Two-dimensional approximately harmonic projection for gait recognition

Ziqiang Wang1 , Xia Sun2 , Lijun Sun3

1, 2, 3School of Information Science and Engineering, Henan University of Technology, Zhengzhou 450001, China

Journal of Vibroengineering, Vol. 15, Issue 2, 2013, p. 693-702.
Received 2 February 2013; accepted 3 June 2013; published 30 June 2013

Copyright © 2013 Vibroengineering This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Creative Commons License
Abstract.

This paper presents a two-dimensional approximately harmonic projection (2DAHP) algorithm for gait recognition. 2DAHP is originated from the approximately harmonic projection (AHP), while 2DAHP offers some advantages over AHP. 1) 2DAHP can preserve the local geometrical structure and cluster structure of image data as AHP. 2) 2DAHP encodes images as matrices or second-order tensors rather than one-dimensional vectors, so 2DAHP can keep the correlation among different coordinates of image data. 3) 2DAHP avoids the singularity problem suffered by AHP. 4) 2DAHP runs faster than AHP. Extensive experiments on gait recognition show the effectiveness and efficiency of the proposed method.

Keywords: dimensionality reduction, approximately harmonic projection, matrix representation, gait recognition.

1. Introduction

Recently, the average silhouettes-based human gait recognition has received extensive attention due to its potential applications in many fields [1-4], such as identity authentication and video surveillance. In general, a binary silhouette image of size 128×88 in the USF HumanID gait database is represented as a vector in the image space R128×88. Consequently, a major challenge of gait recognition is that the captured gait image often lies in a high-dimensional image space. Due to the consideration of the curse of dimensionality, a common way to resolve this problem is to use dimensionality reduction techniques. Once we obtain lower-dimensional representations of the original gait images, the traditional classification methods can be applied in the reduced feature space. Therefore, the main objective of this paper is to find techniques that can introduce lower-dimensional feature representations of gait images with enhanced discriminatory power.

The most representative algorithms for dimensionality reduction are principal component analysis (PCA) and linear discriminant analysis (LDA) [5]. Although PCA and LDA have been successfully applied to face recognition, image retrieval, and gait recognition, they are designed for discovering only the global Euclidean structure, whereas the local manifold structure is ignored. In fact, the global statistics such as variance is often difficult to compute when there are no sufficient samples. In addition, a number of research efforts have shown that the images possibly reside on a nonlinear submanifold and the representation of image is fundamentally related to the problem of manifold learning [6-9]. Given a set of high-dimensional data points, manifold learning techniques aim to discover the geometric properties of the data space. In the past years, a number of manifold learning algorithms have been developed, representative algorithms include locally linear embedding (LLE) [10], ISOMAP [11], and Laplacian eigenmaps (LE) [12]. LLE is designed to maintain the local linear reconstruction relationship among neighboring points in the lower-dimensional space. ISOMAP aims to preserve global geodesic distances of all pairs of samples. LE aims to preserve proximity relationships by manipulations on an undirected weighted graph, which indicates neighbor relations of pairwise samples. These nonlinear methods do yield impressive results on some artificial benchmarks and several real applications. However, they suffer from the out of sample problem, i.e., they can only obtain mappings that are defined on the training data points and how to explicitly calculate the mappings on novel testing data points remains unclear. Therefore, these nonlinear manifold learning algorithms might not be suitable for gait recognition. To cope with the out of sample problem, locality preserving projection (LPP) [13] applies a linearization procedure to construct explicit mappings over new samples. In the recent research, Lin et al. [14] point that utilizing the affine hulls of the manifold and the connected components is more effective for preserving the local geometrical structure and cluster structure of original data, and propose a new algorithm termed approximately harmonic projection (AHP) for dimensionality reduction. AHP is a linear manifold learning method based on the harmonic framework, and the optimal transformation can be obtained by approximating the Dirichlet integral. It is worth noting that all these methods unfold input data into vectors before dimensionality reduction. But images are naturally in the form of second-order or higher-order tensors [15-17]. For example, gray-level images represent second-order (matrix), and Gabor-filtered image represents third-order tensors. Consequently, such kind of vectorization largely increases the computational costs and seriously destroys the intrinsic tensor structure of images. To cope with these issues, multilinear extensions of PCA, LDA, and LPP, namely 2DPCA [18], 2DLDA [19], and 2DLPP [20] are proposed, respectively. These methods aim to conduct subspace analysis by directly encoding images as two-dimensional image matrices rather than one-dimensional vectors. The advantages of using image-as-matrix representation have been indeed consistently pointed out in a number of recent research efforts [15-20], especially when the number of training samples is small. Nevertheless, the multilinear (tensor) extension of AHP and its application to gait recognition are still a research area where few people have tried to explore.

