restoration

Restoration

In this example of blurring a small vertical blurring mask f is convolved with the original signal X to achieve a blurred image f*X. The steps to unblur or restore the image are:

compute the DFT of the blurred image F(f*X) = F(f) . F(X) pointwise product.
guess a form the DFT of F(f) this is called the optical transfer function.
compute a pseudo inverse piF(f) for F(f) i.e., an approximation for 1/F(f) (reciprocal function). The problem is that F(f) has zeros.
compute the DFT of the restored image rX by pointwise multiplcaton by the pseudoinverse

F(rX) = piF(f) . (F(f) . F(X)) = (piF(f) . (F(f)) . F(X))

piF(f) . (F(f) is approxiamtely 1 except being zero in a few small locations.

compute the inverse trasform rX = F^-1((piF(f) . (F(f)) . F(X)) )

Original Blurred Image	Magnitude of DFT of Blurred Image	Magnitude of Optical Transfer Function OTF = DFT of blurring mask
Magnitude of Pseudo Inverse	Magnitude of Pseudo Inverse times OTF	Pseudo- inverse times DFT of Blurred Image
	Restored Image

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