Coupled compressed sensing inspired sparse spatial-spectral LSSVM for hyperspectral image classification

Abstract

Inspired by the recently developed Compressed Sensing (CS) theory, this study advances a sparse Spatial-Spectral Least Square Support Vector Machine (SS-LSSVM) for Hyperspectral Image Classification (HIC). In our work, hyperspectral pixels are redefined in both the spectral domain and spatial domain by adaptively selecting their spatial neighbors according to the edge-map. The weighted sum of spectral and spatial features is utilized to construct an SS-LSSVM model. The SS-LSSVM is regarded as a topology comprised of a large number of support vectors, and a sparse SS-LSSVM is derived from a Coupled Compressed Sensing (CCS) of this topology. The sparsity of our proposed CCS inspired Sparse SS-LSSVM (CCS4-LSSVM) improves the classification accuracy of SS-LSSVM for HIC. Furthermore, by combining spectral information and adaptively extracted spatial information together, CCS4-LSSVM cannot only avoid the speckle-like misclassification of original LS-SVM but also reduce the influence of noisy pixels. The performance of our proposed method is evaluated on some hyperspectral image data, and the results show that it can achieve higher classification accuracy than the Spatial-Spectral SVM (SS-SVM) and Spatial-Spectral LSSVM (SS-LSSVM).

Introduction

The ample information contained in the hyperspectral data can be used to classify classes of the same species, making the Hyperspectral Image Classification (HIC) very attractive in recent years. Among the supervised HIC approaches, Support Vector Machine (SVM) obtains excellent classification result, because it makes a tradeoff between bias and variance via using a compact topology which is established by a small number of carefully selected support vectors [1]. SVM can handle large input spaces efficiently, work with a relatively small number of labeled training samples and deal with noisy samples in a robust way [2], [3]. And various studies are presented to further improve the performance of SVM in recent years. Firstly, multiple-output support vector regression (MSVR) model is designed for addressing the multi-variate application problems [4], [5]. And Fuzzy SVM model is also proposed for efficiently dealing with outliers or noises in classification [6]. Furthermore, due to the importance of the parameters in SVM, many effective parameters optimization methods of SVM are proposed in recent years. The most popular way among them is the grid search, which exhaustively searches on the parameters space for the validation error minimization [7], [8]. Besides grid search, hybrid algorithm such as hybrid comprehensive learning particle swarm optimizer with Broyden–Fletcher–Goldfarb–Shanno (CLPSO–BFGS) algorithm [9], memetic algorithm such as particle swarm optimization and pattern search (PSO–PS) based memetic algorithm [10], and firefly algorithm based methods [4], [11] are proposed to tune the parameters of SVM [9], [10], support vector regression (SVR) [11] and MSVR [4] respectively. These methods greatly improve the stability of the parameter settings, and thus the SVM tuning parameters by them can obtain good generalization performances. Therefore, SVM is well-suited for HIC and has demonstrated excellent performance in terms of accuracy and robustness in recent years [12], [13]. Furthermore, SVM methods for HIC were used in many areas, such as prostate cancer detection [14], land cover classification [15] and plant diseases detection [16], and have obtained good effects.

Although the generalization ability of SVM is very well, the computational complexity of SVM is high, for it solves a set of inequalities constrained quadratic programming. Least Square Support Vector Machine (LS-SVM) is a modified version of the standard SVM, which replaces the inequality constraints with equality constraints when solving quadratic programming [17]. Therefore, LS-SVM is more computationally attractive than SVM and has been applied to HIC [18], [19]. However, LS-SVM does not perform model selection and loses the sparseness of SVM, so the storage cost, computation cost and prediction error of generalization all increase when it is used in HIC. Even though many pruning algorithms have been developed to impose the sparseness on original LS-SVM [20], [21], these approaches need to iteratively omit the training samples and retrain the reduced LS-SVM. Several works aim for alleviating this problem by imposing sparsity on LS-SVM [22], [23]. These techniques are capable of iteratively constructing a sparse LS-SVM when training. However, the iterative training cost is prohibitive. Moreover, the convergence of these algorithms are dependent on the success of optimization algorithms.

In this paper, inspired by the recently developed Compressed Sensing (CS) theory [24], [25], a compact LS-SVM with sparse topology is established to realize accurate classification of hyperspectral images. This sparse model can be considered as a low-dimensional measurement of the original LS-SVM. The sparse topology can be then obtained by learning a compressive measurement matrix from training data and then reducing the useless support vectors by solving a Multiple Measurement Vector (MMV) optimization problem via CS technology. On the other hand, one can observe that neighboring hyperspectral pixels likely belong to the same class. That is, there is spatial homogeneity in the labels of hyperspectral images, which is beneficial for classifying hyperspectral images. Therefore, incorporating spatial information into the classification can improve the classification accuracy [26], [27], [28]. In paper [26], [27], [28], the spatial neighbors set of a given hyperspectral pixel is defined as a small window centered on the given pixel. However, for the pixels lying on the edges of numerous classes, this spatial homogeneity assumption is invalid. In the tiny window, there are some noisy pixels which do not belong to the same class of the center pixel. Taking these noisy pixels as spatial neighbors of the center pixel will involve noisy information in the hyperspectral pixels and decrease the classification accuracy of hyperspectral image. In our work, the LS-SVM is regularized by casting a local adaptive spatial homogeneity assumption on hyperspectral images. The hyperspectral pixel is redefined both in the spectral domain and spatial domain by adaptively selecting its spatial neighbors according to the edge-map. The weighted sum of spectral and spatial features is utilized to construct a Spatial-Spectral Least Square Support Vector Machine (SS-LSSVM) model in this study. And then a Coupled Compressed Sensing inspired Sparse SS-LSSVM (CCS4-LSSVM) for HIC is advanced. By combining spectral information and adaptively extracted spatial information together, CCS4-LSSVM cannot only avoid the speckle-like misclassification of original LS-SVM but also reduce the influence of noisy pixels.

Compared with the available HIC approaches, our proposed CCS4-LSSVM has the following characteristics: (1) CCS4-LSSVM is more computationally attractive than SVM, for it solves a linear system instead of a quadratic programming. (2) The performance of CCS4-LSSVM for HIC is comparable with SVM due to its sparse topology. (3) CCS4-LSSVM is constructed via a one-step strategy and the designed compressive measurement matrix coupled with the dictionary matrix guarantees the high incoherence with dictionary, which avoids the iterative selection of important support vectors and makes a rapid and high-accuracy HIC possible. (4) By combining spectral information with the adaptively extracted spatial neighbors together, CCS4-LSSVM can avoid the influence of noisy pixels and the speckle-like misclassification of the original LS-SVM. Some experiments are conducted on several hyperspectral data to compare the proposed method with its counterparts, and the results show that it can achieve higher classification accuracy than Spatial-Spectral SVM (SS-SVM) and Spatial-Spectral LSSVM (SS-LSSVM).

The remainder of the paper is organized as follows: Section 2 depicts the proposed CCS4-LSSVM. In Section 3, some experiments are conducted to investigate the performance of our proposed method. A conclusion is presented in Section 4.