diff --git a/notes/sections/6.md b/notes/sections/6.md index 17f546b..d4bf380 100644 --- a/notes/sections/6.md +++ b/notes/sections/6.md @@ -92,7 +92,7 @@ of bins default set $n = 150$. In @fig:original an example is shown. $I(\theta)$.](images/6_original.pdf){#fig:original} -## Gaussian noise convolution +## Gaussian noise convolution {#sec:convolution} The sample must then be smeared with a Gaussian noise with the aim to recover the original sample afterwards, implementing a deconvolution routine. @@ -316,7 +316,8 @@ the leght of the vector the same as it was produced by a DFT. This makes it 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 $n$. +are cut away in order to restore the original number of bins $n$. Results are +shown in @fig:convolved. ## Unfolding with Richardson-Lucy @@ -367,6 +368,16 @@ $$ where the division and multiplication are element wise, and $P^{\star}$ is the flipped point spread function. +When implemented, this method results in an easy step-wise routine: + + - create a flipped copy of the kernel; + - elect a zero-order estimate for {$c_i$}; + - compute the convolutions with the method described in @sec:convolution, the + product and the division at each step; + - proceed until a given number of reiterations is achieved. + +In this case, the zero-order was set as $c_i = 0.5 \, \forall i$. Results are +shown in @fig:convolved. ---