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EMD滤波与小波滤波具有一定的相似,区别是小波滤波需要选取某固定小波基,分析过程中小波基的选择对整个分析结果影响很大,一旦确定了小波基,在整个分析过程中将无法替换,有时该小波基在全局上虽是最佳的,但在某些局部上并不适应;而EMD滤波,针对任何一段未知信号,根据数据自身的时间尺度特征,无须预先设定任何基函数,而是从信号数据中直接分解出若干个IMF和一个单调余项[27]。该方法无需人为设置或干预,并自动按模态层次分好,且所分解出的各个IMF分量表示了原信号在不同时间尺度的局部特征。在该过程中,特征时间尺度及IMF都具有一定的经验性和近似性。EMD使非平稳数据进行平稳化处理,然后基于希尔伯特变换的时频谱图,得到有物理意义的频率。由于EMD从信号中分解出一些列的IMF,表示了从高频到低频排列的信号分量,具有主成分分析特点;各分量通过希尔伯特变换,得到时频谱图的瞬时特征量,具有自适应时频分析特点;且单分量信号的瞬时频率强调信号的局部瞬时特性,具有信号局瞬特性表征等优势。相比于傅里叶变换、小波分解等方法,更加直接、直观、后验和自适应。
首先,EMD滤波需要将原始信号分解出能表示各层信号分量的IMF。针对激光测高中回波信号的全波形数据,通常噪声以高频形式存在,那么过滤其高频信号分量也就达到降噪效果。EMD分解方法是基于以下假设条件:(1)数据至少有两个极值,即最大值和最小值;(2)数据的局部时域特性,由极值点间的时间尺度来唯一确定;(3)当数据没有极值点但有拐点,则通过对数据微分一次或多次求出极值,再通过积分来获得分解结果。同时,EMD所分解出的IMF应满足两个约束条件:在整个数据段内,零点数与极点数的个数相等,或至多相差值为1;在任意时刻,由局部极大值点确定的上包络线和由局部极小值点确定的下包络线的平均值都为零[27]。如图1所示,其具体的计算过程有:
(1)对于一个序列信号x(t),确定所有的极值点;
(2)根据局部最大值和最小值,采用三次样条曲线拟合出上包络线emax(t)和下包络线emin(t),并求出两条包络线的平均值m(t),在原始信号中减去平均值得到h(t)=x(t)−m(t);
(3)根据约束条件来判断h(t)是否为IMF;
(4)若否,将h(t)替换x(t),重复前两个步骤直至满足,则该h(t)为信号分量IMFk(t);
(5)每得到一阶信号分量IMF,就从原信号中扣除它,重复以上步骤;直到信号最后剩余部分rn只是单调序列或常值序列。
由此,能被EMD分解的原始信号x(t),由分解出高频到低频的一些列IMF,与一项剩余部分(频率接近零,也称为残差)的线性叠加,可用公式(1)表示。
$$ x\left( {{t}} \right) = \mathop \sum \limits_{k = 0}^n IM{F_k}\left( t \right) + {r_n}\left( t \right) $$ (1) -
EMD依据分解原则,分解出不同尺度下的若干个IMF,并从中选取适当尺度下的IMF进行拟合重构达到降噪效果,由此针对描述噪声IMF的合理筛选尤为重要,目前主要有以下几种方法:
(1) EMD-N:假设信号中噪声存在于高频信号的前N个IMF,则可以选取并去除前N个IMF,并将剩余的低频IMF重新构建波形信号,实现波形信号的降噪[30]。通常情况下,分别选取第一个或前两个IMF作为噪声尺度。
(2) EMD-Threshold:在对信号进行平滑重建时,通过构建阈值函数,并设置阈值参数来筛选IMF,进而实现波形信号的降噪。对于信号中高斯白噪声过滤,Donoho等采用τ值作为通用阈值[31],其计算公式为:
$$ {\tau _j} = {{\tilde \sigma }_j}\sqrt {2{\rm{log}}{_e}\left( T \right)} $$ (2) $$ {{\tilde \sigma }_j} = MA{D_j}/0.6745 $$ (3) $$ MA{D_j} = Median\left\{ {\left| {IM{F_j}\left( t \right) - median\left( {IM{F_j}\left( t \right)} \right)} \right|} \right\} $$ (4) $$ {{\hat f}_j}\left( t \right) = \left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{r}} {IM{F_j}\left( t \right) - {\tau _j},}\\ {0,}\\ {IM{F_j}\left( t \right) + {\tau _j},} \end{array}}&{\begin{array}{*{20}{l}} {IM{F_j}\left( t \right) \geqslant {\tau _j}}\\ {|IM{F_j}\left( t \right)| < {\tau _j}}\\ {IM{F_j}\left( t \right) \leqslant - {\tau _j}} \end{array}} \end{array}} \right. $$ (5) $$ {{\hat f}_j}\left( t \right) = \left\{ {\begin{array}{*{20}{c}} {\begin{array}{*{20}{r}} {IM{F_j}\left( t \right),}\\ {0,} \end{array}}&{\begin{array}{*{20}{l}} {IM{F_j}\left( t \right)| > {\tau _j}}\\ {|IM{F_j}\left( t \right)| \leqslant {\tau _j}} \end{array}} \end{array}} \right. $$ (6) 式中:
$ {{\tilde \sigma }_j}$ 表示对第j个IMF噪声水平的估计;T表示波形数据的个数;MADj表示第j个IMF绝对中位数偏差。对于软阈值和硬阈值分别如公式(5)和公式(6)表示。(3) EMD-Wavelet:将回波信号的全波形数据进行EMD分解之后,对每个IMF降噪后信号进行小波阈值降噪,然后重新构建波形信号,实现波形信号降噪[32-33]。
(4) EMD-DFA:通过DFA确定非稳态信号中长程的幂函数相关性,定量分析IMF序列的关联性质,采用过滤各阶趋势成分来避免噪声和信号的伪关联。也称为线性回归分析的DFA确定拟合阶数K构建Hurst指数[34],估计时间序列上波形分量的自相关性。当指数H=0.5时表明时间序列,可以用随机游走来描述,具有马尔科夫性;当0.5<H<1时序列数据存在状态持续性,是一个有偏的随机游走;当0<H<0.5时序列数据反持续性的,即序列前一个点是向上走的,下一个点多半是向下走的,时间序列处理振荡时刻[30]。
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对激光测高中回波信号的全波形数据,进行EMD降噪后得到平滑的全波形数据。通过降噪前后波形数据分布情况及IMF的拟合精度,可以通过一定的评价指标来描述。通常,采用均方根误差(Mean squared error, MSE)、平均绝对误差(Mean absolute error,MAE)、波形相关性(Correlation coefficient, R2)等,用来表示降噪前后波形间的偏离程度及相似性;采用输出信号的信噪比(Signal noise ratio, SNR)和峰值信噪比(Peak signal noise ratio, PSNR),用来表示降噪后保留有用信号的程度。其具体表示公式如下所示[30]:
$$ {\rm{MSE}} = \sqrt {\frac{1}{N}\mathop \sum \limits_{n = 1}^N {{\left( {Ra{w_{\rm{s}}}\left( n \right) - Smoot{h_{\rm{s}}}\left( n \right)} \right)}^2}} $$ (7) $$ {\rm{MAE}} = \dfrac{{ \displaystyle\sum _{n = 1}^N \left| {Ra{w_{\rm{s}}}\left( n \right) - Smoot{h_{\rm{s}}}\left( n \right)} \right|}}{N} $$ (8) $$ {\rm{SNR}} = 10{\rm{*log}}( {\frac{{ \displaystyle\sum _{n = 1}^N {{\left| {Ra{w_{\rm{s}}}\left( n \right)} \right|}^2}}}{{ \displaystyle\sum _{n = 1}^N {{\left| {Ra{w_{\rm{s}}}\left( n \right) - Smoot{h_{\rm{s}}}\left( n \right)} \right|}^2}}}} ) $$ (9) $$ {\rm{PSNR}} = 10{\rm{*log}}\frac{{ \displaystyle\sum _{n = 1}^N ma{x^2}}}{{ \displaystyle\sum _{n = 1}^N {{\left( {Ra{w_{\rm{s}}}\left( n \right) - Smoot{h_{\rm{s}}}\left( n \right)} \right)}^2}}} $$ (10) $$ {R^2} = \frac{{{{\left[ { \displaystyle\sum _{n = 1}^N \left( {Ra{w_{\rm{s}}}\left( n \right) - \overline {Ra{w_{\rm{s}}}\left( n \right)} } \right)\left( {Smoot{h_{\rm{s}}} - \overline {Smoot{h_{\rm{s}}}} } \right)} \right]}^2}}}{{ \displaystyle\sum _{n = 1}^N {{\left( {Ra{w_{\rm{s}}}\left( n \right) - \overline {Ra{w_{\rm{s}}}\left( n \right)} } \right)}^2}\mathop \sum \nolimits_{n = 1}^N {{\left( {Smoot{h_{\rm{s}}} - \overline {Smoot{h_{\rm{s}}}} } \right)}^2}}} $$ (11) 式中:
