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GIScience & Remote Sensing

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Curve matching approaches to waveformclassification: a case study using ICESat data

Yuhong Zhou, Fang Qiu, Feng Ni, Yifei Lou, Caiyun Zhang, MohammedAlfarhan & Ali A. Al-Dosari

To cite this article: Yuhong Zhou, Fang Qiu, Feng Ni, Yifei Lou, Caiyun Zhang,Mohammed Alfarhan & Ali A. Al-Dosari (2016): Curve matching approaches to waveformclassification: a case study using ICESat data, GIScience & Remote Sensing, DOI:10.1080/15481603.2016.1232147

To link to this article: http://dx.doi.org/10.1080/15481603.2016.1232147

Published online: 16 Sep 2016.

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Curve matching approaches to waveform classification: a case studyusing ICESat data

Yuhong Zhoua, Fang Qiua*, Feng Ni a, Yifei Lou b, Caiyun Zhangc,Mohammed Alfarhand and Ali A. Al-Dosarie

aDepartment of Geospatial Information Science, University of Texas at Dallas, 800 W Campbell Rd.GR31, Richardson, TX 75080-3021, USA; bDepartment of Mathematical Sciences, University ofTexas at Dallas, 800 W Campbell Rd. GR31, Richardson, TX 75080-3021, USA; cDepartment ofGeosciences, Florida Atlantic University, 777 Glades Road, Boca Raton, FL 33431, USA; dNationalCenter for Remote Sensing Technology, King Abdulaziz City for Sciences and Technology, Riyadh11442, Saudi Arabia; eDepartment of Geography, King Saud University, Riyadh 11451, SaudiArabia

(Received 22 March 2016; accepted 31 August 2016)

Light Detection and Ranging (LiDAR) waveforms are being increasingly used in manyforest and urban applications, especially for ground feature classification. However,most studies relied on either discretizing waveforms to multiple returns or extractingshape metrics from waveforms. The direct use of the full waveform, which contains themost comprehensive and accurate information has been scarcely explored. We pro-posed to utilize the complete waveform to test its ability to differentiate betweenobjects having distinct vertical structures using curve matching approaches. Twogroups of curve matching approaches were developed by extending methods originallydesigned for pixel-based hyperspectral image classification and object-based highspatial image classification. The first group is based on measuring the curve similaritybetween an unknown waveform and a reference waveform, including curve root sumsquared differential area (CRSSDA), curve angle mapper (CAM), and Kullback–Leibler (KL) divergence. The second group assesses the curve similarity between anunknown and reference cumulative distribution functions (CDFs) of their waveforms,including cumulative curve root sum squared differential area (CCRSSDA), cumula-tive curve angle mapper (CCAM), and Kolmogorov–Smirnov (KS) distance. Whenemployed to classify open space, trees, and buildings using ICESat waveform data, KLprovided the highest average classification accuracy (87%), closely followed byCCRSSDA and CCAM, and they all significantly outperformed KS, CRSSDA, andCAM based on 15 randomized sample sets.

Keywords: waveform; classification; curve matching; ICESat/GLAS

1. Introduction

Light Detection and Ranging (LiDAR) is an active remote sensing technology which usesvisible, near-infrared, or shortwave infrared laser beams to measure ground elevation,while also providing information on the vertical structure of geographical objects. Thereare two types of LiDAR data: discrete-return LiDAR and full-waveform LiDAR(Ussyshkin and Theriault 2011). Currently, discrete-return LiDAR data are the mainstreamcommercial product and a great number of off-the-shelf tools are available for processing

*Corresponding author. Email: ffqiu@utdallas.edu

GIScience & Remote Sensing, 2016http://dx.doi.org/10.1080/15481603.2016.1232147

© 2016 Informa UK Limited, trading as Taylor & Francis Group

discrete-return LiDAR (Ussyshkin and Theriault 2011). Up to six returns per transmittedlaser pulse are typically recorded, and these typically have x, y, elevation, and intensitymeasured for each (Sasaki et al. 2012; Zhou 2013).

Full waveform, a relatively new LiDAR product, records the quasi-continuous time-varying strength of the return signal from the illuminated area (called the waveformfootprint) using small time intervals (e.g. 1 nanosecond). Waveform LiDAR has becomeincreasingly popular in the last decade and it is able to provide thousands of measure-ments for each transmitted laser pulse (Mallet et al. 2011; Ussyshkin and Theriault 2011;Alexander et al. 2010). Since waveform data provide finer vertical resolution, they offeran enhanced capability to reflect the vertical structures of geographical objects (Wagneret al. 2008; Farid et al. 2008). The shape of a waveform is the result of interactionbetween the transmitted laser pulse and intercepted object surfaces within the footprint(Alexander et al. 2010; Pirotti 2010). For example, waveforms over trees are influencedby characteristics such as tree density, structure, and vegetation phase. Given that discrete-return LiDAR provides no more than six measurements in its vertical profile which maybe separated by a few meters in vertical distance, LiDAR waveform provides a better 3Dprofile of an object (Ussyshkin and Theriault 2011).

