Articulated Statistical Shape Models of the Spine - Spinal Imaging and Image Analysis - page 323

z z

= (0,0,1)

n

y

y

ˀ
4

=

ʸ

x

x

Fig. 3 Rotation vector is de ned as a rotation of radians around an axis of rotation n and

θ

n

represented as their product ( h )

k n k

A rotation vector can be converted into a rotation matrix using Rodrigues

'

formula:

S 2

R ¼ I þ sin ðhÞ S ð n Þþð 1 cos ðhÞÞ ð n Þ ð 1 Þ

2
4

3
5 :

0 n z n y

n z 0 n x

where S ð n Þ ¼

n y n x 0

It is also possible to compute the rotation vector from a rotation matrix using the
following equations:

R T Tr ð R Þ 1 R

h ¼ arccos ð Þ and S ð n Þ ¼

ðhÞ :

ð 2 Þ

2 2 sin

Let ! be an alternate representation of rigid transformation T that uses the
rotation vector instead of a rotation matrix; thus, ! ¼ f

; g

r t . One of the main

advantages of this representation is that it enables us to de ne a left-invariant

T ! ; T ! Þ ¼ T ! T ! ; T ! T ! Þ distance (dðT1!; ð d ð ) between two rigid transformations. This

distance is defined as follows:

T ! ; T ! Þ ¼ T ! 1 T ! Þ d ð N k ð

ð 3 Þ

! Þ

2 2 2 2

with: N k ð ¼ N k ðf r ; t gÞ ¼ kk r þ k t kk :

The parameter k makes it possible to change the relative importance of rotational
changes in comparison to translational changes. It is a parameter worth considering
carefully, because the rotation and translation are measured in different units. It
would be easy to almost completely discard one in favor of the other without a well
chosen value. In our experience, choosing a value that leads to variabilities that are
approximately equally distributed in the translational and rotational parts of the
transformations works well for descriptive studies.