This paper represents a gray-level average silhouette image of size n1×n2 as the matrix (or second-order tensor) in the tensor space Rn1×Rn2. Then a two-dimensional approximately harmonic projection (2DAHP) is proposed by tensorizing AHP. Compared with the original AHP, 2DAHP can directly process gait images in their original matrix form and utilize correlations among pixels within different dimensions (i.e., rows and columns). Moreover, the smaller number of data entries along each data dimension facilitates subspace learning with limited training data. 2DAHP is much more computational efficient since the decomposed matrices are of size n1×n1 or n2×n2, which is much smaller than that of size n×n (n=n1×n2) in AHP. 2DAHP can avoid the singularity problem. In addition, the trace ratio optimization technique is also applied to efficiently solve 2DAHP.

The remainder of this paper is organized as follows. Section 2 briefly reviews AHP. Section 3 introduces our proposed 2DAHP algorithm. Experimental results on gait recognition are presented in Section 4. The concluding remarks are provided in Section 5.

2. Brief review of approximately harmonic projection (AHP)

AHP is a recently proposed linear manifold learning method for dimensionality reduction [14]. It is based on the approximate affine hull and explicitly utilizes the edge length to reflect the geometrical structure of the manifold structure of the data space.

Given a set of data points {x1,,xn}Rm, let X=[x1,,xn]. Let Wc and Wb be two weight matrices defined on the data points. The optimal projection of AHP can be obtained by solving the following minimization problem:

(1)
a o p t = a r g m i n a 1 2 i j e i j f e i j 2 d t = a r g m i n a 1 2 i j 0 d i j a T x i - a T x j d i j 2 d t = a r g m i n a a T X ( D c - W c ) X T a ,

with the constraint:

(2)
3 i j 0 d i j a T x i + t d i j a T x i - a T x j 2 d t = a T X 2 D b + W b X T a = 1 ,

where eij=xj-xi represents an edge vector that has an orientation from xi to xj,dij=xj-xi denotes the length of the edge between xi and xj, t is the arc length of eij. Wc and Wb are two matrices defined as follows: if xi and xj are connected, then Wijc=1/dij and Wijb=dij; otherwise, Wijc=Wijb=0.Dc and Db are two diagonal matrices defined as Diic=jWijc, Diib=jWijb. feij denotes the gradient on each edge, its definition is as follows:

(3)
f e i j = a T x j - a T x i d i j .

Unlike the standard spectral graph methods which mainly consider the connectivity of graph, AHP explicitly makes use of the edge length and edge orientation which reflect the geometrical structure of the manifold. Therefore, AHP can precisely model multiple connected components of the data manifold, which is especially important for discriminating data with different submanifold (cluster) structure.

The objective function in AHP aims to use the approximate affine hull of the graph to separate data points sampled from different components. Therefore, minimizing it is to ensure that if xi and xj lie in the multiple connected components, then yi(=aTxi) and yj(=aTxj) are made close by the optimal projection. Finally, the projection vector a that minimizes (1) is given by the minimum eigenvalue solution to the generalized eigenvalue problem:

(4)
X D c - W c X T a = λ X 2 D b + W b X T a .

Note that, in the appearance-based image analysis, one is often confronted with the fact the dimension of image vector is much smaller than the number of images. Thus, the matrix X(2Db+Wb)XT is singular. To avoid the singularity problem, one may first apply PCA to remove the components corresponding to zero eigenvalues. Thus, the projection vector of AHP can be considered as the eigenvectors of the matrix (X(2Db+Wb)XT)-1X(Dc-Wc)XT associated with the smallest eigenvalues. In addition, since (X(2Db+Wb)XT)-1X(Dc-Wc)XT is not usually symmetric, the AHP projection axes are not orthogonal.