Because of free availability, regular repetition, and global coverage, the large-footprintwaveform data from the Geoscience Laser Altimeter System (GLAS) onboard the Ice,Cloud, and land Elevation Satellite (ICESat) have been widely used. Previous applicationsespecially focused on forest studies (Harding and Carabajal 2005; Lee et al. 2011; Sunet al. 2008; Lefsky et al. 2007; Wang et al. 2012) and building model reconstruction inurban environments (Gong et al. 2011; Cheng et al. 2011). For these studies, the first stepusually involves identifying waveforms over trees or buildings and separating them fromthose over other land cover classes. These analyses can be very time-consuming andlabor-intensive because the first step is usually achieved through visual interpretation(sometimes with the aid of existing land use land cover information) (Gong et al. 2011;Cheng et al. 2011). Our previous study has demonstrated the ability of full waveformsalone to differentiate between trees, buildings, and open space using a Kolmogorov–Smirnov (KS) distance-based curve matching approach (Zhou et al. 2015), which out-performed the widely used metrics-based approach. This current study is a continuingeffort on curve matching approach-based research, but with a focus on the comparison ofdifferent curve matching approaches that we developed for waveform classification. Ifthese approaches are demonstrated to be successful, they could be potentially used toimprove the research in forest and urban studies that primarily employ 2D remotelysensed imagery by incorporating the vertical structure information that the waveformsprovide.

2. Background

In general, a LiDAR waveform may have one or many echoes, where one echo is a peakthat corresponds to an individual target encountered or a cluster of objects too close to beseparated, while multiple echoes often represent either multiple targets or one target withmultiple components that are vertically separable within the footprint (Alexander et al.2010). Consequently, the shapes of waveforms over multiple objects can be very complex.To perform waveform classification, existing studies have relied on simplifying thecomplex waveform data based on either the discretization of the full waveform intomultiple returns or the derivation of metrics to characterize the waveform shapes. Alarge number of studies have attempted to extract more returns from full waveforms

2 Y. Zhou et al.

through discretization to overcome the limitations of discrete-return LiDAR, which hasonly a few returns and fails to adequately capture the vertical structures of geographicalobjects (Wagner et al. 2008; Rutzinger et al. 2008; Mallet et al. 2011; Reitberger et al.2009; Mallet and Bretar 2009; Ussyshkin and Theriault 2011; Guo et al. 2011).Discretization of full waveform offers flexibility in that the number of returns extractedcan be determined by users (Ussyshkin and Theriault 2011). The number of resultantreturns can be set to be many more than that of the traditional discrete- return LiDAR. Thelarger number of extracted returns can better represent the vertical structure of objects.Comparatively, the number of discretized returns is also significantly less than that of theoriginal full waveform. With manageable complexity, the discretized returns can then beanalyzed by algorithms designed for traditional discrete-return LiDAR.

To avoid dealing with the complex shape of a waveform, other studies have utilizedwaveform-derived summary metrics such as the number of echoes, peak amplitude, echowidth, skewness, and kurtosis to represent key characteristics of the waveform’s shapes(Molijn, Lindenbergh, and Gunter 2011; Mallet et al. 2011). These metrics have hadvaried success in differentiating land cover types. As with the discretization of waveforms,these waveform-derived metrics simplify the complexity of the complete waveforms, inthis case by focusing on important shape characteristics. However, by reducing complex-ity, both the discretization of waveforms into multiple returns and the derivation of shape-related metrics are unable to take advantage of the entirety of information that the fullwaveform offers.

The complete curve of a full waveform contains substantially more information thanthe discretized returns or summary metrics extracted from the waveform. However, thecomplete waveform curves have rarely been utilized to perform classification because ofthe lack of appropriate approaches for dealing with their complexity. In an earlier paper(Zhou et al. 2015), we proposed a curve matching approach to directly compare waveformcurves using KS distance as an alternative to the methods based on the simplification ofwaveforms. Preliminary results showed that this approach outperformed a widely adoptedmetrics-based method using waveform-derived metrics. This result suggested that curvematching approaches to waveform classification is a new direction worth in-depthexploration.

In fact, curve matching approaches are not unfamiliar to remote sensing communitysince they have been utilized for both pixel-based hyperspectral image classification andobject-based image classifications. For pixel-based hyperspectral image classification, aspectral signature is a curve of reflectance values across an electromagnetic spectrum(Jensen 2004). The shape of a hyperspectral curve is determined by the absorptivecharacteristics of the material. Various methods have been developed to assess thesimilarity of the hyperspectral curves of pixels with those of different materials in aspectral library for identification and classification purposes, such as the spectral anglemapper (SAM) (Kruse et al. 1993), root sum squared differential area (RSSDA) (Hamadaet al. 2007), and Kullback–Leibler (KL) divergence (Chang 2000).

For object-based image classification, the frequency distribution function (histogram)and cumulative distribution function (CDF) of the pixel values within an image object fora particular band were used to perform object-based feature extraction using methodsextended from existing pixel-based hyperspectral image classification (Stow et al. 2012;Toure et al. 2013; Sridharan and Qiu 2013; Zhou and Qiu 2015). For example, Stow et al.(2012) proposed a Histogram Matching Root Sum Squared Differential Area(HMRSSDA) method, which was extended from the RSSDA approach (Hamada et al.2007), to measure the similarity of object-level histogram curves for land cover and land

GIScience & Remote Sensing 3

use classification. Continuing the work of Stow et al. (2012), Toure et al. (2013) extendedthe SAM approach (Kruse et al. 1993) to develop a Histogram Angle Mapper (HAM) tomeasure the similarity of object-based histograms. Both HMRSSDA and HAM classifiershave consistently outperformed the widely adopted nearest neighbor object-based classi-fiers. We also developed a fuzzy KS-based classifier to assess the similarity of thereflectance values in an unknown object compared to those in a known object by usingthe CDF rather than the frequency histogram distribution. This approach achieved morethan 10% improvements over other popular object- and pixel-based classifiers (Sridharanand Qiu 2013). Recently, we developed a KL divergence-based classifier to quantify thesimilarity of object-level histogram curves, which performed significantly better thanHAM and HMRSSDA, though not significantly better than the KS classifier (Zhou andQiu 2015).

