Deconvolution - Theory

asked 2014-07-06 10:36:08 -0500

JoeMama gravatar image

Hello,

just a quick question about theory: I'm studying deconvolution, specifically, image restoration.

Briefly, we want to calculate an image estimate F_est of the original, unblurred, image f by minimizing:

                     2
T = || I_out - K*f ||
                     2

I_out = our output, blurred image K = point spread function

T measures how close our current estimate is to I_out and we aim at minimize this distance. My question is: if the current estimate is very close to I_out, what does that indicate, why is that a good estimate? What if they would be distant?

Thanks in advance!

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Comments

If T is small, then our deconvolution result is very similar to the original image. Does that make sense/answer your question?

GilLevi gravatar imageGilLevi ( 2014-07-06 11:12:28 -0500 )edit

Hi, thanks for your reply. Yeah, intuitively I had thought of the same thing. My point is, if the residual measures the difference, the "distance", between the output blurry image and the current estimate of the original unblurred image convolved with the PSF, how does that tell us if the current estimate is good or bad? Also, why are we convolving the estimate with the PSF? Shouldn't we calculate the difference between the output blurry image and the current estimate without the PSF?

JoeMama gravatar imageJoeMama ( 2014-07-06 12:22:13 -0500 )edit

But the residual measures the difference between the **reconstructed" (not blurry image) to the original (not blurry image). Do you agree?

GilLevi gravatar imageGilLevi ( 2014-07-06 12:55:18 -0500 )edit

I'm not sure. This is taken from the paper I'm working on: https://dl.dropboxusercontent.com/u/105600602/temp.png The "input image" is the blurry image we want to deconvolve, the "blurred estimate" is the convolution between the current estimate and the PSF (why are we doing this convolution?). The difference between the two gives us the residual image. Help!!! :P

JoeMama gravatar imageJoeMama ( 2014-07-06 13:05:30 -0500 )edit

"currentEstimate" is the result of the deconvolution process?

GilLevi gravatar imageGilLevi ( 2014-07-06 14:34:36 -0500 )edit

Yes, at each step the "currentEstimate" gets deconvolved in "some areas" according to the specific algorithm I'm studying.

JoeMama gravatar imageJoeMama ( 2014-07-06 14:39:17 -0500 )edit

If we have a good estimate, that is close to the original image, then if we would convulve it we the PSF, we would get something close to the blurred input image. Not sure, but that's my intuition.

GilLevi gravatar imageGilLevi ( 2014-07-06 15:37:45 -0500 )edit