Image Processing Reference
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approaches fundamentally differ in the representation of the additional deformation
field. Chen et al. and Fan et al. use spherical harmonics in order to approximate
the residual error between the superquadric estimate of the endocardial LV wall
and the true wall location. Spherical harmonics have the advantage that fine-tuning
can be improved ad infinitum with increasing number of harmonics. However,
adding a new coefficient influences the shape of the model everywhere (nonlocal
basis functions). Bardinet et al. [9] extend the basic superquadric deformations
(tapering and bending) through the use of free-form deformations (FFD), a tech-
nique introduced in computer graphics by Sederberg and Parry [203]. The super-
quadric is attached to a flexible, boxlike frame, inducing a nonrigid deformation
on the superquadric. Bardinet et al. use trivariate B-splines to parameterize this
deformation field. In a later work, Bardinet et al. [122] apply their method to estimate
left-ventricular wall motion. This is accomplished by deforming the full model
(superquadric plus FFD) in the first frame, and modifying only the FFD in the
subsequent frames. By tracking points with the same parametric coordinates along
the cardiac cycle, a number of dynamic parameters such as wall thickening and
twisting motion are computed. Germano et al. [135,136] have developed a system
for automatic quantification of left-ventricular function from gated perfusion
SPECT images. An iterative algorithm fits an ellipsoidal model to a semiautomat-
ically obtained segmentation. This iterative algorithm incrementally adapts the
ellipsoid's parameters and center of mass so that accurate registration of the model
is obtained even in the presence of large perfusion defects. The ellipsoid defines a
coordinate system that is used to refine the model. A Gaussian model of the count
profiles is used to compute radial offsets corresponding to the endocardial and
epicardial walls. Although simple in its formulation, this method has proved very
useful in determining most of the classical cardiac functional parameters [35] from
SPECT images and has been extensively validated in humans [135,136,204].
9.4.1.1.3 Local Approaches
A number of methods have been reported to provide surface reconstruction using
piecewise polynomial surfaces, e.g., B-splines or bicubic Hermite surface patches.
These techniques have appeared mainly in the context of surface reconstruction
from multiple cross sections [31,126,169] or projections [107-110,124]. Given
the ill-posed nature of this problem, most of these techniques require extensive
user interaction. Usually, a set of landmarks or fiducial points are determined
from each cross section or projection and, using high-level knowledge about the
viewpoint and the geometry of the LV, a local surface approximation using surface
patches is performed. One relevant example is the work by Sanchez-Ortiz et al.
[125], in which a tensor B-spline surface model is fitted to multiple planes of a
rotating US probe to recover the 3-D shape of the LV using multiscale fuzzy
clustering features. A very interesting approach is the one of Horkaew and Yang
[128], which uses tensor B-splines to represent a surface model of a full heart.
The technique allows modeling of both the mean shape and the variability around
it using landmarks obtained by optimizing a minimum description length (MDL)
principle and piecewise bilinear maps (PBM).
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