# EM ALGORITHM

The Expectation Maximization (EM) algorithm is based on the Lucy-Richardson Maximum Likelihood method for solving inverse problems. Lucy-Richardson deconvolution is an iterative procedure for recovering a latent image that has been blurred by a known point spread function. It is frequently used with telescopes to obtain the true image from the observed image.

The RHESSI EM algorithm is a direct analog of the Lucy-Richardson method. For RHESSI, the observables are the time-varying (rotation/phase) count rates in each of the nine sub-collimators. The method sets up an iterative scheme which uses the deviations in the predicted counts (derived from the current image) from the observed counts to make changes in the current image.

We start from a gray map and update it in each cycle by multiplying it by a scaling map of the same size. The scaling map is obtained as follows. We first obtain the ratio of the observed count modulation profile to the profile predicted from the current image (here, division by zero returns a zero). We make a back-projection map using this ratio count-rate and scale it by a back-projection map made using a unit count rate. The result is the scaling map. The new map for this iteration is the product of the scaling map and the current map. By computing a statistic that becomes small when the image is deconvolved, we can decide when to stop the iterations.

The EM algorithm uses the ANNSEC coordinate system internally.

*Reference: A&A 555, A61 (2013),"Expectation maximization for hard X-ray count modulation profiles"*

Using the EM algorithm at the command line:

Select algorithm | o->set, image_algorithm='em' | or 'hsi_em' |
---|---|---|

Object Class | HSI_EM | |

Create Object | em_obj = hsi_em() | Creates a stand-alone object |

Extract Object | em_obj = o->get(/obj,class='hsi_em') | Extract the object used in the hsi_image object, o |

Parameter Prefix | em, e.g. | All parameter names specific to this algorithm have this prefix |

Parameter Table | not available yet | List and short description of control and info parameters specific to this algorithm |