Let the column vector of a1,a2,,ad be the solution of (4) ordered according to their eigenvalues λ1<λ2<<λd. Thus, the embedding is given by xiyi=ATxi, where yi is a d-dimensional vector and A=(a1,a2,,ad) is an n×d matrix.

3. Two-dimensional approximately harmonic projection (2DAHP)

Given a set of data points {Xi}i=1n in the second-order tensor (or matrix) space Rn1Rn2, let {ui}i=1n1 be an orthonormal basis of Rn1 and {vj}j=1n2 be an orthonormal basis of Rn2, it has been shown that {uivj} forms a basis of the tensor space Rn1Rn2 [20]. Thus, a second-order tensor X can be uniquely defined as X=i,j(uiTXvj)uivjT.

Given a set of data points Xii=1n in Rn1Rn2, two-dimensional approximately harmonic projection (2DAHP) aims to find two projection matrices URn1×l1 and VRn2×l2 that maps each data point Xii=1,,n to a lower-dimensional matrix representation YiRl1×Rl2(i=1,n,l1<n1,l2<n2) by Yi=UTXiV such that Yi represents Xi.

Let U and V be the projection matrices, according to (1) and (2), the optimal objective function of 2DAHP with the matrix representation can be rewritten as follows:

(5)
( U * , V * ) = a r g m i n U , V 1 2 i j 1 d i j ( U T X i V - U T X j V ) 2 ,

with the constraint:

(6)
3 i j 0 d i j ( U T X i V + t d i j ( U T X i V - U T X j V ) ) 2 d t = 1 ,

where dij is similarly defined as AHP.

Let Yi=UTXiV and Dc be a diagonal matrix, Diic=jWijc. Since A2=Tr(AAT), we have:

(7)
1 2 i j 1 d i j U T X i V - U T X j V 2 = 1 2 i j Tr Y i - Y j Y i - Y j T W i j c = 1 2 i j Tr Y i Y i T + Y j Y j T - Y i Y j T - Y j Y i T W i j c = Tr i D i i c Y i Y i T - i , j W i j c Y i Y j T = Tr U T i D i i c X i V V T X i T - i , j W i j c X i V V T X j T U = Tr U T P V c - Q V c U ,

where PVc=iDiicXiVVTXiT and QVc=i,jWijcXiVVTXjT. Similarly, A2=Tr(ATA), so we can also obtain:

(8)
1 2 i j 1 d i j U T X i V - U T X j V 2 = 1 2 i j Tr Y i - Y j T Y i - Y j W i j c = 1 2 i j Tr Y i T Y i + Y j T Y j - Y i T Y j - Y j T Y i W i j c = Tr i D i i c Y i T Y i - i , j W i j c Y i T Y j = Tr V T i D i i c X i T U U T X i - i , j W i j c X i T U U T X j V = Tr V T P U c - Q U c V ,

where PUc=iDiicXiTUUTXi and QUc=i,jWijcXiTUUTXj. Consequently, we should simultaneously minimize Tr(UT(PVc-QVc)U) and Tr(VT(PUc-QUc)V).

In addition, similar to the above derivation process, the left side of constraint function equation (6) can be converted to:

(9)
3 i j 0 d i j ( U T X i V + t d i j ( U T X i V - U T X j V ) ) 2 d t = Tr ( U T ( 2 P V b + Q V b ) U ) = Tr ( V T ( 2 P U b + Q U b ) V ) ,

where PVb=iDijbXiVVTXiT, QVb=iWijbXiVVTXjT,  PUb=iDijbXiTUUTXi, QUb=iWijbXiTUUTXj, and Db is a diagonal matrix, Diib=jWijb.

Finally, the optimal objective function (5) subject to (6) can be transformed as:

(10)
m i n U , V Tr ( U T ( P V c - Q V c ) U ) Tr ( U T ( 2 P V b + Q V b ) U ) ,
(11)
m i n U , V Tr ( V T ( P U c - Q U c ) V ) Tr ( V T ( 2 P U b + Q U b ) V ) .

