Automated Determination of the Spine-Based Coordinate System for an Efficient Cross-Sectional Visualization of 3D Spine Images - Spinal Imaging and Image Analysis - page 239

2
4

3
5 ;

cos u z ð i Þ sin u z ð i Þ 0

R z ðu z ð i ÞÞ ¼ sin u z ð i Þ cos u z ð i Þ 0 ð 17 Þ

0 0 1

with the center of rotation at point cðiÞ ð i Þ ¼ ð c x ð i Þ; c y ð i Þ; c z ð i ÞÞ . On the other hand, if

uð i Þ ¼ u w ð i Þ (Eq. 14 ), meaning that the axial vertebral rotation is de ned in

transverse planes of measurement that are orthogonal to the spine curve cðiÞ, ð i Þ , then

the modi ed unit normal vector n u ð ^ i Þ equals:

n u ð

ÞÞ e Iy ð

i Þ ¼ R t ð i Þ ðu w ð i i Þ; ð 18 Þ

where e Iy ð i Þ is the unit vector in the direction of the projection of e Iy ¼½ 0 ; 1 ; 0 I to

the plane orthogonal to the spine curve (Eq. 14 ), and matrix R t ð i Þ ðu w ð i ÞÞ represents

ned by the unit tangent vector t^ðiÞ: the rotation for angle u w ð i Þ about the axis de ð i Þ :

R t ð i Þ ðu w ð i ÞÞ ¼ cos ðu w ð i ÞÞ I 3
þ ÞÞ ½t

ð 19 Þ

sin ðu w ð i ð i Þ

ÞÞÞ ½t Þ t þð 1 cos ðu w ð i ð i ð i Þ;

with the center of rotation at point cðiÞ ð i Þ ¼ ð c x ð i Þ; c y ð i Þ; c z ð i ÞÞ . In Eq. 19 , I 3 denotes

½t ð ½t ð Þ t ð the identity matrix of size 3 3, and i Þ and i i Þ are, respectively, the

cross and tensor product matrix of t^ðiÞ: ð i Þ :

2
4

3
5 ;

t z ð

t y ð

0 i Þ i Þ

½t

t z ð

t x ð

ð i Þ ¼ i Þ 0 i Þ ð 20 Þ

t y ð

t x ð

i Þ i Þ 0

2 3

2

ð t x ð

t x ð

Þ t y ð Þ t x ð Þ t z ð

i ÞÞ i i i i Þ

4 5:

½t Þ t

2

ð i ð i Þ ¼ t x ð

Þ t y ð Þ t y ð

t y ð

Þ t z ð

ð 21 Þ

i i i ÞÞ i i Þ

2

t x ð

Þ t z ð Þ t y ð Þ t z ð Þ t z ð i i i i i ÞÞ

In both cases, the unit binormal vector also changes its direction to t the

orthonormal basis, and is therefore equal to b u ð

Þ ¼ t

ð Þ n u ð Þ

. The resulting
axes u, v and w of the spine-based coordinate system are therefore represented by:

i i i

$ b u ð u : ^ e Su ¼½ 1 ; 0 ; 0 S i Þ; ð 22 Þ

v : e Sv ¼½ 0 ; 1 ; 0 S $ n u ð i Þ; ð 23 Þ

$ t w : ^ e Sw ¼½ 0 ; 0 ; 1 S ð i Þ: ð 24 Þ