Automated Determination of the Spine-Based Coordinate System for an Efficient Cross-Sectional Visualization of 3D Spine Images - Spinal Imaging and Image Analysis - page 274

sagittal orthogonal curved-planar cross-sections are obtained by sampling the
3D image on selected sagittal planes, de

￿

ned in the spine-based coordinate
system (section Sagittal Orthogonal Curved-Planar Cross-Sections ),

coronal orthogonal curved-planar cross-sections are obtained by sampling the
3D image on selected coronal planes, de

￿

ned in the spine-based coordinate
system (section Coronal Orthogonal Curved-Planar Cross-Sections ),

axial orthogonal curved-planar cross-sections are obtained by sampling the 3D
image on selected axial planes, de

￿

ned in the spine-based coordinate system
(section Axial Orthogonal Curved-Planar Cross-Sections ).

3

By selecting a point p c ¼ ð u c ; v c ; w c Þ in the spine-based coordinate system

S ,

R

exactly one sagittal w c )

orthogonal curved-planar cross-section can be defined through p c .Ifp c is located
on the spine curve cðiÞ, ð i Þ

(u ¼ u c ), one coronal (v ¼ v c ) and one axial (w ¼

, the
obtained sagittal and coronal cross-sections show, irrespectively of the shape of the
spine, the spinal anatomy at its midline along its entire length, while the obtained
axial cross-section shows the spinal anatomy in the direction orthogonal to its
midline.

, e.g. at point i ¼ i p so that p c ¼ ð c x ð i p Þ; c y ð i p Þ; c z ð i p ÞÞ

Sagittal Orthogonal Curved-Planar Cross-Sections

Sagittal orthogonal curved-planar cross-sections are obtained by sampling the 3D
image on selected sagittal planes that are orthogonal to axis u of the spine-based
coordinate system. The sagittal orthogonal curved-planar cross-section C u¼u c is

therefore obtained by selecting a fixed coordinate u ¼ u c , and sampling the 3D

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

C u¼u c v ð ; w Þ ¼ Iu c ; ð v ; w Þ: ð 75 Þ

In the image-based coordinate system, the sampling plane is represented by a
curved surface that is parallel to the spine curve cðiÞ ð i Þ ¼ ð c x ð i Þ; c y ð i Þ; c z ð i ÞÞ and

follows the axial vertebral rotation uð

Þ

i . If the selected fixed coordinate uðiÞ. c is

represented as u c $ c x ð i Þþ D x, where x is a fixed offset in the left or right

D

direction from the spine curve cðiÞ ð i Þ that corresponds to the sagittal offset of point

p c ¼ ð u c ; v c ; w c Þ from the origin of the spine-based coordinate system, then the

sagittal orthogonal curved-planar cross-section C u¼u c can be obtained as:

C u¼u c ð y ; c z ð i ÞÞ ¼ I ð R t ð i Þ ðuð i ÞÞ R x ðað i ÞÞ ½ c x ð i Þþ D x ; y ; c z ð i ÞÞ; ð 76 Þ

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

ned by t uð i Þ about axis de ð i Þ (i.e. uð i Þ ¼ u w ð i Þ , Eq. 14 ), and matrix R x að ð i Þ Þ

ð t y ð Þ= t z ð (Eq. 52 ) represents the rotation for angle að i Þ ¼ arctan i i ÞÞ about axis x 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