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A more general weighting scheme is required for apodizing the joint
histogram required for the entropy-based cost functions. This is
because the number of entries in each histogram bin becomes
discontinuous as the intensity at a floating image location
(calculated using interpolation) passes through the threshold value
between intensity bins.
As it is the intensity passing through a threshold value that causes
the discontinuities for the joint histogram, we propose a weighting
function that is determined by the intensities and applied to every
location. We choose, once again, a linear weighting function (as
shown in Figure 4) where is the weight for bin
, is the intensity at the location under consideration,
is the intensity threshold between bins and , and is
the smoothing threshold -- the equivalent of in the preceding
section. This weight is then applied to the accumulation of intensity
within the joint histogram bin, as well as the number of entries in
the bin, which is no longer an integer. That is for a point of
intensity , the updating equations for bin are:
where
is the weighting function for the th bin (see figure 4), is the occupancy of bin (a non-integer version of
the number of elements), and is the sum of intensities in the bin.
Figure 4:
Weighting function used for
apodization of the entropy-based cost functions. This weighting
function is equivalent to a fuzzy-bin membership function based on the
intensity .
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This approach is effectively fuzzy-binning, where each intensity bin
no longer has sharp thresholds, but fuzzy membership functions. It
also means that a given location can influence more than one bin
entry. Because of the way the weighting function is calculated
though, each location contributes equally, since the sum of weights
for all bin entries is equal to one.
As changing overlap will still create discontinuities, both the
geometrical weighting and the fuzzy-binning need to be applied to have
a continuous joint histogram. Moreover, the parameter will
give a continuous joint histogram for any value greater than zero,
although the value should not exceed the intensity bin width. The
smoothing capacity of is shown in Figure 5
for the Mutual Information cost function. Results for the Normalised
Mutual Information cost function are very similar. Note that, in
general, a value of
together with equal to the resolution
scale (e.g. 8mm), gives a desirably smooth cost function.
Note that the Partial Volume Interpolation introduced by
Maes [13] also creates continuous joint histogram estimates
if used in conjunction with the geometrical weighting (applied to the
reference locations instead). However, as the name suggests, PVI is
more than just a apodization scheme -- in fact it functions as an
interpolation method too. Therefore, different interpolation methods
cannot be used in conjunction with PVI, whereas for fuzzy-binning the
interpolation method used can be freely chosen. Furthermore, the
fuzzy-binning scheme provides an adjustable parameter, , which
controls the amount of smoothing of the cost function, allowing for
different degrees of smoothing, as desired.
Finally, it can be seen that fuzzy-binning can be made fully symmetric
with respect to the two images (see [2] for a discussion
of symmetry in general registration cost functions). That is, both
floating and reference intensities can have fuzzy-binning applied to
them. However, there is an inherent asymmetry in the way that
interpolation is applied only to the floating image. Therefore,
although such a symmetric approach appears initially attractive, the
simpler and faster approach of only using fuzzy-bins for the floating
image was adopted in practice.
Figure 5:
Plots of the
Mutual Information cost functions versus rotation at different values
for the fuzzy-binning apodization threshold, . The values of
shown are 0.0 for the top row and 0.1, 0.3 and 0.5 for the
three columns (left to right respectively) in the bottom row. In (a)
no geometric apodization is applied, whilst in (b) to (e), the
geometric apodization with mm is applied together with
fuzzy-binning for (c) to (e). The resolution of the image in each
case is 8mm. Each plot was generated by measuring the cost functions
between two real images, about the estimate of the global minimum,
which was a non-identity transformation and so did not correspond to
maximum overlap of the FOVs. This clearly shows that both apodization
methods are required to have the desired smoothing effect on the cost
function, eliminating large discontinuities. In general, a value of
together with equal to the resolution scale
(e.g. 8mm), gives a desirably smooth cost function.
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Next: A Global-Local Hybrid Optimisation
Up: Apodization of the Cost
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Peter Bannister
2002-05-03