Using stereo matching with general epipolar geometry for 2D face recognition across pose.

Abstract

Face recognition across pose is a problem of fundamental importance in computer vision. We propose to address this problem by using stereo matching to judge the similarity of two, 2D images of faces seen from different poses. Stereo matching allows for arbitrary, physically valid, continuous correspondences. We show that the stereo matching cost provides a very robust measure of similarity of faces that is insensitive to pose variations. To enable this, we show that, for conditions common in face recognition, the epipolar geometry of face images can be computed using either four or three feature points. We also provide a straightforward adaptation of a stereo matching algorithm to compute the similarity between faces. The proposed approach has been tested on the CMU PIE data set and demonstrates superior performance compared to existing methods in the presence of pose variation. It also shows robustness to lighting variation.

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Using Stereo Matching with General Epipolar
Geometry for 2D Face Recognition
across Pose
Carlos D. Castillo, Student Member, IEEE, and
David W. Jacobs, Member, IEEE
Abstract—Face recognition across pose is a problem of fundamental importance
in computer vision. We propose to address this problem by using stereo matching
to judge the similarity of two, 2D images of faces seen from different poses. Stereo
matching allows for arbitrary, physically valid, continuous correspondences. We
show that the stereo matching cost provides a very robust measure of similarity of
faces that is insensitive to pose variations. To enable this, we show that, for
conditions common in face recognition, the epipolar geometry of face images can
be computed using either four or three feature points. We also provide a
straightforward adaptation of a stereo matching algorithm to compute the similarity
between faces. The proposed approach has been tested on the CMU PIE data set
and demonstrates superior performance compared to existing methods in the
presence of pose variation. It also shows robustness to lighting variation.
Index Terms—Face recognition, pose, stereo matching, epipolar geometry.
Ç
1INTRODUCTION
FACE recognition is a fundamental problem in computer vision.
There has been a lot of progress in the case of images taken under
constant pose [30]. There are also several approaches to handling
pose variation [24], [15], [17], [8]. However, there is still a lot of
room for improvement. Progress would be important in many
applications, for example, surveillance, security, and the analysis
of personal photos and other domains in which we cannot control
the position of subjects relative to the camera.
Correspondence seems crucial to producing meaningful image
comparisons. The importance of good correspondences is even
greater in the case of face recognition across pose. Standard
systems often align the eyes or a few other features, using
translation, similarity transformations, or perhaps affine transfor-
mations. However, when the pose varies these can still result in
fairly significant misalignments in other parts of the face. Observe,
for example, Fig. 1.
To handle this situation, we use stereo matching. This allows
for arbitrary, one-to-one continuous transformations between
images, along with possible occlusions, while maintaining an
epipolar constraint.
In the process of computing the correspondences between scan
lines in two images, a stereo matching cost is optimized, which
reflects how well the two images match. We show that the stereo
matching cost is robust to pose variations. Consequently, we can
use the stereo matching cost as a measure of similarity between
two face images.
Note that we are not interested in performing 3D reconstruc-
tion, which is the most common purpose of stereo matching. In
reconstruction, the stereo matching costs are discarded and the
correspondences are used along with geometric information about
the camera layout to compute a 3D model of the world. We have
no use for the correspondences except to compute the stereo
matching costs. We are therefore unaffected by some of the
difficulties that make it hard to avoid artifacts in stereo reconstruc-
tion. For example, ambiguities frequently arise when different
correspondences produce similar costs; in this case, selecting the
correct correspondence is essential for reconstruction, but not very
important for judging the similarity of two images.
Prior to stereo matching, we need to estimate the epipolar
geometry. In almost all applications of face recognition, the size of
the face is small relative to its distance to the camera. Therefore, we
can approximate the projection of the face to the camera using
scaled orthographic projection (weak perspective).
We can therefore use four feature points to estimate the
epipolar geometry of the two faces. The images are then rectified
and the similarity score is computed by adding the stereo matching
cost of every row of the rectified images. We also study a specific
case in which the camera is at the same height as the eyes of an
upright subject. In this case, the epipolar lines are parallel to the
lines that connect the two eyes. In this case, we can determine
epipolar geometry using only three points. We also tried obtaining
the epipolar geometry from each pair of images using the method
of Domke and Aloimonos [11], [12]. In this case, our method
requires no hand-clicked points. We verified that there is no
decrease in recognition performance in a fully automatic system.