Because of difficulty in solving the optimal U and V simultaneously, we follows the similar computational methods as [20] to compute U and V iteratively. We first initialize U with an identity matrix, then V can be approximately computed with generalized eigenvalue decomposition (GED) by transforming the optimal objective function (11) into the tractable ratio trace form Tr((VT(2PUb+QUb)V)-1(VT(PUc-QUc)V)). That is, V can be regarded as the eigenvectors associated with the minimum eigenvalues of the following generalized eigenvector problem:

(12)
P U c - Q U c V = λ 2 P U b + Q U b V .

Once V is obtained, similarly, we can update U by solving the following generalized eigenvector problem:

(13)
U T ( P V c - Q V c ) U = λ ( 2 P V b + Q V b ) U

Therefore, we can obtain the final optimal U and V by iteratively solving the generalized eigenvector problems (12) and (13).

In the preceding section, we approximately computed the the optimal objective functions of (10) and (11) by converting them into ratio trace problems, which are solved by GED. However, the obtained solutions may deviate from the original objectives, which may lead to uncertainty in subsequent classification [21]. To address these problems, we describes how to directly solve (10) and (11) with the Iterative algorithm for the Trace Ratio (ITR) optimization problem introduced in [21]. To compute U, we first fix V and initialize U0 as an arbitrary columnly orthogonal matrix. In each iterative step, we solve a trace difference problem Ut=argminUTU=ITr(UT((PVc-QVc)-λt(2PVb+QVb))U), where λt is the trace ratio value calculated from the projection matrix Ut-1 of the previous step, i.e., λt=Tr(Ut-1T(PVc-QVc)Ut-1)/Tr(Ut-1T(2PVb+QVb)Ut-1). Once U is obtained, similarly, we can update V by solving Vt=argminVTV=ITr(VT((PUc-QUc)-λt(2PUb+QUb))V) where λt=Tr(Vt-1T(PUc-QUc)Vt-1)/Tr(Vt-1T(2PUb+QUb)Vt-1). Finally, output the final U and V when the iterative algorithm converges to optimal solutions. The detailed iteration algorithm for solving (10) and (11) can be presented as follows:

Algorithm1: The iteration algorithm for directly solving the optimal problem (10) and (11) in 2DAHP.

Step 1: Initialize U0 and V0 as two arbitrary column-wise orthogonal matrices.

Step 2: For t=1,2,Tmax, do

Step 2.1: Calculate the trace ratio value λt according to the projection matrix Ut-1:

(14)
λ t = Tr ( ( U t - 1 ) T ( P V c - Q V c ) U t - 1 ) Tr ( ( U t - 1 ) T ( 2 P V b + Q V b ) U t - 1 ) .

Step 2.2: Obtain the new Ut by solving the following eigen-decomposition problem:

(15)
P V c - Q V c - λ t 2 P V b + Q V b U i t = τ i t U i t .

Step 2.3: For the given Ut, calculate the trace ratio value λt according to the projection matrix Vt-1:

(16)
λ t = Tr ( ( V t - 1 ) T ( P U c - Q U c ) V t - 1 ) Tr ( ( V t - 1 ) T ( 2 P U b + Q U b ) V t - 1 ) .

Step 2.4: Obtain the new Vt by solving the following eigen-decomposition problem:

(17)
P U c - Q U c - λ t 2 P U b + Q U b V i t = τ i t V i t ,

where τ0tτ1tτd-1t are the d smallest eigenvalues, and Vit is the eigenvector associated with eigenvalue τit, which constitutes the ith column vector of the matrix Vt.

Step 2.5: If Ut-Ut-1<dε and Vt-Vt-1<dε, then break.

Step 3: Output the projection matrices U=Ut and V=Vt.

From the above algorithemic procedure, it can be easily observe that the obtained projection matrices U and V are orthogonal. In addition, following the conclusions in [21], the above iteration algorithm will converge to an optimal value. For more details about proof of convergence, please refer to [21].

