REMOTE sensing hyperspectral sensors, such as HyMap

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1 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 9, NO. 4, JULY Using Hurst and Lyapunov Exponent For Hyperspectral Image Feature Extraction Jihao Yin, Member, IEEE, Chao Gao, and Xiuping Jia, Senior Member, IEEE Abstract Hyperspectral image processing has attracted high attention in remote sensing fields. One of the main issues is to develop efficient methods for dimensionality reduction via feature extraction. This letter proposes a new nonlinear unsupervised feature extraction algorithm using Hurst and Lyapunov exponents to reveal local and general spectral profiles, respectively. A hyperspectral reflectance curve from each pixel is regarded as a time series, and it is represented by Hurst and Lyapunov exponents. These two new features are then used to overcome the Hughes problem for reliable classification. Experimental results show that the proposed method performs better than a few other feature extraction methods tested. Index Terms Feature extraction, Hurst exponent, hyperspectral image, Lyapunov exponent. I. INTRODUCTION REMOTE sensing hyperspectral sensors, such as HyMap and Hyperion, can generate hundreds of images of a scene on the Earth s surface [1]. Each of them detects the reflectance from the imaged area at a narrow waveband between 400 and 2400 nm. Such high spectral resolution data with a complete coverage of the optical reflective wavelength range provide an opportunity to quantitatively identify ground cover materials. Active research works have been conducted in the last three decades on hyperspectral data for various applications, including mineral identification, ecological monitoring, vegetation survey, and precision agriculture. Unlike multispectral data, the major difficulty in hyperspectral image analysis is how to cope with the high dimensionality of its data due to the Hughes problem, where the classification accuracy increases at first and then decreases with the increase of wavebands but limited number of training samples [2] [7]. Therefore, the number of training samples required for reliable parameter estimation becomes very high, and it is hardly met in practice. Feature reduction is one of the hot topics in hyperspectral data processing. It can be done in two ways. One is band Manuscript received April 8, 2011; revised August 1, 2011 and November 12, 2011; accepted November 28, Date of publication January 23, 2012; date of current version May 7, This work was supported in part by National Natural Science Foundation of China and Chinese Academy of Sciences Joint Fund of Astronomy under Grant , by the Research Fund for the Doctoral Program of Higher Education of China under Grant , by the Aero Science Fund of China under Grant , and by the Aerospace Science and Technology Fund of China under Grant CASC J. Yin and C. Gao are with the School of Astronautics, Beihang University, Beijing , China ( yjh@buaa.edu.cn; chao-gao@sa.buaa.edu.cn). X. Jia is with the School of Information Technology and Electrical Engineering, University College, The University of New South Wales, Australian Defence Force Academy, Canberra, ACT 2600, Australia ( x.jia@ adfa.edu.au). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /LGRS selection, in which an optimal subset of bands will be selected directly from the original databased on defined criterions, such as the Bhattacharyya distance [8], the largest simplex in pixel space [9], or mutual information [10]. The other is feature extraction, where a transformation is applied to the highdimensional data and a small set of new features is produced for preserving the original information as much as possible. This letter focuses on feature extraction. Several feature extraction methods from the complete set of spectral bands have been developed and used widely. A popular feature extraction method is principal component analysis (PCA). PCA is easy to implement based on the statistics of the global data of the complete set of wavebands or several groups of segmented wavebands [11]. Retaining the first, a few principal components can best represent the data in the sense of the least squared error. However, PCA could not process nonlinear data effectively. As a useful tool for signal decomposition, wavelet packet analysis (WPA) [12] can convert data sets into the frequency domain for further operation. Other nonlinear methods include Laplacian eigenmap (LE) and kernel PCA (KPCA). LE originated from the concept of manifold and human cognitive science [13], while KPCA combines PCA with so-called kernel tricks [14]. However, negative values may exist in transformed data, which remain difficult to define by exact physical meaning. Nonnegative matrix factorization (NMF) can prevent this dilemma, but its solutions are deeply influenced by the initial values for iterations [15]. Feature extraction can also be conducted on a few selected spectral bands for specific applications via simple arithmetic operations. For example, Normalized Difference Water Index [16] is generated to enhance water content by calculating the ratio between the difference reflectance at the green band and near infrared band and their sum. Normalized Difference Vegetation Index [17] can be derived using the ratio of the difference to the sum of the measurements at the near infrared band and the red band. Furthermore, absorption-band detection [18] can be regarded as another type of feature extraction method, which focuses on certain specific bands and concerns about differences in the first and second derivatives to identify the depth, position, and asymmetry of the absorption [19], [20]. Moreover, there are also various supervised feature extraction methods, such as decision boundary feature extraction [21], nonparametric weighted feature extraction [22], and kernel nonparametric weighted feature extraction [23], aiming at maximizing the separation of interesting classes. When compared with unsupervised methods, these supervised methods directly can find the subset of features that provide the best separation between the classes of interest. However, they are sensitivity to the quality and quantity of the training samples X/$ IEEE