A waveform is not only a curve but also can be considered as a time-varyingfrequency distribution function or histogram (Muss, Mladenoff, and Townsend 2011).Therefore, it is intuitive to extend both pixel-based hyperspectral image classificationmethods and object-based high spatial image classification approaches to waveformcurves. Aside from the KS-based curve matching approach we presented in Zhou et al.(2015), we are unaware of other studies using curve matching approaches to classifycomplete waveforms. In this study, we developed and explored two groups of curvematching approaches by extending methods originally designed for pixel-based hyper-spectral image classification or object-based high spatial image classification. The firstgroup was designed to measure the curve similarity between an unknown waveform andreference waveform, including curve root sum squared differential area (CRSSDA), curveangle mapper (CAM), and the KL divergence. The second group was developed to assessthe curve similarity between an unknown and a reference CDFs of their waveforms,including cumulative curve root sum squared differential area (CCRSSDA), cumulativecurve angle mapper (CCAM), and the KS distance.

3. Study area and data

The study area is located in Dallas, Texas, bounded approximately by 97.031–96.518° Win longitude and 32.553–32.996° N in latitude. The area is characterized by a mixture ofmature residential neighborhoods, and commercial and industrial buildings. ICESat wave-forms are sensitive to surface terrain due to their large footprints. Since the majority of thestudy area is flat (slope ≤10°), the terrain is not expected to significantly influence thewaveform shapes (Gong et al. 2011; Hilbert and Schmullius 2012; Duong et al. 2009) andthe resulting classification.

Due to the large spacing between its footprints, ICESat waveform has not been widelyused for operational land cover mapping, which usually requires contiguous coverage ofthe study area. However, intent of this study is to evaluate the effectiveness of the curvematching approaches for waveform classification by capitalizing the vertical structureinformation embedded in the data. ICESat data is employed here as a case study for thedeveloped approaches. If proved effective, these approaches could be readily applied toother LiDAR waveform data with contiguous coverages, such as those from airbornesystems.

The ICESat program distributes 15 Level-1 and Level-2 data products (ranging fromGLA01 to GLA15), which can be downloaded from the website of the National Snow andIce Data Center (http://nsidc.org/data/icesat/data.html). The full waveform data wereextracted from GLA01 (L1A Global Altimeter Data), the footprint shape information

4 Y. Zhou et al.

were calculated using the parameters in GLA05 (L1B global waveform-based rangecorrection data), and the center location of the waveform footprints were derived fromGLA14 (L2A Global Land Surface Altimetry Data). All datasets were linked by theircommon index and shot number. Figure 1 is an example of ICESat waveform over an areawith trees.

ICESat footprints are elliptical in shape and their sizes vary from 50 to 150 m with anaverage size around 75 m. The approximate areas of the waveform footprints werecomputed using the method proposed by Gong et al. (2011). The resulting waveformfootprint polygons were then converted to keyhole markup language (KML) and importedto Google Earth. The land cover conditions within the selected footprints were determinedthrough visual interpretation of the Google Earth image. We used three general land covercategories with distinct vertical structures: buildings, trees, and open spaces. All landcover types with a flat open surface (e.g. bare land, water, roads, and parking lots) wereassigned to the open space category. They were not identified as separate categories sincethey have generally similar vertical waveform curves, and it is very difficult to classifythem using the waveform LiDAR alone. Similarly, there was no differentiation of buildingsubtypes (such as residential or commercial), or tree subtypes (such as deciduous orconiferous). These very general land cover categories were a necessity because the largesize of the ICESat footprint does not allow a more detailed level of classification (Zhouet al. 2015).

4. Methodology

Curve matching approaches measure the curve similarity between two samples. Onesample is usually with a known class (training sample) and the class of the other ispredicted (testing sample). For this purpose, after applying a set of preprocessing proce-dures, the waveform dataset was divided into training and testing subsets. To compare the

Figure 1. An example of ICESat waveform over an area of trees.

GIScience & Remote Sensing 5

six proposed curve matching approaches, the performance difference between any twocurve matching approaches was assessed with the pairwise McNemar’s chi-square sig-nificance test. The test was based on 15 randomly selected reference waveform sets, eachwith a total of 54 reference waveforms, comprising 19 reference waveforms for buildings,24 for trees, and 11 for open space. The methodological steps required for this study aresummarized in the flow chart in Figure 2.

4.1. Preprocessing

A number of preprocessing steps were needed to make the waveform data suitable for thesubsequent analysis. First, both GLA01 and GLA14 were converted from distributedbinary format to ASCII format, and the unit of GLA01 was converted from count (0–255)to voltage. Then, the ellipsoid of GLA14 was transformed from the default TOPEX/Poseidon to the WGS84 coordinate system to match reference data. Second, unreliableoutlier waveforms, such as those with a reflectivity lower than 0.1(Molijn, Lindenbergh,and Gunter 2011), or those with a signal-to-noise ratio lower than 10 (Hilbert andSchmullius 2012), were excluded because they were primarily caused by dense cloud.Finally, a noise removal procedure was applied to eliminate system noise at the beginningand end of each waveform signal. Any waveform signal value below the threshold wasregarded as noise and was assigned a value of zero. The threshold was determined usingthe method proposed by Lefsky et al. (2007). The starting location of a waveform wasdefined by finding the farthest left location of the waveform where the signal was larger

Figure 2. General flowchart of methodology.

