Image Processing Reference
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consists of three parts: (a) a set of vertices that defines the rest state (the template),
(b) a set of vertices that defines a deformed state (an instance of the template), and
(c) a penalty function that measures the amount of deformation of the template
with respect to its equilibrium shape (the stabilizer). Another solution to the pre-
ceding problem was proposed by Nastar and Ayache [140], who modeled a surface
as a quadrilateral or triangular mesh of virtual masses. Each mass is attached to its
neighbors by perfect identical springs with predefined stiffness and natural length.
The system deforms under the laws of dynamics. In addition to elastic and image
forces, an “equilibrium force” determines the configuration of the mesh in the
absence of external forces.
9.4.1.2.2 Spatiotemporal Models
Several researchers have developed models that explicitly incorporate both spatial
and temporal variations of LV shape. Faber et al. [131] use a discrete 4-D model
to segment the LV from SPECT and MR images through a relaxation labeling
scheme [211]. Endo- and epicardial surfaces are modeled as a discrete template
defined in a mixed spherical/cylindrical coordinate system coaxial with the LV
long-axis. Each point in the template represents a radius connected to this axis.
The model is spatiotemporal because the compatibility functions computed in
the relaxation labeling scheme involve neighboring points both in space and time.
In this way, surface smoothness and temporal coherence of motion are taken into
account. Tu et al. [139] have proposed a 4-D model-based LV boundary detector
for 3-D CT cardiac sequences. The method first applies a spatiotemporal gradient
operator in spherical coordinates with a manually selected origin close to the
center of the LV. This operator is only sensitive to moving edges and less sensitive
to noise compared to a static edge detector. An iterative model-based algorithm
refines the boundaries by discarding edge points that are far away from the global
model. The model is parameterized by spherical harmonics, including higher-
order terms, as the refinement proceeds. An interesting approach to spatiotemporal
3-D segmentation is the work by Gerard et al. [146], which is based on the concept
of active objects (AO). AOs are described by means of simplex meshes [208] and
embedded in a physics-based framework. Based on this approach, the authors
tackle the problem of segmentation of 3-D US image sequences using a statistical
motion model of heart dynamics.
9.4.1.2.3 Polygonal Models
LV
polygonal representations
have been applied by several authors [27,28,129,
132,134,142-144,] in the literature. The approaches differ either in the type of
polygonal primitive (e.g., triangular or quadrilateral meshes) or the details of the
shape recovery algorithm (imaging modality, input data, or recovery features).
Shi et al. [142,143] use a Delaunay triangulation [212] to build a surface descrip-
tion from a stack of 2-D contours obtained with a combined gradient- and region-
based algorithm [213]. This representation is subsequently used for motion anal-
ysis based on point correspondences. Bending energy under a local thin-plate
model is used as a measure of match between models of consecutive frames.
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