In order to complete the design of the restoration algorithm we have to select
the regularization function . In Eq. (8) diagonal weighting
matrices were introduced which means that correlations are neglected. The
result of the convolution operation
should be uncorrelated, or in
other words, ``white''. Therefore
is also called a ``whitening
filter'' (Yaroslavsky 1985). Approximations to this type of filter are
differential operators (Twomey 1963), the Laplace operator (Pratt 1978), and
the linear predictor (Bundschuh 1993):
The application of the optimal whitening filter leads to a constant power density spectrum. Obviously the modulus of the transfer function of the optimal whitening filter must be reciprocal to the modulus of the transfer function of the original signal. Its phase still remains to be determined by some other optimality criterion. The resulting whitening operator is obtained by means of an inverse Fourier transform:
The size of the whitening operator which is used in practice should be as
small as possible in order to save computer time. In this paper an operator
with 55 coefficients is used for the processing of simulated as well as
measured data. Basically this truncation reduces the decorrelation properties.
The optimality criterion for the phase function is minimal spatial dispersion
which minimizes the truncation error. Partial differentiation with respect to
the unknown phase function and equating the result to zero leads to a system
of non-linear equations:
A general solution cannot be found analytically, but some special solutions can easily be obtained. The restriction to real operators enables the use of symmetry properties of the phase function. A special solution which leads to minimal dispersion is
From Eq. (19) we see that basically the power density spectrum and the autocorrelation function
of the
original signal should be known. If available, some a priori knowledge of
the statistics of the original signal can be used. In many cases images can
be modeled quite well as two-dimensional first order Markov processes
(Pratt 1978):
Since the coarse structure of the underlying signal can be seen from the measured data, the power density spectrum of the measured data was used for the processing of simulated as well as measured data instead of the unknown power spectrum of the original signal:
Eventually the self-updating iterative procedure which is used for the
weighting function using the intermediate results of the iteration
can be used for the power density spectrum:
Generally a variation of the regularization function within the image should
enable the adaptation to local statistical image properties. Fig. 2 shows an
example for whitening operator . Obviously it is a high pass
filter which is reasonable for regularization functions. Qualitatively it
shows some similarity to the Laplace operator.
