Biomedical Engineering Reference
In-Depth Information
model for subsequent registration one must select the knot vectors, number of con-
trol points, as well as the degree of spline. These parameters remain fixed for the
registration process for each frame of data. This stems from the fact that material
point correspondence is dictated by the parameterization of the model, and alter-
ing any one of these variables changes the parameterization. In choosing these
initial parameters, we want to avoid oversmoothing of the data using a low degree
of parametric complexity (underfitting) while avoiding undue influence of noise
in our estimation (overfitting). This is the well-known underfitting/overfitting
problem [21]. After the initial model construction, NURBS allow for two de-
grees of freedom for subsequent fitting — control point location and control point
weighting. Nonrational B-splines only allow one degree of freedom (control point
location). Additionally, due to the spatial configuration of displacement data, the
number of spans of the volumetric B-spline model is limited (see the explanation of
the Schoenberg-Whitney conditions in Section 3.1). Therefore, the control point
weight variation inherent with NURBS is essential for increased accuracy over
nonrational B-splines.
Ma and Kruth presented a method for free-form curve or surface fitting using
NURBS in which the optimal homogeneous control point locations were found
using quadratic programming [22]. They found that the ability to vary the control
point weighting associated with NURBS allowed for more accurate fitting over
nonrational B-splines for curves and surfaces with the same parameterization. We
extend this methodology to our research involving volumetric NURBS models in
which the optimal control point coordinates are calculated using tagged MR data.
We give a brief overview of the least-squares method for curve fitting. We also
describe the necessary modifications for fitting curve data to polar B-spline curves.
Extension to surface or volumetric NURBS objects is straightforward.
3.1. NURBSLeast-Squares FittingwithMeasurementConfidences
Solving for the location of the control points of a B-spline curve or surface in
a least-squares sense is a well-studied problem [20]. Suppose a set of s discrete
points, denoted by the set
D = { p i
=( x i ,y i ,z i ) T : i
∈{ 1 ,s
}}
, is given rep-
resenting a free-form curve (extension to volumetric NURBS objects is straight-
forward).
If, in addition to the set
D
, we have a corresponding set of weights
W = { w i : i ∈{ 1 ,s }}
that quantifies the relative confidence in the measurement
of each corresponding point, the weighted least-squares error criterion, E w , for
the x coordinate is
x i
2
j =1
s
N j,d ( u i ) ψ j P j
j =1 N j,d ( u i ) ψ j
E w =
w i
,
(14)
i =1
where u i is the precomputed parametric value corresponding to the point
p i , and
P j
is the x coordinate of the j th control point. Similar equations are used for the
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