Local Clustering Patterns in Polar Coordinate for Face Recognition

Abstract

Facial recognition is an important issue and has various practical applications in visual surveillance system. In this paper, we propose a novel local pattern descriptor called the Local Clustering Pattern (LCP) with low computational cost operating in the polar coordinate system for recognizing face. The local derivative variations with multi-direction are considered and that are integrated on the pairwise combinatorial direction. To generate the discriminative local pattern, the features of local derivative variations are transformed into the polar coordinate system by generating the characteristics of distance (r) and angle ($\theta$). LCP is ensemble of several decisions from the clustering algorithm for each pixel in the polar coordinate system (P.C.S.). Differs from the existing local pattern descriptors, such as local binary pattern (LBP) [1, 8], local derivation pattern (LDP) [11], and local tetra pattern (LTrP) [7], LCP generates the discriminative local clustering pattern with low-order derivative space and low computational cost which are stable in the process of face recognition. The performance of the proposed method is compared with LBP, LDP, LTrP on the Extended Yale B [4, 5] and CAS-PEAL [3] databases.

1 Introduction

Due to the intelligence security monitoring is more popular in recent year, the automatical recognizing face is needed for various visual surveillance systems, for example the accessing control system for personal or company to verify the legal/illegal people, policing system for identifying the thief and the robber who presents the illegal behavior in public or private space. To construct an efficient face recognition system, the facial descriptor with discriminated characteristic is required.

Numerous of methodologies are proposed for recognizing face and those can be classified as global and local facial descriptors. The global facial descriptor describes the facial characteristics with the whole face image, such as Principal Component Analysis (PCA) [6, 10], and Linear Discriminant Analysis (LDA) [2, 9]. PCA converts the global facial descriptor from high-dimension to low-dimension by using the linear transform methodology to reduce the computational cost. Linear Discriminant Analysis (LDA) also called the Fishers Linear Discriminant is similar to PCA, while it is a supervised methodology. Although the global facial descriptor can extracts the principal component from the facial images, reduces the computational cost, and maintains the variance of the facial image, the performance is sensitive to the change of the environment, such as the change of light.

The flexibilities of the local facial descriptors, such as the local binary pattern (LBP), local derivation pattern (LDP), and local tetra pattern (LTrP), are better than the global facial descriptors, because the spatial structure information is successfully and effectively utilized. The local binary pattern (LBP) generates the local facial descriptor by comparing the gray value between reference pixel and its adjacent pixels for each pixel in the face image. The texture information, such as spots, lines and corners, in the images are extracted. Although LBP considers the spatial information to generate the local facial descriptor, it omits the directional information and is sensitivity when light is slightly changed. The local derivation pattern (LDP) analyzes the turnings between reference pixel and its neighborhoods from the derivative values. The derivative values with four directions are considered to generates the local facial discriptor in the high-order derivative space. However, the turnings between reference pixel and its neighboors is discussed in the same derivative direction. The local tetra pattern (LTrP) utilized the two-dimensional distribution with derivative values in four quadrants to describe the texture informance and that can extracts more discriminative information. Although LTrP considers the derivative variations with two dimensions, there exists two problems: the dimension of facial descriptor and the sensitivity of the features. To comparing with LBP and LDP, the dimension of facial descriptor of LTrP is higher. The features of LTrP in the four quadrants of the rectangular(or Cartesian) coordinate system are altered when illumination is changed.

In this paper, we focus on reducing feature length with low computational cost and improving the accuracy of face recognition. To resolve these issues, we develop a novel pattern descriptor, called local clustering pattern (LCP), to describle the facial texture for recognizing face. Moreover, The proposed local clustering pattern considers the derivative variations with various directions on the pairwise combinatorial directions. To overcome the noise, such as light effect and to generate the stable local pattern discriptor, the facial features with derivative variations are transformed from the rectangular coordinate system with derivative variations into the polar coordinate system by generating the characteristics of distance (r) and angle ($\theta$).

