DIMENSIONALITY REDUCTION OF HYPERSPECTRAL IMAGES WITH WAVELET BASEDEMPIRICAL MODE DECOMPOSITION

Esra Tunc Gormus, Nishan Canagarajah, Alin Achim

Department of Electrical and Electronic Engineering University of BristolBristol, BS8 1UB, UK

(Esra.TuncGormus, Nishan.Canagarajah, Alin.Achim)@bristol.ac.uk

ABSTRACT

This paper presents an application of the Empirical Mode Decom-position (EMD) method to wavelet based dimensionality reduction,with an aim to generate the smallest set of features that leads to thebest classification accuracy. Useful spectral information for hyper-spectral image (HSI) classification can be obtained by applying theWavelet Transform (WT) to each hyperspectral signature. As EMDhas the ability to describe short term spatial changes in frequencies,it helps to get a better understanding of the spatial information ofthe signal. In order to take advantage of both spectral and spatialinformation, a novel dimensionality reduction method is introduced,which relies on using the wavelet transform of EMD features. Thisleads to better class separability and hence to better classification.Specifically, the 2D-EMD is applied to each hyperspectral band andthe 1D-DWT is applied to each EMD feature of all bands in or-der to get reduced Wavelet-based Intrinsic Mode Function Features(WIMF). Then, new features are generated by summing up the lowerorder WIMF features. The superiority of the proposed method com-pared to direct wavelet-based dimensionality reduction methods isproven by using the AVIRIS Indian Pine hyperspectral data. Com-pared to conventional direct wavelet-based dimensionality reductionmethods, our proposed method offers up to 65% dimensionality re-duction for the same classification performance.

Index Terms— Empirical Mode Decomposition (EMD), Dis-crete Wavelet Transform (DWT), Dimensionality Reduction, Sup-port Vector Machines (SVMs),Classification.

1. INTRODUCTION

Classification performance generally depends on four factors whichinclude class separability, training sample size, dimensionality re-duction, and classifier type [1]. The focus of our study is dimension-ality reduction, which helps remove redundant information withoutsacrificing significant information and also helps to avoid the curseof dimensionality in order to increase classification accuracy. Thereare two approaches to reduce data dimensions. The first approach isto select a small subset of features directly from the original featurespace according to their contribution to the class separability or clas-sification criteria. This dimensionality reduction process is referredto as feature selection [2]. The other approach is based on using alldata from the original feature space and map the effective featurestogether with useful information to a lower-dimensional subspacelike Principal Component Analysis (PCA), Fast Fourier TransformFFT or Wavelet Transform (WT) [1].

Over the last decade several wavelet based dimensionality re-duction methods have been proposed [3, 4, 1, 5]. Due to the time fre-

quency localization properties of wavelets, Discrete Wavelet Trans-form (DWT) has been exploited in [4] in order to extract the mostdiscriminative multiscale features. In [5], after applying DWT to hy-perspectral data, wavelet coefficients, wavelet energies, and waveletdetail histograms have been used as features for classification. Inanother study [1], WT has been applied to HSI and a sequence ofwavelet coefficients has been obtained. Then, a feature selectionprocedure was used to select the effective features for classification.

EMD was introduced for 1D signal analysis in [6] and extendedto 2D image analysis in [7]. Subsequently, it has been applied toother areas of image processing, such as in image classification [8]and dimensionality reduction of hyperspectral data [9]. In [9], 1DEMD has been applied to class means of hyperspectral data after awhitening process and then it has been followed by another featureextraction in order to increase the classification accuracy.

This study proposes the application of wavelet based EMD fea-tures for dimensionality reduction of HSI. Specifically, first 2D EMDis applied to each band to extract spatial information, then 1D DWTis applied on each Intrinsic Mode Function (IMF) of all the bandsto extract the spectral information in the images. Finally, the newreduced wavelet based EMD features are directly fed into an SVMclassifier in order to be assessed and compared according to theiroverall classification accuracy per number of features.

This paper is organized as follows. In the 2nd section existingwavelet based methods are reviewed, followed in section 3 by somegeneral information about EMD. Then, the proposed method is pre-sented in 4th section. In section 5, the proposed wavelet based EMDapproach is compared with direct wavelet based dimensionality re-duction methods. Finally, conclusions are given in section 6.