Putting these steps together, we have the following simple
algorithm:
.
Prior to recognition, build a gallery of 2D images of faces,
each with three to four landmark points specified.
Given a 2D probe image, find three to four corresponding
landmark points.
Compare the probe to each gallery image as follows:
.
.
-
Using landmark points, rectify the probe and
gallery image.
Run a stereo algorithm on the image pair, using the
enhancements described in Section 4. Discard the
correspondences and use the matching cost as a
measure of image similarity.
Identify the probe with the gallery image that produces the
lowest matching cost.
We will show that this method works very well even for large
viewpoint changes. We evaluate our method using the CMU PIE
data set and the Labeled Faces in the Wild (LFW) data set. Our
results show that with pose variation at constant illumination our
method is more accurate than previous methods of Gross et al.
[17], Chai et al. [8], and Romdhani et al. [24]. While our method is
designed to only handle pose variation, we also test it with pose
and illumination variation to verify that our method does not fall
apart in such a setup. Surprisingly, our method is more accurate
than the method of Gross et al. [15], which is designed to handle
lighting variation, though it is not as accurate as the method of
Romdhani et al. [24]. The experiments on the LFW data set show
reasonable performance in an unconstrained setting (where there
is simultaneous variation in pose, illumination, and expression).
This is an extended version of our conference paper [7]. The
original conference version does not include our method with four
feature points or experiments using four feature points, includes
limited experiments with lighting change, and does not include the
results on the LFW data set. Additionally, in the conference paper
we did not develop a fully automatic system. However, the
conference version of our paper includes an analysis of stereo
matching for face recognition that has been eliminated from this
version due to space constraints.
The rest of the paper is organized as follows: Section 2 discusses
related work. Section 3 discusses issues related to image alignment
-
.
IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,VOL. 32, NO. XX,XXXXXXX 20101
. The authors are with the Department of Computer Science, University of
Maryland, 4420 A.V. Williams Bldg., College Park, MD 20742.
E-mail: {carlos, djacobs}@cs.umd.edu.
Manuscript received 28 Aug. 2008; revised 1 Apr. 2009; accepted 29 Apr.
2009; published online 21 May 2009.
Recommended for acceptance by M.-H. Yang.
For information on obtaining reprints of this article, please send e-mail to:
tpami@computer.org, and reference IEEECS Log Number
TPAMI-2008-08-0575.
Digital Object Identifier no. 10.1109/TPAMI.2009.123.
0162-8828/10/$26.00 ? 2010 IEEE Published by the IEEE Computer Society

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and epipolar geometry, Section 4 presents the details of our face
recognition method, and Section 5 presents and analyzes all
experiments. Section 6 concludes.
2RELATED WORK
Zhao et al. [30] review the vast literature on face recognition.
Although the bulk of this work assumes fixed pose, there have
been a number of approaches that do address the problem of pose
variations. Table 1 presents a summary of existing methods of face
recognition across pose.
Some early approaches compensate for some 2D deformations
in matching, which may partially account for the effects of pose. A
notable example is the work of Wiskott et al. [29]. This work was
among the first to present a face recognition method that was
robust to alignment issues. They developed a method called Elastic
Bunch Graph Matching (EBGM). The comparison function used
Gabor jets at manually clicked feature points and geometric
information of distances between the feature points. Correspon-
dences were obtained for the feature points only.
One of the first methods to study face recognition across pose
was proposed by Beymer and Poggio [4]. In their work, they
generated 2D virtual views from a single image per person using
prior knowledge of the object class (in particular, symmetry and
prototypical objects of the same class) using optical flow (see also
[19] and [23]). Once the virtual view had been generated, the
images were compared. Our method is similar to theirs in the sense
that both are decidedly 2D and stress the importance of finding
good correspondences. In this approach, the correspondences are
obtained using optical flow between the two facial images.
Blanz and Vetter [5] use laser scans of 200 subjects to build a
general 3D morphable model of three-dimensional faces. Then,
with the aid of manually selected features, they fit this model to
images. The parameters of the fit to two different images can be
compared to perform recognition. In their experiments, they show
strong results for a subset of the poses in the PIE database. The
work of Romdhani et al. [24] also focuses on 3D morphable
models. In this work, shape and texture parameters of a 3D
morphable model are recovered from a single image.