The variations of gradient are the important characteristics in face descriptor for recognizing face. Various methods, such as LDP and LTrP, are attempted to find and utilize the variations of gradient in face images by using derivative variations. The derivative variations change with directions, and that results in the clustering phenomenon. The phenomenon of clustering is more obvious in the polar coordinate system than that in the rectangular coordinate system. The ensemble of several decision from the clustering algorithm is applying for encoding local pattern descriptor.

This article is organized as follows: the principles of local clustering pattern (LCP) is presented in Sect. 2. The experimental results are demonstrated in Sect. 3. Finally, conclusions are given in Sect. 4.

Fig. 1.

Overview of generating the proposed local clustering pattern (LCP).

2 The Proposed Local Pattern Descriptor

The proposed Local Clustering Pattern (LCP) mainly aims at addressing the problems of reducing feature length with low computational cost and enhancing the accuracy of face recognition. There are three phases to generate the local clustering pattern: (1) To calculate the local derivative variations with various directions. (2) To project the local derivative variations with various directions on the pairwise combinatorial directions from the rectangular coordinate system into the polar coordinate system. (3) Encoding the facial descriptor which is local clustering pattern, as a micropattern for each pixel by applying the clustering algorithm. The details are described in the following subsections, Local Clustering Pattern (LCP) and Coding scheme.

Fig. 2.

(a) An example of 8-neighbors surrounding reference pixel $P_c$, (b) the adjacent pixels of $P_{c}$ with different distances along four directions (Color figure online).

2.1 Local Clustering Pattern (LCP)

The process of generating the proposed local facial descriptor, Local Clustering Pattern (LCP), is shown in Fig. 1. Firstly, we calculate the derivative variations with four directions, $0^\circ$, $45^\circ$, $90^\circ$, and $135^\circ$. Then the proposed encoding scheme is carried out including the pairwise combinatorial direction of derivative variations, the coordinate system transformation, and the encoding strategy. Finally, the LCP facial descriptor is generated by integrating four results from the proposed coding scheme.

Given a sub-region image I(P), as shown in Fig. 2(a), in which $P_c$ is the reference pixel and $P_i, i = 1,...,8$ are the adjacent pixels around $P_c$. The first-order derivatives of $P_c$ along $0^\circ$, $45^\circ$, $90^\circ$ and $135^\circ$ directions are denoted as $I^{'}_{\alpha }(P_c)$, and can be written as

$$\begin{aligned} I^{'}_{\alpha } (P_c) = I_{\alpha } \left( P_n \right) - I(P_c) \end{aligned}$$

(1)

where $\alpha = 0^\circ , 45^\circ , 90^\circ \text { and } 135^\circ$ are the derivative variation directions.

In this paper, the local clustering pattern is generated based on the derivative variations on the four directions, $0^\circ$, $45^\circ$, $90^\circ$, and $135^\circ$, and those are integrated into four pairwise combinatorial directions of the derivative variations, $0^\circ$–$45^\circ$, $45^\circ$–$90^\circ$, $90^\circ$–$135^\circ$, and $135^\circ$–$0^\circ$. LCP in pairwise combinatorial direction, $\alpha$ and $\alpha + 45^\circ$, at reference pixel $P_c$ is encoded as

$$\begin{aligned} \begin{aligned}&LCP_{\alpha } \left( P_c \right) = \\&\sum _{n = 1}^{N} f_{r,\theta } \left( I^{'}_{\gamma ,D} \left( P_{n} \right) , I^{'}_{\gamma ,D} \left( P_c \right) \right) \times 2^{n-1} |_{\gamma \in \{ \alpha , \; \alpha + 45^\circ \}, \; N = 8} \end{aligned} \end{aligned}$$

(2)

where $f_{r,\theta }(.,.)$ is the proposed coding scheme and that is executed in the polar coordinate system, and $D = 1, 2, 3$ is the distance between reference pixel $P_c$ and its adjacent pixels $P_i$, as shown in Fig. 2(b). The green, blue, yellow blocks indicate the distances between reference pixel and its adjacent pixels are one, two and three, the formula can be formally define as follow

$$\begin{aligned} \begin{aligned}&f_{r, \theta } \left( I^{'}_{\gamma , D} \left( P_n \right) , I^{'}_{\gamma , D} \left( P_c \right) \right) |_{\gamma \in \{ \alpha , \; \alpha + 45^\circ \}} = \\&\left\{ \begin{matrix} 0, &{} \quad if \; I^{'}_{\gamma , D} \left( P_n \right) \; \text {and} \; I^{'}_{\gamma , D} \left( P_c \right) \in C_i\\ 1, &{} else \end{matrix}\right. \end{aligned} \end{aligned}$$

(3)

where $C_i$ is the cluster center.

