Online Improved Eigen Tracking
Subarna Tripathi Santanu Chaudhury Sumantra Dutta Roy
subarna.tripathi@gmail.com schaudhury@gmail.com sumantra@cse.iitd.ac.in
Electrical Engineering Department, IIT Delhi
Abstract
We present a novel predictive statistical framework
to improve the performance of an Eigen Tracker which
uses fast and efficient eigen space updates to learn
new views of the object being tracked on the fly using
candid co-variance free incremental PCA. The
proposed system detects and tracks an object in the
scene by learning the appearance model of the object
online motivated by non-traditional uniform norm. It
speeds up the tracker many fold by avoiding non-
linear optimization generally used in the literature.
1. Introduction
There are numerous tracking algorithms proposed
in the literature like mean-shift or camshift algorithms,
appearance based tracker etc. An appearance-based
tracker (EigenTracker [1]) can track moving objects
undergoing appearance changes powered by
dimensionality reduction techniques. The Isard and
Blake CONDENSATION algorithm [2] can represent
simultaneous multiple hypothesis. There can be several
ways by virtue of which the power of EigenTracker
and particle filter can be combined like [7] and [8]. But
these have the overhead of non-linear optimization. [6]
proposes a fast appearance tracker which eliminates
non-linear optimizations completely but it lacks the
benefit of predictive framework. We enhance the
capabilities of the EigenTracker by augmenting it with
a CONDENSATION-based predictive framework to
increase its efficiency and also make it fast by avoiding
non-linear optimization like [6]. The main features of
our approach are the tracker initialization, presence of
prediction framework, effective subspace update
algorithm [4] and avoidance of non-linear
optimizations.
2. On-Line Prediction in the Tracker
2.1. The Prediction Mechanism
The tracking area is described by a rectangular
window parameterized by [x
t
,y
t,
, w
t
, h
t
, θ
t
], and
modeled by the 7 dimensional state vector X
t
= [x
t
, x'
t
, y
t
, y'
t
, w
t
,h
t
, θ
t
], where (x
t
, y
t
) represents the position
of the tracking window, (w
t
, h
t
) represents the width
and height of the tracking window, (x’
t
,y’
t
) represents
the horizontal and vertical component of the velocity
and θ
t
represents the 2D rotation angle of the tracking
window. These 5 motion parameters can track the
object with its bounding box being an oriented
rectangle. This seed point is needed for sampling
windows around it. The predictive framework helps
generating better seed values for diverse object
dynamics. We use a simple first-order AR process to
represent the state dynamics (t represents time):
X
t
= A
t
X
t-1
+ w
t
, where w
t
is a zero-mean, white,
Gaussian random vector. The measurement is the set of
five motion parameters obtained from the image, Z
t
.
The observation model has Gaussian peaks around
each observation, and constant density otherwise.
We estimate the values of the five motion
parameters based on their predicted values and the
measurements done. These estimated values serve as
seeds to the next frame. For every frame, we get
sampled version of conditional state density (S
t
), and
corresponding weights (
∏
t
) for conditional probability
propagation or CONDENSATION. The state estimate
is used to generate the predictions for the next frame.
The prediction framework we used is motivated by
predictive Eigen tracker [7].
2.2. Initialization of the tracker
Accurate tracker initialization is a difficult problem.
Our coding solution currently can detect the most
moving object automatically by analyzing the first
three frames, i.e. with the overhead of additional two
frames buffering at the beginning of the tracking
process which is quite acceptable. We have used a
moving object segmentation method based on the
improved PCA which is a simplified version of the
methodology used in [3] for moving object detection
and segmentation. For this technique to work the
background should be still or changing slowly such as
grassplot or cloud for the analyzing frames. The
principle component analysis is improved to adapt to
the motion detection. The definition of traditional
covariance matrix is modified to:
C = (X1 – X2)
T
(X1 – X2) + (X2 – X3)
T
(X2 – X3)+
(X1 – X3)
T
(X1 – X3) (1)
Where, Xi is a one dimensional vector obtained by
vectorizing the original image sequence. Secondly, the
calculation result is improved in the following way.
Say, E1 and E2 as the first two eigenvectors
calculated. The element wise product of these two
eigenvectors is:
E = E1 × E2. E effectively eliminates the blur of the
eigen images of the moving object. And after
formation of E, a simple thresholding usually gives a
good initialization of the object’s rectangular bounding
box.
