Image Restoration and Image Degradation Model

Image restoration is the process of recovering an image that has been degraded by some knowledge of degradation function H and the additive noise term {{\eta}(x,y)}. Thus in restoration, degradation is modelled and its inverse process is applied to recover the original image.

Image Restoration and Image Degradation Model
Fig: Image Restoration and Image Degradation Model

Objective of image restoration:

The objective of image restoration is to obtain an estimate of the original image {{f}(x,y)}. Here, by some knowledge of H and {{\eta}(x,y)}, we find the appropriate restoration filters, so that output image {\widehat{f}(x,y)} is as close as original image {{f}(x,y)} as possible since it is practically not possible (or very difficult) to completely (or exactly) restore the original image.


  • {g(x,y)} = degraded image
  • {f(x,y)} = input or original image
  • {\widehat{f}(x,y)} = recovered or restored image
  • {{\eta}(x,y)} = additive noise term
In spatial domain:

{g(x,y) = h(x,y) \circledast f(x,y) + \eta(x,y)}

where, {\circledast} represents convolution

In frequency domain:

After taking fourier transform of the above equation:

{G(u,v) = H(u,v)F(u,v) + N(u,v)}

If the restoration filter applied is {R(u,v)}, then

{\widehat{F}(u,v) = R(u,v)[G(u,v)]}

{\widehat{F}(u,v) = R(u,v)H(u,v)F(u,v) + R(u,v)N(u,v)}

{\widehat{F}(u,v) \approx F(u,v)} (for restoration)

as restoration filter {R(u,v)} is the reverse of degration function {H(u,v)} and neglecting the noise term. Here, {H(u,v)} is linear and position invariant.

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