Finally, the LCP at referenced pixel $P_c$, $LCP(P_c)$, is combinatorial of the four 8-bit binary patterns LCPs, and can be formally as

$$\begin{aligned} LCP \left( P_c \right) = \{ LCP_{\beta } \left( P_c \right) |_{ \beta = 0^\circ , \; 45^\circ , \; 90^\circ , \; 135^\circ }. \end{aligned}$$

(4)

2.2 Coding Scheme

The proposed coding scheme is considered as the problem of classification which is executed in the polar coordinate system based on the characteristics of the derivative variations in the pairwise combinatorial directions. In this paper, we utilize four combinations of the derivative variations in the pairwise directions, $0^\circ$–$45^\circ$, $45^\circ$–$90^\circ$, $90^\circ$–$135^\circ$, and $135^\circ$–$0^\circ$, in the rectangular coordinate system (R.C.S.) and those are transformed into the polar coordinate system (P.C.S.) by calculating the distance (r) and angle ($\theta$) for each pair directions of derivative variations. The distance (r) and angle ($\theta$) of $P_n$ are calculated as

$$\begin{aligned} \begin{aligned} r_\gamma \left( P_n \right) = \sqrt{I^{'}_{\gamma ,D} \left( P_n \right) + I^{'}_{\gamma + 45^\circ ,D} \left( P_n \right) } |_{\gamma \in \alpha } \end{aligned} \end{aligned}$$

(5)

$$\begin{aligned} \begin{aligned} \theta _\gamma \left( P_n \right) = \arctan {\frac{I^{'}_{\gamma + 45^\circ ,D} \left( P_n \right) }{I^{'}_{\gamma ,D} \left( P_n \right) }} |_{\gamma \in \alpha } \end{aligned} \end{aligned}$$

(6)

where $\frac{-\pi }{2} < \theta _\gamma < \frac{\pi }{2}$ is normalized to $0^\circ \sim 360^\circ$.

The feature vectors $\mathbf {v}$ are $r_\gamma$ and $\theta _\gamma$ coordinates in the polar coordinate system and can be written as

$$\begin{aligned} \mathbf {v} = \left[ r_\gamma \left( P_n \right) , \theta _\gamma \left( P_n \right) \right] ^{T} \end{aligned}$$

(7)

where $\gamma \in \alpha$ and $n = 1 \sim 9$ are the pixels in the sub-region image I(P) including the reference pixels and its adjacent pixel in the polar coordinate system.

LCP is ensemble of several decisions from the results of clustering. Each clustering result is considered as a problem of a two-class case, whose center vector $\mathbf {C}$ is written as,

$$\begin{aligned} \mathbf {C} = \left[ C_1, C_2 \right] ^{T} \end{aligned}$$

(8)

where $C_1$ and $C_2$ are the two-class centers, in which $C_1$ is also the center of $P_c$. To classify the feature vectors $\mathbf {v}$ in sub-image I, we randomly initialize two-class centers $\mathbf {C}$ and the k-means clustering algorithm is adopted. The procedure of clustering is repeated T times to find the cluster two-class centers $\mathbf {C}_i$ which has the highest probability $P \left( \mathbf {C}_i | \mathbf {v} \right)$.

The pixels surround with the reference pixel are encoded as the following equation,

$$\begin{aligned} \begin{aligned} C \left( r_\gamma \left( P_n \right) , \theta _\gamma \left( P_n \right) \right) |_{\gamma \in \alpha } =&\left\{ \begin{matrix} 0, &{} \quad if \; P_n \in C_{1} \\ 1, &{}else \end{matrix}\right. \end{aligned} \end{aligned}$$

(9)

where $C_{1}$ is the cluster center which includes $P_c$.

