Palmprint Recognition Using Sparse Representation of Variable Window-Width Real-Valued Gabor Feature

Abstract

This paper proposed a simple but effective palmprint recognition algorithm using improved Real-valued Discrete Gabor Transform (RDGT) and Sparse Representation based Classification (SRC) method. Compared to the existing palmprint recognition methods based on the spatial texture feature of palmprint, the proposed variable window-width real-valued Gabor transform extract the palmprint feature by space-frequency analysis. Given Gauss window as the analysis window, in addition, the window-width is dynamically adjusted according to the local variance of the palmprint image when solving the coefficients of RDGT. Then test sample can be sparsely represented in an overcomplete dictionary composed by training samples. Experimental results on PolyU Palmprint Database and PolyU M_B Database demonstrate the effectiveness of our proposed method.

1 Introduction

Facing the increasing information security problems, biometric technology attracts more and more attention for its unique advantages. Among all, palmprint recognition has been studied deeply in the past ten years. Compared with other biometric technologies, palmprint recognition has the advantages of high recognition rate, simple equipment and easy acceptance by users.

At present, many popular palmprint recognition methods extract spatial texture feature using local feature descriptors. LBP, HOG and WLD are typical local descriptors, for example. Li and Kim [1] improved the Local Tetra Pattern (LTrP), and presents Local Micro-structure Tetra Pattern (LMTrP) by comparing the relationship between the reference pixels and their surrounding pixels with a certain thickness along the horizontal and vertical directions. Hong et al. [2] extract HOG histogram from eight different local coordinate systems. It can overcome the bad effects of image blur, translation and rotation. Jia et al. [3] proposed Histogram of Oriented Lines (HOL), which is a modification of HOG, extracting the line features and direction of palmprint by linear filtering. It robust to small range of translation and rotation. Zhang et al. [4] use block-wise statistics of CompCode as features, and then CRC_RLS (collaborative representation based classification with regularized least square) method for classification. Luo et al. [5] proposed Local Line Directional Patterns (LLDP) to encode the orientation information generated by liner filtering. In [6], WLD method [7] is applied to palmprint recognition for the first time, and proposed Line feature Weber Local Descriptor (LWLD) combining the characteristics of palmprint image. Bounneche et al. [8] used multi-resolution log-Gabor filter to filter palmprint images, which made up for some shortcomings of Gabor filter. Fei et al. [9] proposed DOC (Double-orientation code) method to represent the orientation feature of palmprint and designed an effective nonlinear angular matching scheme.

As you can see, the palmprint recognition methods mentioned above pay more attention to the spatial texture characteristics of palmprint image. However, these methods have a large amount of computation, long recognition time and high feature dimension. Thus, in this paper, the frequency characteristics of palmprint are considered. Observing palmprint images, we find that palmprint has rich palm lines. Therefore, its frequency is spatial-varying. Gabor transform is one of important methods of time (space) frequency analysis. The coefficients reveal the local frequency distribution of a signal or an image. This advantage of Gabor transform has been widely used in various aspects of signal and image processing, such as speech recognition, signal detection, image compression, texture analysis, image segmentation and recognition. The traditional complex-valued discrete Gabor transform (CDGT) algorithm is complex and difficult to implement in hardware and software for complex operations. In this paper, real-valued discrete Gabor transform (RDGT) [10] proposed by Tao et al. is used to solve the transform coefficients.

But the traditional real-valued discrete Gabor transform uses a single window with a fixed width, and has fixed space-frequency resolution. Restricted by the relation between time-width and bandwidth, the spatial resolution and frequency resolution of discrete Gabor transform can not achieve best at the same time. In order to improve the recognition accuracy, in this paper, we adjusted the width of window function adaptively according to the local characteristics of palmprint images and proposed variable window-width RDGT.

The remainder of this paper is organized as follows. Section 2 presents feature extraction based on variable window-width RDGT. Section 3 describes the sparse representation of variable window-width real-valued Gabor feature for palmprint recognition. Section 4 shows the experimental results. Section 5 concludes the paper.

2 Variable Window-Width RDGT

2.1 Real-Valued Discrete Gabor Transform

When image represented by 2-D Gabor transform, the characteristics of human eye and visual system can be effectively combined with image compression coding, and the transform coefficients have less redundancy compared with the original image data. Given an image $\varvec{I}$ $(x,y)$, $x=0,1,\ldots ,X-1$, $y=0,1,\ldots ,Y-1$, and dividing it into $K \times L$ non overlapping lattices of dimensions $M \times N$ such that $X=KM$ and $Y=LN$. Then the image $\varvec{I}$ $(x,y)$ can be expanded as follow

$$\begin{aligned} \varvec{I}(x,y)=\sum _{k=0}^{K-1} \sum _{l=0}^{L-1} \sum _{m=0}^{M-1} \sum _{n=0}^{N-1} a(k,l,m,n)g_{klmn}(x,y) \end{aligned}$$