Basri and Jacobs [3] use a 3D model to generate a low-
dimensional subspace containing all the images that an object can
produce under lighting variation. Pose is determined using
manually selected point features. Correspondences are obtained
by computing a 3D rigid transformation. Georghiades et al. [13]
computed a 3D model for each person using a gallery containing a
number of images per subject taken with controlled illumination at
a constant pose. Pose variation is handled by sampling the set of
possible poses, and building a 2D model for each one.
Gross et al. [15] presented two appearance-based algorithms for
face recognition across pose and illumination. One of them is called
eigenlightfields.Atthecoreofthemethodistheplenopticfunctionor
light field. To use this concept, all of the pixels of the various images
are used to estimate the (eigen) light field of the object. The other
method presented by Gross et al. [15] is called Bayesian Face
Subregions (BFS). The algorithm models the appearance changes of
the different face regions in a probabilistic framework.
There have been several recent approaches to face recognition
across pose that are based on patches. Chai et al. [8] presented a
learning, patch-based rectification method based on locally linear
regression. Lucey and Chen [22] present a patch-based algorithm
for face recognition across pose of sparsely registered images (four
manually selected points). Closely related, the work of Ashraf et al.
[2] presents a new method to discover viewpoint-induced spatial
deformations for general patch-based methods of face recognition
across pose.
All of the methods previously mentioned in this section use
intensity images of the face. This type of face recognition, based on
2D images constitutes the vast majority of face recognition
research. There is, however, a significant amount of work done
acquiring, matching, and performing recognition using 3D
reconstructions of faces (see [6] for a survey).
While progress has been made in handling pose variations,
significant challenges remain. For this problem, current methods
have substantially worse performance than when pose is fixed
between the probe and gallery. In addition, many methods for
handling pose variation require substantially more computation
than other methods and can be very slow. This is in part because
the process of finding a correspondence between the probe and
gallery requires expensive optimization processes.
3ALIGNMENT
In order to perform stereo matching, we first need to know the
epipolar geometry. In the most general case, this requires eight
corresponding points. We can reduce this by assuming that images
are generated by scaled orthographic projection, in which case the
epipoles are at infinity and the epipolar lines are parallel in both
images. This model is valid when the average variation of the
depth of the object along the line of sight is small compared to the
distance of the camera to the object and the field of view is small as
is generally the case with facial images. Even with scaled
orthographic projection, there can be considerable variation in
disparity between two images. See [7] for an analysis of this. For an
excellent overview of epipolar geometry and scaled orthographic
projection, see [18].
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TABLE 1
Existing Methods for Face Recognition across Pose
Fig. 1. Example images from the CMU PIE data set. Observe that no linear
transformation can make corresponding boxes have equal size because a linear
transformation can only linearly scale their size.

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As we will demonstrate, we can calculate the epipolar geometry
under the scaled orthographic model using four feature points. We
will not focus our attention on how these points can be obtained; in
most of our experiments, we specify them by hand. Some
applications involving offline recognition may use such hand-
clicked points directly. At the same time, there is a lot of work on
automatic detection of facial features [28], [14]. By reducing the
number of points needed for recognition, we can make it easier to
use these detectors to build fully automatic recognition systems.
3.1Epipolar Geometry under Scaled Orthographic
Projection
We now want to consider arbitrary viewpoint changes, still using
scaled orthographic projection. Under scaled orthographic projec-
tion,theepipolargeometrycanbecharacterizedasatuple:ð?;?;s;tÞ.
? is the angle of the epipolar lines in the first image. ? is the angle of
the epipolar lines on the second image. s is the relative scale, that is,
scaling the second image by s will cause the distance between two
epipolar lines in the second image to match the distance between
corresponding lines in the first image. Finally, t is the translation
perpendicular to the epipolar lines needed to align corresponding
lines. With four corresponding points, we get a nonlinear system of
equations which we solve in a straightforward way.
