set of number of photons (or digitalization counts proportional to detected light) for expected values for a detector with pixels is given by it is convenient to work with since in the context of maximum likelihood estimation we want to find the position of the maximum of the likelihood function and we are not interested in its absolute value. Again since is a constant, it will not give any additional information regarding the position of the maximum, so let's consider where is something that shares the same maximum position as . Now let's consider that comes from a ground truth and a measurement which we assume to be linear. Then where a matrix multiplication is implied. We can also write this in the form where we can see how , mixes or blurs the ground truth. It can also be shown that the derivative of an element of , with respect to some other element of can be written as: Tip: it's easy to see this by writing a matrix of say (5 x 5) and two arrays and of 5 elements and check it. This last equation can interpreted as how much one element of , say element influences the other elements (and of course the case is also taken into account). For example in a typical case an element of the ground truth will influence nearby elements in but not the very distant ones (a value of is expected on those matrix elements). Now, the key and arbitrary step: we don't know but we want to estimate it with , let's call and the estimated ground truths while we are using the RL algorithm, where the hat symbol is used to distinguish ground truth from estimator of the ground truth Where stands for a -dimensional gradient. If we work on the derivative of we get and if we now use () we get But we can also note that by definition of transpose matrix. And hence Then if we consider spanning all the elements from to this equation can be rewritten in its vectorial form where is a matrix and , and are vectors. Let's now propose the following arbitrary and key step where is a vector of ones of size (same as , and ) and the division is element-wise. Using () and () we can rewrite () as which yields Where division refers to element-wise matrix