1
$\begingroup$

I got an assignment where it asked to implement (in MATLAB) the gradient descent algorithm in order to resolve an ill posed least square problem:

$$ \min_u \Vert Gu - f \Vert $$

where $u$ is the reconstructed image, $G$ is the blur applied to $u$ and $f$ is the blurred image.

I have successfully implemented it with MATLAB, it is effectively removing a portion of the applied blur, but I do not understand why applying a gradient descent method to this ill posed least square problem can effectively remove the blur from an image.

$\endgroup$
7
  • $\begingroup$ shouldn't this be $\min_u$ not $\min_f$? $\endgroup$ – ogogmad Dec 12 '20 at 13:58
  • $\begingroup$ The answer is because you're solving the equation $G u = f$. In other words, you're finding an image $u$ that when blurred produces $f$ $\endgroup$ – ogogmad Dec 12 '20 at 14:00
  • $\begingroup$ I assume it should also minimize the square of the norm. $\endgroup$ – Wolfgang Bangerth Dec 12 '20 at 19:02
  • 2
    $\begingroup$ I have to admit that I don't understand the question. Why would it not remove the blur? $\endgroup$ – Wolfgang Bangerth Dec 12 '20 at 19:02
  • $\begingroup$ How badly conditioned is the $G$ matrix? If it's not badly conditioned, then gradient descent should work (albeit slowly) to minimize the difference between Gu and f. $\endgroup$ – Brian Borchers Dec 12 '20 at 21:51

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.