Face Detection and Modeling for Recognition - Biometrics Research ...

adius of a cluster ‘i’ w.r.t. another cluster ‘j’ as follows.
R i = max(R p i , Re i ) + k · R c i ,
(B.1)
R p i = a i|cos(θ ij )| ,
(B.2)
(
) 1/2
Ri e 1
=
cos 2 (θ ij )/a 2 i + sin2 (θ ij )/b 2 ,
i
(B.3)
R c i = (N i /π) 1/2 ,
(B.4)
where R i is the effective radius of the cluster i; R p i is its projection radius; R e i is
its elliptical radius; R c i
is the circular radius used in [84]; the constant k (equals
0.1) is used to prevent the effective radius from vanishing when two clusters are thin
and parallel; a i and b i are the lengths of the major and minor axes of the cluster i,
respectively; θ ij is the angle between the major axis of the cluster i and the segment
connecting the centroids of clusters i and j; and N i is the area of the cluster i. The
major and minor axes of the cluster i are estimated by the eigen-decomposition of
the covariance matrix
C =
⎡
⎣ σ2 x
σ xy
σ xy
σ 2 y
⎤
⎦ ,
(B.5)
where σ x , σ y , and σ xy are the second-order central moments of the skin cluster i. The
eigenvalues of the covariance matrix C and the lengths of the major and minor axes
158