Computational Intelligence - August 2015 - 12
The remainder of the paper is organized as follows. In Section II, we present the proposed sub-sequence-based integrated
CSF approach. Experimental results and the related discussions
are given in Section III, and our concluding remarks are presented in Section IV.
struction of CSF-based weaker estimators, the selection of
weaker estimators, and the computation of the weighting coefficients for the selected weaker estimators via a linear programming (LP) model. A detailed description of these four steps is
presented in the following subsections.
II. Methodology
A. Sub-Sequence Extraction
Figure 1 shows the flowchart of the proposed sub-sequencebased integrated CSF approach for missing-data estimation of
small-size image sequences. There are four main steps in the
proposed method: the extraction of sub-sequences, the con-
Assume that there are F frames in a sequence, and n feature
points are marked on each frame in the tracking process.
Denote [x t, j, y t, j] T (t = 1, 2, g, F, and j = 1, 2, g, n) as the 2D
projection of the j th 3D point in the t th frame of the
sequence. Then, the observation matrix w t of this frame is a
2 # n matrix as follows:
x t, 1 x t, 2 g x t, n
m.
wt = c
(1)
y t, 1 y t, 2 g y t, n
Correspondingly, the observation matrix W of the sequence
with F image frames can be represented as a 2F # n observation matrix as follows:
Begin
Given a 2D Observation
Matrix W with Missing
Entries, Set i = 1.
J x 1, 1
K
K y 1, 1
T
T T
K h
=
=
W [ w 1 , g, w F ]
K
KKx F, 1
L y F, 1
j=1
j=j+1
Construct the SubSequence Wsj
g
g
j
g
g
j 2 F-2
Yes
The Estimated Values of the Missing
Entries Obtained from the F-2 Weaker
Estimators for the ith Frame
i=i+1
Weigh the Outputs of the Selected
Weaker Estimators via the LinearProgramming Model
The Estimated Values
of the Missing Entries
for the ith Frame
Jx 1, 1
K
K y 1, 1
W =K h
K
KK #
L#
#
#
h
x F, 2
y F, 2
g
g
j
g
g
x 1, n N
O
y 1, n O
h O,
O
x F, nO
O
y F, n P
W r = [w T2 , g, w TF ] T .
i2F
No
End
Figure 1 Flowchart of the proposed sub-sequence-based integrated
CSF approach for missing-data estimation of small-size image
sequences.
12
(2)
(3)
where the x and y coordinates of those invisible feature points
represented by the symbol # are considered as missing entries
in the observation matrix W.
In order to estimate the missing values in w t, the first step
of our proposed method is to extract a set of sub-sequences
W sk that are used as the inputs of the CSF algorithm. Take the
first observation w 1 for example. Figure 2 shows the procedure of sub-sequence extraction. We first extract w 1 from the
original sequence W, and put it aside for the moment. We
then denote the remaining observations as W r , i.e.,
Select the Weaker Estimators
According to the Estimation Errors
of the Non-Missing Entries
Yes
x 1, n N
O
y 1, n O
h O.
O
x F, nO
O
y F, n P
For the problem of NRSFM with occlusion, some of the feature points are invisible due to motion or deformation of the
target. Under such a condition, the observation matrix can be
rewritten as follows:
Estimate the Missing Entries of the
ith Frame Image via the jth CSFBased Weaker Estimator
No
x 1, 2
y 1, 2
h
x F, 2
y F, 2
IEEE ComputatIonal IntEllIgEnCE magazInE | august 2015
(4)
Furthermore, a smaller sequence is obtained by removing w T2
from W r , and is then merged with w 1 to form the first subsequence W s1, i.e.,
W s1 = [w T1 , w T3 , g, w TF ] T .
s
2
(5)
Similarly, the second sub-sequence W can be obtained by
removing w T3 from W r , and then by merging the remaining
observations with w 1 . As a result, for each frame w t, in all,
Table of Contents for the Digital Edition of Computational Intelligence - August 2015
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