Imaging mathematics fundamentals

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SUMMARY

This discussion focuses on the fundamentals of imaging mathematics, specifically linear stochastic inverse problem theory. Participants seek clarification on the method of least squares as referenced in equation 47 of "image 1" and the derivation of equations 52 and 53 in "image 2". Recommended resources include an MIT lecture on inverse problem theory and Tarantola's book on optimal estimation, which provide essential insights into the topic.

PREREQUISITES
  • Understanding of linear stochastic inverse problem theory
  • Familiarity with the method of least squares
  • Basic knowledge of mathematical equations and derivations
  • Access to educational resources such as MIT OpenCourseWare
NEXT STEPS
  • Watch the MIT lecture on inverse problem theory (Lecture 4)
  • Read Tarantola's book on optimal estimation for in-depth understanding
  • Explore "Liebelt, P. B., 1967, An introduction to optimal estimation" for an engineering perspective
  • Research additional resources on linear stochastic processes and their applications
USEFUL FOR

This discussion is beneficial for students and professionals in mathematics, engineering, and data science who are looking to deepen their understanding of imaging mathematics and inverse problem theory.

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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.
 

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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)
 
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This is really amazing. Thanks.
 

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