Earth Sci Inform DOI 10.1007/s12145-014-0168-0
RESEARCH ARTICLE
A stable and accurate wavelet-based method for noise reduction from hyperspectral vegetation spectrum Ladan Ebadi & Helmi Z. M. Shafri
Received: 15 January 2014 / Accepted: 9 June 2014 # Springer-Verlag Berlin Heidelberg 2014
Abstract Hyperspectral vegetation spectrum is normally contaminated with noise and the presence of noise affects the results of vegetation studies, such as species discrimination and classification, disease detection, stress assessment and the estimation of vegetation’s biophysical and biochemical characteristics. Additionally, hyperspectral signals are usually studied using the derivative analysis method that is very sensitive to noise in the data. This study investigates denoising of the hyperspectral vegetation spectrum using different wavelet-based methods. A test signal and several real-world vegetation spectra are denoised using four wavelet methods: traditional discrete wavelet transform (DWT); stationary wavelet transform (SWT); lifting wavelet transform (LWT); and a combination of SWT and LWT, which in this paper is called stationary lifting wavelet transform (SLWT). SLWT incorporates the advantages of both SWT and LWT methods, including a translation invariance property and a fast simple algorithm. Experimental results show that SLWT highly outperforms other wavelet-based methods in terms of accuracy and visual quality. Furthermore, this research reveals the following novel results: SLWT 1) for different levels of Communicated by: H. A. Babaie L. Ebadi (*) : H. Z. M. Shafri Geomatics Engineering Unit, Department of Civil Engineering, Faculty of Engineering, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia e-mail:
[email protected] H. Z. M. Shafri e-mail:
[email protected] H. Z. M. Shafri e-mail:
[email protected] H. Z. M. Shafri Geospatial Information Science Research Centre (GISRC), Faculty of Engineering, Universiti Putra Malaysia (UPM), 43400 Serdang, Selangor, Malaysia
decomposition of the wavelet transform gives similar results and its denoising results is independent to the selection of decomposition level; 2) generates stable statistical results; 3) can make use of mother wavelets with small filter size (i.e., low-order mother wavelets) that are suitable for preserving subtle features in vegetation spectrum; and 4) its denoising results do not depend on the selection of the mother wavelet when applying low-order mother wavelets. Keywords Hyperspectral remote sensing . Vegetation spectrum . Noise reduction . Noise removal . Denoising . Stationary wavelet transform . Second generation wavelet
Introduction Hyperspectral remote sensing technology generates a variety of data types, such as high resolution aerial images, hyperspectral satellite images and spectroradiometric signals, by using airborne, space-borne and field or laboratory sensors. The sensors gather data using hundreds of narrow spectral bands at different wavelength channels to generate continuous spectra of the Earth’s features (Plaza et al. 2010). These data are contaminated by noise from different sources, such as atmospheric conditions, the oscillation of light intensity, and self-generated noise in the sensors. Also the hyperspectral sensors that make the measurements based on point spectroscopy method produce photon noise that has Poisson distribution (Uss et al. 2011). Noise supresses the small spectral features in the spectra and reduces the efficiency of spectral discrimination among different features; while noise-free spectra increases the classification accuracy considerably (Kusuma et al. 2010). Moreover, many hyperspectral remote sensing applications such as spectral detection and measurement, target recognition, land cover classification and
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clustering need high signal-to-noise ratio (SNR) to obtain good results (Bilgin et al. 2008). One-dimensional hyperspectral signals, acquired by spectroradiometry or extracted from hyperspectral images, are usually studied using derivative analysis. In vegetation studies, the first and second derivatives are used to estimate foliar chemical concentrations, like foliage nitrogen concentration (Huang et al. 2004), and also for crop stress detection (Estep and Carter 2005). In the frequency domain, derivatives act as high-pass filters and thus they magnify highfrequency noise in the data (Bruce and Li 2001); therefore, the derivative analysis method is very sensitive to the noise in data. The vegetation reflectance spectra acquired by hyperspectral remote-sensing techniques have been widely used for various applications, such as species discrimination (Vaiphasa et al. 2007; Banskota et al. 2011; Adjorlolo et al. 2012; Pu et al. 2012), vegetation and agricultural crop classification (Thenkabail et al. 2004; Thenkabail and Wu 2012), crop mapping (Biradar et al. 2009), disease detection (Shafri and Hamdan 2009; Shafri et al. 2011; Lelong et al. 2010; Chávez et al. 2009), stress assessment (Liu et al. 2011; Kempeneers et al. 2005), developing a vegetation spectral library (Sharma et al. 2011), modelling crops’ biophysical and biochemical characteristics (Thenkabail et al. 2013), the estimation of vegetation biophysical and biochemical properties like chlorophyll content (Huang and Blackburn 2011; Wu et al. 2008; Blackburn 2007) and dry matter content (Wang et al. 2011a; Jia et al. 2011), and defoliation assessment (Ge et al. 2011). Noisy data hamper these applications and produce false results. Therefore, a noise reduced form of the hyperspectral vegetation spectrum is essential before performing any derivative analysis or information extraction. The spectral features of the vegetation reflectance spectrum are delicate and tend to overlap. One of the important spectral features in vegetation spectrum is