3.2Epipolar Geometry and Horizontal Movement
We will now consider a special case of the general setup: an
upright person with both images taken with the camera located at
the same height as the person’s head (in fact, our reasoning applies
to any situation in which the eyes and both camera focal points are
coplanar). In that case, we know that the epipolar lines are parallel
to the lines connecting the eyes. For this case, we can determine the
epipolar geometry using three feature points. The two eyes will
define the direction of the epipolar lines. This tells us ? (an
unknown ? still allows for in plane rotation of the images). Given a
correspondence between three points we can solve for the epipolar
geometry linearly. Moreover, our experiments show that, in many
practical situations, even when the cameras are not perfectly at eye
level, these alignments work reasonably well.
4STEREO MATCHING AND FACE RECOGNITION
There exist a wide variety of stereo algorithms. We require an
efficient stereo algorithm appropriate for wide baseline matching
of faces. Since faces are very slanted objects, we require the
algorithm to have excellent support for surfaces that are not
frontoparallel planes. A number of methods might be suitable.
We have decided to use a 1D dynamic-programming-based
algorithm, which is quite fast. We have used the method given by
Criminisi et al. [10]1which has been developed for video
conferencing applications and so seems to fit our needs. It is
not obvious that it will work for the large changes in viewpoint
that can occur in face recognition, but we will show that it does.
It is important to stress that, provided we get good correspon-
dences, we are relatively unaffected by some of the difficulties that
make it hard to avoid artifacts in stereo reconstruction. For
example, when many matches have similar costs, matching is
ambiguous. One weakness of dynamic programming stereo
algorithms is that, when matching is ambiguous, it can be difficult
to produce correspondences that are consistent across scan lines.
Selecting the right match is difficult, but important for good
reconstructions. Since we only use the cost of a matching, selecting
the right matching is unimportant to us in this case. Also, errors in
small regions, such as at occluding boundaries, can produce bad
artifacts in reconstructions, but that is not a problem for our
method as long as they don’t affect the cost too much.
4.1 Stereo Matching
The core of the stereo method calculates a matching between two
scan lines (rows of each face). The algorithm accounts for exactly
one pixel in one image with each step taken. Each step involves a
transition from one point to another in four planes (or cost
matrices) called CLo, CLm, CRo, and CRm. Each point in a matrix
represents the last point in each image that has been accounted for,
along with the nature of the last step used to account for a point.
Points are accounted for by matching (m) and occlusions (o) in the
left (L) and right (R) images. The planes naturally define the
persistence of states. By setting the state transition costs ade-
quately, many state transitions can be favored or biased against.
For example, long runs of occlusions can be favored over many
short runs by setting a high cost for entering or leaving an
occluded state. This formulation handles slanted surfaces well
(because it allows many-to-one matches) and offers better control
over the occlusion costs than traditional one plane models [9]. See
[10] for a complete presentation of the matching algorithm. In the
rest of the section, we point out some of the details we use in our
image comparison algorithm.
The cost of matching the two scan lines l1 and l2, denoted
costðl1;l2Þ, is: CRo½l ? 1;r ? 1?. The optimal matching solution will
be a sequence of symbols in the alphabet: ? ¼ fCLo;CLm;CRo;CRmg
which can be obtained by following a backward step. A solution (a
word in ??) that encodes the optimal matching to a given matching
problem between scan lines I1;i and I2;i has length equal to
jI1;ij þ jI2;ij. We have no use for the optimal matching itself, we
only use its cost and its length to normalize it.
One of the key ingredients to the flexibility of this method is the
ability to match multiple pixels in one scan line to one pixel in the
other. This is done by concatenating several consecutive CLm(or
CRm) in the word that encodes the solution.
4.2
When we match a probe image to different gallery images, we
obtain different rectifications. While the original thumbnails are
axial rectangles, the rectified thumbnails will be arbitrarily rotated
rectangles that will contain varying numbers of rows with valid
pixels, and different numbers of valid pixels in each row. It is
therefore important to avoid any bias in our image comparisons
which favor some thumbnail orientations over others. In this
section, we explain how to adapt the method presented by
Criminisi et al. [10] to match rectified images in which the length
of scan lines varies.
All solutions found by the method of Criminisi et al. have length
equal to the sum of both scan lines being matched. However, since
each cost is going to be compared to other costs matched over scan
lines of potentially different lengths, we need some normalization
strategy. The cost used weighs every match equally:
Pn
The cost expressed in (1) is a sensible measure of similarity since it
is not dependent on the relative scale of the images, it just
calculates the average cost per match made over all scan lines.
