Automated Determination of the Spine-Based Coordinate System for an Efficient Cross-Sectional Visualization of 3D Spine Images - Spinal Imaging and Image Analysis - page 281

Fig. 24 A sagittal oblique curved-planar cross-section CT u¼u c ;/ ¼ / p of a 3D CT image of a scoliotic
spine, shown in a 3D view, b left sagittal view, c posterior coronal view and d superior axial view
of the image-based coordinate system (Note The image-based coordinate system and the spine
curve correspond to Figs. 1 and 7 )

C u¼u c ;/ ¼ / p ð y ; c z ð i ÞÞ ¼ I ð R t ð i Þ ðuð i Þþ/ p Þ R x ðað i ÞÞ ½ c x ð i Þþ D x ; y ; c z ð i ÞÞ; ð 85 Þ

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

ned by t angle uð i Þþ/ p about axis de ð i Þ (i.e. uð i Þ ¼ u w ð i Þ , Eq. 14 ), and matrix

ð t y ð Þ= t z ð

R x ðað i ÞÞ (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 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 rotation for every point i on the spine curve cðiÞ. . Figure 24 displays

the sagittal oblique curved-planar cross-section that passes through the spine curve
(i.e. D x ¼ 0) and is rotated for / ¼ / p ¼ 25 .

ð i Þ