Eyebrow Shape Analysis by Using a Modified Functional Curve Procrustes Distance Yishi Wang, Cuixian Chen, Midori Albert, Yaw Chang, Karl Ricanek University of North Carolina Wilmington {wangy, chenc, albertm, changy, ricanekk}@uncw.edu Abstract To tackle the problem of automatic recognition of hu- man eyebrow, a novel approach for shape analysis based on frontal face images is proposed in this paper. First, eye- brow curves are acquired by fitting cubic splines based on landmark points. Next, we propose to use a modified func- tional curve procrustes distance to measure the similarities among the cubic splines, and finally a multidimensional scaling method is adopted to evaluate the effectiveness of the distance. This work extends previous work in analyzing the eyebrow for both human and machine recognition by providing a framework based on shape contours. Further this work demonstrates the effectiveness of eyebrow shape for discrimination when teamed with the appropriate metric distance. 1. Introduction As the most common facial hairs among men and women, eyebrows can convey both subtle and bold expres- sions such as fear, anger, surprise, contempt, happiness, sadness, etc. However, the human eyebrow is an often over- looked facial component thought to not to have much value in for automatic recognition. In the area of identity sciences including biometrics, where the major interests lie in iris, ear, palm, fingerprint, and the face as a whole, there have been very few studies focused on the eyebrow for recogni- tion and/or soft-biometrics. There has been some interest in the periocular recognition and soft-biometrics, which may or may not include the eyebrow. It is this inconsistency of definition for the periocular region, which has led to this study. In a forensic context, faces of suspects captured in surveillance photographs may be partially covered, such as by masks or sunglasses. An interesting question is: when occlusions exist, based on an exposed part, such as eye- brows, would it be possible to use eyebrow as a reliable tool to identify a person? Figure 1. Example of importance of Eyebrow in face recognition. 1.1. Prior work As indicated in Fig 1, It is a challenge to examine the magnitudes of difference in the eyebrow between different individuals as well as among different images of the same individual. The role of eyebrows in face recognition has been studied in [10], where the work revealed that the eye- brow was far more important than eyes for human recog- nition. However, in [10] only human perception was dis- cussed, rather than using biometric modeling. Bharadwaj et al. [1] demonstrated that better automatic recognition was possible with the periocular region if the eyebrow was in- cluded. Since the focus of this work was on the texture from the periocular region, it is unclear whether shape was a factor in this work. In [4], eyebrow region was manually segmented, and features were extracted and calculated ac- cording to three categories: global shape feature, local area feature, and critical point feature. Classifications were then conducted based on the three feature categories. These fea- tures, especially the global shape feature, maybe subjective and not general enough to be finely quantified. Li et al. [8] studied an automatic human eyebrow recognition system via fast template matching and Fourier spectrum distance, and concluded that eyebrow can serve as an independent biometric for human recognition. 1.2. Contribution of work In this work, we study the shape of the human eyebrows by using a modified functional curve procrustes distance to 1
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Eyebrow Shape Analysis by Using a Modified Functional Curve Procrustes
Distance
Yishi Wang, Cuixian Chen, Midori Albert, Yaw Chang, Karl Ricanek
The organization of this paper is laid out as follows: Sec-
tion 2 presents landmarking scheme and cubic-spline curve
fitting. Section 3 introduces the properties of the new func-
tional curve shape procrustes distance to measure the dis-
tance between functional curves. Section 4 presents the
techniques of Multidimensional Scaling (MDS). The exper-
iment results are presented in Section 5, and conclusions are
drawn in final section of this paper.
2. Eyebrow shapes
In our preliminary study, we find out that it is very
difficult to determine the boundaries of human eyebrows.
Some people have very bushy eyebrows, while it is com-
mon for the seniors to have sparse ones. Therefore, we
develop a descriptor of the landmarks shown in Fig 3 and
their location for the quantitative shape analysis with the
top left eyebrows of n subjects, (xi, yi), for i = 1, · · · , n,
where xi = (xi,0, ..., xi,4)T be the x coordinates and yi =
(yi,0, ..., yi,4)T be the y coordinates of the labeled landmark
points 0-4. Due to the nature of the human eyebrows, it is
easier to consider the general eyebrow shapes as intersec-
tions of two continuous curves.
