Automated Determination of the Spine-Based Coordinate System for an Efficient Cross-Sectional Visualization of 3D Spine Images - Spinal Imaging and Image Analysis - page 278

Axial Orthogonal Curved-Planar Cross-Sections

Axial orthogonal curved-planar cross-sections are obtained by sampling the 3D
image on selected axial planes that are orthogonal to axis w of the spine-based
coordinate system. The axial orthogonal curved-planar cross-section C w¼w c is

therefore obtained by selecting a w c , and sampling the 3D

image I along coordinates u and v in the spine-based coordinate system:

fixed coordinate w ¼

ðÞ ¼ ; ð ; ; Þ: ð Þ

C w¼w c u v Iu v w c 81

If the selected fixed coordinate w c is represented as w c $ c z ð i p Þ , where i ¼ i p

de , then the

sampling plane is, in the image-based coordinate system, orthogonal to the spine
curve at point c ð i p Þ

nes the point cðiÞ, ð i p Þ ¼ ð c x ð i p Þ; c y ð i p Þ; c z ð i p ÞÞ on the spine curve cðiÞ. ð i Þ

.
The axial orthogonal curved-planar cross-section C w¼w c can be therefore obtained as:

and rotationally aligned with the axial vertebral rotation uð i p Þ

C w¼w c x ; ðÞ ¼I ð R t ð i Þ ðuð i p ÞÞ R y ðbð i p ÞÞ R x ðað i p ÞÞ ½x ; y ; c z ð i p ÞÞ; ð 82 Þ

where matrix R t ð i p Þ ðuð i p ÞÞ (Eq. 19 ) represents the axial vertebral rotation for angle

ned by t

uð

i p ÞÞ
(Eq. 52 ) represents the rotation for angle að i p Þ ¼arctan ð t y ð i p Þ= t z ð i p ÞÞ about axis x
of the image-based coordinate system, and matrix R y ðbð

i p Þ

ð

i p Þ (i.e. uð i p Þ ¼ u w ð i p Þ , Eq. 14 ), matrix R x ðað about axis de

i p ÞÞ (Eq. 53 ) represents the

ð t x ð i p Þ= t z ð

rotation for angle bð i p Þ ¼ arctan i p ÞÞ about axis y of the image-based

t ð i Þ ¼ t x ð i Þ; t y ð i Þ; t z ð i Þ

coordinate system, considering that is the unit tangent

vector to the spine curve and c ð i p Þ ¼ ð c x ð i p Þ; c y ð i p Þ; c z ð i p ÞÞ is the center of rotation

(Eq. 57 ) at the selected point i . Axial orthogonal

curved-planar cross-sections in general show a geometrically correct shape of the
vertebral anatomy, because sampling planes cut through vertebrae at the same
anatomical locations (Fig. 23 ).
In the case the axial vertebral rotation uð

¼ i p on the spine curve cðiÞ. ð i Þ

ned in transverse planes that are
orthogonal to axis z of the image-based coordinate system (i.e. uð i Þ ¼ u z ð i Þ ,
Eq. 13 ), then the axial orthogonal curved cross-section C w¼w c can be obtained as:

i Þ is de

C w¼w c ð x ; y Þ ¼ I ð R z ðuð i p ÞÞ R y ðbð i p ÞÞ R x ðað i p ÞÞ ½ x ; y ; c z ð i p ÞÞ

ð 83 Þ

¼ I ð R ðað i p Þ; bð i p Þ; uð i p ÞÞ ½ x ; y ; c z ð i p ÞÞ;

where matrix R z ðuð i p ÞÞ (Eq. 54 ) represents the rotation for angle uð i p Þ about axis z,

and matrix R ðað i p Þ; bð i p Þ; uð i p ÞÞ (Eq. 55 ) represents the composition of extrinsic

rotations for angles að i p Þ , b; ð i p Þ and uð i p Þ about axes x, y and z, respectively, of the

image-based coordinate system.