Imaging mathematics fundamentals

[electronic engineer]

Hello. I can't understand some things in these two files attached with this thread.
Firstly,in file "image 1",i don't know about the method of least sequares mentioned in equation 47.

Secondly, in file "image 2",from where the equation 52 comes and how do we get the next equation 53.

I want to discuss about these two pages if there's anyone already knows about this field or has special experience with it.

thanks.

 

[maze]

This is linear stochastic inverse problem theory. Here is a great MIT graduate video lecture that explains it pretty well (lecture 4). The geometric perspective he explains at around 17 mins in really helped my intuition about it.
http://ocw.mit.edu/OcwWeb/Mathematics/18-085Fall-2007/VideoLectures/index.htm

If you want to really understand inverse problem theory, I recommend you Tarantola's book (you can download the pdf from the authors site):
http://www.ipgp.jussieu.fr/~tarantola/Files/Professional/Books/index.html

"Liebelt, P. B., 1967, An introduction to optimal estimation" is also alright (this one is much more of an engineering perspective)

 

[electronic engineer]

This is really amazing. Thanks.