Automated Determination of the Spine-Based Coordinate System for an Efficient Cross-Sectional Visualization of 3D Spine Images - Spinal Imaging and Image Analysis - page 234

Fig. 5 The geometrical
torsion sð i Þ of the spine curve
c ð i Þ against the independent
parameter i 2 ½ 0 ; 1 . Labels
C7, T1, … , L3 indicate
vertebral segments (Note The
spine corresponds to Fig. 1 )

7 corresponds to a regular parametrization of the curve by
location i on the curve (Eq. 1 ). In the case the curve is reparameterized by its arc
length s, the natural parametrization cðiÞ

The form of Eqs. 2 -

ð Þ ð Þ

s of cðiÞ i is yielded:

Z

i

dc ðkÞ

d k

s 1

c ð s Þ ¼ c ð i ð s ÞÞ; i ð s Þ ¼ ð i Þ; s ð i Þ ¼ d k: ð 8 Þ

i sp

Considering the natural parametrization of the curve, the unit tangent vector t ð s Þ ,

and unit binormal vector b

^

unit normal vector n ð s Þ ð s Þ are computed as:

d t ð s Þ
d s
d t ð s Þ
d s

dc ð s Þ

t b

Þ ¼t ð s Þ ¼ ; ^ n ð s Þ ¼ ; ð s ð s Þ^ n ð s Þ; ð 9 Þ

ds

the corresponding geometrical curvature jð s Þ and torsion sð s Þ are computed as:

¼ ;

dt db d 2 c ð s Þ ð s Þ ð s Þ

jð s Þ ¼ sð s Þ ¼ n ð s Þ ; ð 10 Þ

ds ds 2 ds

and the Frenet-Serret frame in the matrix form is:

2 3 2 3 2 3

t t

0 jð s Þ 0

ð s Þ ð s Þ

d
ds

4 5 ¼ 4 5 4 5 :

n ð Þ jð s Þ 0 sð s Þ n ð Þ ð 11 Þ

s s

b b

ð Þ 0 sð s Þ 0 ð Þ

s s

However, the natural parametrization is in the case of spine curves rare, as it is
often based on the axial coordinate z at the start and end point of the spine curve (i.e.
i sp ¼

z 1 and i ep ¼ z 2 , where p 1 ¼ ð x 1 ; y 1 ; z 1 Þ is the start point and p 2 ¼ ð x 2 ; y 2 ; z 2 Þ is

the end point), or on pre-de ned constant values (e.g. i sp ¼ 0 and i ep ¼ 1).