red-edge position (REP) that is the position of the inflection point of the red-near infrared slope or the point where the first derivative of the spectral reflectance curve reaches its maximum value (Liang 2004). REP has been used for the evaluation of vegetation health and other environmental effects, such as water stress, and has also been used to differentiate plant species (Shafri et al. 2006). For a sick plant, REP shifts to the left side of the electromagnetic spectrum. The vegetation spectral features have being applied for different applications such as disease detection (Shafri and Hamdan 2009), species classification, estimation of vegetation indices (Pu et al. 2003) and the extraction of biochemical information such as water content and chlorophyll content (Ju et al. 2010). Therefore, a denoising method should preserve the wavelength positioning of absorption features, inflection points and local minima or maxima. Also, the algorithm’s computation should be straightforward. This means that, to preserve spectral details,
there should be a trade-off between resolving fine spectral details and denoising (Schmidt and Skidmore 2004). In the literature, several traditional smoothing methods have been applied for denoising hyperspectral signals. However, these methods are reported to have some shortcomings when dealing with the hyperspectral vegetation spectrum. A mean filter over-smoothes and changes the details of the original spectrum (Tsai and Philpot 1998; Rollin and Milton 1998), a Savitzky-Golay filter disturbs and masks many details of interest in the spectrum (Curran et al. 1992) and should be used with caution (Rollin and Milton 1998), a KawataMinami algorithm does not eliminate much noise (Tsai and Philpot 1998), spline curve-fitting covers certain details, and a Fourier transform has unstable effects (Curran et al. 1992). A wavelet is a waveform function that is bounded in both frequency and time (Buaba et al. 2011). Wavelet transform, having joint representation of time-frequency, generates wavelet coefficients that are localized in scale and space and can clearly reveal the details concealed in a signal with certain scales. A wavelet transform has the inherent ability to vary the width of the wavelet function to separate fine-scale and largescale information in a hyperspectral data set. Wavelet function, a.k.a. the mother wavelet, is an oscillatory small wave with finite length (Danandeh Mehr et al. 2013). The width and scale of the mother wavelet change systematically, therefore a wavelet transform is a useful tool to separate and identify features that are associated with a phenomenon or variable that may be present in different scales (Sreekala and Subodh 2011). In contrast to traditional smoothing methods, a wavelet offers an efficient and statistically accurate approach to noise reduction. It reduces the noise level while preserving important properties of the original data (Shafri and Mather 2005). For hyperspectral data, wavelets are computationally efficient, practical and feasible techniques for noise reduction application prior to derivative analysis (Bruce and Li 2001). A review of the literature about the application of wavelet methods for denoising remote-sensing data is reported in (Ebadi et al. 2013). The wavelet transform can be continuous or discrete. Continuous wavelet transform (CWT) analyzes the signals across a continuum of scales while with discrete wavelet transforms (DWT) signals are analyzed over a discrete set of scales. The CWT and the DWT have been applied to various areas of remotely sensed signal processing including data compression (Ebadi and Shafri 2014), texture analysis (Liu and He 2007), edge detection (Xiu-bi 2009), denoising (Parrilli et al. 2012), and data fusion (Yu et al. 2012). However, DWT is more suitable for dealing with discrete signals and its implementation is much easier than CWT. There are different types of discrete wavelet transforms including stationary wavelet transform (SWT), lifting wavelet transform (LWT) and a combination of SWT and LWT, which
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in this paper is called stationary lifting wavelet transform (SLWT). This combination of lifting scheme and stationary wavelet transform was first introduced in 2000 (Lee et al. 2000). In the literature, SLWT has been used for signal denoising (Lee et al. 2000; Ma et al. 2009), signal compression (Li et al. 2006), features extraction (Chendong and Qiang 2008; Bao et al. 2009; Gao et al. 2011; Yang et al. 2012; Duan et al. 2012), fault diagnosis (Li et al. 2012; Li and Zhou 2012; Zhou 2010; Lu and Wang 2011), spike detection (Yang et al. 2010; Yang and Mason 2011), image fusion (Chai et al. 2010; Wang et al. 2011b), real-time sleep quality assessment (Bsoul et al. 2010), real-time sleep apnea monitoring (Bsoul et al. 2011), spectral estimation (Knight et al. 2012), point detection (Bsoul and Tamil 2011) and damage detection (Chen et al. 2013). There are, however, no reports of applying SLWT in the remote sensing field. Especially for the denoising of hyperspectral signals, most research has concentrated on traditional wavelet methods such as DWT and SWT and no research has considered applying SLWT. Therefore, this work investigates the effects of the SLWT method on noise reduction in hyperspectral vegetation spectra to discover its efficiency in relation to DWT, SWT and LWT methods. The rest of the paper is organized as follows. Section 2 describes the concept of wavelet methods. Section 3 gives some insights into wavelet thresholding selection rules. The mother wavelets and test signal that are used in this study are described in Sections 4 and 5, respectively. The statistical evaluation methods are explained in Section 6. Experimental results are drawn in Section 7, and finally Section 8 concludes the paper.
