Types of Restoration Filters

Restoration Filters are the type of filters that are used for operation of noisy image and estimating the clean and original image. It may consists of processes that are used for blurring or the reverse processes that are used for inverse of blur. Filter used in restoration is different from the filter used in enhancement process. Types of Restoration Filters: There are three types of Restoration Filters: Inverse Filter, Pseudo Inverse Filter, and Wiener Filter. These are explained as following below. 1. Inverse Filter: Inverse Filtering is the process of receiving the input of a system from its output. It is the simplest approach to restore the original image once the degradation function is known. It can be define as:

H'(u, v) = 1 / H(u, v) 

Let,
F'(u, v) -> Fourier transform of the restored image
G(u, v) -> Fourier transform of the degraded image
H(u, v) -> Estimated or derived or known degradation function
then F'(u, v) = G(u, v)/H(u, v)
where, G(u, v) = F(u, v).H(u, v) + N(u, v)
and F'(u, v) = f(u, v) - N(u, v)/H(u, v) 

Note: Inverse filtering is not regularly used in its original form. 2. Pseudo Inverse Filter: Pseudo inverse filter is the modified version of the inverse filter and stabilized inverse filter. Pseudo inverse filtering gives more better result than inverse filtering but both inverse and pseudo inverse are sensitive to noise. Pseudo inverse filtering is defined as:

H'(u, v) = 1/H(u, v), H(u, v)!=0
H'(u, v) = 0, otherwise

3. Wiener Filter: (Minimum Mean Square Error Filter). Wiener filter executes and optimal trade off between filtering and noise smoothing. IT removes the addition noise and inputs in the blurring simultaneously. Weiner filter is real and even. It minimizes the overall mean square error by:

e^2 = F{(f-f')^2}
where, f -> original image
f' -> restored image
E{.} -> mean value of arguments

H(u, v) = H'(u, v)/(|H(u, v)|^2 + (Sn(u, v)/Sf(u, v))
where H(u, v) -> Transform of degradation function
Sn(u, v) -> Power spectrum of the noise
Sf(u, v) -> Power spectrum of the undergraded original image 

No blur only additive noise:

H(u, v)=1
W(u, v) = 1 / (1 + Sn(u, v)/Sf(u, v))
W(u, v) = SNR/(1 + SNR)
where, SNR = Sf(u, v)/Sn(u, v) 

No noise only blur:

Sn(u, v)=0
W(u, v) = 1/H(u, v) 

Drawbacks of Restoration Filters:

• Not effective when images are restored for the human eye.
• Cannot handle the common cause of non-stationary signals and noise.
• Cannot handle spatially variant blurring point spread function.