Biomedical Engineering Reference
In-Depth Information
and can be defined as ill-conditioned if the reciprocal of its condition number
approaches the computer's floating-point precision. This can occur if the prob-
lem is overdetermined (number of sample points,
x
d
greater than number of
coefficients
C
) and underdetermined (ambiguous combinations of the coeffi-
cients
C
work equally well or equally bad). To avoid such numerical problems,
a singular value decomposition (SVD) linear equation solver is recommended
for use in combination with the moving least-squares method. The SVD solver
identifies equations in the matrix
A
that are, within a specified tolerance, re-
dundant (i.e., linear combinations of the remaining equations) and eliminates
them thereby improving the condition number of the matrix. We refer the reader
to [54] for a helpful discussion of SVD pertinent to linear least-squares problems.
Once we have the expansion coefficients
c
, we can readily express the Hes-
sian matrix and the gradient vector of the combined input volumes as
C
(0)
001
100
,
C
(0)
010
,
C
(0)
∇
V
=
(8.24a)
,
⎛
⎝
⎞
⎠
2
C
(0)
200
C
(0)
110
C
(0)
101
C
(0)
110
2
C
(0)
020
C
(0)
011
H
V
=
(8.24b)
C
(0)
101
C
(0)
011
2
C
(0)
002
evaluated at the moving expansion point
x
0
. This in turn is used in Eq. (8.13) to
compute the edge information needed to drive the level set surface.
8.4.1.3 Algorithm Overview
Algorithm 1 describes the main steps of our approach. The initialization rou-
tine, Algorithm 2, is called for all of the multiple nonuniform input volumes,
V
d
. Each nonuniform input dataset is uniformly resampled in a common coordi-
nate frame (
V
0
's) using trilinear interpolation. Edge information and the union,
V
0
, of the
V
d
's are then computed. Algorithm 1 calculates Canny and 3D direc-
tional edge information using moving least-squares in a narrow band in each
of the resampled input volumes,
V
d
, and buffers this in
V
edge
and
V
grad
. Next
Algorithm 1 computes the distance transform of the zero-crossings of the Canny
edges and takes the gradient of this scalar volume to produce a vector field
V
edge
, which pulls the initial level set model to the Canny edges. Finally the level
set model is attracted to the 3D directional edges of the multiple input volumes,
V
grad
, and a Marching Cubes mesh is extracted for visualization. The level set
