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Methods
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5.2.5 Energy
Displacement fields define how the atlases are deformed. The aim is to match the measured
data while taking into account certain restrictions. The deformation is estimated by minimiz-
ing a global energy function, which is a weighted sum of several components, e.g., one com-
ponent calculates the difference between a deformed atlas image and the corresponding meas-
ured phase image. The energy is calculated in a way so that it decreases if the deformed at-
lases become better. What is better? There are various objectives. The deformed atlases
should be similar to the corresponding measured data. However, sometimes there are high
levels of noise in the measured data and the atlas should not follow every local variation in the
measured image. The displacement fields should also be smooth. The total (overall) energy
has to take into account all of these and other objectives.
Therefore, the overall energy is a weighted sum of different energy components. The user
can specify the weights of these components and it is therefore possible to emphasize certain
objectives. Currently, there are four energy components. They are explained next. For more
information, please refer to section 6.6, which explains all energy components as well as the
calculation of the total energy in detail. Please also refer to the source code.
The wedge energy is determined by calculating a difference measure between the de-
formed phase-wedge atlas image and the corresponding measured phase-wedge image. For
this to work, the range of phase values has to be the same in both images (automatic or man-
ual phase offset estimation). The coherence value at the location of the current pixel is used to
weight the squared difference for that pixel. The sum of weighted squared differences de-
creases when the non-NaN region shrinks, because a NaN entry leads to an energy of zero for
that pixel. To ensure that the area that is not NaN in the atlas does not shrink too much, the
sum of squared differences is divided by the number of non-NaN pixels to calculate the aver-
age non-NaN-pixel energy. For more information, please refer to the next chapter.
The ring energy is calculated in the same way as the wedge energy; the only difference is
that atlas and measured image are for the ring stimulus experiment.
As mentioned before, if data is very noisy, which is the case for many flat maps, the de-
formation energy can help to ensure that the local atlas deformation does not change too
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