2 706 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 9, NO. 4, JULY 2012 A complete new nonlinear unsupervised method for feature extraction is proposed in this letter, which generates two generic indices from the complete set of spectral measurements. We introduce Hurst [24] and Lyapunov [25] exponents to reveal local and general spectral profiles, respectively. A hyperspectral reflectance curve from each pixel is regarded as a chaotic series, and it is represented by these two measures. We denote this method as Hurst Lyapunov feature extraction (HLFE). Details of new feature generation are given in Section II. In Section III, experiments and results will be presented, followed by conclusion in Section IV. II. HLFE The Hurst exponent (H) and the largest Lyapunov exponent (L) are the two measures used in nonlinear time series analysis. H is one classical value to detect long memory in time series. When the series is persistent, H will be greater than 0.5, and when it is antipersistent, H should be less than 0.5 [24]. In other words, the smoother the local neighborhood is, the larger the H should be. Thus, it can reveal the local trend of series among adjacent successive samples. L is an important quantitative indicator to measure system dynamics, which characterizes the average rate of the convergence or divergence in phase space between adjacent tracks [25]. The most notable characteristics of chaotic systems are the system s extreme sensitivity to initial values. If L is greater than zero, two trajectories generated by almost the same initial values will separate exponentially over time. Therefore, L can reveal the general trend of series. For hyperspectral data, there should be similar pattern among the hyperspectral reflectance curves of the same materials. This inspires us to treat a spectrum as a time series. However, different bands of hyperspectral data often vary greatly, and it is hard to guarantee the homogeneity of variances for all bands, which can be called heteroscedasticity in statistics if the variables have different variances or volatilities. Hyperspectral data are typical heteroscedastic variables, which makes it inappropriate to apply the normal/classic time series analysis, such as the autoregressive-integrated-moving-average model. In contrast, there is no such limit when calculating H and L in heteroscedastic series, and thus, it can avoid the potential side effect of heteroscedastic disturbances theoretically. H and L, as used in nonlinear time series analysis, have been found very powerful for revealing local and general spectral profiles. Moreover, the calculation of H and L can be implemented to any series or curves. On this note, it is not an issue to apply these two indices onto hyperspectral curves whether hyperspectral data are chaotic time series. The main idea of the proposed feature extraction method is to treat the hyperspectral reflectance curves as heteroscedastic time series and use their H and L to replace original spectrums. The dimensionality is therefore reduced from a couple of hundreds to two. The methods for finding H and L are given in the next two sections. A. Hurst Exponent The well-known method to estimate H is R/S analysis [24], where R is the local mean-removed cumulative range and S is the local standard deviation of the series. The procedure is as follows. 1) Divide the given series {x k k =1,,K} into D subseries, where K is the total number of bands. The dth subseries is denoted as X d = {x k k =(d 1) n +1,..., d n}, d =1,,D. Generally, the length of each subseries n is set between ln K and K. For convenience, rename the samples in each subseries as g d,i, where g d,i = x (d 1) n+i, d =1,,D, and i =1,,n. 2) For each subseries, find the mean m d and the standard deviation S d n m d = 1 n g d,i i=1 S d = 1 n n. (1) (g d,i m d ) 2 i=1 3) Normalize each subseries by subtracting its sample mean g d,i = g d,i m d, i =1,...,n. (2) 4) Find the mean-removed cumulative series {c d,i } and its range R d for each subseries c d,i = i gd,j, i =1,...,n j=1 R d =max 1 i n {c d,i} min 1 i n {c d,i}. (3) 5) Calculate the mean value (R/S) n of the rescaled range R d /S d for all subseries with length n (R/S) n = 1 D D R d /S d. (4) d=1 6) Repeat the aforementioned steps for all possible n. 7) The H is estimated by the following empirical equation: (R/S) n = c n H. (5) The value of H can be obtained by running a simple linear regression over least square method after taking the logarithm of both sides of the above equation ln(r/s) n =lnc + H ln n. (6) The slope of the regression line is H. B. Largest Lyapunov Exponent There are several feasible methods for solving L. We select Wolf s algorithm [26] for its concision. The major steps are listed as follows. 1) Determine the mean period p, the delay time τ, and the embedding dimension m of the hyperspectral series {x k k =1,,K}, where K is the total number of bands. In our experiments, p =20, τ =3, and m =8. 2) Reconstruct the subspace of size m with waveband gap τ Y j = ( x j,...,x j+(m 1)τ ) R m, j =1,...,M (7) where M = K (m 1)τ is the number of subspaces. 3) Define Y J as the initial evolved point, and select Y 1 as the first initial evolved point Y J1.

