Biomedical Engineering Reference
In-Depth Information
field over the whole image volume. Thus a deformation function for each voxel
is estimated. This means that a line in one image volume can be registered to
a curve in the other.
The elastic registration algorithms can be classified according to the under-
lying model that they use to estimate the deformations. Following this scheme
of classification, there are three main categories of such algorithms. These are
based on:
•physical models
•
interpolation methods
•
transformation constraints
The algorithms of the first category use such physical models as stress-
strain deformation or fluid flow equations. Optical flow algorithms, which will
be discussed in detail here, also belong to this category. Algorithms of the sec-
ond category use basis functions such as splines, polynomials or wavelets whose
coecients are adjusted to give an approximation of the displacement field.
The third category includes methods that use transformation constraints.
These include the transformation consistency; i.e., the registration of A with
B should correspond to registration of B with A or the transformation should
be bijective, etc. These methods may actually use one of the algorithms of the
first two categories and add a constraint to it.
Simple elastic registration methods show disadvantages in the case of lo-
calized deformations [69]. Interpolation- and approximation-based algorithms
such as the splines based methods encounter problems at finer resolution scales
[13] in addition to the fact that they do not model any physical or biological
process. The optical flow methods allow a higher degree of freedom in esti-
mating the motion of different organs simultaneously and model the motion
of a fluid.
8.3 Optical flow
Optical flow is the change in the intensity pattern of the voxels between
two image volumes. A vector eld, called \ow eld," can be estimated by
modeling the voxel intensity differences between the two image volumes on
the pattern of the flow of a fluid (see Figure 8.4).
8.3.1 Image constraint equation
If two image volumes are given at times t and t + t. A voxel at location
(x;y;z;t) with intensity I(x;y;z;t) will move by x, y, z and t between the
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