# file - BioMed Central

```Appendix I:
Denoting a facial mesh composed of N points by F = {pi} for i = 1, …, N. Suppose S is
the set of M points that are within distance R around the point pj(1  j  N ). The best
fit sphere T around pj is therefore determined by two parameters, namely the center
O  a, b, c  and radius r. The squared distance from each point p k  ( xk , yk , zk ) ,
1  k  M in S to the surface of T is defined by
d 2 (p k , T)  ( x k  a) 2  ( y k  b) 2  ( z k  c) 2  r 2
(s1)
Let us denote d 2 (p k , T) as  k , then the above equation can be expressed as
 k  ( x k2  y k2  z k2 )  (2ax k  2by k  2cz k )  (a 2  b 2  c 2  r 2 )
(s2)
And the square distance vector is
ε   1 , ,  M 
T
(s3)
Our goal is to minimize the following error function
M
E    k2  ε T ε  ( A  B T W ) T ( A  B T W )
(s4)
k 1
Where
 x12  y12  z12 


A


 x k2  y k2  z k2 



W  2a,2b,2c, r 2  a 2  b 2  c 2
 x1
y
B 1
 z1

1





T
xk 
y k 
zk 

1

This is a simple least squares problem. The solution is W  BBT

1
BA ，and the radius
is r  W(4)  W(1) 2  W(2) 2  W(3) 2 / 4 . The radius r is a key measurement for
nose-tip recognition. In order to assess how close the point set S matches the sphere T,
we introduce another measurement: the mean fitting residual, defined as e  E M .
The smaller e is, the better S fits to a sphere.
Appendix II:
The vector P is defined as in the main text equation 2. Denote the mean of the P
vectors across the training set as Pt , the covariance matrix is calculated as
C
1 m
Pt ,m  Pt Pt ,m  Pt T

m  1 i 1
(s5)
The eigen space U is then constructed by the eigenvectors u i such that
Cu i  i u i
(s6)
where i is the ith largest eigen value of C. And U is given by U  u1 , u 2 ,..., u k  .
Here k is the actual number of eigen vectors to be used, which is set to 16 in our case.
U therefore defines an eigen space where the sample P patches can be evaluated for
similarity.
For a sample face, every point in the 2D grid is given a 21mm&times;21mm patch and a
sample patch vector Ps is similarly derived following equation (2). Ps is then subtracted
by Pt and projected into the eigen space U to give the weight vector
w  UT ( Ps  Pt )  1 , 2 ,, k 
T
(s7)
Ps can be reconstructed using w as Ps '  Pt  wU  Pt  U T ( Ps  Pt )U . The
reconstruction error can be described as
e  ( Ps  Ps ' )T ( Ps  Ps ' )
(s8)
A valid landmark point should lie close to the origin point in the U space; we
therefore use only points satisfying  3 i  i  3 i , where λi is the variance
along ωi across the training set. We also calculate the Mahalanobis distance from Ps '
to Pt .
i 2
d 
i 1 i
k
which can be another indicator of pattern similarity.
(s9)
```