Automated Determination of the Spine-Based Coordinate System for an Efficient Cross-Sectional Visualization of 3D Spine Images - Spinal Imaging and Image Analysis - page 245

axial vertebral rotation can be based on axial pixel coordinates z at the start and
end point of observation, resulting in i sp ¼ 1 and i ep ¼ Z, respectively, with the

corresponding number of samples equal Z, where Z is

the number of axial cross-sections in the 3D image. Such parametrization is, con-
sidering the usual in-plane resolution and slice thickness of CT and MR spine
images, in general suf

to N ¼ i ep i sp þ 1 ¼

cient for a smooth and continuous description of the spine

curve c ð i Þ and axial vertebral rotation uð i Þ .

Automated Determination of the Spine Curve and Axial Vertebral Rotation
in CT Images

Vrtovec et al. [ 83 ] proposed a method for automated determination of the spine
curve and axial vertebral rotation in CT images. If the spine curve cðiÞ is repre-

sented by a curve that passes through the centers of vertebral bodies, then its
determination can be based on the anatomical property that vertebral bodies are
locally the largest bone structures of the spine, and on the geometrical property that
the center of each vertebral body is represented by the point that is most distant
from corresponding edges of the vertebral body. To obtain a quantitative repre-
sentation of these properties, a distance transform function based on Euclidean
metrics is applied twice to image I ¼ I ð p Þ

ð i Þ

, resulting in distance map D I ¼ D I ð p Þ :

; p ð p

Þ ¼ þ

d \ ð p Þ; I ð p Þ T ; I Þ \ T ;

D I ð p ð 28 Þ

; p ð p

d ð p Þ; I ð p Þ \ T ; I Þ T ;

where d \ ð p ; p Þ and d ð p ; p Þ are the Euclidean distances between the observed

point p ¼ ð x ; y ; z Þ and point p ¼ ð~ x ; ~ y ; ~ z Þ , which represents the closest point to p

with I T, respectively. The image intensity threshold T, which

separates the bone structures from the background, can be in the case of CT images
determined from the corresponding Houns

ð~ p Þ \ T and I ð~ p Þ

eld values. Each value at point p in the
resulting distance map D I , which is of the same size as image I, represents the
Euclidean distance from p to the edges of the bone structures, and this distance is
positive when p is located inside and negative when p is located outside the bone
structures (Fig. 9 ). As vertebral bodies are locally the largest bone structures of the
spine, distance map values are expected to be the highest in geometrical centers of
vertebral bodies and smoothly decrease by moving away from the centers. The
optimal polynomial parameters b c that de

ne the spine curve cðiÞ ð i Þ (Eq. 25 ) are

finally obtained in an optimization procedure that searches for the combination of
parameters that corresponds to the maximal sum of distance map values along the
spine curve:

!
:

X

N

b c ¼

argmax
b c

D I ð c ð i Þj b c Þ ð 29 Þ

i¼1