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w.r.t. d C ; ij = d h ; ij , where a higher value of p D indicates a
Fig. 3 Distance constraints p D X i ; X j
with a higher probability
configuration X i ; X j
nodes in the component chain to adaptively determine the number of vertebrae
during the inference procedure. The details about
the vertebra number
determination will be explained in details in part 2.5.
2.4 Optimization
The optimization procedure aims to
nd the con
guration X
¼f
X 0 ;
X 1 ; ...;
X i ; ...;
X N 1 g
that maximizes
Y
Y
ð
j
Þ/
ð
j
Þ
p X i ;
ð
Þ
p
X
I
p I
X i
X j
8
i
e i ; j ¼ 1
i.e., to obtain the MAP estimation of the model con
guration that can
t the
observed data.
In [ 1 ], the optimization is achieved by a generalized Expectation-Maximization
(EM) algorithm given the known disc number and a proper initialization. In [ 2 ], the
candidate con
guration for each object can be detected by searching the whole data
volume using trained random classi
cation trees and the inference is achieved by
the A* algorithm. Both of their optimization methods are not suitable for our
graphical model. This can be explained brie
y as follows. Firstly, we do not have a
proper initialization as in [ 1 ]. Secondly, the con
fl
guration of each object in our case
is high dimensional so that the complete search for candidate con
gurations of each
object as presented in [ 2 ] is computationally expensive. In [ 5 ], the optimization is
achieved by a joint application of a decision forest to detect the vertebral centers
and a graphical model to re
ne the detection results.
Our optimization procedure to
find the solution of Eq. ( 1 ) consists of two levels:
1. An inner iteration to
nd the con
guration X i of each individual component Vi i
by a particle
filtering.
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