# How applying the gradient descent method for solving a least square problem can remove the blur from an image?

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.

• shouldn't this be $\min_u$ not $\min_f$? – ogogmad Dec 12 '20 at 13:58
• 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 $f$ – ogogmad Dec 12 '20 at 14:00
• I assume it should also minimize the square of the norm. – Wolfgang Bangerth Dec 12 '20 at 19:02
• I have to admit that I don't understand the question. Why would it not remove the blur? – Wolfgang Bangerth Dec 12 '20 at 19:02
• 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. – Brian Borchers Dec 12 '20 at 21:51