ex-6: went on writing about the RL deconvolution

notes/sections/6.md

@@ -318,6 +318,55 @@ necessary to rearrange the two halfs of the final result.
 At the end, the external bins which exceed with respect to the original signal
 are cut away in order to restore the original number of bins $n$.
 
+
+## Unfolding with Richardson-Lucy
+
+The Richardson–Lucy deconvolution is an iterative procedure usually used for
+recovering an image that has been blurred by a known point spread function.
+
+It is based on the fact that an ideal point source does not appear as a point
+but is spread out into the so-called point spread function, thus the observed
+image can be represented in terms of a transition matrix
+$P$ operating on an underlying image:
+
+$$
+ d_i = \sum_{j} P_{i, j} u_j
+$$
+
+where $u_j$ is the intensity of the underlying image at pixel $j$ and $d_i$ is
+the detected intensity at pixel $i$. In general, a matrix whose elements are
+$P_{i,j}$ describes the portion of signal from the source pixel $j$ that is
+detected in pixel $i$.  
+In one dimension, the transfer function
+can be expressed in terms of the distance between the source pixel $j$ and the
+observed $i$:
+
+$$
+  P_{i, j} = \widetilde{P}(i-j)
+$$
+
+In order to estimate $u_j$ given the observed $d_i$ and a known $\widetilde{P}$,
+the following iterative procedure for the estimate  $\hat{u}^t_j$ of $u_j$ can
+be applied. The $t^{\text{th}}$ iteration is updated as follows:
+
+$$
+  \hat{u}^{t+1}_j = \hat{u}^t_j \sum_i \frac{d_i}{c_i} \, P_{ij}
+  \with c_i = \sum_j P_{ij} {\hat{u^t}}_j.
+$$
+
+It has been shown empirically that if this iteration converges, it converges to
+the maximum likelihood solution for $u_j$.  
+Writing this more generally in terms of convolution with a point spread function
+$\tilde{P}$ it becomes:
+
+$$
+   \hat{u}^{t+1} = \hat {u}^{t} \cdot \left( \frac{d}{{\hat{u}^{t}} \otimes
+   \widetilde{P}} \otimes \widetilde{P}^{\star} \right)
+$$
+
+where the division and multiplication are element wise, and
+$\widetilde{P}^{\star}$ is the flipped point spread function.
+
 ---
 
 <div id="fig:convolved">