Automated Determination of the Spine-Based Coordinate System for an Efficient Cross-Sectional Visualization of 3D Spine Images - Spinal Imaging and Image Analysis - page 233

being a straight line relative to the osculating plane is measured by the geometrical
curvature jð i Þ (Fig. 4 ):

d c ð i Þ
d i

d 2 c ð i Þ
d i 2

1
r j ð

jð i Þ ¼

Þ ¼

; ð 5 Þ

3

i

d c ð i Þ
d i

where r j ð is the radius of curvature that represents the radius of the osculating

circle in the osculating plane. On the other hand, the deviation of the curve from
being a plane curve, represented by the rotation of the unit binormal vector b^ðiÞ

i Þ

ð i Þ

about the unit tangent vector t^ðiÞ, ð i Þ , is measured by the geometrical torsion sð i Þ

(Fig. 5 ):

d c ð i Þ
d i

d 2 c ð i Þ
d i 2

d 3 c ð i Þ
d i 3

1
r r ð i Þ ¼

sð Þ ¼ ; ð Þ

i 6

2

d c ð i Þ
d i

d 2 c ð i Þ
d i 2

denotes the dot vector product, and r r ð

Þ

where is the radius of torsion. By using

the geometrical curvature and torsion, the resulting Frenet-Serret frame can be
written in matrix form as:

i

2
4

3
5 ¼

2
4

3
5

2
4

3
5 :

t t

0 jð i Þ 0

ð i Þ ð i Þ

d
di

dc ð i Þ

jð i Þ 0 sð i Þ ð 7 Þ

^ n ð i Þ n ^ ð i Þ

di

b b

0 sð i Þ 0

ð i Þ ð i Þ

Fig. 4 The geometrical curvature jð i Þ of the spine curve cðiÞ ð i Þ against the independent parameter

i 2 ½0 ; 1 . Labels C7, T1, … , L3 indicate vertebral segments (Note The spine corresponds to Fig. 1 )