Automated Determination of the Spine-Based Coordinate System for an Efficient Cross-Sectional Visualization of 3D Spine Images - Spinal Imaging and Image Analysis - page 254

for the opposite edge point p m
exist along the search direction (Fig. 12 c). By computing the absolute dot vector
product between the corresponding normalized gradient vectors g ð p m Þ and g ð p m ; n Þ :

multiple candidate points f p m ; n ; n ¼ 1 ; 2 ; ...; N g

p m ; n Þj r ;

X

Q

S ð p m ; p m ; n Þ ¼ g ð p m Þj q : g ð ð 42 Þ

q ; r ¼ 0

the congruence S between the edge point p m and its candidate opposite

point p m ; n is evaluated, i.e. the more parallel the normalized gradient vectors g ð p m Þ
and g

ð p m ; p m ; n Þ

, the larger the congruence between the corresponding points, as edge
points on the opposite sides of the vertebral body walls have gradient vectors of
similar magnitudes and approximately opposite directions. However, the determi-
nation of such edge points is obstructed by edges that do not represent vertebral
body walls, and an opposite edge point may not always exist for each edge point.
To increase the robustness of the computation, gradient vectors in neighbourhoods
of edge points, represented with Q

ð p m ; n Þ

1 points on planes orthogonal to the gradient
vectors, are taken into account. As a result, the summation in Eq. 42 is performed
over every qth point in the neighborhood of edge point p m and the corresponding
rth mirror point in the neighborhood of edge point p m ; n :

þ

g ð p m Þj q : g ð p m ; n Þj r ¼ Þ

þ g y ð y m þ q Þ g y ð y m ; n þ r Þ
þ g y ð y m þ q Þ g y ð y m ; n þ r Þ:

g x ð x m þ q Þ g x ð x m ; n þ r

ð 43 Þ

As the size of vertebral bodies varies due to the biological variability of human
anatomy (e.g. age, gender, pathology), among all candidate opposite edge points
f

p m ; n ;

¼ ; ; ...; g

for edge point p m , only points that are less than d vb ; min ¼ n 1 2 N 7

mm and up to d vb ; max ¼ 60 mm distant from p m are taken into account (values are

determined according to the average size of the human vertebral body [ 47 , 52 , 53 ]):

p 0 m ; n 7! p m ; n ; 8 p m ; n : d vb ; min d ð p m ; p m ; n Þ d vb ; max ; n ¼ 1 ; 2 ; ...; N ; ð 44 Þ

where d is the Euclidean distance between points p m and p m ; n . Among the

resulting candidate points

ð p m ; p n Þ

p 0 m ; n ; N 0 g ; N 0 f n ¼ 1 ; 2 ; ...; N, the point that results in

p 0 m ; n Þ is selected as the opposite edge point p m to edge the largest congruence S ð p m ;

point p m :

p m ¼ p 0 m ; n ÞÞ; N 0 : argmax

p 0 m ; n

ð S ð p m ; n ¼ 1 ; 2 ; ...; ð 45 Þ

The search for opposite edge points is performed in both the positive direction

þ of the gradient vector, as it may point

inwards or outwards vertebral body walls due to variations in the distribution of

g ð p m Þ and the negative direction g ð p m Þ