Biomedical Engineering Reference
In-Depth Information
as
variational
, defining a scalar criterion to minimize, or using partial differen-
tial equations (PDE). The continuously defined deformation function minimizes
a given criterion, or solves a given PDE. The essence of these methods is thus
entirely in the criterion
(resp., PDE). The PDE come from the
optical flow
approach (gradient methods) [26], viscous fluid model [27-29], elastic deforma-
tions with physical analogs [7, 30] or without them [31], or from the variational
criterion [32]
The deformation function can be also modeled
indirectly
, e.g., as a potential
field [33]. This reduces the dimensionality of the problem, at the expense of
reducing the generality of the deformation. Displacement might be
quantized
(limited) to integer number of pixels [34].
9.2.2.2
Global Models
At the other end, we have
parametric, global methods
that describe the cor-
respondence function using a global model with a relatively small number of
parameters [35]. The model mostly consists of expressing the warping function
in a global linear [36], polynomial [37] or harmonic basis [38, 39]. For these
methods, the deformation model corresponding to a specific warp space is as
important as the criterion being minimized.
9.2.2.3
Semi-Local Model
In between the two extremes are semi-local models, using a moderate number
of parameters with local influence. A grid of
control points
is usually placed
over the image and a basis function associated with each of them. Their spacing
corresponds loosely to knot or landmark density. By changing the spacing, we
can approach either local or global models or choose the best trade-off.
Semi-local models are instrumental for the B-spline based approach de-
scribed in section 9.4 and were also used, for example, for motion estima-
tion [40].
9.2.2.4
Image Dependent Models
It is sometimes useful to adapt the warping model to the images considered.
Hierarchically structured semi-local models, based on splines, wavelets, or
