Biomedical Engineering Reference
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parameters in the figure caption. The histogram after registration (Fig. 4.4b) is
more aggregated than that before registration.
Those joint histograms were normalized before being displayed. The maxi-
mum histogram value may be larger or smaller than 255, depending on the image
contents and the image size. The maximum histogram value is always normal-
ized to 255 with a nonlinear transformation. The nonlinear transformation is
relatively simple. First a linear transformation (
x
/max) is used to normalize the
histogram such that the maximum histogram value is 1.0. A nonlinear transform
x
0
.
25
is used to change the histogram such that the small values are enhanced.
The resultant histogram values are multiplied by 255 afterwards. Alternatively
one can use a logarithmic operation to rescale the dynamic range of the his-
togram, as employed to display the Fourier transform of an image.
The joint histogram aggregation at registration can be studied and charac-
terized by entropy. Entropy is a measure of randomness. A higher disordered
system has larger entropy. If the histogram is well structured, then the entropy of
the joint pdf of the pixel values has smaller entropy. In fact, entropy minimization
was exploited as a measure for image registration. However, it is too sensitive
to the overlapping size of two images. To overcome that mutual information is
now used instead. In practice, mutual information maximization proves to be a
robust measure for image registration.
For retinal image registration there are four registration parameters. It is
difficult to visualize how the mutual information as a registration measure be-
haves. We use mutual information maximization to register images (b) and (d)
(temporal registration) and calculate the mutual information in the vicinity of
the optimized solution. For this particular image pair the registration parameters
are:
x
translation of 65.00 pixels,
y
translation of
−
4.56 pixels, rotation angle of
0.31
◦
and scaling factor of 0.9858. To visualize the mutual information surface in
the hyperspace, we fix three registration parameters and change the other one
in the neighborhood of the optimal.
Figure 4.5 shows the mutual information values in the vicinity of the op-
timal
x
translation while other three registration parameters are fixed. Fig-
ure 4.6 shows the results when the
y
translation is varied around the opti-
mal value. The rotation angle dependent behavior is illustrated in Fig. 4.7. Fig-
ure 4.8 displays how the mutual information changes when the scaling factor
varies. It can be seen that the mutual information indeed has a maximum value
around a good registration. Note that the mutual information is very sensitive