Fig. 3.

Example of the stability of the existing methods including LBP, LDP, LTrP. (a) The original Image, (b) the image with noise.

Figure 3 shows an example of an original image and the corresponding image after adding Gaussian noise. LBP is encoded by comparing the gray value of the reference pixel with the adjacent pixels, the $7^{th}$ bit of LBP is changed from 1 to 0. In LDP, we takes two-direction as an example, $0^\circ$ and $45^\circ$. The $6^{th}$ bit of LDP in $0^\circ$ direction is changed from 0 to 1, the $3^{th}$ and $7^{th}$ bits of LDP in $45^\circ$ direction are changed from 1 to 0. In LTrP, two directions, $0^\circ$ and $90^\circ$, are considered as an example, the six eighths bits are changed. These methods are susceptible to noise, the encoding results are unstable. Figure 4 demostrates the encoding results of the LCP which takes Fig. 3 with $0^\circ$ and $45^\circ$ directions as an example, Fig. 4(a) and (b) are the distributions of the original and that of the noised images, in which $C_1$ and $C_2$ are the cluster centers. The characteristics of distance (r) and angle ($\theta$) are more stable than derivative variation and gray values. LCP provides the corresponding pattern even in the situation of that noise and non-monotonic illumination changes.

Fig. 4.

Example of the stable of LCP takes Fig. 3 as an example (the derivative variations along $0^\circ$ and $45^\circ$). (a) The distribution of the original image of LCP in P.C.S., (b) The distribution of the noise image of LCP in P.C.S.

3 Experimental Results

In this section, we first describe the evaluation of similarity. Then we analyze the accuracy of local clustering patterin (LCP) in the rectangular coordinate system (R.C.S.) and that in polar coordinate system (P.C.S.). After the analysis, we demonstrate the accuracy of LCP by comparing with LBP, LDP, and LTrP. Two publicly available face databases, CAS-PEAL, and Extended Yale B, are used to evaluate the accuracy of these methodologies.

All the original facial images are cropped according to the location of the two eyes and normalized into $64\times 64$ pixels. Various derivative modes are compared, $M = [-1, 1]$ means the approximation of the forward derivative and $M = [-1, 0, 1]$ means the approximation of the central derivative.

3.1 Evaluation of Similarity

To measure the similarity, the spatial histogram is adopted for modeling the distribution of each methodology. Three phases are considered: (1) Uniform quantization, (2) Histogram intersection, (3) 1-NN classifier. Firstly, each image is divided into $4 \times 4$ sub-regions and the uniform quantization method is applied to reduce the number of histogram bins in each sub-region from 256 to 8. Secondly, the histogram intersection is calculated to evaluate the similarity of two histograms,

$$\begin{aligned} K \left( H_P, H_G \right) = \sum _{i = 1}^{N} \min \left( H_{P_i}, H_{G_i} \right) \end{aligned}$$

(10)

where K(., .) is the kernal of histogram intersection to find the minimum value between two histograms, $H_{P_i}$ and $H_{G_i}$, $H_{P_i}$ is the histogram of the probe image, $H_{G_i}$ is the histogram of the gallery images, and i is the index of the total bins N for each image.

Finally, we utilize 64 run of tests with the 1-NN classifier to evaluate the performance of each methodology.

3.2 Experimental Results of LCP in Various Coordinate System

The concept of clustering in LCP can be applied in both the rectangular coordinate system (R.C.S.) and the polar coordinate system (P.C.S). We first use the Extended Yale B and the CAS-PEAL face databases to evaluate the accuracy of LCP in both R.S.C. and P.S.C.

Fig. 5.

Experimental results of LCP in both R.C.S. and P.C.S. with various illumination variations. (a) The Extended Yale B database, (b) The CAS-PEAL Light database.

Figure 5 demonstrates the performance of LCP with various illumination variations in both R.C.S. and P.C.S., in which D is the distance between reference pixel and its adjacent neighbors. From Fig. 5, comparing the performance of LCP in R.C.S. with that in P.C.S., the LCP is the best in the $1^{st}$ order in P.S.C.

Fig. 6.