$ Raw_{\rm{s}}$ 表示原始的回波信号;$ {Smoot{h_{\rm{s}}}}$ 表示降噪后的回波信号。 -
高分七号激光测高系统的主要组成部分有[14]:1台主接收望远镜、2台二级扩束镜、4台±0.7°的主备份激光器、2套发射光路调整机构、2台±0.7°的足印相机、1台0°监视相机、2套足印相机调焦机构及4套主备光路切换机构、激光驱动组件、全波形组件等。激光测高仪利用垂轨方向±0.7°的两台激光器及对应的足印相机,在沿轨方向上形成两条相平行的激光脚点轨迹,提供地表高程及沿轨和垂轨方向的局部坡度信息,并在激光发射瞬时拍摄地表上激光脚点区域的可见光影像(即激光光斑足印图像)。黄庚华等对其系统设计及采集数据形式进行了详细的描述[14],其中激光测高仪的主要参数如表1所示。
表 1 高分七号激光测高仪的主要参数
Table 1. Main parameters of the GaoFen-7 laser altimeter
Item Value Number of laser beams 2 Laser wavelength/nm 1 064 Laser energy/mJ 100-180 Pulse width/ns 4-8 Divergence/μrad 30-40 Emission frequency/Hz 3, 6 Digitization interval/ns 0.5 为验证EMD方法对高分七号激光测高仪全波形回波信号的降噪效果,文中选取两个比较复杂的低信噪比波形数据进行测试与比较,其全波形数据为2019年11月11日采集于我国西南部高山区域,两个激光脚点分别位于约(33°18′20.15″N, 104°30′10.15″E)的森林区域(如图2(a)所示)和位于约(33°7′1.10″N, 104°27′18.39″E)的坡地(如图2(b)所示)。(a1)、(b1)为遥感影像;(a2)、(a3)、(b2)、(b3)为足印图像与回波数据。其中,A区域内呈现出复杂回波波形是因为该区域地形与植被比较复杂,存在密集的森林;B区域内呈现出复杂回波波形是因为该区域地形复杂,通过遥感影像与足印图像来分析区域内地表粗糙,另外观测时刻的云层也导致回波数据中噪声比较大。对于激光测高的足印图像,内部自检光在足印相机及光轴监视相机中均有成像光斑[14],如图2足印图像中绿圈标注;但该成像光斑与激光在地面的实际落点位置间存在一定几何关系,根据实验室标定出的转换关系,其转换的实际光斑位置,如图2足印图像中蓝圈标注。对激光测高的全波形数据降噪,文中中仅选取回波信号中的低增益全波形数据进行测试。
针对高分七号激光测高采集到的低信噪比复杂全波形数据,依据全波形数据的自身时间尺度,采用EMD方法分解之后得到6个分量,依次为从高频到低频的系列本征模函数,以及最后的残差,其典型高分七号全波形数据中EMD分解出的IMF及残差如图3所示。
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为测试EMD所分解出不同尺度IMF的筛选方法,实验中对激光脚点A和B区域的全波形数据,分别采用EMD-N、EMD-Threshold、EMD-Wavelet和EMD-DFA等筛选方法对IMF分量进行选取并滤波。其中,对于EMD-N方法,选取了第1个和前2个IMFs作为噪声IMF,分别标记为EMD-1IMF和EMD-2IMFs;对于EMD-Threshold方法,选取了软阈值和硬阈值对前2个IMFs进行筛选,分别标记为EMD-soft和EMD-hard;对于EMD-Wavelet方法,选取“db4”小波基和软阈值进行IMF降噪和重构;对于EMD-DFA采用拟合阶数K为1对时间序列Hurst指数进行估算,标记为EMD-DFA1。
针对激光脚点A和B区域的全波形数据,通过对几种基于EMD的降噪方法的降噪效果定性和定量比较。如图4所示,在全波形的EMD降噪中,(1) EMD-DFA1指数与EMD-1IMF降噪效果最好,且基本上保持一致,都是去除了第1个高频成分;(2) EMD-Wavelet降噪效果较好,但存在降低峰值幅值、展宽波形等偏离现象;(3) EMD-2IMFs由于直接去除前两个IMF存在过降噪现象,消除了有效的波形信号;(4) EMD-soft和EMD-hard与EMD-2IMFs相比均取得了较好的降噪效果,但EMD-soft仍存在过降噪现象。另外,对于同一个全波形数据,由于在滤波过程中选用不同的IMF筛选方法,得到的降噪后波形数据间存在一定的差异,其原因可能是不同尺度的混淆,自动识别IMF的阈值不稳健。