3 YIN et al.: USING HURST AND LYAPUNOV EXPONENT FOR HYPERSPECTRAL IMAGE FEATURE EXTRACTION 707 Fig. 1. Results of HLFE for real objects from Washington, DC Mall. 4) Locate Y J1 s nearest neighbor Y J1 among those having a waveband separation greater than the mean period p of the spectrum, and denote the distance between these two points as l J1 l J1 =miny J1 Y J1, J1 J 1 >p. (8) 5) At a later subspace t, the initial length will evolve to Y J1 +t Y J 1 +t. Chose the minimum of t which makes { } T 1 =min t : Y J1 +t Y J 1 +t >ε (9) where ε is the average distance of all points in the reconstructed subspace ε = 2 M(M 1) M M j=1 j =j+1 Y j Y j. (10) Denote the corresponding evolved length l J1 as lj1 = Y J1 +T 1 Y J1 +T 1. (11) Fig. 2. False color image of Washington, DC Mall. 6) Look for a new initial evolved point Y J2 that has a small separation from Y J1 +T 1 and simultaneously has a small angular separation between the evolved and replacement elements. Repeat the aforementioned steps with the new initial evolved point until reaching the end of spectral series. 7) Denote N as the total number of replacement steps, and Y J1,,Y JN are all initial evolved points, so we can estimate 1 L = J N J 1 N n=1 log 2 ljn l Jn. (12) We can see that the delay τ controls the waveband gap when a subspace series is created. When it is selected as three, every fourth spectral band is used to form the new subspace. In this way, the correlation between the neighboring bands in the subspace series is lower than that in the original space. The imbedding parameter m determines the number of samples in each subspace. Considering the spectral resolution of 10 nm for AVIRIS data, each subspace will cover the wavelength range of Fig. 3. False color image of Indian Pine = 240 nm when the delay is set to three. The mean period p avoids the selection of the neighboring subspaces as the closest subspaces. In practice, users may select based on their understanding of the function of each parameter, via crossvalidation or by trial and error. C. H LMeasures for Earth s Surface Materials As we can see from the methods for calculating H and L, they reflect the inherent characteristics of each hyperspectral

4 708 IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 9, NO. 4, JULY 2012 TABLE I PRODUCER S ACCURACY (PA), USER S ACCURACY (UA), AND OVERALL ACCURACY (OA) OF WASHINGTON, DCMALL DATA (%) TABLE II PRODUCER S ACCURACY (PA), USER S ACCURACY (UA), AND OVERALL ACCURACY (OA) OF INDIAN PINE DATA (%) reflectance curve. In particular, H can describe the local change in the ratio between the ranges of accumulated mean-removed values to the original standard deviation and thus represent the diversity of spectral values, while L can concern differences in pairs of reconstructed neighboring spectral bands greater than the mean period and thus represent the changes in the curvature of the spectral curves. A full spectral curve at a pixel can now be reduced to these two features to capture the local and general nonlinear features, respectively, which allow the classification of hyperspectral images more reliably. Fig. 1 illustrates the normalized results of HLFE for some classes in the Washington, DC Mall data. Detailed information of this data set will be provided in the next section. From Fig. 1, we can see that different classes tend to gather in different regions. III. EXPERIMENTS A. Data Sets In this letter, two real data sets are applied to compare the performances of HLFE with other unsupervised featureextraction methods. They are the urban area of Washington, DC Mall and the mixed forest/agricultural field of Indian Pine [27]. The data set of the Washington, DC Mall was collected by the Hyperspectral Digital Imagery Collection Experiment Sensor (HYDICE) on August 23, The original data set includes 210 bands in the range of μm. Those bands from infrared spectrums have more noise than bands from visible wavelengths due to water absorption, which leads to the removal of 19 noisy bands. A thematic map with groundtruth labels is available with the original data. The candidate region is the subset including 542 rows by 307 columns by 191 bands from the original image. Its false color image is depicted in Fig. 2. In terms of this selected image, there are six types of objects. They are Roof, Street, Path, Grass, Tree, and Water. The Indian Pine hyperspectral image (145 by 145 pixels), gathered by the Airborne Visible/Infrared Imaging Spectrometer (AVIRIS), has 220 spectral bands with a spatial resolution of 20 m. Since there is no plan to provide a thematic map for