Since we do not know which image is left and which image is
right we have to try both options. One of them will be the true cost,
the other cost will be noise and should be ignored.
8
>
Rectification and Matching Costs
costðI1;I2Þ ¼
i¼1costðI1;i;I2;iÞ
Pn
i¼1jI1;ij þ jI2;ij:
ð1Þ
similarityðI1;I2Þ ¼ min
costðrectifyðI1;I2ÞÞ;
costðrectifyðI2;I1ÞÞ;
costðrectifyðflipðI1Þ;I2ÞÞ;
costðrectifyðI2;flipðI1ÞÞÞ:
>
>
>
:
<
ð2Þ
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1. We also tried the method described by Cox et al. [9] and found it to be
about twice as fast but less accurate (the accuracy was about 8 percent lower
on average on several gallery-probe experiments with a gallery of
68 individuals).

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Additionally, flip produces a left-right reflection of the image and
adjusts the hand-clicked positions of the four points accordingly.
flip is helpful when two views see mainly different sides of the
face. In this case, a truly correct correspondence would mark most
of the face as occluded. However, since faces are approximately
vertically symmetric, flip approximates a rotation about the y axis
that creates a virtual view so that the same side of the face is visible
in both images. For example, if we viewed a face in left and right
profile, there would be no points on the face visible in both images,
but flipping one image would still allow us to produce a good
match. rectify performs the rectification described in the four-point
case or, in the three-point case, does nothing at all since all images
are already partially rectified to handle this case.
Finally, we perform recognition simply by matching a probe
image to the most similar image in the gallery. For the method
to work well, all of the images in the gallery should be in the
same pose. Our C implementation of the method can compare
two 60 ? 72 facial images in 1 second. The number of operations
done to compare two w ? h images is Oðhw2Þ.
Before closing this section it is important to note how simple
the proposed approach is. It is a two-step process: 1) alignment
according to assumptions regarding the viewing conditions and
2) similarity computation using stereo matching. In the next
section, we will see that this very straightforward approach
demonstrates excellent performance.
5EXPERIMENTS
We have tested our algorithm using the CMU PIE database [25].
This database consists of 13 poses of which 9 have approximately
the same camera altitude (poses: c34, c14, c11, c29, c27, c05, c37,
c25, and c22). Three other poses have a significantly higher camera
altitude (poses: c31, c09, and c02) and one last pose has a
significantly lower camera altitude (pose c07). We say that two
poses have aligned epipolar lines if they are both from the set:
{c34, c14, c11, c29, c27, c05, c37, c25, c22}. If not, we say that two
poses have misaligned epipolar lines.
The thumbnails used were generated as described in Section 3.2.
All images have a height of 72, a pose-dependent width, and a
distance between the eyes and the mouth of d ¼ 50, and the eyes
are horizontally located in ye¼ 13. For the three-point Stereo
Matching Distance (3ptSMD) this is all the image processing
performed, the stereo matching cost was then computed and
normalized and this cost is the image similarity between the two
faces. For the four-point Stereo Matching Distance (4ptSMD) the
epipolar rectification was then performed on the thumbnail. After
rectification, the stereo matching cost was computed and this cost is
the image similarity between the two faces.
A number of prior experiments have been done with pose
variation using the CMU PIE database, but somewhat different
experimental conditions. We have run our own algorithm under a
variety of conditions so that we may compare to these. For example,
to compare results with those of Gross et al. [15], [17] and Chai et al.
[8], we need to use a subset of 34 people because they use 34 people
for training and the remaining 34 for testing. We do not require
training, but we are interested in comparing the methods in equal
conditions so we tested on individuals 35-68 from the PIE database.
To compare with the method of Romdhani et al. [24], we used
68 people as a test set. Then, to illustrate that our method works in
more realistic situations, we evaluated simultaneous variation in
pose and illumination. This too is done in two separate experi-
ments, one to compare with the method of Gross et al. [15], [17] and
one to compare with the method of Romdhani et al. [24].
5.1PIE Pose Variation: 34 Faces
We conducted an experiment to compare our method with four
others. We compared with two variants of eigen light fields [15],
eigenfaces [26] and FaceIt as described in [15], [17]. FaceIt2is a
commercial face recognition system from Identix which finished
top overall in the Face Recognition Vendor Test 2000. Eigenfaces
is a common benchmark algorithm for face recognition. Finally,
eigen light fields is a state-of-the-art method for face recognition
across pose.