如表2所示,通过对EMD分解量的几种IMF筛选方法定量比较,可以看出:(1)在R2系数上,EMD-DFA1与EMD-1IMF的值最高,而EMD-2IMFs与EMD-soft值分别最小与次最小,表明前两种方法降噪后相关性最好,而后两种方法相关性较差;(2)在MSE和MAE上,EMD-2IMFs最大,其次为EMD-soft,表明该方法降噪后的偏离程度较大;(3)在SNR和PSNR上,EMD-DFA1与EMD-1IMF最大,其次为EMD-Wavelet和EMD-hard,表明其各自降噪后保留有用信号的程度由大到小。另外,EMD- DFA1与EMD-1IMF由于都是去除了第1个高频成分,降噪后的MSE、MAE、SNR、PSNR和R2量值一致。
表 2 不同基于EMD降噪效果定量对比
Table 2. Quantitative evaluation of denoised effect based on EMD
Spot Methods MSE MAE SNR PSNR R2 Spot A EMD-1IMFs 0.000403 0.017172 21.954108 33.944032 0.994883 EMD-2IMFs 0.000741 0.020372 19.313580 31.303504 0.990584 EMD-soft 0.000688 0.020099 19.634734 31.624658 0.991269 EMD-hard 0.000589 0.019156 20.312091 32.302015 0.992524 EMD-Wavelet 0.000452 0.017191 21.462847 33.452771 0.994255 EMD-DFA1 0.000403 0.017172 21.954108 33.944032 0.994883 Spot B EMD-1IMFs 0.000428 0.017824 21.528431 33.686438 0.991504 EMD-2IMFs 0.003198 0.033598 12.793858 24.951865 0.936689 EMD-soft 0.001459 0.028087 16.201364 28.359371 0.970888 EMD-hard 0.000923 0.023394 18.189807 30.347814 0.981577 EMD-Wavelet 0.000598 0.019404 20.071557 32.229564 0.988085 EMD-DFA1 0.000428 0.017824 21.528431 33.686438 0.991504 另外,为验证EMD-DFA1滤波对高分七号激光测高全波形数据的适应性,采用多组不同波峰分布情况的复杂波形进行测试。同时,采用传统高斯滤波与该方法进行比较,这里采用高斯模板和标准差分别是19和6。如图5所示,对于单个主波峰的光斑A和光斑B,降噪后的波形与原始波形的形状保持一致,并且保留了明显的混叠波峰;对于多个波峰的光斑C和光斑D,降噪后的波形与原始波形的形状和波峰个数都保持一致。另外,高斯滤波降噪后波形更加平滑,但是在混叠波峰情况下会丢失细节波峰,如图5(D)所示。由此可见,该方法可适用于单个波峰、混叠波峰、多个波峰的全波形数据,在混叠波峰情况下比高斯滤波效果更好。
Noise reduction based on empirical mode decomposition for full waveforms data of GaoFen-7 laser altimetry
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摘要: 针对具有多个高度层的复杂场景,全波形激光测高系统记录的回波信号中往往带有较高的噪声,采用合适的降噪方法将有助于提高计算激光测距的精确性、反演地物垂直结构和构建目标特征参数的准确性。根据高分七号激光测高在轨探测的低信噪比全波形数据的特性,采用经验模态分解(Empirical mode decomposition,EMD)方法来构建典型的本征模函数(Intrinsic mode function, IMF),对于分解出多个不同尺度IMF的筛选,比较了使用去除高频分量,阈值选取、Wavelet选取和去趋势波动分析(Detrended fluctuation analysis, DFA)等方法与策略,通过降噪效果及定量评价,测试结果表明EMD-DFA1与EMD-1IMF对高分七号激光测高的全波形数据具有较好的降噪效果,其次为EMD-Wavelet和EMD-Threshold。另外通过EMD-DFA1对单个波峰、混叠波峰、多个波峰等不同情况的全波形数据测试,结果表明该方法具有较好的自适应性。