Automated Determination of the Spine-Based Coordinate System for an Efficient Cross-Sectional Visualization of 3D Spine Images - Spinal Imaging and Image Analysis - page 282

However, in the resulting cross-sections, anatomical deformations are present that
result from the intersections of sagittal pro about

axis x of the image-based coordinate system. To avoid anatomical deformations, the
sagittal oblique curved-planar cross-section C u¼u c ;/ ¼ / p can be obtained as:

les due to the rotation for angle að i Þ

C u ¼ u c ;/ ¼ / p ð y ; c z ð i ÞÞ ¼ I ð R z ðuð i Þþ/ p Þ ½c x ð i Þþ D x ; y ; c z ð i ÞÞ; ð 86 Þ

(Eq. 54 ) represents the axial vertebral rotation for

where matrix R z uð i Þþ/ p

angle uð Þþ/ p about axis z of the image-based coordinate system. In this case, the

axial vertebral rotation uð

i

i Þ has to be de 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 ). As a result, the resulting cross-sections are no longer de ned on the basis

of the spine-based coordinate system, and therefore the spine curve is no longer
represented by a straight line.

Coronal Oblique Curved-Planar Cross-Sections

The coronal oblique curved-planar cross-section C v¼v c ;/ ¼ / p is obtained by sampling
the 3D image I on the coronal orthogonal plane at the selected

fixed coordinate

v v c that is additionally rotated for angle / ¼ / p about axis w of the spine-based

coordinate system:

¼

C v ¼ v c ;/ ¼ / p ð u ; w Þ ¼ I ð R w ð/ p Þ ½ u ; v c ; w Þ; ð 87 Þ

with the center of rotation at point p c ¼ ð u c ; v c ; w c Þ . The rotation angle must be on

4 ; þ 4

the closed interval / p 2 ½ , otherwise the cross-section turns into a sagittal

oblique curved-planar cross-section. In the image-based coordinate system, the
sampling plane is represented by a curved surface that is parallel to the spine curve
c

and follows the axial vertebral rotation uð

ð i Þ ¼ c x ð i Þ; c y ð i Þ; c z ð i Þ i Þ with an offset

of / ¼ / p . If the selected fixed coordinate v c is represented as v c $ c y ð

Þþ D

i y,

where y is a fixed offset in the anterior or posterior direction from the spine curve

D

c from the origin

of the spine-based coordinate system, then the coronal oblique curved-planar cross-
section C v¼v c ;/ ¼ / p can be obtained as:

ð i Þ that corresponds to the sagittal offset of point p c ¼ ð u c ; v c ; w c Þ

C v¼v c ;/ ¼ / p ð x ; c z ð i ÞÞ ¼ I ð R t ð i Þ ðuð i Þþ/ p Þ R y ðbð i ÞÞ ½ x ; c y ð i Þþ D y ; c z ð i ÞÞ; ð 88 Þ

where matrix R t ð i Þ ðuð i Þþ/ p Þ (Eq. 19 ) represents the axial vertebral rotation for angle
uð ned by t i Þþ/ p about axis de ð i Þ (i.e. uð i Þ ¼ u w ð i Þ , Eq. 14 ), and matrix R y ðbð i ÞÞ

ð t x ð i Þ= t z ð i ÞÞ (Eq. 53 ) represents the rotation for angle bð i Þ ¼ arctan about axis y of the

image-based coordinate system, considering that t^ðiÞ Þ ¼ t x ð Þ; t y ð Þ; t z ð

ð i i i i Þ is the unit

tangent vector to the spine curve, and cðiÞ ð i Þ ¼ ð c x ð i Þ; c y ð i Þ; c z ð i ÞÞ is the center of