Wavelet transform The Mallat algorithm, based on the connection between wavelets and filter banks, introduces a fast decomposition and reconstruction structure for DWT (Mallat 1989). Fig. 1 shows a two-channel filter bank for DWT; the top part represents the low-pass channel while the bottom part is the high-pass channel. The decomposition or analysis part is done by convolution of the signal, X,with two filters which are a low-pass decomposition filter ha and a high-pass decomposition filter ga, resulting in low-frequency and high-frequency components that are approximation coefficients a1 and detail coefficients d1, respectively. The convolutions are followed by dyadic decimation (a.k.a. downsampling) ↓2 which preserves only those elements with an even index; therefore, each part has approximately half the length of the original signal. The synthesis or reconstruction part starts with inserting zeroes between the approximation a1 and detail d1 coefficients (i.e. upsampling ↑2), then filtering with a low-pass reconstruction filter hs and a high-pass reconstruction filter gs, respectively, and finally summing to obtain the output signal Y.
The approximation coefficient a1 acquired from one level of decomposition, as shown in Fig. 1, still contains some details; therefore, it can be fed into the filter bank again. The number of iterations, or so-called levels of decomposition, is limited by the length of the input signal. In theory, the highest possible value for the level of decomposition is: J ¼ log2 L
ð1Þ
where L is the signal length. Based on the experimental results from this study, for denoising purposes the optimum level of decomposition is considered to be half of this value, i.e. J/2, as levels higher than that result in over-smoothing. In the DWT structure, down-sampling or decimation is not invariant by translation in time and it is not an invertible operator (Strang and Nguyen 1996). This means that the DWT of a translated version of the signal X is not, in general, a translated version of the DWT of X. This loss of the translation-invariance property of DWT creates artefacts around the boundaries and discontinuities in the denoised signal. In denoising methods that are based on the thresholding of wavelet coefficients, these artefacts, known as the Gibbs phenomenon, affect the estimation of the threshold value and degrade the results, but by using a translation-invariant transform they can be avoided (Ma et al. 2009). The translationinvariant version of the DWT is known by a variety of names, including stationary wavelet transform (SWT), redundant wavelet transform, algorithme à trous, quasi-continuous wavelet transform, translation-invariant wavelet transform, shift invariant wavelet transform, cycle spinning, maximal overlap wavelet transform and undecimated wavelet transform. SWT is achieved by removing the down-sampling and upsampling operators in the DWT structure. To keep the multiresolution analysis consistent, the coefficients of the filters are up-sampled at each level of decomposition. As a result, for N levels of decomposition, at the j th decomposition level the filters are up-sampled by a factor of 2j-i: h j ¼ ha ↑2 j−1 ; g j ¼ ga ↑2 j−1 ; j ¼ 1; 2; 3; …; N
ð2Þ
Figure 2 illustrates the decomposition structure of SWT. The reconstruction of SWT is not unique and can be done by computing the average of several possible inverse transforms. This causes a smoothing effect which is helpful for
Fig. 1 Two-channel filter bank representation of DWT
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The inverse lifting transform is simply obtained by inverting each step of the forward transform. However, similar to DWT, LWT is not translation invariant. SLWT, by combining SWT and LWT, retains the simplicity and fast performance of the lifting scheme as well as the translation invariant property of SWT; therefore, it is a superior technique for denoising. The undecimated version of the polyphase analysis matrix is acquired by up-sampling the filter coefficients which in the z-transform are equivalent to squaring the variable z. As a result, the polyphase analysis matrix of the SLWT becomes:
Fig. 2 Three decomposition levels of SWT
noise reduction. Moreover, this redundancy makes the transform much more robust against noise, such that the main application of SWT is denoising (Misiti et al. 2007). In SWT, however, as the approximation and detail coefficients have the same size as the input signal, the computation is more expensive and has O (N log N) arithmetic complexity rather than the O (N) operations for DWT. In contrast to SWT, second-generation wavelets (SGWs) introduce a simple and fast algorithm. The lifting scheme, introduced by Sweldens (Sweldens 1996) as the core function of SGWs, allows for the in-place implementation of wavelet transforms so the signal can be transformed without requiring additional memory for temporary results. This property reduces the computation time by a factor of at least 2 and lowers memory usage significantly. The next section gives more details about the lifting scheme. Lifting wavelet transform (LWT) In the two-channel filter bank of DWT shown in Fig. 1, if there is no intermediate step between the decomposition and reconstruction parts, a necessary and sufficient condition links the four filters, ha, ga, hs, and gs, and that is known as a perfect reconstruction (PR) condition. Sweldens (Sweldens 1998) showed that by using a quadruplet solution of the PR condition, an unlimited number of new solutions can be built that are known as elementary lifting steps. These elementary lifting steps are dual lifting, a.k.a. update, and primal lifting or so called predict step. The factorization theorem (Daubechies and Sweldens 1998) proves that all quadruplet perfect reconstruction filters have a polyphase matrix that can be factorized into a sequence of predict P(z) and update U(z) steps. The analysis matrix acquired via the polyphase method is given as follows: n k 0 1 U i ðzÞ 1 0 E ðzÞ ¼ ∏ 0 k −1 0 1 −Pi ðzÞ 1 i¼1
S ðzÞ ¼ E z2 ¼ N U z2 P z2 1 0 1 1 0 k 0 1 S z2 ¼ 2 −1 −1 1 −T z 0 z 0 k 0 0 1
ð4Þ The delay parameter z−1 and advance parameter z assure a perfect reconstruction. Figure 3 illustrates SLWT. The sum of the coefficients is halved to put the final signal in the correct scale.
