Human Age Estimation With Regression on Discriminative Aging Manifold

Abstract—Recently, extensive studies on human faces in the
Human–Computer Interaction (HCI) field reveal significant
potentials for designing automatic age estimation systems via
face image analysis. The success of such research may bring in many
innovative HCI tools used for the applications of human-centered
multimedia communication. Due to the temporal property of age
progression, face images with aging features may display some
sequential patterns with low-dimensional distributions. In this
paper, we demonstrate that such aging patterns can be effectively
extracted from a discriminant subspace learning algorithm and
visualized as distinct manifold structures. Through the manifold
method of analysis on face images,the dimensionality redundancy
of the original image space can be significantly reduced with sub -
space learning. A multiple linear regression procedure, especially
with a quadratic model function, can be facilitated by the low
dimensionality to represent the manifold space embodying the
discriminative property. Such a processing has been evaluated
by extensive simulations and compared with the state -of-the-art
methods. Experimental results on a large size aging database
demonstrate the effectiveness and robustness of our proposed
framework.
Index Terms —Age estimation, conformal embedding analysis,
manifold, multiple linear regression, subspace learning.

I. INTRODUCTION

Fig. 1. Framework of age estimation via face image analysis.

models. Moreover, these methods require the size of training
data to be large enough so that it is statistically sufficient to learn
the faithful manifolds.

 In this paper, we propose a novel framework of age estimation via multiple
linear regression on the discriminative aging
manifold of face images. The basic idea is to first use the con-
formal embedding analysis to find a sufficient low-dimensional
aging manifold subspace that embodies the discriminative property.
Then we model the manifold representation with a multiple
linear regression procedure based on a quadratic function. For
a new testing image, we fit the extracted low-dimensional feature with
the learned regression model to estimate the exact age
or an age interval. Extensive experiments on our large size internal
data base demonstrate the effectiveness and robustness of
our proposed framework compared with several state-of-the-art
methods.
  In the rest of the paper, we first introduce the details of our
proposed age estimation framework in Section II. The experimental
result sand discussions are then presented in Section III.
We conclude the paper and provide some future directions in the
last section.

II. AGING MANIFOLD LEARNING AND REGRESSION

A. Framework of Human Age Estimation

  in the same way. The age label is predicted through fitting the
regression model on the learned manifold representation.

B. Discriminative Aging Manifold Learning

  Suppose the image space is represented by a set of aligned face images
of subjects in the order of subjects' ages. A ground
 truth set associated with the images provides the age
 label. Our goal is to learn a low-dimensional discriminative manifold embedded
in as well as  a low-dimensional representation y=
which is a one-to-one mapping to X.More specifically,the projection from the
 image space to  the manifold space can be modelled as (1):

where P(•) denotes the projection function, which can be either linear or
nonlinear.  Considering the label information in this modeling, we adopt
a supervised learning approach to find P(•)  Due to the limited generalization
capability of nonlinear methods in the test cases of real-world applications
[19], we adopt a linear strategy in this paper to learn the manifold embedding.

  Since the final age estimation comes down to classification,
we want to learn a manifold embedding that is sufficiently discriminative [3],
[24], [29]. Hence, based on the graph embedding theory[23],we define two -node graphs,
and their corresponding affinity matrices, The i-th node of the graph
 represents the  data point x_i For the graph we only consider each pair of
data x_i and x_j from the same  class with For the graphwe only consider
 each pair of data x_i and x_j from the different class with An edge is constructed
 between nodes i and j if x_j is among the k_s or k_d nearest neighbors of x_i  and vice
versa, where parameters k_s and k_d are chosen empirically. If node i and j are connected,
 the weight of the edge between x_i and x_j is set by where t
is a free parameter to be tuned empirically.Otherwise,w_ij=0 if node i and j are not
connected.Following our previous work of Conformal Embedding Analysis(CEA)[3]
on face images,the Dist(x_i,x_j) is defined by a linear function of the centered
Pearson correlation coefficient[30] between x_i and x_j that is Dist(x_i,x_j)=1-ρ_i,j. we have


Fig. 2. Geometric interpretation of CEA. (a) CEA perspective with the data
normalization. (b)  Correlation measure of the objective function.

  In a geometric sense, the correlation metric projects the
original data points onto a high-dimensional unit hypersphere
with data centering and normalization. Fig. 2(a) illustrates
the CEA perspective on a toy data set. Note that the distance
metric is changed from the Euclidean to cosine-angle. Such
a measurement provides the image feature extraction with a
tolerance against gray-level variations, which has been amply
demonstrated by [27], [29] and [3].