2. WAVELET BASED DIMENSIONALITY REDUCTION

An object may be more easily identifiable based on the shape of itsspectral signature rather than on the magnitude of the spectral values.Consequently, in wavelet based approaches, the 1D DWT has gener-ally been applied to pixel spectral signatures rather than to the mag-nitude of the spectral values. Then, the fewest wavelet coefficientsrequired to perform dimensionality reduction are selected. Waveletbased dimensionality reduction methods show better classificationperformance compared to PCA and FFT because of wavelets’ abil-ity to detect the local energy variation of hyperspectral signals indifferent spectral bands at each scale.

The wavelet transform decomposes a signal into approximationand detail coefficients, which represent the low and high frequencycomponents respectively. In the literature, DWT based dimension-ality reduction methods differ according to the criteria used for theselection of wavelet coefficients. Linear wavelet feature extraction

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(Linear WFE) and non-linear wavelet feature extraction (Non-LinearWFE) methods are two of these and have been introduced in [1, 3]respectively. Linear WFE decomposes the hyperspectral signatureusing DWT and then selects the fewest R wavelet coefficients re-quired to perform dimensionality reduction. Later, either approxi-mation [1, 3] or detail coefficients [1] have been respectively usedas features for classification. On the other hand, in non-linear WFEmethod, after decomposing the signatures, wavelet coefficients havebeen sorted in decreasing order and the first largest R coefficientshave been taken as the important features for classification [1]. Thedecomposition level has been decided by the similarity metrics (cor-relation) between the original spectra and the reconstructed spectralapproximation in [3], whereas in [1], the decomposition level is se-lected according to an user defined R value. In this study we willshow that these 2 methods can be significantly improved by usingthe 2D EMD framework.

3. EMPIRICAL MODE DECOMPOSITION

Limitations due to filters and cost functions in combinations withdifferent signals make the wavelet transform difficult to use in adap-tive systems that help to get a better understanding of the physicsbehind the signals [7]. In order to achieve this purpose, EmpiricalMode Decomposition (EMD) was introduced in [6]. Complicatedsignals can be decomposed into Intrinsic Mode Functions (IMF) anda residue with none or with only a few extreme points. The EMDsorts the spatial frequency along the characteristic scales of the pro-cess. The first IMF in 2D EMD extracts the highest local spatialfrequencies in the image while the second IMF holds the next high-est spatial frequencies and so on. There are two conditions for anIMF. First, the number of zero crossing and the number of extremepoints must be equal or differ at most by one. Then the mean valuesof the envelopes defined by the local maxima and the local minimamust be zero at any point. EMD is deterministic which means itdoes not depend on signal statistics and is based only on the localproperties of the signal. It gives information about not only the interfrequencies generated by the construction of the signal but also theintra frequencies [7].

The sifting process is an iterative process which is used to findIMFs of hyperspectral bands. It starts with the first hyperspectralband itself I(1)l,1 = Bl(i, j), as input signal. (n) is the iteration num-ber, l = 1, 2, ....L is the band index, where L is the total number ofbands in the hyperspectral data cube, (i, j) is the spatial location andm = 1 : M , M is the number of IMFs. More background theoryon EMD can be found in [6]. In the following the sifting process of2D-EMD for finding the first IMF of lth spectral band IMF1,m issummarised.

1. Find the 2D positions and values of all local maxima and localminima in the input image,

2. Create the upper envelope (Emax(i, j)) by 2D spline interpo-lation of the local maxima and the lower envelope Emin(i, j)by 2D spline interpolation of the local minima,

3. For each spatial position (i, j), calculate the mean of the up-per and the lower envelopes,

EM nm(i, j) =

Emax(i, j) + Emin(i, j)

2(1)

4. Subtract the envelope mean from the input signal

Snm(i, j) = In

1m(i, j)− EM nm(i, j) (2)

This is one iteration of the sifting process,

5. Check whether the obtained 2D image from step 4 is an IMF.The stopping criterion for the checking process is for the en-velope mean to be close to zero [8];

P

(∑i=l

R∑j=l

|EM nm(i, j)|)/(PxR) < τ, (3)

where P andR are the dimensions of the mean envelope, andτ is a small threshold close to zero. If the stop criterion is ful-filled at N th (n = N) iteration, the current IMF is obtainedas IMFl,m(i, j) = SN

m(i, j), if not, repeat the process fromstep 1 with the resulting image from step 4 as the input signaluntil finding the first IMF; In+1

l,m (i, j) = Snm(i, j),

6. If the current IMF obtained successfully, the residue signalRm(i, j) = In

l,m(i, j) − IMF l,m(i, j) is computed. If theresidue doesn’t contain any more extreme points, the EMDprocess is stopped. Otherwise the next IMF is obtained start-ing from step 1, using the current residue as the next input;In+1

l,m (i, j) = Rm(i, j).

