Two-dimensional locality preserving projections (2DLPP) with its application to palmprint recognition

doi:10.1016/j.patcog.2006.06.022

Pattern Recognition

Volume 40, Issue 1, January 2007, Pages 339-342

https://doi.org/10.1016/j.patcog.2006.06.022 Get rights and content

Abstract

This paper proposes a novel algorithm for image feature extraction, namely, the two-dimensional locality preserving projections (2DLPP), which directly extracts the proper features from image matrices based on locality preserving criterion. Experimental results on the PolyU palmprint database show the effectiveness of the proposed algorithm.

Introduction

Locality preserving projections (LPP) is a recently proposed method in image recognition for feature extraction and dimension reduction. The objective of LPP is to preserve the local structure of the image space by explicitly considering the manifold structure, which is in fact to solve a generalized eigenvalue problem [1] ${XLX}^{T} a = λ {XDX}^{T} a .$ A difficulty in using the LPP method for image recognition is the high-dimensional nature of the image space, in such a space, the ${XDX}^{T}$ matrix is always singular, which makes the direct implementation of the LPP algorithm impossible.

One possible solution to attack this problem is to utilize the principal component analysis (PCA) as a preprocessing step to reduce the dimensionality of the vector space, which is known as Laplacianface algorithm and has been applied successfully to face representation and recognition [1]. However, in the existing Laplacianface ( $PCA + LPP$ ) algorithm, several disadvantages should be pointed out:

(1) The 2D image matrices must be previously transformed into 1D image vectors. The resulting image vectors usually lead to a high-dimensional image vector space, where it is difficult to calculate the bases to represent the original images, which is also called the “curse of dimensionality” problem. This problem is more apparent in small-sample-size cases such as image recognition.

(2) Such a matrix-to-vector transform may cause the loss of some structural information residing in original 2D images.

(3) In the PCA step of the Laplacianface algorithm, how to determine the numbers of principal components is a hard problem.

(4) In the Laplacianface algorithm, after all the image vectors are projected into the subspace spanned by the principal components, the LPP algorithm is then performed. However, since the objective of the PCA and that of LPP are essentially different, the preprocessing procedure to reduce the dimensionality using the PCA could result in the loss of some important information for the LPP algorithm that follows the PCA. To illustrate this problem, a brief proof is given:

In [2], a locality preserving function $f$ was defined as follows: $f (a) = \frac{a^{T} {XLX}^{T} a}{a^{T} {XDX}^{T} a} .$ The locality preserving function $f (a)$ evaluates the locality preserving power of the projective map $a$ . Actually, in the LPP algorithm $a^{T} {XDX}^{T} a = 1$ , then Eq. (2) can be reduced to $f (a) = a^{T} {XLX}^{T} a .$ In fact, ${XLX}^{T}$ is also singular [1], which implies that the null space of ${XLX}^{T}$ contains valuable discriminatory information; however, the PCA step may discard such projection directions which satisfy $f (a) = 0$ . Now we can see that some important information for the following LPP algorithm may have been lost in the PCA step.

Inspired by Yang et al. [3], in this paper, an alternative way is proposed to handle the above problems by directly projecting the image matrix under a specific projection criterion, rather than using the stretched image vector. Our algorithm proposed here is a straightforward manner based on locality preserving criterion and the image matrix projection. Experimental results on the PolyU palmprint database show that the 2DLPP algorithm outperforms the conventional PCA, $PCA + LDA$ and $PCA + LPP$ algorithms in terms of the recognition performance rate. Our work will fit into the scene for a better picture about LPP-based methods for image recognition.

Section snippets

The algorithm of 2DLPP

Like that of the vector-based LPP [1], the objective function of 2DLPP is defined as $\min \sum_{i, j} ∥ Y_{i} - Y_{j} ∥^{2} S_{ij},$ where $Y_{i}$ is the $n$ -dimensional representation of $m \times n$ matrix $X_{i}$ , the matrix $S$ is a similarity matrix, and $∥ \cdot ∥$ means the $L_{2}$ norm. A possible way of defining $S$ is as follows: $S_{ij} = \exp (- ∥ X_{i} - X_{j} ∥^{2} / t)$ , if $X_{i}$ is among k nearest neighbors of $X_{j}$ or $X_{j}$ is among k nearest neighbors of $X_{i}$ , otherwise, $S_{ij} = 0$ .

Here, k defines the local neighborhood. The objective function with this choice of symmetric weights $S_{ij}$

Experimental results

Now we test the 2DLPP algorithm on the PolyU palmprint database [4]. There are 100 different palms in the database released in March 2004, and six samples from each of these palms were collected in two sessions, where three samples were captured in each session. Since the images contain not only the palmprints, but also other parts of the palm and background, a coordinate system proposed in [5] is used to align different palmprint images for further processing. Fig. 1(a) and (d) show the

Conclusions

In this paper, an efficient image representation and recognition method called 2DLPP is proposed. The main difference between our 2DLPP method and the existing $PCA + LPP$ method is that the latter works on the stretched vector of images, while the former works directly on the image matrix of images. The main advantage of 2DLPP over LPP lies in that the dimensionality of the bases derived by the former for image representation is much smaller than the latter, which means a more accurate

Acknowledgements

The authors would like to thank the anonymous reviewers and the editor for their helpful comments and suggestions. This work is partially supported by the National Program 973 (2003CB716104), Specialized Research Fund for the Doctoral Program of Higher Education of China (20049998012), Distinguished Young Scholars Fund of China (60225015), and Ministry of Education of China (TRAPOYT Project). The authors would like to thank Dr. Xiaofei He and Dr. Deng Cai for discussions on the topic of LPP and

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