Next, we use statistical curve fitting to approximate the
boundaries of eyebrows. The general idea of curve fitting
with given landmark points is to find a continuous function
that passes through those landmark points while being rel-
atively smooth. The way to achieve such a goal is to find a
twice differentiable function such that among all functions
f(x), the following objective function is minimized:
L(f, λ) =5
∑
i=1
(yi − f(xi))2 + λ
∫
(f′′
(x))2dx.
2
According to Green and Silverman (1994) [6], under certain
conditions, the above loss function has a unique minimizer:
a natural cubic spline that passes through (xi,j , yi,j) for
each fixed i and j = 0, · · · , 4. In our preliminary study, we
test on multiple curve fitting techniques and find out that the
cubic spline is effective and the resulting curves are good
approximation of the true eyebrow curves. With a group
of curves, our next goal is to define the distances between
these curves, such that the distances can be used to measure
the similarities and differences of the shapes among the eye-
brows.
Let F = {fi : fi is the top-left eyebrow spline function
of the ith subject} be a set of continuous function both end-
ing points provided. Without loss of generality, we may as-
sume that the slopes of the straight lines connecting the two
ending points are zero and the Euclidean distance is one.
3. Distances of planar functional curves
We propose to use Modified Functional Curve Procrustes
Distance, to measure the similarities between different con-
tinuous curves. The definitions were proposed as following:
Definition 1. (Equally spaced heights with zero ending
points). Let f : [0, 1] 7→ R be a continuous function
with f(0) = f(1) = 0, and let 0 = t1 < ... <tn = 1 be n equally spaced points on [0, 1] , then F :=(f(t1), f(t2), · · · , f(tn))T is called equally spaced heights
of f on [0, 1].
Hereafter, we assume that all curves considered are zero
at ending points of their domain. Therefore, when the
equally spaced heights with zero ending points of a function
is a constant, it indicates that the constant is zero. Gener-
ally, the equally spaced heights may be positive or negative,
depending on the curve f and where the sampled points are
located.
Definition 2. (Equally spaced distance (ESD) between two
continuous functions) Let g and h be two continuous curves
on closed domains, and let U = (u1, ..., un)T and V =(v1, ..., vn)T be the equally spaced heights with zero ending
points of g and h respectively. When ||U ||2 · ||V ||2 > 0, the
equally spaced distance between curves g and h is
Dn(U, V ) :=
√
1 −< U, V >2
||U ||22 · ||V ||22, (1)
where < ·, · > is the inner product of two vectors, and || · ||2is the L2 norm.
Let f1, f2 and f3 be continuous functions on [0, 1] with
both ending points equal to zero, let X = (x1, ..., xn)T ,
Y = (y1, ..., yn)T and Z = (z1, ..., zn)T be their corre-
sponding equally spaced heights, with ||X|| · ||Y || · ||Z|| >
0, it is obvious that Dn(f1, f1) = 0, and D(f1, f2) =D(f2, f1) ≥ 0. If D(f1, f2) = 0, it indicates that∑n
i=1 |xiyi| = ||x||||y||. By Cauchy-Schwartz inequality,
we have X = rY , where r is a scalar. As to the triangle
inequality, it can be proved that
Theorem 3. With the previous notations ,
Dn(f1, f2) ≤ Dn(f1, f3) + Dn(f2, f3).
Therefore Dn is a pseudo metric.
Notice that for (1), if we replace U and V by function gand h respectively, change the inner product and L2 norm
of vectors to the ones for functions, we have
D(g, h) :=
√
1 −< g, h >2
||g||22 · ||h||22
, (2)
where
< g, h >=
∫ 1
0
g(x)f(x)dx, and ||g||22 =
∫ 1
0
g2(x)dx.