The original hyperspectral image bandBl(i, j) can be exactly recon-structed by adding all corresponding IMFs and the final residue.

Bl(i, j) =

M∑m=1

IMFl,m(i, j) +RM (i, j) (4)

4. PROPOSED METHOD

In [8], 2D-EMD has been used to obtain the IMFs of hyperspec-tral image bands. New features have been reconstructed by addingthe lower order IMFs of all bands. According to the results in [8],the sum of lower order IMFs increases the between-class distanceswhich also leads to better classification accuracy in the spatial do-main.

In this study we propose a dimensionality reduction methodwhich uses both spatial information by exploiting 2D-EMD, andspectral information, by exploiting 1D-DWT on the new IMF fea-tures. It has been claimed in previous studies that mixing the spectraland spatial information is an efficient way of achieving better clas-sification [10]. Motivated by this idea, wavelet based IMF features(WIMF) are introduced to get better classification accuracy.

Specifically, the proposed method is described as follows;

1. 2D-EMD is applied to each hyperspectral image band in thespatial domain and M orders of IMFs are obtained,

2. 1D-DWT is applied along each pixel of each IMF feature inall bands IMFl,m(i, j),

3. Obtain M order of IMF features as a result of step 2,WIMFm, m = 1 :M ,

4. Generate new wavelet based IMF features WIMF (1 : M)by summing up lower order WIMFm. e.g. When summingup WIMF1 and WIMF2, the new feature will be denotedas WIMF12, etc. as formulated in Formula 5.

WIMF1..M(i, j) =

M∑m=1

WIMFm(i, j) (5)

5. According to the type of wavelet decomposition, selectR fea-tures from new wavelet based IMF featuresWIMF (1 :M).

6. Classify R features using an SVM classifier.

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Fig. 1. Proposed method in flowchart

The proposed method is also presented as a flowchart in Fig. 1.The drawback of this method is that the computational cost of

EMD is very high and decomposition time also depends on the com-plexity of the signal. By contrast, in DWT case, the signal decom-position time depends only on the signal length and the number oflevels extracted.

5. EXPERIMENTAL RESULTS

AVIRIS Indian Pine data set [11] is used for the experiments in thisstudy. This image (145x145) was taken over the northwest Indi-ana’s Indian Pine test site in June 1992 and has sixteen classes. Thedata has 220 spectral bands, about 10 nm apart between 0.4 to 2.45µm with a spatial resolution of 20 m. The 20 water absorptionbands (numbered 104-108, 150-163, and 220) were removed fromthe original image. The original ground truth has 16 classes, butsome classes have a very small number of elements. Therefore, nineclasses that have a higher number of samples have been used (Fig.2) and the number of labelled samples per class is given in Table 2.

SVM is used with Radial Basis Function (RBF) kernel becauseof their high classification performance. The penalty parameter C ofSVM is set to 1000 and the γ parameter of the RBF kernel is testedbetween [0.1-2] using a five fold cross validation (as in [8]). Theone-against-one multi-class classification algorithm is used becauseof its ability to train fast. Half of the pixels from each class are usedfor training, with the remaining 50% forming the test set on whichperformance is assessed.

It has been experimentally found that the best parameters for2D-EMD are 17x17 size window, 0.006 for stopping criteria, and4 IMFs are selected. The 1D-DWT is implemented along 3 levelDaubechies-3 (db3) wavelet. The feature number R is chosen as 29which corresponds to the number of approximation features after 3decomposition levels. The proposed method is performed both forLinear WFE and Non-Linear WFE. The overall classification resultsof SVM with EMD based Non-Linear wavelet features and EMDbased Linear wavelet features for both approximation and detail co-

Fig. 2. Classification results of W_A and the proposed WIMF12 forthe same number of features

efficients are compared with the classification results of the originalwavelet method [1] in Table 1.

