*Bounty: 50*

*Bounty: 50*

I make confusions in the using of **(1) Maximum Likelihood to find (approximately) the original signal knowing the data observed** and **(2)the using of Maximum Likelihood to find estimations of parameters of the PSF**.

**First task 1) Find (up to some point) the original signal :**

I start from this general definition (in discretized) : $$y=H,x+wquad(1)$$ (with $w$ a white noise)

Question 1.1) **How can I demonstrate this relation ? (it seems that we should start from a discrete convolution product, doesn’t it ? Then, the correct expression would be rather : $y=H x+w$ with $$ the product covolution**

)

For estimation, I have to maximalize the likelihood function :

$$phi(x) = (H cdot x – y)^{T} cdot W cdot (H cdot x – y)$$

with $H$ the PSF, $W$ the inverse of covariance matrix (of data $y$), and $y$ data observed : the goal is to find $x$ original signal.

So the estimator is given by : $$x^{text{(ML)}} = (H^{T}cdot Wcdot H)^{-1}cdot H^{T}cdot W cdot y$$

Question 1.2) Is $x$ vector really the original image (I mean the real image that we want to determinate) ?

Question 1.3) For this task, I don’t know practically how to compute this estimator $x^{text{(ML)}}$ ?

2) **Second task : next step, I have to find the parameters on the function with $theta=[a,b]$ parameter vector** : this allow to find the best parameters of PSF and gives the best fit as a function of data $y$

Question 2.1) Is this step well formulated ?

I am working with the following PSF :

And for this second task, I have to find the parameters $a$ and $b$ of :

knowing ($r_0,c_0$)

**In practise, I have used on Matlab the function to perform the Least Mean squares method between the data observed (PSF with noise) and the raw data (the PSF without noise).**

In this way, I can find the two affine parameters $a$ and $b$ (actually, I think this is called a linear regression).

I saw that we could take the vector of parameters $theta[a,b]$ and use the following relation with a matricial form :

$$y=Htheta + wquad (2)$$

Question 2.2) **What is the link between $H$ and the PSF used above (Moffat PSF) ?**

($theta$ is the vector of parameters to estimate and w the white noise).

Question 2.3) **How to demonstrate this important relation ?** and what’s the difference between $(1)$ and $(2)$ ?

Question 2.4) I saw that I have to write $(2) under matricial form :

$$y=Htheta + wquad (2)$$

But how to produce this matrix $H$ from $text{PSF(r,c)}$ ?

Finally, I need remarks or help to knowing what the differences between the 2 tasks and what is the right method to apply for each one.

Sorry if there are multiple questions in this post but I need to grasp all the subtilities of this kind of problem.

**UPDATE 1 :**

You can find here the Matlab script that generates a typical output image (with Moffat PSF and white noise) : Matlab script

Here this typical output image :

Here the estimated image with Maximum Likelihood method (parameters of Moffat are fixed and I want to estimate the original image) :

As you can see, the reconstructed image is very bad.

For this inversion problem, I have taken a $y$ matrix (2D array) in the formula :

$$x^{text{(ML)}} = (H^{T}cdot Wcdot H)^{-1}cdot H^{T}cdot W cdot y$$

I don’t know if it is correct to do this.

This would be fine if someone could help me, Regards.