Wavelet thresholding After the wavelet decomposition of a noisy signal, the noise energy is heavily concentrated in high-frequency detail coefficients while the energy of the signal is concentrated in lowfrequency approximation coefficients and these carry more signal information than noise (Han and Chang 2013). Replacing the small detail coefficients whose absolute values are lower than a threshold value by zero along with a backwards wavelet transform on the results leads to a reconstruction with the essential signal characteristics and with less noise. This is called hard thresholding and is defined as follows: x; j xj > λ f ð xÞ ¼ ð5Þ 0; jxj ≤ λ where λ is the threshold value, f is a thresholding function and x is the wavelet coefficient. Another wavelet thresholding method is soft thresholding. It sets to zero those elements whose absolute values are lower than the threshold λ but shrinks non-zero coefficients towards zero. 8 < x − λ; x ≥ λ f ðxÞ ¼ x þ λ; x ≤ −λ ð6Þ : 0; jxj≤ λ
¼ N U ðzÞPðzÞ ð3Þ where n is the number of lifting steps and k is the scaling constant unequal to zero to form the normalization matrix N.
0 z
Fig. 3 Decomposision and reconstruction of SLWT
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Soft thresholding makes the thresholding function continuous by shrinking the retained coefficients. It leads to smoother estimators that are less sensitive to noise. Hard thresholding is unstable and more sensitive to small changes, and exhibits some discontinuities, while soft thresholding is stable and avoids discontinuities (Han and Chang 2013). As a result, this study uses soft thresholding instead of hard thresholding. A significant advantage of the wavelet thresholding method is that it reduces the noise level while still preserving important features of the original signal (Shafri and Mather 2005).
Thresholding selection rule There are several methods for choosing the threshold value λ. The most famous threshold selection rule is universal thresholding (Donoho and Johnstone 1994): λU ¼ σ b
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 loge N
ð7Þ
where N is the length of the wavelet coefficients and σ b is an estimate of the noise level:
median dbJ −1;k ð8Þ σ b¼ 0:6745 n o Where dbJ −1;k : k ¼ 0; 1; …; 2 J −1 −1 are wavelet coefficients at the finest resolution level (Antoniadis et al. 2001). The universal threshold is simple with low cost software implementation; however, it largely depends on the length of the data sample. For small N it gives noisy results, and if N is too large important features in the signal will be removed (Han and Chang 2013). Another popular method is a threshold estimator based on minimizing Stein’s unbiased risk estimator (SURE), and this is known as SureShrink. Stein’s unbiased
(a)
estimate of the risk corresponding to the soft shrinkage function is shown as: SUREðλ; X Þ ¼ s−2⋅#fi : jX i j≤ λg þ ½minðjX i j≤ λÞ2
ð9Þ
where #B denotes the cardinality of any set B σ ¼ X i and s=2j the SureShrink By considering dbjk =b threshold at any level j=j0,…,J−1, is given by: ! dbjk s λ j ¼ arg min0 ≤ λ ≤ λU SURE λ; ð10Þ ; k ¼ 0; :::2 j −1 σ b where λU is the universal threshold with N=2j. The SureShrink threshold has a serious drawback in situations of extreme sparsity of wavelet coefficients. If the signalto-noise ratio (SNR) is very small, the SURE estimate is very noisy. To avoid this drawback a hybrid scheme using the SureShrink threshold can be considered, as in the following heuristic idea: if the set of empirical wavelet coefficients is judged to be sparsely represented, then the hybrid scheme defaults to the level-wise universal threshold, otherwise the SURE criterion is used to select a threshold value. The heuristic SureShrink threshold is expressed as: (
2 2 λUj ; if Σ 2 j −1 b HS b 2 j=2 2 j=2 þ j3=2 k¼0 d jk ≤ σ λj ¼ ð11Þ λSj ; otherwise
Where λjU is the universal threshold, λjS is the SureShrink threshold and dbjk are wavelet detail coefficients. Spectroradiometers perform the spectral measurements in the form of point spectroscopy, i.e. the spectra are measured one point at a time (Garfagnoli et al. 2013). This type of measurement causes photonic noise or Poisson-distributed noise that is dominant type of noise generated by hyperspectral sensors (Uss et al. 2011). In the wavelet transform of a Poisson-distributed signal, the coefficients in some
(b)
Fig. 4 Normalized correlation coefficient between vegetation spectrum and mother wavelets of (a) Daubechies; and (b) biorthogonal families
Earth Sci Inform Table 1 Biochemical parameters of corn leaves Vegetation name
Leaf structure parameter N
Chlorophyll a+b content Cab
Equivalent water thickness Cw
Dry matter content Cm
corn
1.518
58.0
0.013100
0.003662
places are more intense than elsewhere; accordingly, it is necessary to have a higher threshold value for the related coefficients (Charles and Rasson 2003). Also, in the Poisson problem, the distribution of coefficients is not symmetric, therefore there needs to be a pair of thresholds (Kolaczyk 1999). As a result, this paper applies the heuristic SureShrink threshold for the denoising process.