2.3. On-the-fly Eigen space Updates
In most tracking problems, the object of interest
undergoes changes in appearance over time. It is not
feasible to learn all possible poses and shapes even for
a particular domain of application, off-line. Therefore,
one needs to learn and update the relevant Eigen
spaces on the fly. Since a naive O(mN
3
) algorithm (for
N images having m pixels each) is time-consuming, we
use an efficient-estimation motivated by optimal
incremental principal component analysis of O(mNk)
algorithm (for k most significant singular values)
proposed by Juyang Weng et al. [4].
At each time frame F
i+1
, the IPCA method
iteratively computes the new principal components
vj(i+1) (for j = 1, 2, ...d), as follows:
1. u1(i + 1) = Oi+1.
2. For j = 1, 2, ...,min(d, i + 1) do,
(a) If j = i + 1,
initialize the jth eigenvector as vj(i + 1) = uj(i + 1);
(b) Otherwise,
|| vj(i)||
) vj(i
1) (iuj' 1) uj(i
1i
l1
vj(i)
t
1-i
1) vj(i ++
+
+
+=+
(2)
||1) vj(i||
) 1vj(i
||1) vj(i||
) 1vj(i
)1('1)uj(i 1) (i1
+
+
+
+
+−+=++ ijuuj
(3)
where l is the amnesic parameter giving larger weights
to newer samples, and ||v|| is the eigenvalue of v.
Intuitively, eigenvectors v
j
(i) are pulled towards the
data u
j
(i+1), for the current eigenvector estimate v
j
(i +
1) in eq (3). Since the eigenvectors have to be
orthogonal, therefore eq (4) shifts the data u
j+1
(i+1)
normal to the estimated eigenvector vj(i+1). This data
u
j+1
(i + 1) is used for the estimating the (j +1) th
eigenvector v
j+1
(i + 1). The IPCA method converges to
the true eigenvectors in fewer computations than PCA
(proof in [5]).
Since the real mean of the image data is unknown, we
incrementally estimate the sample mean m’(n) by
)(
1
)1('
1
)(' nx
n
nm
n
n
nm +−
−
=
(4)
Where x(n) is the nth sample image. The data entering
the IPCA algorithms are the scatter vectors,
u(n) = x(n) – m’(n) for n=1,2,…
2.4. The Overall Tracking Scheme
The following section outlines our overall tracking
scheme. In the first frame, we initialize the tracker
(Section 2.2). For all subsequent frames, the next step
is to obtain the measurements – taking the minimum
distant prediction from the learnt sub-space (in RGB
plane) as the description of the tracked object. We then
update the eigen-spaces incrementally. Finally, we
predict the motion parameters values for the next
frame. The idea behind the subspace construction for
the appearance based tracking is the uniform L2
reconstruction error norm
Error
∞
(L, {x1, · · · xN}) = max
i
d
2
(L, xi) (5)
To define the quality of approximation, we use the
uniform reconstruction error norm Error
∞
introduced
in Equation 5 in our approach. If N denotes the number
of previous frames whose tracking results are retained
and δ > 0 is a threshold parameter, we can specify a
pair of input parameters (N, δ). We can define the
subspace L to be any subspace such that the uniform
reconstruction error norm between L and {x1, · · · , xN}
is less than the threshold δ. i.e.
Error
∞
(L, {x1, · · · xN}) < δ. (6)
This definition of L is general and the solution is
generally not unique. As along as δ is greater than
zero, there exists at least one L that satisfies the
inequality in Equation 6, the subspace L spanned by
the entire collection of samples {x1, · · · , xN}. One of
the great advantages of this non-uniqueness of the
solution is that we only need to find one such L, and it
allows us to design a simple and computationally
inexpensive algorithm to find just one such L. Having
a computationally inexpensive update algorithm is
necessary if the tracking algorithm is expected to run
in real-time.
4. Remark and Discussions
The computational complexity of the algorithm is
dominated by the number of windows generated from
the sampling. Like all appearance-based tracker it
cannot handle situation like sudden pose or
illumination changes or fully occlusion, but it can
handle partial occlusion and gradual pose or
illumination changes (Figures 1, 2, 3). There are three
important free parameters in our algorithm, N, the
number of samples to pick and l, amnesic parameter
for the subspace update and k, the number of principal
components. In the experiments we reported below, we
let l range from 2 to 6 and N range from 150 to 200
and k range from 3 to 10.
6. Experiments and Results
We implemented the proposed method in MATLAB 7.