4. Experimental results

In this section, to investigate the performance of our proposed 2DAHP algorithm for gait recognition, we compare the 2DAHP algorithm with 2DPCA [18], 2DLDA [19], 2DLPP [20], and the original AHP [14] algorithms for gait recognition on the well-known USF HumanID gait database, where 2DPCA, 2DLDA, and 2DLPP are three popular tensor methods in face recognition and gait recognition, and the original AHP algorithm is a vector-based algorithm. The settings of these compared algorithms are identical to the description in the corresponding papers. In addition, to cope with the singular problem existed in the original AHP, we apply PCA to remove the components corresponding to zero eigenvalues before carrying out AHP. For 2DAHP, we empirically set the optimal iteration number Tmax as 5 for each probe set, since the latter experimental results show that the 2DAHP algorithm converges quite quickly.

The USF HumanID gait database is constructed by Sarkar et al. [1], it contains 1870 sequences from 122 individuals walking on an elliptical path in front of two cameras. This database provided one gallery set containing the sequences from 122 individuals and 12 probe sets containing different numbers of individuals varying from 33 to 122 for algorithm training and testing, respectively. More details information about the USF HumanID gait database can be found in [1]. In this gait database, we consider sequences of binary silhouette images. As in [22] and [23], we construct the average silhouette-based gait image representation: First, a complete sequence is partitioned into several subsequences according to the gait period length Ngait provided by Sarkar et al. [1]. Then, the binary silhouette images within each gait cycle of a sequence are averaged to acquire several gray-level average silhouette images according to:

(18)
A T i = 1 N g a i t k = ( i - 1 ) N g a i t + 1 k = i N g a i t T ( k ) , i = 1 , , F / N g a i t ,

where {T(1),,T(F)} represents the binary images for one sequence with F frames, F/Ngait denotes the largest integer less than or equal to F/Ngait. Since numerous researches have experimentally shown that the average silhouette image is more effective and efficient than the original binary silhouette image for human gait recognition, we also utilize the average silhouette image for gait recognition. Fig. 1 shows some original binary images and the average silhouette images of two different individuals, where the first seven images and the last image in each row denote the binary silhouette images and the average silhouette images, respectively. As can be seen, different individuals have different average silhouette images.

To perform gait recognition, we first obtain the average silhouette image subspaces by dimensionality reduction algorithms. Then, all the averaged images from both the gallery set and probe sets are projected into the image subspaces. Finally, the nearest-neighbor classifier is adopted to identify new average silhouette images, where the distance measure uses the median operator for its robust to noise [1, 23]:

(19)
Dist L S P , L S G = Median i = 1 N p Min j = 1 N g L S P i - L S G j 2 ,

where LSP(i), i=1,,Np and LSG(j), j=1,,Ng are the lower representations from one probe sequence and one gallery sequence, respectively, Np and Ng denote the total number of average silhouette images. For each dimensionality reduction algorithm, we only show its performance in the l- or (l×l)- dimensional subspace. For each case, we average the results over 20 random splits of training and testing sets.

Fig. 1. Some original binary images and the average silhouette images of two different individuals in the USF HumanID gait database

 Some original binary images and the average silhouette images  of two different individuals in the USF HumanID gait database

The recognition accuracies are shown in Table 1 and Table 2, where Rank-1 indicates that the correct subject is ranked as the top candidate, Rank-5 means that the correct subject is ranked among the top five candidates, and Average denotes the recognition rate among all the probe sets. Moreover, we also plot the recognition rate variance with different numbers of iterations for probe sets A, B, C, D, E, F, G, H, I, J, K and L in Fig. 2. Finally, we report the running times of 2DAHP and AHP in Table 3.

From the experimental results listed in Table 1-3 and Fig. 2, we can have the following observations:

1) Our proposed 2DAHP consistently outperforms the 2DPCA, 2DLDA, 2DLPP, and AHP algorithms, which demonstrates that it is beneficial to use simultaneously two-dimensional matrix representation as well as the local geometrical structure and cluster structure for gait recognition.

