Biomedical Engineering Reference
In-Depth Information
homologous points or surfaces) based on the mechanical properties of a thin
metal plate with minimum bending energy.
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Nonlinear mappings can also
be constrained with elastic material properties
18,19,58,59
or the dynamics of a
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viscous fluid.
It is important to note that the underlying physical model is
used only to constrain the spatial mapping function in order to yield the
desired properties mentioned above. These mappings are not meant to model
the true physical properties of brain tissue deformation since we are not actu-
ally physically deforming one brain into the shape of another.
14.2.3
The Similarity Measure
Ideally, the spatial mapping function will align all features of interest between a
given brain volume with the corresponding features of the target in the refer-
ence space. We term the mapping that puts homologous features into corre-
spondence
is the practical
implementation of the homology function and is used to measure the goodness
of alignment between two brain volumes (for more information, see the similar-
ity measure concept introduced in Chapter 3).
The actual definition of the homology function is highly task dependent
since it relies completely on the features of interest, and many different fea-
tures can be selected. For example, one might be interested only in the align-
ment of gross anatomical structures such as the cerebral lobes, or in the detail
of particular gyri. Anatomists may be interested in the alignment of cytoar-
chitectonic regions across subjects. Some neuropsychologists are interested in
alignment of functional regions. Even when the features have been selected,
the homology function may be impossible to define in certain situations, e.g.,
when a certain brain region (like Heschl's gyrus) may be represented by one
gyrus in one subject and by two gyri in another subject.
It is important to note that the similarity measure is only an approximation
to the homology function. For example, in brain mapping, the goal is to align
both anatomical and functional regions. In practice, cross-correlation,
homology function.
The
similarity measure
28
a cor-
61
62
relation ratio,
may be used to measure similarity
between brain volumes. Since it is only an approximation, a successful regis-
tration (i.e., one that maximizes the similarity measure) may not represent the
best mapping for the homology function.
or mutual information
14.2.4
The Optimization Procedure
The goal of the optimization procedure is to find the global maximum of the
similarity measure given the domain of possible mapping functions. While
not as important as the reference space, mapping function, or similarity
measure, the optimization procedure must be nonetheless practical and
robust—especially when one takes into account the complex shape of the
similarity measure. The different procedures trade off speed, robustness,
and reliability. We have found that hierarchical methods that function with