  Directly measuring the correlation between two data vectors
leads to iteratively solving a nonconvex objective function[30].
For simplicity, we want to find an approximate measure which
can lead to a closed-form solution. Consider the simplest case
of a pair of data points x₁ and  x₂ in Fig.2(b). Intuitively, meauring θ
is highly related to simultaneously measuring
and because the length values of the two adjacent sides
and and the included angle,,can exactly determine the
triangle constructed by x₁,x₂,and θ.Therefore, following our previous
work in [3], CEA objective function is defined as follows:

where δ is a constant number, such as 1. The weight matrices
are symmetric and nonnegative
They are defined over all data points to model the local
correlational affinity structure of the manifold.The basic criteria
for setting the weights is to penalize the distance measure between
each two data points via the pairwise weight to infer the
embedding.The above objective function achieves the discriminative
learning by preserving the pairwise correlation metric
between the data points in the same class,while minimizing the
pair wise correlation metric between the data points from the different
classes at the same time.
  Define the projection matrix The basic idea for
learning the manifold space is to find the matrix P satisfying
Following the matrix formulation
in [3], (3) can be rewritten as

whereEquation
(4) can be solved in a closed form through the following reformulation,

   The column vectors of the projection matrix indicate the eigenvectors of
corresponding to the large steigen values,which
can be solved by SVD.

  The above CEA manifold learning algorithm has the following properties:
1) this is a supervised subspace learning
method, which is provided with boosted discriminative power
by incorporating the labeling in formation of both neighborhood
and class to each sub-manifold. Such a strategy conforms to
the Fisher Criterion [7] so as to be more reliable for the multiclass
classification problems; 2) since the high-dimensional
data are modeled and embedded with a local affinity graph
in the correlational mapping manner, the learned manifold
embodies a different geometric intuition from related methods,
such as LDA [28], LDE [24], or MFA [23]; 3) the CEA also
differentiates itself from CDA [30] with a different objective
function and a different solution;4)the learned image subspace
is gray-level variation tolerant [3] since the correlation metric
enhances the robustness of the feature extraction [30]. 5) the
CEA performance is not sensitive to the two parameters
and , when they are relatively large. The parameter may
distinctly influence the CEA performance.But,acommon constant,
 without too much tuning, is good enough for reasonable
CEA classification.
 

C. Regression on Discriminative Aging Manifold

  After finding the low-dimensional representation of the original
image space via manifold embedding,we define the agees-
timation as a multiple linear regression problem[6] in the manifold space.
Equation (6) specifically describes this regression
model,

where denotes the estimated age label,ƒ(•) the unknown regression function,
and ƒ(•) the estimated regression function.In
a matrix formulation, we have


Fig.3. The2-Dand3-D aging manifold visualization using PCA,NPP,LPP,OLPP,and CEA algorithms
respectively.The color of the points representsage with blue being the youngest and red the
oldest.

where L is the age label vector. is a known matrix including
a column of 1 for the intercept and all the observed values for
the rest. The vector B is the unknown parameter vector which
we need to estimate during the learning stage. The error vector
consists of unobservable random variables, assumed to have
zero mean and be uncorrelated, with common variance σ².To
fit the model,B is estimated by Ordinary Least Squares (OLS)
or robust regression, and the fitted
value of is then given by

The vector of residuals is and
Suggested by the previous work in [11],
[12],we adopt a quadratic function for the regression.Taking the
corresponding low-dimensional representation of each image in
the manifold space as the aging feature,(9) formulates the fitted
age estimation model. For each of y_i of x_i

where denotes the estimated age for the image x_i,the estimated
intercept term, the estimated parameter vectors, and the y_i²
arraywise square of  y _i.Hence, we have

Compared with the quadratic function for regression in such
a scenario, the linear function may lead to an under-fitting of
the result; and the cubic or higher order function may over-fit
the data. The evidences to support this conclusion are provided
by the experiments in the next section.

III. EXPERIMENTS

  As we mentioned in the beginning, data collection is crucial
for the precise age estimation. However, it is extremely hard in
practice to collect a large size aging database, especially when
one want to collect the chronometric image series from an indi-
vidual. In addition to the novel age estimation framework, this
paper also introduces a new real-world aging data base collected
in our project, which satisfies the particular experimental pro-
tocol for estimating precise human ages through face images.

A. Data Collection

  The age estimation performance is tested on our UIUC-IFP
internal aging database.1 The database contains 8000 high-res-
olution RGB color face images in total. Specifically, there are
1600 different voluntary Asian subjects, 800 female and 800
male, with ages ranging from 0 to 93 years (we have newborn
subjects).Each subject has about five near frontal images and a
ground truth label of his or her approximate age as an integer.
Since the photo graphs are taken in an outdoor environment with
spontaneous subject attentions, the data have significant vari-
ances on illumination,facial expression,and makeup.In the ex-
periments,we first run a face detector introduced in [25] to find
the face area in each image; and then label the eye corner loca-
tions of each subject. The face images are finally cropped, re-
sized,and transformed to 60×60 gray-level patches according
to the detection and labeling results. After reshaping the image
matrices to vectors, each datum has a dimension of 3600.