(1)

The real-valued Gabor basis function is defined as

$$\begin{aligned} g_{klmn}(x,y)=\tilde{g}(x-kM,y-kN)\cdot cas\left\{ 2\pi \left[ \frac{mx}{M}+\frac{ny}{N} \right] \right\} \end{aligned}$$

(2)

where $cas(x)=cos(x)+sin(x)$ is Hartley’s cas function. The Gabor transform coefficients a(k, l, m, n) can obtained by

$$\begin{aligned} a(k,l,m,n)=\sum _{x=0}^{X-1}\sum _{y=0}^{Y-1}\varvec{I}(x,y)h_{klmn}(x,y) \end{aligned}$$

(3)

where the real-valued auxiliary biorthogonal function is given by

$$\begin{aligned} h_{klmn}(x,y)=\tilde{h}(x-kM,y-lN)\cdot cas\left\{ 2\pi \left[ \frac{mx}{M}+\frac{ny}{N}\right] \right\} \end{aligned}$$

(4)

Once given a synthesis window as shown in Fig. 1, its biorthogonal analysis window can be obtained by numerical method [11, 12], shown in Fig. 2. With the analysis window, we can calculate coefficients a(k, l, m, n) using the fast 2D-DHT

$$\begin{aligned} a(k,l,m,n) =&\sum _{x=0}^{X-1}\sum _{y=0}^{Y-1}\varvec{I}(x,y)\tilde{h}(x-kM,y-lN)\cdot \left\{ 2\pi \left[ \frac{mx}{M}+\frac{ny}{N} \right] \right\} \nonumber \\ =&\sum _{j_1=0}^{M-1} \sum _{j_2=0}^{N-1} \left\{ \sum _{i_1=0}^{K-1}\sum _{i_2=0}^{L-1}R_{mn}\left( i_1M+j_1,i_2N+j_2 \right) \right\} \nonumber \\ \cdot&\,\, cas\left\{ 2\pi \left[ \frac{mj_1}{M}+\frac{nj_2}{N}\right] \right\} \end{aligned}$$

(5)

where $R_{mn}(x,y)=\varvec{I}(x,y)\tilde{h}(x-kM,y-lN)$, $x=i_1M+j_1$, $y=i_2N+j_2$.

Fig. 1.

Synthesis window

Fig. 2.

Analysis window

2.2 Variable Window-Width RDGT

From the Fig. 2, the analysis window obtained by numerical method is disperse, which is not suitable for extracting palmprint feature. Fortunately, the synthesis window and analysis window can exchange, and we only need the RDGT coefficients. So we can define a more centralized analysis window directly. Since the Gauss function has the smallest product of the effective time-width and bandwidth in the Heisenberg Uncertainty Principle, its distribution is most concentrated in the time-frequency plane, so the Gauss window is chosen in this paper.

The window-width q in Gauss window function is used to adjust the spatial resolution and frequency resolution. When q is fixed, the space-frequency resolution is fixed too. For palmprint image, some areas have abundant lines and complex frequency components; while some regions are relatively smooth, and the frequency characteristics are not obvious. Therefore, the q should be dynamically adjusted in order to get more discriminative features. In this paper, we adopt variance to measure the local variations of palmprint image, as shown in Fig. 3. The larger variance is, the more dramatic palmprint image changes, and the smaller variance is, the more slowly palmprint image changes. In this way, the q can be changed according to the variance of local image. Now, we redefine the window function as follow

$$\begin{aligned} h_{klmn}=\tilde{h}_{kl}(x-kM,y-lN)\cdot cas\left\{ 2\pi \left[ \frac{mx}{M}+\frac{ny}{N}\right] \right\} \end{aligned}$$

(6)

$$\begin{aligned} \tilde{h}_{kl}(x,y)=\sum _{i}\sum _{j} h_{kl}(x+iX,y+jY)=\tilde{h}_{kl}(x+X,y+Y) \end{aligned}$$

(7)

$$\begin{aligned} h_{kl}(x,y)=\frac{\sqrt{2}}{q}\cdot exp\left\{ -\pi \left[ \left( x-\frac{X}{2} \right) ^2+\left( y-\frac{Y}{2}\right) ^2 \right] \Big / q^2\right\} \end{aligned}$$

(8)

The window-width q is defined as follow

$$\begin{aligned} q=F\left( \varvec{I}_m(u,v)\right) =\sum _{u}\sum _{v}\left| \varvec{I}_m(u,v)-\lambda \right| /MN \end{aligned}$$

(9)

$$\begin{aligned} \lambda =\sum _{u}\sum _{v}\varvec{I}_m(u,v)/MN, \quad (u=0,1,\ldots ,M-1, v=0,1,\ldots ,N-1) \end{aligned}$$

(10)

$$\begin{aligned} \varvec{I}_m(u,v)=\varvec{I}(s,t) \end{aligned}$$

(11)

where

$$\begin{aligned}&s =(kM+X/2)\%X,\ldots ,(kM+X/2+M-1)\%X \nonumber \\&t =(lN+Y/2)\%Y,\ldots ,(lN+Y/2+N-1)\%Y \end{aligned}$$

(12)

Once the width of window function adjusted adaptively according to the local characteristics of palmprint, more palmprint feature data can be extracted. In this way, we make improve on the traditional RDGT. In this way, we make improve on the traditional RDGT.

