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h
c
condition 0
:
5
N
1 and determines the number of ordered residuals used in the
\
computation.
Basing on the obtained spine curve cðiÞ,
, planes that are orthogonal to the spine
curve can be extracted from the MR image for each location i on the spine. In these
planes, axial vertebral rotations
ð
i
Þ
fu
i
;
;
; ...;
g
i
¼
1
2
N
are computed by
finding the
fc
i
;
¼
;
; ...;
g
in-plane lines of symmetry (Eqs.
34
as
initialization values. A continuous representation of the axial vertebral rotation
uð
-
36
) and using angles
i
1
2
N
i
Þ
is then obtained by
fitting a polynomial function to the resulting angles
fu
i
g
.To
determine the optimal polynomial parameters b
u
(Eq.
26
), the non-linear LTS
regression can be again applied:
!
X
h
u
b
u
¼
r
u;
½i
j
;
ð
Þ
argmin
b
u
b
u
41
i¼1
2
where r
u;
½i
¼
ðu
i
uð
i
ÞÞ
represent the ordered squared residuals in increasing
order; r
2
u;
½
r
2
...
r
2
, and h
u
is the trimming constant that satis
es the
1
u;
½
2
u;
½
N
h
N
condition 0
1 and determines the number of ordered residuals used in the
computation. Two thirds of ordered residuals can be used to determine the optimal
polynomial parameters for the spine curve (i.e. h
c
¼
:
5
\
2
3
N in Eq.
40
) and axial
vertebral rotation (i.e. h
u
¼
3
N in Eq.
41
).
The method was evaluated [
86
] on 21 MR images of the thoracic and lumbar
region of the spine, and the reported mean difference between the obtained spine
curve in 3D and manually de
2
1 mm, while
the reported mean difference between the obtained axial vertebral rotation and
manually de
ned ground truth points was 2
:
5
1
:
9
.
ned ground truth angles was 1
:
7
0
:
Automated Modality-Independent Determination of the Spine Curve and
Axial Vertebral Rotation in 3D Images
Š
tern et al. [
78
] proposed a method for automated determination of the spine curve
c
that is applicable to both CT and MR images of the spine, and can be therefore
regarded as a modality-independent method. The method is based on the anatomical
property that the walls of each vertebral body usually form a cylindrically shaped
structure, which can be represented by a closed surface that contains the edges of
the vertebral body in 3D, and on the geometrical property that any line orthogonal
to vertebral body walls intersects with these edges in two points located on the
opposite sides of vertebral body walls. The spine curve, de
ð
i
Þ
ned as a curve in 3D
that passes through the center of each vertebral body, is therefore located in the
middle of any pair of opposite edge points on vertebral body walls.
The directions of lines that de
ne the pairs of opposite edge points can be
represented by image intensity gradient vectors in 3D, which are orthogonal to the