B. Aging Manifold Visualization

 To observe the aging manifolds,we choose the 4000 face images
of female subjects to visualize the embedded low-dimensional
manifold.Fig.3 displays the2-D and 3-D manifold visualizations
of the PCA[26],NPP[22],LPP[19],OLPP[21],and
CEA[3] algorithms respectively.Each data point represents one
faceimage.The data point sofagefrom0to93arecoloredfrom
blue to red. We choose four nearest neighbors for all the graph
embedded learning algorithms. We can see that, since PCA is
un-supervised, it produces a mapping with no clear manifold


Fig.4. Age estimation results on the UIUC-IFP internal aging  database using PCA+regression,NPP+regression,LPP+regression,OLPP+regression,andCEA+re-
gression algorithms.(a)–(d)are for the female subjects,and(e)–(h)forthemalesubjects.(a),
(e)MA Eversus dimensionality.(b),(f)Variance versus dimensionality.(c), (g) Accuracy score
versus error level. (d), (h) Accuracy score versus error level and dimensionality.
 Red implies better performance.

trend or structure. NPP does not yield obvious improvement
over PCA. Compared with PCA and NPP, both LPP and OLPP
achieve much better visualization results with distinct manifold
trend and structure in a chronological series.LPP forms the ap-
proximate manifold into a curve, while OLPP forms a straight
line. The manifold formed by CEA is in a dispersal type and
following a curved chronological trend. We can see that LPP,
OLPP,and CEA all reveal clear clusters and discriminative pat-
terns in the visualization.

C. Age Estimation

D. Discussion

  From the above evaluations, we can see that PCA and NPP
perform the worst in both manifold visualization and age es-
timation for all the dimensionality reduction cases. LPP and

Fig. 5. Age estimation comparisons on the linear, quadratic, and cubic regression functions
for female face images using CEA+regression. (a) MAE versus dimensionality. (b) Variance
versus dimensionality. (c) Accuracy score versus error level. (d) Accuracy score versus error
level and subspace dimensionality using CEA+quadratic regression.

OLPP achieve superior performance in both manifold visual-
ization and age estimation in very low-dimensional subspaces,
such as twoor three dimensions.CEA shows its superiority for
most of the dimensionality reduction cases,especially when the
subspace dimensionality is a little higher.The seresults indicate
that the human aging manifold is not sufficiently represented by
such a low number of dimensions. Obviously, there are person,
gender, illumination, and occlusion variations in the face im-
ages. The visualization of CEA manifold shows not only the
main trend of the aging manifold like OLPP ,but also some local
sub-trends that describe the other incidental variations. This is
one possible reason to explain why CEA+regression outper-
forms the other methods in the comparison forage estimation.

 To demonstrate the effectiveness of the regression using a
quadratic function,we compare its age estimation performance
with the regression using the linear or cubic function for the fe-
male face images. Fig. 5(a)–(c) show the experimental results
yielded by CEA+regression.(a) and (b) show the MAE and the
variance of the absolute error against the reduced dimensions for
the three regression cases. (c) shows the accuracy score of the
three regressions against the estimation error levels from one
to 15 years. It is obvious that CEA+quadratic regression con-
sistently outperforms CEA+linear regression and CEA+cubic
regression with significant improvements.These overall results
are consistent with the reports from [11], [12]. We can see that
the cubic regression has an over-fitting while the linear regres-
sion has an under-fitting compared with the quadratic regression
in such a specific scenario. (d) shows the accuracy score of the
three regressions against the estimation error level and subspace
dimensionality (50, 100, 300, 700, 800) using CEA+quadratic
regression.We observe that a subspace dimensionality of about
300 yields the best performance. This observation can provide
an empirical guidance for choosing the size of the manifold
space.

IV. CONCLUSIONS

REFERENCES

Yun Fu (S’07) received the B.E. degree in informa- tion engineering from the School of Electronic and Information Engineering, Xi’an Jiaotong University, China, in 2001, the M.S. degree in pattern recogni- tion and intelligence systems from the Institute of Artificial Intelligence and Robotics, Xi’an Jiaotong University, in 2004, and the M.S. degree in statis- tics from the Department of Statistics, University of Illinois at Urbana-Champaign, in 2007. He is currently pursuing the Ph.D. degree in the Electrical and Computer Engineering Department, University of Illinois at Urbana-Champaign.

His research interests include human computer interaction,machine learning,
image processing, multimedia and computer vision.
Mr. Fu is the recipient of a 2002 Rockwell Automation Master of Science
Award, two Edison Cups of 2002 GE Fund “Edison Cup” Technology Inno-
vation Competition, 2003 HP Silver Medal and Science Scholarship for Excel-
lent Chinese Student,2007 Beckman Graduate Fellowship,2007 DoCoMo USA
Labs Innovative Paper Award (IEEE ICIP’07 best paper award), 2007 Chinese
Government Award for Outstanding Self-financed Students Abroad, and 2008
M.E.VanValkenburg Graduate Research Award.He is a student member of In-
stitute of Mathematical Statistics (IMS), and a 2007–2008 Beckman Graduate
Fellow.

Thomas S.Huang(S’61-M’63-SM’76-F’79-LF’01) received the B.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., and the M.S. and Sc.D. degrees in electrical engineering from the Massachusetts Institute of Technology (MIT), Cambridge. He was at the faculty of the Department of Elec- trical Engineering at MIT from 1963 to 1973, and at the faculty of the School of Electrical Engineering and Director of its Laboratory for Information and Signal Processing at Purdue University, West