Biomedical Engineering Reference
In-Depth Information
In
template-based
methods, the template can be compared with a specific
region in the target image using any of the similarity measures suitable for
pixel-based methods. For
transform-based
methods, a (semi-)norm in the trans-
formed domain such as
L
2
(least-squares measure) is usually used.
9.2.3.2
Regularization
In most applications, it appears to be necessary to add an additional
regulariza-
tion term
to the criterion, mainly to make the problem well-posed and to stabilize
the algorithm. Regularization is also used to express our a priori knowledge. For
continuous, local deformation models, regularization defines the warping space
in the variational sense. For instance, in the case of landmark interpolation,
minimization of the norm of the Laplacian
g
is often used in practice [1, 19],
leading to a
thin-plate spline
solution [50]. Minimizing other similar measures
leads to generalized splines [51] determined either directly or using PDEs. See
also section 9.3.
Other regularizers are constructed by applying a non-linear (often quadratic
and sometimes image dependent) function on the derivative operator. This is
done mainly to preserve discontinuities [52]. Regularizers based on thresholding
in wavelet domain are also used. Implicit regularization for iterative methods
works by alternatively driving the intermediate solution towards the data, and
applying a smoothing operator.
9.2.4
Search Strategy
Given a cost function, there are several basic ways of minimizing it.
9.2.4.1
Direct Solution
In some cases, notably if the cost function is quadratic or if higher order terms
in the Taylor expansion of the criterion can be neglected, then the solution can
be found in one step [38, 53]. Transform methods using directional filters are
often engineered in this way [54, 55].
9.2.4.2
Exhaustive Search
If the search space is finite, exhaustive search can be used. For example, when
small templates are extracted from the reference image, their positions in the