Selection of mother wavelet The optimum mother wavelet for denoising can be selected using the correlation coefficient between the signal and the low-pass decomposition filter of the wavelet (Singh and Tiwari 2006). A high correlation ensures that the approximation coefficients resulting from convolving the wavelet and signal properly resemble the overall shape of the signal. Therefore, there will be better separation between approximation coefficients and noisy detail coefficients, and consequently better denoising results. This technique has given good results for denoising ultrahigh frequency (UHF) signals (Singh and Tiwari 2006) and electrocardiogram (ECG) signals (Singh and Tiwari 2006) that have high correlation with some wavelets. Although there is no mother wavelet that resembles the vegetation reflectance spectrum (Shafri and Yusof 2009), in this study the correlation coefficient method is applied to select a suitable mother wavelet. Daubechies (abbrev. db) and biorthogonal (bior) wavelet families are the most popular and practical wavelet types in the literature. Figure 4 shows the correlation coefficient results between the synthetic vegetation spectrum (See section 5) and different mother wavelets.
Fig. 5 Green vegetation spectrum
In Fig. 4, the highest correlation values for the vegetation spectrum are generated by mother wavelets with short-length filters, or low-order mother wavelets. Moreover, the reflectance spectrum of vegetation contains narrow spectral features, e.g. the red-edge position (REP) has a spectral width of only 50 nm. To preserve spectral details after denoising, the smoothing width or filter length should not be much larger than the width of spectral features. Hence, in this study four low-order mother wavelets that have the highest correlation coefficient values are chosen. The selected mother wavelets are db1 and bior1.1, which are the same, and known as Haar wavelet (haar), db2, bior1.3 and bior3.1.
Test signal The test signal applied in this study is a synthetic reflectance spectrum of the green leaves of corn created using the PROSPECT leaf model (Jacquemoud and Baret 1990; http://teledetection.ipgp.jussieu.fr/opticleaf/models.htm). PROSPECT allows the construction of leaf reflectance and transmittance spectra in the 400–2,500 nm wavelength range by introducing input variables as follow: leaf structure parameter (N), chlorophyll a+b content (Cab), equivalent water thickness (Cw) and dry matter content (Cm). The test signal is created by running the PROSPECT leaf model program in MATLAB (PROSPECT) and introducing the biochemical parameters of corn leaves (Hosgood et al. 2005) as shown in Table 1. The test signal was interpolated by cubic interpolation to have a sampling interval of 1 to become compatible with the wavelength of the real-world vegetation spectrum.
Fig. 6 RMSE of the curve using haar mother wavelet
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Fig. 7 RMSE for green maxima using haar wavelet
Fig. 9 RMSE for REP using haar wavelet
Afterwards, the signal was smoothed by moving average filter to compensate for the jaggedness caused by interpolation. Subsequently, a noisy signal was generated by adding Poisson noise to the test signal.
RMSE
Noise reduction evaluation indicators In order to determine the efficiency of wavelet-based methods in noise reduction, the results are evaluated using various statistical methods including SNR, root mean square error (RMSE) and correlation coefficient. SNR SNR shows the ability of wavelet methods to reduce the noise rate. A greater value reflects a better result. The SNR of the denoised signal is calculated as: Xn ! x2i i¼1 SNR ¼ 10log10 Xn ð12Þ y2 i¼1 i
RMSE indicates the degree of deviation of the denoised signal from the original signal and is shown as: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 1 n 0 Σ i¼1 xðiÞ−x ðiÞ RMSE ¼ n
ð13Þ
where x(i) is the original signal and x0 ðiÞ is the denoised signal. A lower value indicates a better result. Correlation coefficient The correlation coefficient measures the quality of fitting the denoised signal to the original signal and has a value between +1 and −1, inclusive. A value of +1 indicates a perfect fit. Its formula (King et al. 2011) can be written as: X X X
y n xy − x r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X X 2 X X 2ffi n x2 − n y2 − x y
ð14Þ
where xi is the original signal, yi is noise that is the difference between the original signal and the denoised signal, and n is the signal length (Tan 2007).
where x and y denote the original and denoised signals, respectively
Fig. 8 RMSE for red minima using haar wavelet
Fig. 10 SNR at five levels of decomposition for a haar wavelet
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Experimental section To perform noise removal, the noisy test signal was decomposed using DWT, SWT, LWT and SLWT methods. According to Section 2, for a test signal with a length of 2,101, the optimum value for the level of decomposition is 5. The denoising result is evaluated by calculating the RMSE between the noisy test signal and its denoised version. It is called the RMSE of the curve. Moreover, to determine the ability of denoising methods to preserve the position of absorption features and reflectance peaks, the RMSE for red minima, green maxima and REP are also computed. Figure 5 presents the spectrum of green vegetation. Other statistical evaluation methods are SNR and correlation coefficients. To obtain better accuracy for RMSE and SNR, the process of adding noise, followed by denoising and calculating RMSE and SNR, is repeated 30 times and the averages calculated.