6 Y. Zhou et al.

than the threshold value, and the ending location was identified by the rightmost locationwhere the signal was larger than the threshold value. The earliest starting location of thereliable waveforms was the 651st ns, and the last ending location was close to the 1000thns value (Zhou et al. 2015). Therefore, only signals over the last 350 ns (from 651st to1000th ns) were analyzed in this study.

After preprocessing, only waveforms having a dominant type were identified assamples for each class in this study because the waveform shapes were primarilyinfluenced by the vertical structures of the dominant land cover type within the footprints.This was relatively easy to achieve for the open space category due to common presenceof large open spaces in Dallas. However, due to the likely presence of bare land, grasswith a mixture of buildings and trees inside the large ICESat footprints, only waveformswhere buildings or trees covered more than 50% of the area based on Google Earth imagewere considered as buildings or trees, respectively. The waveform footprints that did nothave a single dominant land cover type, but instead had a mixture of several dominanttypes with none exceeding 50% in coverage were excluded from the training and testingprocesses because they could not be easily labeled with a single class in a hard classifica-tion such as the one used in this study. Like a mixed pixel, the classification of mixedfootprints should be investigated using a fuzzy classification algorithm in the future,which, however, is not the focus of this study. If the potential of waveform curvesusing curve matching approaches can be demonstrated, the same methods could beextended to small footprint airborne LiDAR by segmenting point clouds or image intoobjects first and then synthesizing object-level pseudo-waveforms using a histogram ofdiscrete LiDAR returns or waveforms (Zhou and Qiu 2015). Since the footprints of theseobjects are derived from HSR imagery through image segmentation, the mixed footprintproblem would not be an issue anymore because each image segment usually has adominant land cover type as a result of segmentation. When all the samples wereidentified for each class, around half of the samples for each class were randomlyseparated as training samples and the other half as testing samples, as shown in Table 1.

4.2. Reference waveform selection

For the majority of existing curve matching-based classifications, each class is assumed tohave a single end-member or reference sample. An end-member/reference is one of thetraining samples that is identified to represent relatively pure material (Jensen 2004) or isobtained by averaging all the training samples (Ghiyamat et al. 2013; Stow et al. 2012;Toure et al. 2013). This approach is meaningful when differences among the curves withinthe same class (within-class variation) are negligible. However, the training dataset herehas a high-degree of within-class variation, especially for the tree and the building classes.This is due to varied height levels, vertical structures, or net coverages within thefootprints. As a consequence, using a single end-member/reference sample is not appro-priate. For complexity and resource reasons, it is also not appropriate to use all the

Table 1. Number of training and testing samples.

Class No. of training samples No. of testing samples

Building 91 92Tree 135 135Open space 195 196

GIScience & Remote Sensing 7

training samples as reference waveforms since many waveforms in a training class havesimilar profiles. Instead, a subset of reference waveforms needs to be selected for eachclass. This should provide better discrimination ability than using a single end-memberdue to the capturing of within-class variation of the training data (Cho et al. 2010;Ghiyamat et al. 2013), while avoiding the complexity and data duplication associatedwith the use of all training waveforms as references.

In our previous study (Zhou et al. 2015), a principal component analysis (PCA) wasused to select a small number of references from the training dataset. This avoided dataduplication while also reflecting within-class variation. However, it is not appropriate inthe current context to use PCA-selected reference waveforms to evaluate the performanceof the six proposed curve matching approaches. The PCA-based reference selection usedKS distance to measure the curve similarity when selecting references. The referencesselected using the KS distance may favor the performance of the KS-based approach overthe other five curve matching approaches when the actual classification is performed.Therefore, to avoid biasing any of the six curve matching approaches, the referencewaveform selection process should not utilize any similarity measure that is employedfor subsequent classification.

For this reason, the reference waveforms were simply randomly selected from thetraining dataset. Our previous study (Zhou et al. 2015) showed that 10 sets of randomlyselected references were able to produce results with an average overall accuracy of 82%using the KS-based approach. To increase the representativeness of the random samples,we introduced additional random sample sets in this study. Each time when a new randomsample set was introduced, the average overall accuracy was calculated over all theexisting sample sets and was compared with the average overall accuracy of the previousiteration. The iteration stopped when the difference of the average overall accuracies wasbelow 0.1% for two consecutive iterations, which indicated a stable average overallaccuracy. As a result, a total of 15 sets of randomly selected reference waveforms wereused to evaluate the final average performance of the six curve matching approaches. Inorder to maintain consistency with our earlier study, each set contained the same numbersof reference waveforms for each class as those determined by the PCA analysis in ourprevious study (Zhou et al. 2015).

4.3. Curve matching approaches

The classification of each testing waveform was accomplished by measuring the degree ofcurve similarity between the waveform and each in the randomly selected set of referencewaveforms. Each testing waveform was assigned to the class of the reference waveformwith the highest curve similarity. Generally, to calculate the curve similarity between twowaveforms, let r ¼ ðr1; :::rnÞT and t ¼ ðt1; ::: tnÞT be a reference waveform and a testingwaveform respectively, each with n time bins (n = 350 in this study), and

Cr ¼ ðCr1; :::C rnÞT , Ct ¼ ðCt1; :::C tnÞT be their corresponding CDFs. Each of sixdifferent curve matching measurements was used to assess the similarity between r andt, as follows.

4.3.1. Curve root sum squared differential area

CRSSDA is an area-based classifier that computes curve similarity by integrating the areabetween a testing waveform and a reference waveform. Similar to HMRSSDA (Stow et al.

