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variables from the image intensities, i.e., the disc-label variables will capture the
local pixel-level intensity models while the disc variables will capture the high-level
geometric and contextual models of the full set of discs. This approach is simpler
and more robust than the model by Schmidt et al. [
82
] where they had a particular
label for each disc. Next, we present more details about this highly cited work. This
approach marginalizes over the possible disc-labelings since these are auxiliary
variables giving the following optimization function:
X
D
¼
argmax
D
P
ðL
;
Dj
I
Þ
ð
2
Þ
L
X
P
ð
I
jD
;
LÞ
P
ðD
;
LÞ
¼
argmax
D
ð
3
Þ
P
ð
I
Þ
L
X
¼
argmax
D
P
ð
I
;
LÞ
P
ðLjDÞ
P
ðDÞ
ð
4
Þ
L
where the second equality follows from the multi-level nature of the model (the disc
variables are assumed independent of the intensities). Note the summation is over a
very large set of possible assignments 2
jKj
. Then, the authors model it as a Gibbs
distribution:
½
b
1
X
s
2K
1
Z
exp
P
ð
I
;
LÞ
¼
U
I
ð
l
s
;
I
ð
s
ÞÞ
ð
5
Þ
½
b
2
X
s
1
Z
exp
P
ðLjDÞ
¼
U
D
ð
l
s
;
DÞ
ð
6
Þ
2K
½
b
3
X
d
i
2D
d
i
Þb
4
X
ð
1
Z
exp
P
ðDÞ
¼
U
L
ð
V
D
ð
d
i
;
d
j
Þ
ð
7
Þ
i
j
Þ
where
b
k
0
;
k
¼
f
1
;
...
;
4
g
are tunable parameters and Z
½
are the partition
ð Þ
functions. The
notation denotes the set of neighboring elements on the disc
chain. The potentials U
I
and U
D
model the pixel (low)-level intensity and spatial
models, respectively. The potentials U
L
and V
D
model
the object (high)-level
location and context, respectively.
The exact inference is infeasible for this model because of the dependencies of
D
on all
is a Markov chain. They used the generalized Expectation
Maximization (gEM) algorithm to optimize Eq. (
4
). Whereas an EM algorithm
requires maximization in the M step, a generalized EM algorithm only requires an
improvement over the current state. This particular method has a high disc local-
ization rate. However, the spatial information assumes higher locality in low spine
area within the MRI. Moreover, it does not incorporate other aspects such as angles
within the disc chain and most importantly, it does not take into consideration the
L
despite that
D