Fig. 12 Denoised spectra using a haar wavelet and LWT and DWT methods
These results imply that the DWT and LWT methods displace the position of spectral features critically. For the RMSE for REP, DWT displays a uniform behaviour but with a high value. Somehow, SWT shows better results compared to DWT and LWT, but with variable results for different decomposition levels. The SLWT method, yet again, shows a uniform trend over different levels of decomposition with very low RMSE values.
RMSE results SNR results Figure 6 shows the results for RMSE of the curve for denoising a noisy synthetic vegetation spectrum by four wavelet methods and using a haar mother wavelet at levels of decomposition of 1 to 5. The red line is the averaged RMSE of the noisy test signal that has a value of 0.00056. In Fig. 6, from levels 1 to 5, the DWT, SWT and LWT methods show an ascending slope for the RMSE of the curve. It means that by increasing the level of decomposition, the separation between the original and denoised signals increases. Moreover, these wavelet methods have great variations over different decomposition levels. As it can be seen from the plots, only SLWT and SWT for the first two levels provid RMSE values less than the initial RMSE of the curve and are able to improve the RMSE. Compared to other wavelet methods, SLWT produces the least RMSE of the curve with a uniform trend over all decomposition levels. Figures 7, 8, and 9 show the results of RMSE for green maxima, red minima and REP for a haar mother wavelet.
Fig. 11 Correlation coefficient at five levels of decomposition for a haar wavelet
Figure 10 presents the results of SNR for four wavelet methods by applying a haar wavelet at five levels of decomposition. The SNR of the noisy test signal before denoising equals 54 dB. According to Fig. 10, the DWT, SWT and LWT methods degrade the SNR of the signal after denoising, except for the SWT method at the first two levels. SLWT shows high values for SNR, and similar to the results for RMSE, it has a uniform trend over different levels of decomposition. Correlation coefficient results Figure 11 shows the results of the correlation coefficient for four wavelet methods, using a haar wavelet and at five levels of decomposition.
Fig. 13 Denoised spectra using a haar wavelet and SWT and SLWT methods
Earth Sci Inform Table 2 Mean and variance of RMSE and SNR results for (a) haar (b) db2 (c) bior1.3 and (d) bior3.1 mother wavelets DWT (a) RMSE of the curve RMSE for green maxima RMSE for red minima RMSE for REP SNR (b) RMSE of the curve RMSE for green maxima RMSE for red minima RMSE for REP SNR (c) RMSE of the curve RMSE for green maxima RMSE for red minima RMSE for REP SNR (d) RMSE of the curve RMSE for green maxima RMSE for red minima RMSE for REP SNR
SWT
LWT
SLWT
Mean Variance Mean Variance Mean Variance Mean Variance Mean Variance
0.0055 2.91e-05 8.066 76.82 6.99 63.65 1.92 0.032 37.99 90.69
0.0015 3.57e-06 1.138 0.58 3.34 18.82 1.84 0.776 50.55 98.34
0.0055 2.89e-05 5.666 24.01 10.99 291.77 4.37 13.01 37.99 90.52
0.0003 1.23e-09 1.094 0.02 1.42 0.046 1.29 0.10 59.87 1.35
Mean Variance Mean Variance Mean
0.0018 5.56e-06 4.32 13.31 6.30
0.0009 1.05e-06 1.40 0.40 2.24
0.0019 6.57e-06 4.43 20.22 4.79
0.0003 1.68e-09 1.09 0.041 1.45
Variance Mean Variance Mean Variance
27.86 15.06 181.76 49.50 113.28
2.43 1.81 1.93 53.49 63.28
11.29 14.13 116.09 49.33 119.77
0.02 1.32 0.04 59.72 1.02
Mean Variance Mean Variance Mean Variance Mean Variance Mean Variance
0.0056 3.05e-05 6.66 67.88 5.79 54.83 0.306 0.24 37.93 92.58
0.0009 1.05e-06 1.36 0.36 2.35 2.22 1.912 1.73 53.61 64.52
0.005 3.04e-05 5.59 24.85 4.56 16.57 4.226 13.99 37.94 92.39
0.0003 1.35e-09 1.19 0.03 1.70 0.03 1.314 0.05 59.33 0.55
Mean Variance
0.0010 9.35e-07
0.0009 1.05e-06
0.0013 3.01e-06
0.0003 8.68e-10
Mean Variance Mean Variance Mean Variance Mean Variance
2.03 1.05 4.30 23.99 1.40 0.32 51.56 47.19
As Fig. 10 shows, for DWT and LWT by increasing the level of decomposition, the value of the correlation coefficient greatly decreases. This means that the DWT and LWT
1.32 0.74 2.30 2.27 1.95 1.60 53.46 62.51
1.32 0.74 3.52 5.41 4.47 12.40 50.55 73.65
0.98 0.074 1.17 0.07 1.22 0.05 59.85 0.81
Before Denoising
0.00056 1.60 2.18 2.77 54
0.00056 1.60 2.18 2.77 54
0.00056 1.60 2.18 2.77 54
0.00056 1.60 2.18 2.77 54
denoising results strongly divide the original signal. The SWT method gives good results for the first three levels, but from level four the correlation coefficient declines. The SLWT
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method gives correlation coefficient values quite near to 1. Therefore, a denoised signal using the SLWT method fits the original signal very well. Once again, SLWT displays a uniform trend over different levels of decomposition.
