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automated determination of the spine curve and axial vertebral rotation in MR
images [
86
] (section
Automated Determination of the Spine Curve and Axial
Vertebral Rotation in MR Images
),
automated modality-independent determination of the spine curve and axial
vertebral rotation in 3D images [
78
,
83
,
86
] (section
Automated Modality-
Independent Determination of the Spine Curve and Axial Vertebral Rotation in
3D Images
).
In all of the presented approaches [
78
,
83
,
86
], the spine-based coordinate
system is determined from 3D spine images of normal and scoliotic subjects by
parameterizing the spine curve cðiÞ
ð
i
Þ
¼
ð
c
x
ð
i
Þ;
c
y
ð
i
Þ;
c
z
ð
i
ÞÞ
and axial vertebral rotation
uð
i
Þ
with polynomial functions:
!
K
y
X
K
x
X
X
K
z
b
x
;
k
b
k
b
y
;
k
b
k
b
z
;
k
b
k
i
k
i
k
i
k
c
ð
i
Þ
¼
;
;
;
ð
25
Þ
k¼0
k¼0
k¼0
X
K
u
b
u
;
k
b
k
i
k
uð
i
Þ
¼
;
ð
26
Þ
k
¼
0
where b
x
¼
f
b
x
;
k
;
k
¼
0
;
1
; ...;
K
x
g
, b
y
¼
f
b
y
;
k
;
k
¼
0
;
1
; ...;
K
y
g
and b
z
¼
f
b
z
;
k
;
k ¼
0
;
1
; ...;
K
z
g
are the parameters of polynomial functions c
x
ð
i
Þ
, c
y
ð
i
Þ
and c
z
ð
i
Þ
of
degrees K
x
, K
y
and K
z
, respectively, corresponding to the spine curve cðiÞ,
ð
i
Þ
, and
are the parameters of the polynomial function
uð
b
u
¼
b
u;
k
;
k
¼
0
;
1
; ...;
K
u
i
Þ
of
the degree K
u
corresponding to the axial vertebral rotation
uð
i
Þ
. The normalization
cients b
k
:
coef
i
ep
Z
b
k
¼
i
k
j
j
di
;
ð
27
Þ
i
sp
where i
i
ep
represent the locations on the spine at its start and end
point of observation, respectively, regularize the impact of each polynomial
parameter to the absolute variation of the corresponding term. With such parame-
trization,
¼
i
sp
and i
¼
the goal
is to automatically determine polynomial parameters b
c
¼
b
x
[
b
y
[
b
z
that describe the spine curve cðiÞ,
ð
i
Þ
, and polynomial parameters b
u
that
de
in the given 3D spine image.
Moreover, it is assumed that the 3D image is cropped to a volume of interest
according to the start and end point of observation along axis z of the image-based
coordinate system, so that the resulting cropped 3D image contains only axial cross-
sections that display the observed anatomy of the spine. Although the presented
examples may be therefore labeled as semi-automated, manual determination of the
volume of interest does not represent a very demanding or time-consuming task. An
advantage of such an assumption is that the parametrization of the spine curve and
ne the axial vertebral rotation
uð
i
Þ