8 Y. Zhou et al.

2012), which was successfully used to match object-based histogram curves to differ-entiate between various image objects, CRSSDA is also extended from RSSDA (Hamadaet al. 2007), which was originally designed for pixel-based hyperspectral classification.The method first calculates the squared difference between two waveform curves at eachtime bin, then sums the squared differences across all the bins, and finally computes thesquare root of the total squared difference to obtain waveform curve similarity(Equation (1)).

CRSSDA ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXmax

i¼minðri � tiÞ

2r

; (1)

where min and max are the start and end location of the waveform. In this study, minequals 651 and max equals 1000, because only the signals from the 651st ns to the 1000thns are considered. A smaller value of CRSSDA indicates a higher curve similaritybetween two waveforms.

Additionally, we generated the CDF curves of the waveforms and used these as inputsto Equation (1). We call this new approach CCRSSDA. The performance of CRSSDA andCCRSSDA was compared to see whether the CDF curve or the original waveform curveachieved better accuracy when the area-based approaches were used for waveformclassification.

4.3.2. Curve angle mapper

As with HAM (Toure et al. 2013), the CAM approach is also extended from the SAMalgorithm that was originally designed to quantify the curve similarity of hyperspectralpixels. The SAM algorithm compares the curve similarity of a testing spectral t to areference spectral r by calculating the angle α between t and r in n spectral dimensionsusing Equation (2). A smaller value of α indicates a higher similarity between the twospectra. In this study, CAM is employed to measure the curve similarity between a testingwaveform t and a reference waveform r using the same equation, where i refers to the ith

ns and n is the total number of bins. Theoretically, α can vary from 0�to 90

�, where 0

�

stands for maximum similarity and 90�stands for minimum similarity.

α ¼ cos�1Xn

i¼1tiri

� Xn

i¼1ti2

� �12Xn

i¼1ri2

� �12

� �� : (2)

Similarly, when the CDF curves of waveforms are used as inputs to Equation (2), themethod becomes a new approach named CCAM. The performance of CAM and CCAMwas compared to determine which achieved better discrimination accuracy.

4.3.3. Kolmogorov–Smirnov distance

The KS distance is often used to measure curve similarity by calculating the maximumabsolute distance between the CDFs of two frequency distributions, as described inEquation (3) (Burt and Barber 1996).

D ¼ Maxmin�i�max CrðiÞ � CtðiÞj j: (3)

GIScience & Remote Sensing 9

Since a waveform can be considered as a frequency distribution function (Zaletnyik,Laky, and Toth 2010), the CDF of the waveform was first generated and the KS distancewas then calculated between each testing waveform and individual reference waveformsas in Zhou et al. (2015). The value of D can vary from 0 to 1, where 0 indicates maximumsimilarity; while 1 indicates minimum similarity. KS distance can be applied to the CDFsbut not their original waveforms.

4.3.4. Kullback–Leibler divergence

The KL divergence is another technique that measures curve similarity between twoprobability distribution functions (Kullback and Leibler 1951). It has been widely andsuccessfully used in image pattern recognition (Olszewski 2012) and hyperspectral imageclassification (Ghiyamat et al. 2013). Since an object-level spectral histogram can beconsidered a probability distribution function, in Zhou and Qiu (2015) we developed a KLdivergence approach to quantitatively measure the curve similarity between the spectralhistogram of two objects. This achieved the highest classification accuracy compared withall other object-based curve matching approaches investigated. Since a waveform also canbe normalized as a probability distribution function, we applied the KL divergenceapproach to measure the curve similarity between r and t using Equation (4).

DðrjjtÞ ¼Xn

ilog rðiÞ=tðiÞð Þ � rðiÞ: (4)

A smaller value of DðrjjtÞ indicates a higher curve similarity. Since KL divergence isonly between two probability functions, we cannot apply it to the CDF of a waveform.

4.4. Accuracy assessment

The overall performance of each curve matching approach was evaluated in terms of totalpercentage accuracy. A pairwise McNemar’s chi-square test (Agresti 1996) was thenconducted between each pair of approaches to evaluate if the improvement of oneapproach over the other was statistically significant for each of the 15 trials of randomlyselected references.

Since each curve matching approach uses different scales or units, the resultantmeasures cannot be compared directly. To address this issue, all measures were normal-ized using a relative curve discriminatory probability (RCDPB) as in Equation (5). Thiswas originally employed to evaluate the performance of different hyperspectral classifica-tion methods (Ghiyamat et al. 2013).

RCDPBðt;riÞ ¼sðt; riÞPmj¼1 sðt; rjÞ

; (5)

where sðt; riÞ represents the curve similarity between t and the ith reference waveform rusing any one of the curve matching approaches and m is the number of selected referencewaveforms. A smaller value of RCDPB indicates a higher curve similarity between twowaveforms. RCDPB was used to assess and compare the performance of the six curvematching approaches for each of the classes.

10 Y. Zhou et al.

5. Results and discussion

As explained earlier, 15 sets of randomly selected reference waveforms were used toevaluate the performance of the 6 curve matching approaches. In this section, thecharacteristics of the waveform curves were first examined using one set of randomlyselected reference waveforms as an example. The relative performance of the 6 curvematching approaches was then compared using all 15 sets of randomly selected referencewaveforms.

5.1. Waveform curve characteristics

For each random sampling, a total of 54 reference waveforms were selected from 421training samples, including 19 for buildings, 24 for trees, and 11 reference waveforms foropen space. Figure 3 presents the original waveform curves and their corresponding CDFcurves for one set of randomly selected reference samples.

Figure 3. Original waveform curves and their corresponding CDF curves from one randomsampling result (a, c, and e are original waveforms of open space, tree, and building, respectively,and b, d, and f are their corresponding CDF).

