Biomedical Engineering Reference
In-Depth Information
quadtrees [41] can be refined only where needed. In feature based methods, the
basis functions of the warping model can be placed where the features are and
the deformation field interpolated in regions where no information is available.
Typical examples are
radial basis functions
such as thin plate splines [1,19,42],
to be discussed more in section 9.3.
9.2.3
Cost Function
The quality of a registration result is assessed by a cost function. It is
mainly composed of a
data term
, measuring the similarity of the images af-
ter warping. Sometimes a
regularity term
is added to privilege likely (smooth)
deformations.
9.2.3.1
Data Term
For methods based on geometric features, we can use a (mean) distance between
corresponding features in the source and target images. Note that
landmark
interpolation
(section 9.3) is a limit case with infinite weights given to this
distance, constraining it effectively to zero. If the pairing between the source
and target features is unknown, the iterative closest point algorithm [43], can
be used to determine it.
For
pixel-based
methods, the data term is a
similarity measure
on the two
images. The simplest and fastest criterion is the appropriate (e.g.,
l
1
or
l
2
) norm
of the pixel-wise difference, such as in the
SSD
(sum of squared differences)
criterion. However, it assumes an equivalency of intensities in both images.
If the intensities only correspond up to a possibly varying linear relationship
perhaps, then it is appropriate to use
correlation
, respectively, local normalized
correlation [7, 44]. More general and perhaps non-functional relation between
the intensities warrants the use of the
mutual information
criterion [45-48].
This is encountered, for example, in intermodality registration [49].
All three criteria lead to statistically optimal estimates of the registration
under corresponding image and noise models. Their complexity, computational
cost, and fragility increases in the order in which they were presented. For local
criteria, such as local normalized correlation, or local mutual information, the
neighborhood size must be properly chosen.
Image interpolation
is used to
calculate the warped image, see also section 9.2.1.1.