2) 2DPCA performs the worst among the compared algorithms. A possible explanation is as follows: similar to the traditional PCA, the 2DPCA is simply achieves object reconstruction and it is not necessarily useful for discriminating gait images with different subjects which is the ultimate goal of gait recognition.

3) The 2DLDA performs comparatively to 2DLPP. This demonstrates that it is hard to evaluate whether local manifold structure or class label information is more important, which is consistent with existing studies.

4) The 2DAHP algorithm converges quite quickly for probe sets A, B, C, D, E, F, G, H, I, J, K and L, and recognition rates changes slightly with different iteration numbers. After about 5 iterations, 2DAHP can converge to the optimal solution in all 12 probe sets.

5) 2DAHP achieves significant speed up comparing to AHP. Theses results are consistent with the theoretical analysis of the efficiency, i.e., 2DAHP can utilize the intrinsic tensor structure of gait images to improve running efficiency.

Table 1. Performance comparison in terms of Rank-1 recognition results (%)

Probe
A
B
C
D
E
F
G
H
I
J
K
L
Average
2DPCA
86
87
75
26
27
18
19
56
61
53
10
11
44.1
2DLDA
89
91
82
33
33
23
25
67
78
67
19
19
52.2
2DLPP
90
92
81
34
36
22
24
69
82
65
18
20
52.8
AHP
88
90
78
32
40
21
20
64
79
61
13
15
50.1
2DAHP
93
94
85
45
48
26
33
84
84
69
27
22
59.2

Fig. 2. Some original binary images and the average silhouette images of two different individuals in the USF HumanID gait database

 Some original binary images and the average silhouette images  of two different individuals in the USF HumanID gait database

a)

 Some original binary images and the average silhouette images  of two different individuals in the USF HumanID gait database

b)

 Some original binary images and the average silhouette images  of two different individuals in the USF HumanID gait database

c)

 Some original binary images and the average silhouette images  of two different individuals in the USF HumanID gait database

d)

 Some original binary images and the average silhouette images  of two different individuals in the USF HumanID gait database

e)

 Some original binary images and the average silhouette images  of two different individuals in the USF HumanID gait database

f)

 Some original binary images and the average silhouette images  of two different individuals in the USF HumanID gait database

g)

 Some original binary images and the average silhouette images  of two different individuals in the USF HumanID gait database

h)

 Some original binary images and the average silhouette images  of two different individuals in the USF HumanID gait database

i)

 Some original binary images and the average silhouette images  of two different individuals in the USF HumanID gait database

j)

 Some original binary images and the average silhouette images  of two different individuals in the USF HumanID gait database

k)

 Some original binary images and the average silhouette images  of two different individuals in the USF HumanID gait database

l)

Table 2. Performance comparison in terms of Rank-5 recognition results (%)

Probe
A
B
C
D
E
F
G
H
I
J
K
L
Average
2DPCA
89
94
88
51
53
44
42
80
79
64
22
17
60.3
2DLDA
97
99
95
58
57
50
50
86
93
77
43
40
70.5
2DLPP
95
98
93
59
60
48
51
88
91
75
45
41
70.3
AHP
91
95
89
60
59
49
48
85
82
69
29
26
65.2
2DAHP
99
100
96
75
77
56
60
94
94
83
46
38
76.4

Table 3. Running time(s) comparison on the USF HumanID gait database

Probe
A
B
C
D
E
F
G
H
I
J
K
L
AHP
2.51
1.17
1.18
2.64
1.29
2.48
1.32
2.41
1.31
2.40
1.06
1.08
2DAHP
0.32
0.15
0.15
0.34
0.21
0.30
0.19
0.30
0.18
0.28
0.10
0.11

5. Conclusions

This paper introduces a tensor dimensionality reduction algorithm called two-dimensional approximately harmonic projection (2DAHP). Compared with the original AHP, 2DAHP can directly conducts subspace analysis by encoding an image as a two-dimensional matrix and has higher computational efficiency. Experimental results on gait recognition have demonstrated the effectiveness and efficiency of our proposed approach.

Acknowledgements

This work is supported by NSFC (Grant No. 70701013), the National Science Foundation for Post-Doctoral Scientists of China (Grant No. 2011M500035), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No.20110023110002).

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