GIScience & Remote Sensing 11

For the open space class, the within-class variation of the waveform curves is thesmallest. The majority of waveforms for open space have similar single-echo shapes withlate starting locations due to their low elevation and narrow echo widths because of theirflat surface (Figure 3(a)). However, they differ slightly from each other in terms of thestarting and ending location, the echo width, or the peak amplitude because of differencesin surface elevation, slope, or material. The CDF curves of waveforms for open spacegenerally exhibit a late rising location (where the CDF value begins to rise) and a steeplyincreasing edge that quickly reaches 1, with minor differences in the rising locations andthe slopes of the increasing edges (Figure 3(b)). These characteristics make the waveformsfor open space highly distinguishable from those for buildings and trees.

The tree class is characterized by a large within-class variation in the original wave-form curves (Figure 3(c)) as well as in their corresponding CDF curves (Figure 3(d)). Theoriginal waveform curves for trees vary markedly in terms of starting location, number ofechoes, and/or width and peak amplitude for each echo. This results in differences in theircorresponding CDFs in terms of the rising locations, the slopes of increasing edges, and/orthe areas under the CDF curves. These differences are primarily determined by treeheights, tree crown shapes, and/or the net coverage of trees within the footprints. Forexample, waveforms over tall trees usually have earlier starting locations, while wave-forms over short trees often have relatively later starting locations, which contributes tothe different rising locations in their corresponding CDF curves. Waveforms with afootprint containing multiple trees with bigger tree crowns usually have several widerechoes at high altitude, while waveforms with footprint covering trees with smaller treecrowns have several narrower echoes. Trees usually have gradually rising edges withvaried degrees of slopes in their CDF curves. Overall, waveforms and their CDFs for thetree class have the highest within-class variation among the three classes. This makes itdifficult to separate them visually from the waveforms in the other classes.

For the building class, Figure 3(e,f) shows moderate within-class variation, primarilydue to the different types of buildings from which the waveforms were returned. There arethree major types of buildings in the study area: flat-roofed, slope-roofed, and buildingssurrounded by trees. Waveforms over flat-roofed buildings usually have two individualnarrow echoes separated by a distance of a few meters to about 15 m. The top echocorresponds to the flat roof of the building, the bottom one corresponds to the flat ground,and the vertical interval between the two corresponds to the height of the building. Jointly,this generates a stair-step pattern (i.e. steep–flat–steep–flat) in their corresponding CDFcurves, which makes it easy to separate waveforms over flat-roofed buildings from thoseover trees. Waveforms over slope-roofed buildings present different waveform patternscompared with those over flat-roofed buildings. Slope-roofed buildings generally haveone wider echo resulting from scattering caused by the sloping roof. Consequently, theCDF curves of waveforms over slope-roofed buildings show a gradual–flat–steep–flatpattern, which are again clearly different from those over trees. However, for the buildingsclosely surrounded by trees, the waveform curves and their corresponding CDF curvesexhibit similar patterns to those over trees, which can cause confusion between the treeand the building classes.

5.2. Performance comparison of different curve matching approaches

The average total percentage accuracy of the classification results over the 15 trials ofrandomly selected reference waveforms, in ascending order, was: 70% for CAM, 75% forCRSSDA, 79% for KS, 85% for CCAM and CCRSSDA, and 87% for KL.

12 Y. Zhou et al.

5.2.1. Comparison of waveform-curve-based approaches

Among the three approaches that used the original waveform curve as inputs (CRSSDA,CAM, and KL), KL achieved the highest average classification accuracy in terms of totalpercentage accuracy. The results of McNemar’s chi-square test indicated that the perfor-mance of KL was significantly better than that of CAM and CRSSDA at a significancelevel of 0.0001 for all 15 trials (Figure 4). This result reconfirms the superiority of KLover HAM and HMRSSDA as in our previous study (Zhou and Qiu 2015). With respectto CRSSDA compared with CAM, Figure 4 shows that CRSSDA significantly outper-formed CAM at a level of 0.01 for most of the 15 trials. This suggests that theperformance of CRSSDA is in general significantly better or at least comparable to thatof CAM.

To achieve better insight, the within-class and between-class curve similarities mea-sured by these three methods are shown in Figure 5 with all the measures normalizedusing RCDPB. The within-class curve similarities are obtained when the testing wave-forms are from the same class as that of the reference waveforms. For example, if a testingwaveform is from the building class, the within-class curve similarity for this testingwaveform is the smallest RCDPB value between this testing waveform and all thereference waveforms in the building class, using a given method. After the within-classcurve similarities for all the testing waveforms in the class are calculated for a givenmethod, the boxplot of the within-class curve similarities (e.g. Building–Building) is then

Figure 4. Boxplot of the p-values of McNemar’s chi-square test among waveform-curve-basedapproaches across the 15 trials.

Figure 5. Analysis of within-class and between-class curve similarity for the three waveform-curve-based approaches. For full colour versions of the figures in this paper, please see the onlineversion.

GIScience & Remote Sensing 13

drawn. Similarly, the between-class curve similarities are obtained when the testingwaveforms and the reference waveforms are from different classes. If a testing waveformis from the building class, the between-class curve similarity for this testing waveform tothe tree class is the smallest RCDPB value between this testing waveform and all thereference waveforms in the tree class. Again, after the between-class curve similarities forall the testing waveforms in the class are calculated, the boxplot of the between-classcurve similarities (e.g. Building–Tree) is produced.

