Automated Determination of the Spine-Based Coordinate System for an Efficient Cross-Sectional Visualization of 3D Spine Images - Spinal Imaging and Image Analysis - page 267

coordinate x ¼ x c that is additionally rotated for angle b ¼ b p about axis y and/or
for angle c ¼ c p about axis z of the image-based coordinate system:

M x¼x c ;b ¼ b p ;c ¼ c p ð y ; z Þ ¼ I ð R ð 0 ; b p ; c p Þ ½ x c ; y ; z Þ; ð 58 Þ

with the center of rotation at point p c ¼ ð x c ; y c ; z c Þ (Eq. 57 ). The rotation angles

2 ½

must be on the closed interval b p ; c p

4 ; þ 4

, otherwise the cross-section
turns into an axial or coronal oblique multi-planar cross-section. However, in
practice only one rotation is usually applied:

M x¼x c ;b ¼ b p ð y ; z Þ ¼ I ð R ð 0 ; b p ; 0 Þ ½ x c ; y ; z Þ ¼ I ð R y ðb p Þ ½ x c ; y ; z Þ; ð 59 Þ

M x¼x c ;c ¼ c p ð y ; z Þ ¼ I ð R ð 0 ; 0 ; c p Þ ½ x c ; y ; z Þ ¼ I ð R z ðc p Þ ½ x c ; y ; z Þ: ð 60 Þ

Figure 17 displays the sagittal oblique multi-planar cross-section M x¼x c ;c ¼ c p
(Eq. 60 )atx

25 .

¼ x c and c ¼ c p ¼

Coronal Oblique Multi-planar Cross-Sections

The coronal oblique multi-planar cross-section M y¼y c ;a ¼ a p ;c ¼ c p is obtained by
sampling the 3D image I on the coronal orthogonal plane at the selected xed

coordinate y ¼ y c that is additionally rotated for angle a ¼ a p about axis x and/or
for angle c ¼ c p about axis z of the image-based coordinate system:

M y¼y c ;a ¼ a p ;c ¼ c p ð x ; z Þ ¼ I ð R ða p ; 0 ; c p Þ ½ x ; y c ; z Þ; ð 61 Þ

with the center of rotation at point p c ¼ ð x c ; y c ; z c Þ (Eq. 57 ). The rotation angles

2 ½

4 ; þ 4

must be on the closed interval , otherwise the cross-section

turns into an axial or sagittal oblique multi-planar cross-section. However,

a p ; c p

in

practice only one rotation is usually applied:

M y¼y c ;a ¼ a p ð x ; z Þ ¼I ð R ða p ; 0 ; 0 Þ ½x ; y c ; z Þ ¼ I ð R x ða p Þ ½x ; y c ; z Þ; ð 62 Þ

M y¼y c ;c ¼ c p ð x ; z Þ ¼I ð R ð 0 ; 0 ; c p Þ ½x ; y c ; z Þ ¼ I ð R z ðc p Þ ½x ; y c ; z Þ: ð 63 Þ

Figure 18 displays the coronal oblique multi-planar cross-section M y¼y c ;c ¼ c p
(Eq. 63 )aty ¼ y c and c ¼ c p ¼ 25 .

Axial Oblique Multi-planar Cross-Sections

The axial oblique multi-planar cross-section M z¼z c ;a ¼ a p ;b ¼ b p ð x ; y Þ is obtained by