## 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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