Figure 5 shows that CRSSDA, CAM, and KL differ considerably in the size of theirinterquartile boxes for the three within-class curve similarities (Building–Building, Tree–Tree, and Open–Open), the closeness of these boxes to the x axis, and the overlap of theboxes for within-class and between-class curve similarities. For KL, the boxes of theBuilding–Building, Tree–Tree, and Open–Open were much closer to the x axis and theranges of their interquartile boxes were substantially smaller compared with those ofCRSSDA and CAM. This explains why KL achieved the highest classification accuracyfor all three classes. KL also had smaller overlaps of the boxes between the individualwithin-class and their corresponding between-class curve similarities than CRSSDA andCAM. Taking the building class as an example, there were smaller overlaps of the boxesbetween Building–Building and Building–Tree and between Building–Building andBuilding–Open for KL, while the corresponding overlaps of the boxes for CRSSDAand CAM were considerably larger. As a result, the probability that a testing waveform forthe building class was misclassified as tree or open space when using KL was much lowerthan using CRSSDA and CAM.

In general, CRSSDA had similar patterns in their interquartile boxes to CAM in termsof their closeness to the x axis and ranges of the boxes for all the three within-class curvesimilarities. However, compared with CAM, CRSSDA had smaller overlaps of the boxesbetween the within-class and their corresponding between-class curve similarities for allthe classes. Taking the open space class as an example, CRSSDA had a smaller overlap ofthe boxes between the within-class (Open–Open) and the between-class curve similarities(Open–Building) than CAM, even though the range of its box for the Open–Open within-class curve similarities was slightly larger than that of CAM. As a consequence, theprobability of a testing sample of open space being misclassified when using CRSSDAwas lower than using CAM.

5.2.2. Comparison of CDF-curve-based approaches

CCRSSDA, CCAM, and KS are all based on the CDF curves of the original waveforms.CCRSSDA and CCAM had similar performance, while KS performed slightly worse interms of the average total percentage accuracy. The results of McNemar’s chi-square testindicated that there were no significant differences between the performance ofCCRSSDA and CCAM and both performed significantly better than KS at the level of0.05 (Figure 6). This is probably due to the fact that KS measures a single maximumabsolute difference between two CDFs, while CCAM sums the angle differences andCCRSSDA sums the area differences of all the time bins between two CDFs.

The within-class and between-class curve similarities measured by the three CDF-based methods are shown in Figure 7. As with the waveform-based curves, the measureswere normalized using RCDPB. CCRSSDA and CCAM had similar patterns in theinterquartile boxes in terms of their closeness to the x axis and the ranges of the within-class similarity boxes for all three categories. Similarly, both approaches had no majordifferences in the overlaps of the boxes between the within-class and the corresponding

14 Y. Zhou et al.

between-class curve similarities. These were consistent with the results of the McNemar’stest which showed no significant differences in performance between CCRSSDA andCCAM for all 15 trials.

Compared with CCRSSDA and CCAM, KS had larger ranges of the boxes for the threewithin-class curve similarities and these boxes were farther away from the x axis. For thebuilding class, KS had a larger overlap of the boxes between the within-class (Building–Building) and the between-class (Building–Tree) curve similarities. Consequently, the prob-ability of a testing waveform in this class being misclassified as tree using KS was higherthan using CCRSSDA and CCAM. For the tree and the open space classes, there were nooverlaps of the boxes between the within-class and their corresponding between-class curvesimilarities for all three methods. However, compared with CCRSSDA and CCAM, KS hadmore extreme values above the within-class interquartile boxes that overlapped with thecorresponding between-class interquartile boxes. This indicated that KS may not perform aswell as CCRSSDA and CCAM for these two classes.

5.2.3. Comparison of waveform-curve-based and CDF-curve-based approaches

As mentioned above, CRSSDA, CAM and KL are based on the original waveform curves,while CCRSSDA, CCAM, and KS are based on their derived CDFs. Figure 8 showsboxplots of the results of the pairwise McNemar’s test between the three waveform-curve-based approaches and the three CDF-curve-based approaches for all 15 trials.

Figure 6. Boxplot of the p-values of McNemar’s chi-square test among CDF-curve-basedapproaches across the 15 trials.

Figure 7. Analysis of within-class and between-class curve similarity for the three CDF-curve-based approaches. For full colour versions of the figures in this paper, please see the online version.

GIScience & Remote Sensing 15

To assess the role of curve type (noncumulative versus cumulative), we specificallycompared the performance of CRSSDA versus CCRSSDA and CAM versus CCAM.CRSSDA and CCRSSDA use the same root sum squared differential area-based algorithmand CAM and CCAM use the same angle mapper-based algorithm. With the CDF curvesof waveform data as inputs, CCRSSDA and CCAM approaches achieved significantimprovements at the level of 0.001 in classification accuracy compared with their non-cumulative counterparts (Figure 8). For example, CCAM improved over CAM in averagetotal accuracy by 15%. Regarding the within-class and between-class curve similarities(Figures 5 and 7), the within-class interquartile boxes for CCRSSDA and CCAM hadsmaller ranges and were much closer to the x axis for all three classes compared withCRSSDA and CAM, respectively. CCRSSDA and CCAM also had smaller overlapsbetween the within-class boxes and their corresponding between-class boxes for eachclass. Taking the building class as an example, CCRSSDA and CCAM had considerablysmaller overlaps of the boxes between Building–Building and Building–Tree and betweenBuilding–Building and Building–Open compared with CRSSDA and CAM, respectively.This resulted in fewer building waveforms misclassified as trees or as open space. For aspecific class, waveforms usually have similar patterns in their shapes, but may differfrom each other in the local peaks and troughs, as well as the peak location and the widthof each echo. CRSSDA and CAM are very sensitive to these differences, which causesthese two approaches to perform relatively poorly. However, as Zhou et al. (2015) noted,the aggregation process inherent in creating CDFs tends to minimize these differences bycancelling the local peaks and troughs, thus their corresponding CDFs are more similar toeach other in shape. As a consequence, the two CDF-curve-based approaches are prefer-able to their waveform-curve-based counterparts when the same algorithm is used forclassification.