Our current implementation runs at about 0.25 to 0.5
frames/sec with 320x240 and 176x144 video input
respectively on a standard Intel centrino P4 1.8 MHz
machine and thus it is quite expected that C
implementation easily can run on real time. Our test
cases contain scenarios which a real-world tracker
encounters, including changes in appearance, large
pose variations, significant lighting variation and
shadowing, partial occlusion, object partly leaving
field of view, large scale changes, cluttered
backgrounds, and quick motion resulting in motion
blur.
Frames tracked Avg Time/frame
video
No
predicti
on
With
predictio
n
No
predictio
n
With
predicti
on
Coast
guard
80 100 4.2 sec 4.2 sec
hall 82 112 4.5 sec 4.6 sec
Table 1: comparison of predictive and non-predictive
framework ( N = 150 windows sampled for each case)
It is evident from the above table that incorporation of
predictive framework makes the tracker more robust.
Coastguard sequence has presence of the boat up to
frames 100 out of total 300 frames and then it
disappears (figure 1). Hall is the sequence where a
person (tracking object) appears in frame 25 and
disappears after 140
th
frame, and in that interval it
changes poses heavily. If we increase the number of
windows to be sampled by 250, no prediction
framework (with almost double time complexity)
shows almost similar robustness that of predictive
framework with 150 samples.
7. Summary and conclusions
In this paper, we have introduced a technique for
predictively learning the statistical distribution on-line
with an Eigen subspace representation of an object that
is being tracked with a fast EigenSpace update
technique. The resulting tracker is both simple and
fast. The method can robustly track an object in the
presence of large viewpoint changes, partial occlusion,
lighting variation, changes to the shape of the object
shaky cameras, and motion blur. Moreover avoidance
of non-linear optimization makes our tracking task
faster than that of [7].
8. References
[1] M. J. Black and A. D. Jepson, “EigenTracking: Robust
Matching and Tracking of Articulated Objects Using a View-
Based Representation”, International Journal of Computer
Vision, vol. 26, no. 1, pp. 63 - 84, 1998.
[2] M. Isard and A. Blake, “CONDENSATION –
Conditional Density Propagation For Visual Tracking”,
International Journal of Computer Vision, vol. 28, no. 1, pp.
5 - 28, 1998.
[3] Chun-Ming Li, Yu-Shan Li, Qing-De Zhuang, Qiu-Ming
Li, Rui-Hong Wu, Yang Li
. “Moving Object Segmentation
and Tracking In Video”, Proceedings of the Fourth
International Conference on Machine Learning and
Cybernetics, Guangzhou, 18-21 August 2005, pp. 4957-
4960
[4] J. Weng, Y. Zhang, and W. Hwang,. “Candid covariance-
free incremental principal component analysis” IEEE
Transactions on Pattern Analysis and Machine Intelligence,
Vol.25(8), pp.1034-1040, 2003.
[5] Y. Zhang and J. Weng, “Convergence Analysis of
Complementary Candid Incremental Principal Component
Analysis,” Technical Report MSU-CSE- 01-23, Dept. of
Computer Science and Eng., Michigan State Univ., East
Lansing, Aug. 2001.
[6] Jeffrey Ho, Kuang-Chih Lee, Ming-Hsuan Yang, David
Kriegman, “Visual Tracking Using Learned Linear
Subspaces”, Proceedings of the 2004 IEEE Computer
Society Conference on Computer Vision and Pattern
Recognition (CVPR’04), Vol 1 pp. 782-789
[7] Namita Gupta, Pooja Mittal, Kaustubh S. Patwardhan,
Sumantra Dutta Roy, Santanu Chaudhury and Subhashis
Banerjee, “On Line Predictive Appearance-Based Tracking”.
Proc.IEEE Int’l Conf. on Image Processing (ICIP 2004), pp
1041 - 1044
[8] Kaustubh Srikrishna Patwardhan, Sumantra Dutta Roy,
“Hand gesture modelling and recognition involving changing
shapes and trajectories, using a
Predictive EigenTracker”,
Pattern Recognition Letters, vol. 28, no. 3, pp. 329 - 334,
February 2007
Frame 1 Frame 21 Frame 35
Frame 67 Frame 86 Frame 108
Figure 1: Sequence of tracking a boat (sequence coastguard) which shows high background motion,
background clutter as well as object partly going out of the field of view
Frame 1 Frame 210 Frame 237
Frame 261 Frame 264 Frame 271
Figure 2 Sequence of tracking a helicopter in a changing background and which goes under partial occlusion
Frame 1 Frame 25 Frame 84
Figure 3: Sequence of tracking a woman’s face (sequence Renata) which shows apparent pose changes
