Inverse Filtering  We have degraded image g(x,y) by degradation function H(which we already estimated)  This is the simplest approach, where we calculate F^(u,v) by,  As we know, 

Inverse Filtering (cont…)  So our equation reduces to,  i.e. even if we know the degradation function we cannot recover the original image because Noise N is the random function whose Fourier X’form is not known.  As H(u,v) = tends to zero / very small then F^(u,v) can be easily determine. 

Pros & Cons of Inverse Filter  Saves the considerable amount of calculation i.e. it is simple.  Noise in the image can leads to distortion in image. 

Weiner Filter / Min. Mean Square Error Filter  Limitation of Inverse Filter:  Makes no provision for handling noise  In this approach – that incorporates both the degradation function & also statistical property of the noise into restoration process.  Method is to find f^ of original image f such that mean square error between them is minimized. This error measure is, where E is expected value of the argument 

Weiner Filter (cont…)  Here it is assumed that noise and image are uncorrelated; that any of them have zero mean. 

Weiner Filter (cont…) The terminology used in the equation, 

Weiner Filter (cont…)  The previous result is known as Winer filter  The term in the bracket is referred as,  Minimum means square error filter or  Least square error filter  Winer filter is not having problem like inverse filter in degradation function, unless H(u,v) & Sn (u,v) both are zero for the same value of the u & v. 

Weiner Filter (cont…)  When we are dealing with the white/complex noise then |N(u,v)|2 is constant.  As the power transform of the undegraded image seldom is know so one approach is used, where K is constant. 

Drawbacks of Weiner Filter  It requires the basic prior knowledge of the power spectrum density of original image. 

Result of Inverse & Weiner Filter 

Inverse Filter vs Weiner Filter  “A weiner filter is better than the inverse filter in presence of noise because a weiner filter uses the prior knowledge of the noise field. The transfer function of weiner filter is chosen to minimize the mean square error using statistical information on both image & noise fields.” - [Digital Image Processing By Subramania Jayaraman, S. Esakkirajan, T. Veerakumar] 

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