technique requires a spatially invariant ps£ and also an ergodic 
Stationary random field image model. As to the psf requirement the 
problem is not so Serious. The second condition, however, is serious, 
Most images are highly nonstationary having large flat areas Separated 
by sharp edges (Fig 4.2), In the best case the image may be locally 
Stationary (small fluctuations in grey-level), Furthermore, the Wiener 
filter requires uncorrelated signal and noise which is hardly the case 
for many noise Sources, 
4,3 Constrained Filtering 
In the Wiener filter discussed in the previous section it was assumed 
that the undegraded image and noise belonged to two independent and 
Separate stationary random fields and their power Spectra were known. 
In many situations a proiri information is not aviable.  Constrained 
least square filtering requires information about the variance of the 
noise only and allows the designer additional control over the restor- 
ation process (Andrews and Hunt, 1977), Consider 
g=HTf +n (4-17) 
The problem can be expressed as minimization of some linear operator Q 
on the object f, which is Subject to some Side-constraint that is known 
a priori or measurable a posteriori. Suppose that norm(n)2, the 
variance of noise n , Satisfies that side constraint of knowledge, 
Then the constrained least Square problem could be formulated as 
minimize norm(Qf)2 = 
£ToTo£ 
subject to norm(n)2 = norm(g - Hf)2 = 
nn = (g - H£)T(g - H£) 
which is an optimisation problem'with a side constraint, It is also a 
restoration process that is both deterministic and Stochastic using a 
deterministic criterion function with a side constraint based on 
Statistical assumptions, The solution can be obtained by the method of 
Lagranges multipliers, 
UCE) = norm(Qf)2 - L(norm(g - H£)2 - norm(n)2) 
Minimizing by taking derivatives of the objective function U with 
respect to f gives 
OUC£) = 20TQ£ + 2L(HT(g - Hf)) = 0 
à f 
Solving for that f which provides the minimum for U yields 
f - (HTH « v QTg)-1gTq (4-18) 
where V - 1/L. The Lagrangian multiplier V must be adjusted such that 
the constant norm( g- Hf)'is' satisfied. This is often done in an 
iterative fashion (Andrews and‘ Hunt; 1937730 By assigning the linear 
operator Q ditferent properties the method allows à variety of 
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