Automatic Quantification of Congruity from Knee MRI - Computational Biomechanics for Medicine - page 52

2.4.2 Second Order Features

The second order features are the curvatures as formulated in Sect. 2.1 .We
computed the normal curvature along the principal direction of knee motion
(flexion/extension). We computed the normal curvature along the principal direc-
tion of motion because we would like to compute whether the knee is locally
congruent along principal knee motion direction. Since the principal knee motion
is flexi on /extension, we therefore approximated the local principal direction of
motion v as the cross product of sagittal axis (given as [1 0 0], the other axes are
coronal and axial directions as defined in the scan) wit h the local normal vector. The
normal curvature at t on TibProx in the direction v is comput e d by using ( 1 ).
Similarly, the normal curvature at f on FemProx in the direction v is computed by
using ( 2 ). We elaborated the computation of these normal curvatures below.
The Hessian H T at t on the TibProx and the Hessian H F at f (the corresponding
voxel location) on the FemProx are constructed by using second order partial
derivatives of f t and f f respectively.

0
@

1
A

f txx f txy f txz

H T ¼ f txy f tyy f tyz

f txz f tyz f tzz

0
@

1
A

f fxx f fxy f fxz

f fxy f fyy f fyz

H F ¼

f fxz f fyz f fzz

Since, we are interested in surface principal normal curvatures, the orthonormal
basis for H T and H F were say b t and b f . The transformed/projected hessian matrices
on to a plane perpendicular to the normal become, H Txy ¼ b t T * H T * b t and H Fxy ¼
b f T * H F * b f . The eigen values of H Txy that are divided with gradient magnitude ( jf t j )
are the pri nci pal c urv atures k t 1 and k t 2 , corresponding eigen vectors are the principal
directions p t and p t .
Using ( 1 ), the normal curvature at on th e TibProx in the direction of v becomes,
k t ¼ k t 1 cos 2

+ k t 2 sin 2

a¼ cos 1

ðp t vÞ .
Similar ly , by ( 2 ) the normal curvature at f on th e Fe mProx in the direction of v is
given by k f ¼ k f cos 2

, where

a a

b þ k f sin 2 b¼ cos 1 ðp f vÞ . b with

2.4.3 Congruity Over Contact Area

We computed the congruity indices locally at every voxel location using ( 3 ) over
CA from TibProx to FemProx and vice versa and then mean value is reported as the
overall CI. Let TibProx has N 1 voxels and FemProx has N 2 voxels. The overall CI in
the contact region mathematically is given by

P N 1
p¼ 1 CI p
N 1

P N 2
q¼ 1 CI q
N 2

þ

CI ¼ (5)