Abstract: The complex full waveforms from laser altimetry, mixed with high noise, are usually reflected by the object with multiple height elevations. To accurately analyze the decomposition, vertical structure and characteristic parameters from these waveforms, a noise reduction method based on empirical mode decomposition (EMD) was investigated and tested with the full waveform of nonlinear and nonstationary signals obtained by GaoFen-7 space-borne laser altimetry. The reconstruction of an effective waveform signal was implemented through reverse superimposition of its intrinsic mode functions (IMFs) and the residual. And then different selection methods for these IMFs were compared, such as removed high frequency, threshold, wavelet and detrended fluctuation analysis (DFA). The results show that EMD-DFA1 and EMD-1 IMF have a higher noise reduction effect on these full waveforms, followed by EMD-Wavelet and EMD-Threshold. Finally, EMD-DFA1 was performed on the full waveforms with single peak, mixed peaks and multiple peaks. And the results show that EMD-DFA1 does well adaptability.
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Key words:
- GaoFen-7 /
- laser altimetry /
- full waveform /
- empirical mode decomposition /
- noise reduction
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表 1 高分七号激光测高仪的主要参数
Table 1. Main parameters of the GaoFen-7 laser altimeter
Item Value Number of laser beams 2 Laser wavelength/nm 1 064 Laser energy/mJ 100-180 Pulse width/ns 4-8 Divergence/μrad 30-40 Emission frequency/Hz 3, 6 Digitization interval/ns 0.5 表 2 不同基于EMD降噪效果定量对比
Table 2. Quantitative evaluation of denoised effect based on EMD
Spot Methods MSE MAE SNR PSNR R2 Spot A EMD-1IMFs 0.000403 0.017172 21.954108 33.944032 0.994883 EMD-2IMFs 0.000741 0.020372 19.313580 31.303504 0.990584 EMD-soft 0.000688 0.020099 19.634734 31.624658 0.991269 EMD-hard 0.000589 0.019156 20.312091 32.302015 0.992524 EMD-Wavelet 0.000452 0.017191 21.462847 33.452771 0.994255 EMD-DFA1 0.000403 0.017172 21.954108 33.944032 0.994883 Spot B EMD-1IMFs 0.000428 0.017824 21.528431 33.686438 0.991504 EMD-2IMFs 0.003198 0.033598 12.793858 24.951865 0.936689 EMD-soft 0.001459 0.028087 16.201364 28.359371 0.970888 EMD-hard 0.000923 0.023394 18.189807 30.347814 0.981577 EMD-Wavelet 0.000598 0.019404 20.071557 32.229564 0.988085 EMD-DFA1 0.000428 0.017824 21.528431 33.686438 0.991504 -
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