EUV Snapshot Imaging Spectrograph (ESIS) [Parker et al., 2022].

In this package, our implementation of MART will generally follow the version described in Parker et al. [2022], with some slight adaptations to make it more work on genereal, curvilinear meshes.

Vanilla MART#

The basic version of MART starts with an initial guess at the solution, \(\hat{u}_0\), which can be all ones, or some other informed choice. Given this boundary condition, we then loop through the following steps until the convergence criterion is reached:

  • Compute the images corresponding to the current guess, \(d_i = P \hat{u}_i\), where \(P\) is a projection operator representing the forward model of a CTIS instrument, and \(i\) is the current iteration index.

  • Compute the mean chi squared, \(\langle \chi_i^2 \rangle = \biggl\langle \left( \frac{d_i - d}{\sigma_i} \right)^2 \biggr \rangle\), where \(d\) are the actual images measured by the CTIS, and \(\sigma_i\) is the uncertainty of the predicted images, \(d_i\).

  • Determine if the algorithm has converged by checking if \(\langle \chi^2 \rangle\) has stopped decreasing, \(\langle \chi_{i-1}^2 \rangle - \langle \chi_{i}^2 \rangle < T\), where \(T\) is some threshold close to zero.

  • If convergence has not been reached, compute the correction factor for each channel, \(C_i = \frac{P^* d}{P^* d_i}\), where \(P^*\) is a deprojection operator, similar to \(P^T\), which spreads the intensity gathered by each CTIS channel evenly along the projection direction.

  • Generate an effective correction factor for each channel, \(C_i' = C_i^\gamma\), where \(0<\gamma<1\) is the learning rate.

  • Find the total correction factor, \(\overline{C}_i\) by taking the geometric average of each channel’s correction factor.

  • Finally, generate a new guess by applying the correction factor to the current guess, \(\hat{u}_{i+1} = \overline{C}_i \hat{u}_i\)

The main difference of this implementation from the one described in Parker et al. [2022] is that there is no contrast-enhancement filtering yet. Another difference is that the correction factor is calculated in the coordinate system of the scene instead of the sensors. This is to allow us to conserve flux on both the forward and backward passes, potentially increasing the stability of the algorithm.