In fact, the distance D(g, h) in (2) is the limit of Dn(g, h)in (1) as n approaches to infinity. It can be verified that
D(g, h) is also a pseudo metric with the following theorem.
Theorem 4. With the conditions and notation in Theorem 3,
D(f1, f2) ≤ D(f1, f3) + D(f2, f3).
The distance D(g, h) is similar with the the distance pro-
posed in [7], in which the functions take a complex format
to accommodate both the x and y variables. In this work,
because of the way we register our function, we can only
focus on the y variable, which is exactly the reason why
all we need is the vertical heights used in the definition of
Dn in (1). This simplification should reduce the compu-
tation time since only y coordinates are involved. Similar
with the distance proposed in [7], all the above mentioned
distances do not allow constant functions or vectors. How-
ever, in many real world application, such a limitation can
be a big hurdle. There is hardly a function that is constant,
but there are many planar curves that are very similar with
constant curves. For example, in our eyebrow classification
experiment, some eyebrows demonstrate relatively flat cur-
vature. In order to overcome this difficulty, we propose to
add a constant parameter k > 0 to U , V , g and h in (1) and
(2), and therefor the revised distance of (1) and (2) are:
Dkn(U, V ) :=
√
1 −< U + k, V + k >2
||U + k||22 · ||V + k||22, (3)
which we name as modified discrete procrustes distance,
and
3
Dk(g, h) :=
√
1 −< g + k, h + k >2
||g + k||22 · ||h + k||22, (4)
which we name as modified functional curve procrustes dis-
tance. It can be proved that both (3) and (4) are metric dis-
tance and now it allows for U or V , or g or h to be con-
stant(s).
Let fi : [0, 1] 7→ R for 0 ≤ i ≤ n be n cubic splines (or
let F1, ..., Fn be the corresponding equally spaced heights),
we may calculate the modified functional curve procrustes
in (4) for each pair, and hence there aren(n−1)
2 distances
among all pairs. A n × n matrix S is often used with the
(i, j) element represents the distance between the ith and
jth functions, i.e., si,j = Dk(Fi, Fj). Obviously, S is sym-
metric with diagonal elements equal to zero.
4. Multidimensional scaling
Because of the nature of the matrix S, it is difficult to
have a visual idea on the relationships among all functions,
in terms of the similarities of shapes. The challenge is
then to reduce the dimension of the distances fromn(n−1)
2to two or three dimensions where visual understanding of
the similarities is easier. In this work, we use Multidimen-
sional Scaling (MDS) [11] to reduce the dimension. With
the aforementioned distance matrix Sn×n = (si,j), MDS
searches vectors z1, · · · , zn ∈ Rd to minimize the follow-
ing objective function:
LM (z1, · · · , zn) :=∑
i 6=j
(si,j − ||zi − zj ||2)2. (5)
The major idea of MDS is to use zi, for i ∈ 1, · · · , n as a
lower dimensional representation of the data that preserves
the pairwise distances, si,j , as much as possible. For vari-
ous objective functions for different types of MDS, [2] con-
tains a great amount of information. For (5), essentially we
are looking for the approximation that
s2i,j ≈ ||zi − zj ||
22 = ||zi||
22 + ||zj ||
22 − 2zT
i zj
Since it is the inner product of zi and zj that we are most
interested in, the above expression is equivalent with
−1
2s2
i,j ≈ zTi zj − ||zi||
22/2 − ||zj ||
22/2.
Let A = (− 12s2
i,j) and let B be the centered version of A in
the following sense
B = HAH where H = In − Jn/n,
and Jn is an n × n matrix of one.
Thus, the optimization of objective function in (5) be-
comes an eigenvalue problem. The minimizer of (5) has an
explicit solution by using the largest d eigenvalues of matrix
B and the corresponding eigenvectors. Let λ1, · · · , λd be
the largest eigenvalues of matrix B with associated eigen-
vectors e1, · · · , ed, then
(zT1 , · · · , zT
n )T = (√
λ1e1, · · · ,√
λded).
In the following experiments, we will present the effec-
tiveness of the distance in (4) by using MDS with d = 2.