Fig. 3.

Local variance

Fig. 4.

The energy distribution

2.3 Variable Window-Width Real-Valued Gabor Feature of Palmprint

Dividing the palmprint image into $K \times L$ non overlapping lattice of dimension $M \times N$ (critical sampling), obtaining four-dimensional coefficients through the variable window-width RDGT. Let k and l stay the same, m takes from 1 to M, n takes from 1 to N. Then we will obtain a coefficient matrix of dimensions $M \times N$, representing the energy distribution of the (k, l) lattice. For example, Fig. 4 shows the energy distribution of the RDGT coefficients, when $K=L=8, M=N=8$. Let

$$\begin{aligned} e(k,l)=\sum _{m=1}^M\sum _{n=1}^{N}\left| a(k,l,m,n) \right| ^2, (k=1,\ldots ,K, l=1,\ldots ,L) \end{aligned}$$

(13)

Calculate e(k, l) and form a $h-$dimensional vector $d=(e_1,e_2,\ldots ,e_h)(h=K \times L)$. In order to eliminate the influence of image resolution, the vector d is normalized by the $l_2$ norm $d_N=d/\left\| d \right\| _2$. In summary,the steps of extracting real-valued Gabor features for palmprint image are as follow

• 1. Dividing the $\varvec{I}(x,y)$ into $K \times L$ non overlaping lattice of dimensions $M \times N$, then the variance $q_i(i=1,2,\ldots ,K \times L)$ of each lattice is calculated according to Eqs. (9)–(12).

• 2. The window function $h_{kl}(x,y)$ is generated with the window-width $q_i$ according to Eq. (8).

• 3. Calculating the coefficients according to Eqs. (5)–(7).

• 4. Each $e_i$ is computed to form a feature vector d.

• 5. Normalizing the vector d by the $l_2$ norm.

3 Sparse Representation of Variable Window-Width Real-Valued Gabor Feature

Sparse Representation based Classification (SRC) method has a series of excellent advantages such as high recognition rate and robustness [13,14,15]. In this work, we present a new method for palmprint recognition. Figure 5 shows the process of the proposed method. The steps are as follow

• 1. Given a training sample set (including k class), extract the palmprint features according to Sect. 2.3 and form the matrix $\varvec{A}$

  

  (14)

  where, h denotes the dimension of the feature vector, $n_i$ denotes the number of training samples for each class.

• 2. Given the test sample, extract the real-valued feature $\varvec{y}$.

• 3. The following $l^1$ minimization problem is solved to obtain the sparse representation coefficients

  $$\begin{aligned} \hat{\varvec{x}}=argmin\left\| \varvec{x} \right\| \quad \varvec{A}\varvec{x}=\varvec{y} \end{aligned}$$

  (15)

• 4. Obtain the reconstruction residual

  $$\begin{aligned} r_i\left( \varvec{y} \right) =\left\| \varvec{y}-\varvec{A}_i\varvec{x}_i \right\| _2\quad (i=1,..,k) \end{aligned}$$

  (16)

• 5. Obtain the label of the test sample

  $$\begin{aligned} \hat{i}=\mathop {\arg \min }_{i}r_{i}\left( \varvec{y}\right) \end{aligned}$$

  (17)

In this paper, DALM (Dual Augmented Lagrangian Method) is used to solve the sparse representation coefficients.

Fig. 5.

The proposed algorithm processing

4 Experiments

Experiments are conducted on PolyU Palmprint Database and PolyU M_B Database. PolyU Palmprint Database contains 386 different palmprint.The palmprint images were collected in two sessions, and in each session, about 10 palmprint images were captured from each palm. PolyU M_B consists of 6000 images of 500 palms. The palmprint images were also collected in two sessions, and in each session, about 6 palmprint images were captured from each palm. In the experiments, the palmprint collected in the first stage is used as the training set, and the palmprint collected in the second stage is used as the test set.

