Image Processing Reference
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real and imaginary parts of log[ F ( u )] can be similarly related, provided F has
no zeros within the contour of integration. This statement also defines the
minimum phase condition. Function F can be written in terms of magnitude
and phase as
FFe i
Taking the complex logarithm of (8.10) yields
+ ϕ
where log| F | is the real part and φ is the imaginary part. One can compute
the phase of F
] =
′ −
The above integral only works if F has no zeros in the upper-half plane. If
F is not a minimum phase function, then the result of the above integral will
be to produce a phase that when applied to | F | generates a minimum phase
function. An important feature of a minimum phase function is that the phase
is a continuous function bounded between −π and π, and “minimum” in this
sense can be interpreted to mean that the phase is already unwrapped. Using
this property of minimum phase functions, we can execute the step defined
in Equation (8.2) in a satisfactory fashion.
In the 1-D problem, it is possible to enforce the minimum phase condi-
tion on a function by applying Rouche's theorem (Fiddy, 1987). A 2-D ver-
sion of Rouche's theorem has been validated in Shahid et al. (2005). Suppose
h = F ( z ) is analytic in a domain D where F = ( F 1 , F 2 ,… , F N ) and the boundary
of D is smooth and contains no zeros of F , then if for each point z on the
boundary, there is at least one index j ( j = 1, 2,…, N ) such that | F j ( z )| > | G j ( z )|
then G ( z ) + F ( z ) have the same number of zeros in D as the number of zeros in
F ( z ) (it actually suffices that Re{ G j ( z )} < Re{ F j ( z )}). In other words, if a function
A has N number of zeros; B has M number of zeros, and | A | > | B | on same
contour, then A + B will have N number of zeros in that contour contrary for
| A | < | B |, then A + B will have M number of zeros (see Figure 8.1). The sum
of the two functions will have the number of zeros equal to the number of
zeros of the larger magnitude function. Consequently, adding a sufficiently
large background or reference wave to a band-limited function A , where we
If | A | > | B | then A + B N zeros
N zeros
M zeros
If | B | > | A | then A + B M zeros
Figure 8.1
Pictorial description of Rouche's theorem.
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