We compare the methods according to their improvement interms of the dimensionality reduction of the data and the increasein classification accuracy for a given number of features. Fromthe direct WT features in Table 1, it can be seen that the highestclassification accuracy (93.58%) is obtained by classifying 25 ap-proximation (A) features only and second highest classification isobtained by Non-Linear WFE (ranking A and D) with 79.62% ac-curacy for 29 features. The worst classification result obtained byclassifying D (details) features only give 74.44% accuracy with 25features. For the proposed method, the classification accuracy ob-tained by using the first and the first two Non-Linear WIMF featuresis under 57%, but it increases dramatically to 94.43% by summingup the first three Non-Linear WIMF features. The performanceincreases to 96.08% by adding the fourth Non-Linear WIMF featureto the first three Non-Linear WIMF (WIMF1234) features with 29features. Thus, we can see that the classification result with theEMD based Non-Linear WFE is 2.5% better than all direct WFEmethods and the highest accuracy obtained in direct WFE (93.58%)is captured by less than 20 Non-Linear WIMF1234 features with94.31% accuracy. This difference is significantly higher in the EMDbased Linear WFE. By exploiting only the first linear WIMF feature(WIMF1) 98.29% accuracy is obtained by classifying 25 A fea-tures and 97.07% is obtained by classifying 25 D features. Furthercombination with the second, third and fourth WIMF slightly in-creases even more the classification accuracy of EMD based LinearWFE. The highest classification accuracy obtained with the pro-posed method (98.59%), corresponds to the summation of the firstthree WIMF (WIMF123)’s 29 A features. In terms of dimension-ality reduction, the highest accuracy for direct WFE (93.58%) isachieved by less than 10 A features of WIMF1234 with 96.57% ac-curacy (exactly with 6 D -WIMF1234 features), resulting in around65% reduction compared to conventional WT based dimensionalityreduction methods. The Fig. 2 shows the enhanced classificationresults of WIMF12-A for the same feature number compared toW_A method.

The proposed method combines the advantages of both EMD

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Table 1. Overall classification result of W_A (Approximation coefficients of Linear-WFE), W_D (Detail coefficients of Linear-WT), W_NL(Largest NoF coefficients of NonLinear-WT), A (approximation) and D (detail) of Linear and Non-linear WIMF1 (The first wavelet basedIMF feature ), WIMF12, WIMF123, and WIMF1234.

WFE Methods Proposed EMD based Linear WFE Proposed EMD based Non-Linear WFE

NoF W_A W_D W_NL WIMF1

WIMF12

WIMF123

WIMF1234

WIMF1

WIMF12

WIMF123

WIMF1234

A D Both A D A D A D A D Both Both Both Both

5 53.67 48.64 53.86 47.51 57.70 56.35 68.80 84.05 91.82 83.82 92.29 35.41 38.81 72.43 72.23

10 80.24 66.17 64.35 88.70 91.74 92.44 94.82 95.40 96.42 95.42 96.57 41.02 45.54 86.98 88.10

15 89.45 68.85 67.39 96.40 94.50 96.55 96.57 97.69 96.79 97.50 96.85 44.06 50.25 91.18 93.08

20 92.44 72.70 69.24 97.54 96.27 98.05 97.39 98.07 97.11 97.99 97.07 47.12 53.11 92.76 94.31

25 93.58 74.44 76.13 98.29 97.07 98.63 98.52 98.50 97.02 98.44 97.07 48.11 54.66 93.68 95.25

29 93.04 73.97 79.62 98.03 95.74 98.57 97.71 98.59 97.09 98.39 97.24 49.26 56.18 94.43 96.08

Av. 83.74 67.46 68.43 87.75 88.84 90.10 92.30 95.38 96.04 95.26 96.18 44.16 49.76 88.58 89.84

Table 2. Number of samples for different classes of the Indian Pine data set used in the experimentsClass Name Corn no

tillCorn-min

tillGrass /Pasture

Grass /Trees

Hay -windrowed

Soybean -no till

Soybean -min till

Soybean -clean till

Woods Total

NoS 1434 834 497 747 489 968 2468 614 1294 9345

and DWT in reducing the dimensionality of HSI because it incorpo-rates both spatial information and spectral information. The advan-tage of using IMF features stems from their ability to represent fur-ther detailed information and more features of the image [7]. DWTfeatures bring not only sharp localization but also sharp frequency[12], which leads to better classification with less number of fea-tures.

6. CONCLUSION

This paper described an application of the 2D-EMD method towavelet based dimensionality reduction and introduced waveletbased IMF features (WIMF) for dimensionality reduction in orderto get enhanced classification accuracy. Our proposed method of-fers up to 65% dimensionality reduction for the same classificationaccuracy when compared to direct wavelet-based dimensionalityreduction methods. Future work will consider the application ofthe proposed method in conjunction with multidimensional wavelettransform for the classification of HSI.

7. ACKNOWLEDGEMENT

The first author would like to thank Dr. Begum Demir for usefuldiscussions and for her inspirational work in hyperspectral imaging.This work has been partly supported by the Institute of Materials,Minerals and Mining’s scholarships (Bosworth Smith Trust, EdgarPam Fellowship, and the G. Vernon Hobson Bequest).

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