4.1 Improvement Analysis

In this section, we compare the traditional RDGT and variable window-width RDGT on palmprint recognition. The result is shown in Table 1. The parameters are set as $K=L=8$, $q=16$. Table 1 indicates that the recognition rate has been greatly improved by using the variable window-width RDGT. The results show that the method proposed in this paper can better represent space-frequency characteristic of palmprint image compared with the traditional RDGT. In addition, the reconstruction residuals in the SRC method actually measure the similarity between the given test samples and the training samples. We analyze the distribution of the within-class reconstruction residuals and between-class reconstruction residuals. Figures 6 and 7 demonstrate that within-class reconstruction residuals and between-class reconstruction residuals are separated. The within-class reconstruction residuals is concentrated in the left half, the between-class reconstruction residuals is concentrated in the right half. It indicates that, the proposed method can effectively classify different palmprint.

Table 1. The effect of proposed method on PolyU

Fig. 6.

PolyU

Fig. 7.

PolyU M_B

4.2 Parameter Setting

When extracting variable window-width real-valued Gabor features, K and L are important parameters. In order to achieve the best recognition rate, different parameters are selected for experiments to find the optimal parameter settings. When $K=L=8$ or $K=L=16$, the dictionary $\varvec{A}$ formed by the feature vectors of training samples meet the requirement of sparsity. While $K=L=32$, $\varvec{A}$ does not satisfy the sparsity requirement, dimensionality reduction must be carried out. PCA method is used to reduce the dimension to 600, 500, 400 and 300 respectively, and make recognition under different dimension. Table 2 shows, when $K=L=16$, the recognition rate will up to 100% on PolyU palmprint database; when $K=L=8$, the recognition rate will up to 99.97% on PolyU M_B database. The proposed method in this paper can meet the demand of palmprint recognition at present.

Table 2. The test results under different parameters

4.3 Performance Comparison of Palmprint Recognition Methods Based on SRC

In this section, we will extract palmprint features using some popular methods. The subspace based method such as Eigenpalms [16], Fisherpalms [17]. The statistically based method DCT [18]. The local descriptor based method such as LBP [19], HOG [3], HOL [3], LLDP [5], Gabor Wavelet [20], LGBP [19], WLD [7]. Then the SRC method is used to identify the test samples. Due to the high dimensionality of the feature vectors, the dictionary $\varvec{A}$ formed by the feature vectors of training samples does no meet the requirement of sparsity. Therefore, dimensionality reduction is also needed. The experimental results are given in Table 3 and Fig. 8. From these results we can see that the proposed method usually achieves a higher recognition rate against the other methods under the same dimension. Eigenpalms and Fisherpalms are two important methods of palmprint feature extraction. These two methods analyze the spatial structure of palmprint images and map the high-dimensional data into low dimensional vectors. However, this method is easily affected by noise, resulting in low recognition rate. DCT method extracts the frequency characteristics of palmprint images, but the spatial information is omitted. HOL, LLDP and WLD methods use the directional information, and the feature dimension is higher. During the feature dimension reduction, a lot of information is lost. The proposed method takes the spatial and frequency characteristics into account. This method has obtain the higher recognition rate and the better noise robustness compared with traditional methods.

Table 3. The recognition rate of different methods on PolyU

Fig. 8.

The recognition rate of different methods on PolyU M_B

4.4 Computational Complexity

In this paper, all experiments are carried on MATLAB 2010 in PC with CPU 3.20 GHz, RAM 4 GB. In order to analyze the computational complexity, we compare the computational cost of the proposed method with state-of-the-art palmprint recognition methods. In Table 4, the computational time of the feature extraction and matching for different palmprint recognition methods are listed. Due to the simple feature extraction scheme, the feature extraction speed of the proposed method is faster than that of other methods.

Table 4. Comparison of recognition time of different palmprint recognition methods

4.5 Robustness Experiment

In this section, we will design some translation and rotation experiments to test the robustness of the proposed method. First, we carry out a simple translation test. The test palmprint is moved with left-ward shift 5 pixels, right-ward shift 5 pixels, up-ward shift 5 pixels, down-ward shift 5 pixels, shown in Fig. 9(a). Then rotation test is carried out. The test palmprint is rotated with 3$^{\circ }$ and 5$^{\circ }$, as shown in Fig. 9(b). Only 4 palmprint images are not correctly identified in these experiments. The experimental results show that the proposed method is robust to small range translation and rotation variations.

Fig. 9.

The translation and rotation changes

5 Conclusion

In this paper, unlike many palmprint recognition methods based on spatial texture features, proposed a simple and effective palmprint recognition method using sparse representation of variable window-width real-valued Gabor feature. In order to analyze palmprint image in space-frequency domain, the traditional RDGT has been improved according to the characteristic of palmprint. Extensive experimental results demonstrate that the proposed method can achieve a competitive performance comparing with the state-of-art palmprint recognition methods. But in this paper, the processing of the coefficients of RDGT is relatively simple. In the next step, we will study how to make better use of the information to further improve the performance of the algorithm.