4. IMAGE RESTORATION 
Consider the signal independent analytical degradation model 
g(x) = h(x)*£(x) + n(x) (4-1) 
where the shift invariant linear psf h(x) operates on the original 
input function f(x) and together with the additative noise function 
n(x) produces the output function g(x), 
The goal of image restoration is to recover (deblur) the original 
function f(x) (image) as well as possible, There are several methods 
of recovering the image both linear and non linear, Non linear 
methods are beyond the scope of this paper. Linear restoration methods 
are usually mentioned collectively as deconvolution. Depending on the 
underlying assumed image model, the restoration techniques in the 
following sections are based on either Statistical or deterministic 
image models, or both. The inverse filter will mainly be treated in 
the deterministic approach. Further on, the Wiener smoothing filter 
will be derived in the least square sense based on a statistical 
approach, It is possible to combine some elements of the deterministic 
and statistic approach, Constrained least Square restoration techniqe 
is an example of that mixed approach, 
4,1 Inverse Filtering 
The inverse filter derived in this section is based on the 
deterministic image model, This approach reduces eq.(4-1) to the 
problem of Solving a system of linear equations (Andrews and Hunt, 
1977). 
4.1.1 Algebraic Approach 
  
In matrix formulation Eq. (4-1) may be written as | 
g=Hf +n (4-2) 
where g, f and n are column vectors and H is a matrix, For a shift 
invariant system matrix H is symmetric with respect to the diagonal. 
Each row is the same as the row above except that it is shifted one 
element to the right, Under this condition the psf H is called a 
Toeplitz matrix. The image restoration problem is to estimate the 
object f given samples of the recorded image g. An approximate 
Solution of eq. (4-2) ig 
Y = n1 g (4-3) 
ot 
By substituting eq. (4-2) in eq. (4-3) the solution can be expressed as 
— 
fzfzaH]p (4-4) 
Thus, the estimate of the object f consists of two parts: the actual 
object distribution and the term involving the inverse acting on the 
noise. If S/N is small (noisy image), the second term in eq. (4-4), 
the error term, is very large. The reason for this is that H, which 
represents the psf, has small eigenvalues, and causes H"1 to have very 
large elements, 
- 228 - 
  
  
BETEN tea SEE