In this experiment, we selected each gallery pose as one of the
13 PIE poses and the probe pose as one of the remaining 12 poses,
for a total of 156 gallery-probe pairs. We evaluated the accuracy of
our method in this setting and compared to the results in [15], [17].
Table 4 summarizes the average recognition rates. Table 2 presents
detailed results for this experiment using 3ptSMD, and Table 3
presents detailed results for this experiment using 4ptSMD. Fig. 2
shows several cross sections of the results with different fixed
gallery poses.
The fact that 3ptSMD performs solidly both when the epipolar
lines fit (with an average of 81.4 percent) and when they don’t
(with an average of 75.4 percent) and overall (with an average of
78.5 percent, as reported in Table 6) shows that assuming
horizontal epipolar geometry is not a bad approximation for real
applications of face recognition across pose, even when this
assumption does not hold perfectly.
Fig. 2 shows a comparison with the results presented in the
paper of Gross et al. [15], [17]. In this experiment, we observe that
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2. Version 2.5.0.17 of the FaceIt recognition engine was used.
TABLE 2
Results for Pose Variation with 3ptSMD
The cell format is accuracy with 68 faces/accuracy with 34 faces. The diagonals are not included in any average. The global averages are 74.5 percent for 68 faces and
79.8 percent for 34 faces.

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in all gallery poses our method outperforms all the other methods
for the extreme probe poses (c34, c31, c14, c02, c25, and c22).
Observe that the 4ptSMD method is considerably better than
3ptSMD at the poses where there is considerable misalignment (the
poses marked with ?).
Table 5 shows a comparison with the method of Chai et al. [8],
using the experimental conditions described in their paper. The
gallery pose is c27 and contains 34 faces; the probe poses are c05,
c29, c37, c11, c07, and c09. Note that this is a slice of data from
Table 2. Our 3ptSMD method produces nearly perfect results in
these conditions, results that are much better than those reported
by Chai et al.
5.2PIE Pose Variation: 68 Faces
WealsocomparedourresultswiththeonespresentedbyRomdhani
et al. [24]. These results are, to our knowledge, the best reported on
the whole PIE database for pose variation. In this work all 68images
were used, so, for this part, we report our results using all 68 faces.
Table 4 summarizes the results of this experiment.
The global average for the method of Romdhani et al. [24] is
74.3 percent, the global average for our 3ptSMD method is
almost the same, at 74.5 percent. For the subset of poses in which
the epipolar lines fit perfectly, our average performance is
80.8 percent, while theirs is 71.6 percent. We consider the case
where all epipolar lines fit to be the best possible scenario for the
3ptSMD. When the epipolar lines are misaligned, the average for
3ptSMD is 69.2 percent. Our 4ptSMD achieves overall accuracy of
82.4 percent, which is considerably higher than the performance
reported by Romdhani et al. Our method runs about 40 times
faster than the method presented in [24], requires fewer
manually specified points, and is much simpler. Detailed results
are presented in Tables 2 and 3.
We also tried the fully automatic method (using probabilistic
egomotion [11], [12] to compute the epipolar geometry) on the PIE
data set on a small subset of the pose combinations obtaining the
same results as with four hand-clicked points. Because the
egomotion computation is a bit slow and must be computed for
each pair of images, it is not practical to use it on the entire PIE
data set.
5.3
We also evaluated the performance of the method across pose and
illumination. Although our method is not designed to handle
PIE Pose and Illumination Variation
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TABLE 4
A Comparison of Our Stereo Matching Distance
with Other Methods across Pose
Fig. 2. Cross sections with fixed gallery pose for the results presented in Table 4. Probe poses marked with?have a vertical misalignment of about 10 degrees with the
corresponding gallery pose. (a) Gallery Pose c27. (b) Gallery Pose c22. (c) Gallery Pose c31.
TABLE 3
Results for Pose Variation with 4ptSMD
The cell format is accuracy with 68 faces/accuracy with 34 faces. The diagonals are not included in any average. The global average is 82.4 percent for 68 faces and 86.8
for 34 faces.