Experimental results of LCP in both R.C.S. and P.C.S with CAS-PEAL database.

Figure 6 shows the average performance of LCP with CAS-PEAL database in both R.C.S. and P.C.S. From Fig. 6, comparing the performance of LCP in R.C.S with that in P.C.S., the gap of LCP with $1^{st}$ order derivative in P.S.C. is more obvious than that with other order derivative in both R.C.S. and P.C.S.

From Figs. 5 and 6, with the level of the derivation is increasing, the performance is decreasing either in R.C.S. or in P.C.S. The features of distance (r) and angle ($\theta$) perform better than that of the derivative variations. The best performance of LCP in P.C.S. is obviously better than that in R.C.S. The best performance of LCP in P.C.S. appears with $1^{st}$ order derivative which has the lowest computation cost. It reveals that the clustering approach applied in P.S.C. is more preferable than that in R.S.C.

Fig. 7.

Expterimental results on Yale database. (a) LCP with various derivative variations in different distance between neighbor, (b) Comparison results of LCP with other methods.

Fig. 8.

Expterimental results on CAS-PEAL Light database. (a) LCP with various derivative variations in different distance between neighbor, (b) Comparsion results of LCP with other methods.

3.3 Experimental Results on Extended Yale B Database

The Extended Yale B database contains 2,432 frontal facial images of 38 subjects under 64 different illumination variations. The frontal face images distinctly illuminated from various angles are evaluated.

Figure 7(a) shows the performance of the proposed method in various derivative variations with different distance of neighbor. From Fig. 7(a), the best result is presented in $1^{st}$ order derivation with distance 2 and also has small standard derivation. Figure 7(b) demonstrates the performance of LCP compares with LBP, LDP, LTrP. From Fig. 7(b), LCP performs well in all derivatives with various derivative modes and has the best result in the $1^{st}$ order derivative.

3.4 Experimental Results on CAS-PEAL Database

The CAS-PEAL light database has 9,060 images of 1,040 subjects, Fig. 8 firstly demonstrates the experimental results in CAS-PEAL light database. The CAS-PEAL light database has 2,450 frontal images with various illumination from different angles and generates from 233 people. Figure 8(a) demonstrates the performance of LCP in various derivative variations with different distance between neighbor. From Fig. 8(a), LCP performs well in $1^{st}$ order derivative with distance 3. Figure 8(b) shows the comparison of LCP with various methodologies, LBP, LDP, and LTrP, in various derivatives. LCP performs well in all derivatives with various derivative modes and has the best result in $1^{st}$ order derivative.

Fig. 9.

Comparsion result of LCP with other methods on the CAS-PEAL database.

Figure 9 shows the performance of LCP in CAS-PEAL database. LCP performs well in both derivative modes, the approximation of the forward derivative and the approximation of the central derivative, and has the best result in $1^{st}$ order derivative with distance 3.

Table 1 shows the feature length of various methodologies. The LBP can be encoded into N-dimensional features depends on the number of neighbors it used. In our study, the 8 dimensional features of LBP is adopted. Although the feature length of LBP is the shortest, the performance of LBP is not ideal in all databases. The feature length of LCP is equal or lesser than LDP and LTrP, while the performance of LCP is the best.

Table 1. Feature length of various methods

4 Conclusion

In this paper, we design a novel local facial descriptor, Local Clustering Pattern (LCP), for recognizing face. The local derivative variations with various directions on the pairwise combinatorial direction are calculated. To encode the local clustering pattern, LCP, the derivative values with the pairwise combinatorial direction are projected into the polar coordinate system with distance (r) and angle ($\theta$) values. The features, distance (r) and angle ($\theta$), have significantly discriminative for recognizing face. In the polar coordinate system, the clustering phenomenon is obvious, the k-means is applied for clustering the features to generate the ensemble LCP for each pixel.

In the experimental results, we analyze the accuracy of the proposed method (LCP) to compare with LBP, LDP, LTrP on the sets of the Extended Yale B and CAS-PEAL databases. In the condition of the changing light, the proposed method has the best performance, either in the Extended Yale B or CAS-PEAL Light database. Moreover, LCP has the low computational cost and the best performance in both databases.