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first modeling the vertebral foramen in 2D axial cross-sections with circles, and
then
fitting a cubic B-spline function to the centers of the obtained circles.
Besides modeling the spine curve in 3D, different geometrical descriptors of
spinal curvature were derived from mathematical functions. Poncet et al. [
58
]
proposed the geometrical torsion as a measure for classifying spinal deformities,
and Kadoury et al. [
29
,
32
] further showed that it can be potentially used to
discriminate among different types of thoracolumbar deformations of the spine.
Vrtovec et al. [
85
] showed that clinically relevant features of the spine can be
identifi-
ed in 3D by observing the geometrical curvature as well as the curvature
angle, which was de
ned as the angular magnitude of the geometrical curvature on
an arbitrary spine section, and was as such independent of the size of the spine. Hay
et al. [
24
] observed both the geometrical curvature and geometrical torsion that
were scaled to a subject-independent coordinate system, and showed that they can
be used to detect and quantify pathological spinal curvatures.
2.3.2 Automated Determination of the Axial Vertebral Rotation
Similarly as for the spinal curvature, measurements of axial vertebral rotation were
in the past possible only by examining the location of pedicles and spinous pro-
cesses in relation to corresponding vertebral bodies in antero-posterior radiographs.
As a result, the axial vertebral rotation in 3D was observed as its projection in 2D,
and several methods based on indices (e.g. the Cobb method, the Nash-Moe
method, the Fait-Janovec method) or actual angles (e.g. the Bunnell method, the
Drerup method, the Stokes et al. method) were proposed. With the introduction of
3D imaging techniques, cross-sectional imaging in the axial plane became possible
and stimulated the development of methods that were based on manual identifi-
-
cation of distinctive anatomical reference points (e.g. the tip of spinous process, the
center of the vertebral body, etc.). A detailed review of methods for the determi-
nation of axial vertebral rotation was performed by Lam et al. [
40
] and Vrtovec
et al. [
88
].
The measurement of axial vertebral rotation was also approached by comput-
erized techniques based on image processing and analysis, although manual ini-
tialization was still required. Haughton et al. [
23
] and Rogers et al. [
66
,
67
]
proposed a method that required manual determination of the axial CT [
66
]orMR
[
23
,
67
] cross-section, the center of rotation and the circular area that encompassed
the observed lumbar vertebra. After initialization, the method automatically mea-
sured the axial vertebral rotation relative to the cross-section of a different vertebra
by searching for the maximal correlation of image intensities between the circular
areas determined in both cross-sections. Oblique CT cross-sections were used by
Adam and Askin [
2
], who determined the axial vertebral rotation from the line that
bisected the thresholded image of the vertebral body according to the symmetry
ratio, de
ned by the maximal correlation of image intensities in the bisected
regions. Kouwenhoven et al. [
37
,
38
] manually selected axial cross-sections
through the centers of vertebral bodies in CT [
38
] and MR [
37
] images of normal