Table 3 Real-world vegetation spectra
Visual evaluation
Live oak
Spectral feature analysis of the hyperspectral vegetation spectra has been used for several purposes, including vegetation mapping, classification and the accurate identification of vegetation cover types (Kokaly et al. 2003). The visual quality of vegetation spectra is important for spectral feature analysis and applications related to visual interpretation. So, in addition to statistical analysis, the denoised spectra were compared to the original signal visually. A derivative of the spectrum magnifies small details in the spectrum; therefore, a visual comparison was performed using the first derivative of denoised signals. Figures 12 and 13 show the first derivative of denoised signals using DWT, SWT, LWT and SLWT methods, by applying a haar wavelet at a decomposition level of five. As shown in Fig. 12, the DWT and LWT methods were unable to remove the noise, disturbed the signal significantly and produced spiky signals. In Fig. 13, the SWT method has
Madrone
Vegetation name
Source
Arroyo Willow
ENVI spectral library Stanford University Jasper Ridge Biological preserve Laboratory Spectra measured by CDE Field spectrometry
Oil palm
removed most of the noise but has over-smoothed the signal and suppressed all the useful details. The spectral peaks around 520 and 720 nm are flattened and the depth of the pits around 1,390 and 1,880 nm is decreased. SWT has removed all the small features that can be useful for vegetation studies. In contrast, SLWT has removed the noise perfectly and at the same time kept the small spectral features in the spectrum. SLWT has maintained the wavelength positions of spectral features and the denoised signal resembles the test signal perfectly. So far, the results show that denoising the synthetic vegetation spectrum using the SLWT method strongly outperforms
Fig. 14 Statistical denoising results using different mother wavelets. (a) RMSE of the curve; (b) RMSE for green maxima; (c) RMSE for red minima; (d) RMSE for REP; (e) SNR
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(b)
(b)
(d)
(e)
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Fig. 15 Denoised spectra of Arroyo Willow vegetation
other wavelet methods in terms of both statistical results and visual evaluation. Moreover, SLWT shows consistent and uniform results over different levels of decomposition; therefore, it is not dependent on the selection of the decomposition level. To support this theory, the denoising process was repeated for three more mother wavelets. The same synthetic vegetation spectrum was denoised using DWT, SWT, LWT and SLWT wavelet methods and by applying haar, db2, bior1.3 and bior 3.1 mother wavelets. The mean and variance of RMSE and SNR results for decomposition levels of 1 to 5 were obtained. Table 2 shows these results. According to the RMSE and SNR values of the noisy test signal before denoising, the results in Table 2 show that SLWT constantly gives superior results. Compared to other wavelet methods, SLWT provides very low mean values for all the RMSE results. An RMSE of the curve of as low as 0.0003 and very low values for the RMSE for spectral features (between 0.98 and 1.70) indicate that SLWT is able to denoise the vegetation spectrum effectively while still preserving the positions of spectral features. SWT shows better RMSE results
than LWT and DWT but the values are still high. The DWT and LWT methods provide very high RMSE results and critically change the spectrum after denoising. The mean values of SNR before denoising for haar, db2, bior1.3, and bior3.1 are 54.05, 54.18, 54.29 and 54.19, respectively. As Table 2 shows, only SLWT was able to improve the SNR after denoising while other wavelet methods degraded the SNR value. In Table 2, the variance measures how far the results from levels 1 to 5 are spread out. A small variance indicates that the data points are close to the mean (expected value) and hence to each other, while a high variance indicates that the data points are very spread out from the mean and from each other. The SLWT method shows a very low variance for SNR and all RMSE results. It indicates that for each mother wavelet, from level one to level five, SLWT provides almost identical statistical results. This leads to the novel conclusion that SLWT is a level-independent wavelet method. Meanwhile, the high values of variance for other wavelet methods show that they generated extremely varying statistical results over different levels; therefore, their results are highly dependent on the selection of the level of decomposition. To demonstrate the effect of selection of different mother wavelets on denoising, Fig. 14 shows the RMSE and SNR results from Table 2. As displayed in Fig. 13, unlike DWT, SWT and LWT methods, SLWT always produces uniform results for all four mother wavelets. As a result, among these low-order mother wavelets, denoising of the vegetation spectrum using SLWT is independent of the selection of mother wavelet. This characteristic and the level-independency property of SLWT greatly lower the complexity of wavelet application, so it can be used more easily.