the Indian Pine area, Corn-min, Corn, Grass, Hay, Wood, and Stone are selected for investigation. Its false color image is depicted in Fig. 3. All the labeled points are evenly divided into two subsets. Half of the labeled points are chosen as training samples, and the rest become test samples. The back-propagation artificial neural network (BPANN) [28] is used as the classifier. The numbers of neurons for input layer (IN), hidden layer (HI), and output layer (OU) are set to be 2, 11, and 3, respectively. The symmetric sigmoid function and linear function are used as the transfer functions from IN/HI to HI/OU, respectively. The gradient descent weight/bias function is used as the learning function, and the mean square error is used as the evaluation criterion in the performance function. PCA, WPA, LE, KPCA, and NMF were tested for comparative analysis. The wavelet basis function of WPA is Haar, and the kernel function of KPCA is Gauss. B. Results and Discussions Tables I and II give the producer s accuracy (PA), user s accuracy (UA), and overall accuracy (OA) of HLFE and other comparative algorithms for Washington, DC Mall and Indian Pine, respectively. Considering those samples classification results, we can see that HLFE has relatively higher overall accuracy than other comparative algorithms. Therefore, HLFE is considerably effective for feature extraction. For the data set of Washington, DC Mall, it is clear that Water has a strong divisibility when considering Table I. However, there are small overlaps among Roof, Street, and Path. The main reason for this phenomenon may result from the similar materials contained in those classes. More importantly, the

5 YIN et al.: USING HURST AND LYAPUNOV EXPONENT FOR HYPERSPECTRAL IMAGE FEATURE EXTRACTION 709 training samples of Path are so limited that most comparative algorithms fail to get the complete information about Path, which leads to relatively lower accuracy when compared with HLFE. Thus, HLFE performs less sensitive than other algorithms for objects with small samples, since it involves local and general statistics directly, controlled by the parameters including the imbedding dimension, delay constant, and mean period. For the data set of Indian Pine, the results are consistent with the previous one. Most samples are classified correctly, shown in Table II. Part of Corn-min and Corn are erroneously recognized to others. So do Grass and Wood. The main reason may be that they all belong to plant or vegetation. Again, the number of pixels of Corn is small, but HLFE can still give relatively satisfactory results. In summary, HLFE achieves the highest classification accuracy on average in this data set. From Tables I and II, we have the following findings. 1) HLFE can outperform other classical unsupervised feature extraction methods with the classifier of BPANN. 2) As a nonlinear unsupervised feature extraction technique, HLFE is more powerful than linear transformation and can deal with nonlinear data structure efficiently. 3) HLFE is insensitive to the size (number of pixels) of the class of interest and thus achieves a good performance both to the low and high populated classes. IV. CONCLUSION This letter addresses the issue of unsupervised feature extraction for hyperspectral image classification and introduces Hurst and Lyapunov exponents as new extracted features for each spectrum. They are two generic indices to reveal local and general spectral profiles, respectively, are not limited to defined applications, and do not require training data. On the other hand, as the proposed method is unsupervised, it does not directly address the separation between the classes of interest. Another important characteristic is that the two features are extracted from the complete set of spectral measurements. Experimental results show that the two new features have the capacity to separate different ground cover types. High classification accuracy is achieved, showing that HLFE can not only largely reduce the dimension of data space but also simultaneously keep reliable information about objects. Extra computational load is imposed when HLFE is performed as a preprocessing step. This extra computational load will be partly offset by the large saving in the classification stage when only two features are used. The remaining cost is spent on gaining the advantage of overcoming the Hughes problem. ACKNOWLEDGMENT The authors would like to thank the reviewers for their valuable comments and suggestions. REFERENCES [1] J. A. Richards and X. 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