Biomedical Engineering Reference
In-Depth Information
(intrasubject registration). As described above, the issues of correspon-
dence are rather different.
2.5.1
Intersubject Registration
Establishing correspondence of an atlas with a set of images or a number of
images from a cohort of individuals requires a transformation that reflects the
variation in anatomy between the atlas and the individual patient. There will
be changes in shape and size as well as grosser changes in topology. This
remains an area of active research with several approaches under investiga-
tion. Approaches include extending the rigid-body method to incorporate
deformations that follow quadratic and higher order polynomial curves, or a
wide range of other, more complicated functions such as Fourier or wavelet
basis functions and splines, including radial basis functions such as the thin-
plate spline and B-splines. These methods are described in Chapter 13.
Registration algorithms have been devised based on some approximation
to the physical process inducing the deformation, including the elastic prop-
erties of solids and the dynamics of viscous fluids. These transformations
produce physically plausible transformations and are possible to compute,
although they take some time. Such algorithms, however, do not directly
model the underlying causes for the differences in shape, and hence results
should be interpreted with care.
In the optimization process used in all nonrigid algorithms, the goodness
of match is balanced against some constraint prohibiting implausible defor-
mations. This constraint may be provided by some estimate of the energy
required to physically induce the deformation, as if the structures to be reg-
istered were made of elastic material, or may be couched in probabilistic
terms. Multiscale approaches may be used in which a rigid-body transforma-
tion is computed for a coarse or blurry image, followed by multiple rigid-
body transformations for arrays of volume elements at progressively finer
detail with interpolation between these elements.
37
One intriguing algorithm,
aptly named the “Demons” algorithm, models the deformation on the phys-
ical process of diffusion.
38
The mathematics are analogous to Maxwell's
demons in statistical physics.
The problem might be made easier by using statistical shape models based
on principal component analysis of the variations observed across a popula-
tion of individuals.
2
In these models the variation in shape of a structure
between one individual and another is captured by a small number of param-
eters or so called “modes of variation.” In the original paper by Cootes et al.,
2
the outline of the hand is used as an example and the different modes corre-
spond to the individual movements of each finger. While somewhat con-
trived, this shows how a very small number of parameters can capture quite
complex variations in shape. In a study of fetal liver shape, only five modes
are required to capture 89% of the variation in shape.
39
In principle, a tissue growth model might be used to model differences
between different individuals. Study of tissue growth and links with gene