Fig. 16 Visual comparison of SWT and SLWT denoising methods in different portions of the spectrum. (a) 400–700 nm (b) 700–1,200 nm (c) 1,100– 1,400 nm (d) 1,400–2,000 nm
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(b)
(c)
(d)
Earth Sci Inform Table 4 RMSE results from denoising the spectrum of Arroyyo Willow using a haar mother wavelet
Green maxima Difference from vegetation spectrum
Red minima Difference from vegetation spectrum
REP Difference from vegetation spectrum
Wavelength Reflectance Wavelength Reflectance Total RMSE Wavelength Reflectance Wavelength Reflectance Total RMSE Wavelength Reflectance Wavelength Reflectance Total RMSE
Vegetation spectrum
DWT
SWT
LWT
SLWT
566 0.1352
559 0.1257 7 0.0094 7.0094 656 0.08383 7 0.00406 7.00406 719 0.4334 4 0.41374 4.41374
557 0.128 9 0.0071 9.0071 669 0.08307 6 0.0033 6.0033 719 0.01569 4 0.00611 4.00611
571 0.1303 5 0.0048 5.0048 667 0.08592 4 0.00615 4.00615 715 0.2472 8 0.22754 8.22754
567 0.1309 1 0.0043 1.0043 666 0.08288 3 0.00311 3.00311 722 0.01667 1 0.00299 1.00299
663 0.07977
723 0.01906
Denoising of the real-world vegetation spectrum For real-world data, several vegetation spectra were selected from different sources. Table 3 shows the list of applied vegetation spectra. These spectra were denoised using DWT, SWT, LWT and SLWT methods and applying four mother wavelets. Figure 15 shows the denoised spectra of Arroyo Willow vegetation using different wavelet methods and a haar mother wavelet. The spectra are offset to distinguish them better. Evidently, LWT and DWT were unable to denoise the spectrum and produced jagged denoised spectra. SWT has
removed the noise but also over-smoothed the spectrum. Figure 16 compares SWT and SLWT results in different portions of the spectrum. According to Fig. 16, there is always some separation between the denoised spectrum using SWT and the vegetation spectrum, especially at absorption features and peaks. SWT has removed small spectral features from the vegetation spectrum, e.g. the details at 550, 800 and 1,000 nm wavelengths. In contrast, SLWT has successfully removed the noise but at the same time maintained subtle features. The denoised spectrum using SLWT resembles the vegetation spectrum perfectly, and
Fig. 17 RMSE for the displacement of spectral features of denoised real-world vegetation spectra (a) Arroyo Willow (b) Live Oak (c) Madrone (d) Oil Palm
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Earth Sci Inform
the shape of minima, maxima and inflection features are preserved. In addition to visual evaluation, the denoising effects of different wavelet methods were examined by comparing denoised spectra to original spectra based on the RMSE for green maxima, red minima and REP in terms of reflectance and wavelength displacements. Table 4 shows the RMSE results from denoising the spectrum of Arroyyo Willow using a haar mother wavelet. The total RMSE were acquired by summing the reflectance difference between the original and denoised spectra with the wavelength displacements of spectral features. The same process were repeated to achieve RMSE results for denoising the Live Oak spectrum using db2, the Madrone spectrum using bior1.3 and the Oil Palm spectrum using a bior3.1 mother wavelet. Figure 17 demonstrates these results. Unlike DWT, SWT and LWT, the RMSE results show very low values for SLWT. Therefore, SLWT preserves the wavelength positions of spectral features very well and gives predictable and reliable results; as a result it is a stable method for denoising the vegetation spectrum.
Conclusion This paper has investigated denoising of the hyperspectral vegetation spectrum using different wavelet methods, including discrete wavelet transform (DWT), lifting wavelet transform (LWT), stationary wavelet transform (SWT) and stationary lifting wavelet transform (SLWT). Although SWT, DWT, and LWT methods yield detail and approximation coefficients that are localized in scale and space, they are not translation invariant. Therefore, lifting translated data generated using SLWT method can result in entirely different detail and approximation coefficients than the coefficients corresponding to the non-translated data. Compared to the DWT, LWT and SWT methods, SLWT generated remarkably better statistical results with very low root mean square error (RMSE) and high signal-to-noise ratio (SNR) and correlation coefficient values. In terms of visual quality, SLWT successfully removed noise while preserving spectral features perfectly; however, DWT and LWT were unable to remove noise and SWT oversmoothed the spectrum and suppressed subtle details. Also, experimental results have revealed these novel outcomes: SLWT 1) is a statistically stable denoising wavelet method; 2) is level-independent; and 3) its denoising results do not depend on the selection of the mother wavelet when applying low-order mother wavelets. These characteristics greatly reduce the complexity of applying a wavelet transform, so it can be used more easily. In conclusion, SLWT is a reliable and stable denoising method for denoising hyperspectral vegetation spectra. Since the SLWT method does not manipulate the
spectrum shape after denoising, we suggest testing its efficiency on applications other than denoising, inside which preserving the signal shape is vital such as classification and spectral feature extraction.
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