However, the CDF-curve-based approaches do not always perform better than thewaveform-curve-based approaches when different algorithms are adopted. For example,KL is based on the original waveform curves, while KS is based on the CDF curves ofthe waveform data. KL performed significantly better than KS at the level of 0.005 forall 15 trials (Figure 8). For KL, its three within-class interquartile boxes had muchsmaller ranges and were closer to the x axis compared with those of KS (Figures 5 and7). For the building class, KL did not have any overlap between the within-class andbetween-class interquartile boxes, while KS had a large overlap between within-classBuilding–Building and between-class Building–Tree. For the tree and open spaceclasses, the within-class and between-class boxes for both KL and KS did not overlap,but KL had fewer extreme values above the within-class boxes that overlapped thebetween-class boxes than did KS.

Figure 8. Boxplot of the p-values of McNemar’s chi-square test among waveform-curve-based andCDF-curve-based approaches across 15 trials.

16 Y. Zhou et al.

Additionally, KL also achieved slightly higher average performance than CCRSSDAand CCAM, but these differences were not statistically significant at the 0.05 level(Figure 8). Finally, KS performed better or at least comparable with CRSSDA, andsignificantly better than CAM at the 0.005 level (Figure 8).

6. Conclusions

ICESat waveform data have been widely used for forest and urban studies by numerousresearchers. Recently several studies have been conducted on ICESat waveform-basedclassifications. These studies were primarily based on waveform discretization andmetrics derivation. In this study, we developed two groups of curve matching approachesthat utilized the complete information provided by waveforms. As a case study, theseapproaches were applied to ICESat waveform data and were demonstrated to be effectivein differentiating different classes in an urban environment. However, the set of curvematching approaches developed in this study could also be useful to process other LiDARwaveform data, including those acquired by airborne sensors with continuous coverage.To the best of our knowledge, besides our previous study (Zhou et al. 2015), we have notseen any research using complete waveform curve for land cover classification, whichmakes our study novel.

The shapes of the waveforms over different classes, such as tree, building, andopen space, are primarily influenced by their vertical and horizontal compositionwithin the waveform footprints. There exist considerable within-class variations dueto different height levels, vertical structures, and footprint net coverage for each class.To reduce the impact of this high within-class variability on classification accuracy,multiple reference waveforms were selected by using random selection. A performancecomparison of six curve matching approaches showed that KL provided the highestaverage classification accuracy, which was closely followed by CCRSSDA andCCAM, but there were no significant differences between any two of these methods.However, they all significantly outperformed KS, CRSSDA, and CAM. Anotheroriginal finding in this study was that the two approaches based on the CDF-curveof waveforms are preferred over those based on the original waveform curves whenthe same algorithm is used for waveform classification. However, the CDF-curve-based approaches are not always superior to those based on original waveform curveswhen different algorithms are used. Future research is needed to assess if these CDF-curve-based approaches used for waveform classification would also achieve superiorresults when applied to pixel-based hyperspectral curves and object-based imagehistogram.

The use of ICESat waveforms alone for land use classification is unlikely a commonpractice due to the large spacing between its footprints and the presence of mixed landcovers within its footprints. The primary goal of this research was to verify that the novelcurve matching approaches developed in this research are able to capture vertical structureinformation of full waveforms for land-cover classification. The approach is likelyappropriate for feature extraction from small footprint waveforms. With more and moreairborne LiDAR systems now providing full waveforms with small footprints and con-tiguous coverage, curve matching approaches will play a more important role in com-plementing existing approaches to feature extraction and land cover classification. Whencombined with traditional remotely sensed imagery, LiDAR waveforms are expected toenhance the accuracy of traditional classification that relies on only 2D spectral informa-tion (Zhou and Qiu 2015).

GIScience & Remote Sensing 17

However, the findings in this study may only apply to relatively flat areas similar tothis study, since ICESat waveforms are sensitive to underlying terrain variation within thelarge footprints (Hilbert and Schmullius 2012). Future study may focus on removing theterrain effects from the ICESat waveforms in advance and then evaluate the performanceof the six curve matching-based approaches to waveform-based classification over differ-ent types of terrain. In addition to the terrain effect, due to limitation of large footprint sizeof ICESat waveform data, waveforms with a mixture of several types with none exceeding50% in coverage were excluded from this study. Classification of mixed ICESat footprintsshould be investigated using a fuzzy classification algorithm in the future. It is alsoimportant to point out that, if our proposed curve matching methods are applied tosmall footprint airborne LiDAR by segmenting the point clouds or the image into objectsfirst and then synthesizing object-level pseudo-waveform using a histogram of discreteLiDAR returns or waveforms (Zhou and Qiu 2015), the mixed footprint problem wouldnot be an issue anymore because each segment would have a dominant land cover type asa result of the segmentation process.

AcknowledgmentsThe authors would like to thank the National Snow and Ice Data Center by distributing ICESat data.

Disclosure statementNo potential conflict of interest was reported by the authors.

ORCID

Feng Ni http://orcid.org/0000-0001-7731-7266

Yifei Lou http://orcid.org/0000-0003-1973-5704

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