Automated Determination of the Spine-Based Coordinate System for an Efficient Cross-Sectional Visualization of 3D Spine Images - Spinal Imaging and Image Analysis - page 232

curve c ð i Þ can be then obtained by continuous interpolation of v ð k Þ between

c ð i sp Þ ¼v ð 1 Þ and c ð i ep Þ ¼v ð K Þ (Fig. 3 ).

, the geometric properties of the curve can be
described in terms of differential geometry by the Frenet-Serret frame, which is
de

For a differentiable curve cðiÞ, ð i Þ

ned as an orthonormal basis by the unit tangent, normal and binormal vectors to
the curve. The unit tangent vector t^ðiÞ represents the direction of the curve that

corresponds to increasing values of parameter i:

ð i Þ

t ð i Þ dc ð i Þ

t

ð i Þ ¼

kk ;

t ð i Þ ¼ ; ð 2 Þ

t ð i di

ð Þ ð Þ

where t i is the tangent vector to the curve cðiÞ, i , obtained as the first derivative of

ð Þ kk

denotes the vector norm. The unit normal vector n^ðiÞ

ð Þ

c i with respect to i, and i

represents the deviation of the curve from being a straight line:

dt d 2 c n ð i Þ Þ

di ¼

ð i dc Þ

di

ð i ð i Þ dc ð i Þ

^

n ð i Þ ¼

k ;

n ð i Þ ¼ ; ð 3 Þ

n ð i di 2 di

ð Þ ð Þ

where n i is the normal vector to the curve cðiÞ, i , obtained as the first derivative of

t

denotes the cross vector product. To satisfy the
orthonormality of the basis, the unit binormal vector b^ðiÞ

ð i Þ with respect to i, and

ð Þ

i is orthogonal to both the

unit tangent vector and the unit normal vector:

d 2 c

b ð i Þ dc Þ

di

ð i ð i Þ

b

Þ ¼t

Þ ^

ð i ð i n ð i Þ ¼

k ;

b ð i Þ ¼ ; ð 4 Þ

b ð i di 2

where b ð i Þ is the binormal vector to the curve cðiÞ, ð i Þ , obtained as the cross vector

product of the first and the second derivative of the curve cðiÞ. ð i Þ .

The unit tangent vector t^ðiÞ

and the unit normal vector n^ðiÞ

ð i Þ ð i Þ at location i on curve

ð Þ

c i de ne the osculating plane at that location. The deviation of the curve from

Fig. 3 The sagittal c x ð i Þ ,
coronal c y ð i Þ and axial c z ð i Þ

component 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 )