Biomedical Engineering Reference
In-Depth Information
registration [20]. Other methods are based entirely on non-rigid registration,
such as active deformation models (ADM) [19, 63] and a method described by
Guimond
et al.
[22]. Most of these, however, require not only non-rigidly regis-
tering several individuals, but also inverting the transformations between them.
In general, non-rigid transformations are not easily inverted. Even for bijec-
tive mappings there is typically no closed-form inverse. An iterative method for
generating average shape images that does not require inverse computations
was first suggested by Ashburner [2]. It also does not require the explicit com-
putation of an average transformation, the definition of which is not trivial for
non-linear mappings. Instead, the average deformation is generated by the itera-
tive process itself. This technique was later extended to segmented atlas images
and applied to generate an average shape atlas of the bee brain [51].
The central idea is to first map all original individual images onto a common
reference and then generate an average image. After that, the original images are
mapped onto the average, and a new average image is generated. This process
produces a sequence of average images that converges to an average shape
image. Note that convergence and average shape in this context are not defined
in a strict mathematical sense. However, the result of this iteration is sufficient
for the purpose of obtaining an average atlas for atlas-based segmentation, as
we will demonstrate below.
In the first step of the iteration, there is not yet an average image to register
the original images to. One of the latter is therefore chosen as the reference for
the initial registration. In order to avoid bias of the averaging iteration by the
choice of the initial reference, the first registration is affine only, thus correcting
for pose and size differences but leaving object shape unchanged. For the sub-
sequent steps, the average image resulting from the previous registration step
is chosen as the reference image of non-rigid registrations, while the individual
images are used as floating images, one at a time. As a result, all floating im-
ages are independently mapped into the same reference space, thus enabling
the generation of the next average image.
11.4.3.2
Propagation of Transformations
It is well known that intensity-based image registration algorithms, both rigid
and non-rigid, fail when started with an initial transformation estimate outside
the “capture range” of the desired local optimum of the similarity measure. A
