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A Comparison of Signal Deconvolution Algorithms Based on Small-Footprint LiDAR Waveform Simulation Jiaying Wu, Member, IEEE, J. A. N. van Aardt, and Gregory P. Asner
Abstract—A raw incoming (received) Light Detection And Ranging (LiDAR) waveform typically exhibits a stretched and relatively featureless character, e.g., the LiDAR signal is smeared and the effective spatial resolution decreases. This is attributed to a fixed time span allocated for detection, the sensor’s variable outgoing pulse signal, receiver impulse response, and system noise. Theoretically, such a loss of resolution can be recovered by deconvolving the system response from the measured signal. In this paper, we present a comparative controlled study of three deconvolution techniques, namely, Richardson–Lucy, Wiener filter, and nonnegative least squares, in order to verify which method is quantitatively superior to others. These deconvolution methods were compared in terms of two use cases: 1) ability to recover the true cross-sectional profile of an illuminated object based on the waveform simulation of a virtual 3-D tree model and 2) ability to differentiate herbaceous biomass based on the waveform simulation of virtual grass patches. All the simulated waveform data for this study were derived via the “Digital Imaging and Remote Sensing Image Generation” radiative transfer modeling environment. Results show the superior performance for the Richardson–Lucy algorithm in terms of small root mean square error for recovering the true cross section, low false discovery rate for detecting the unobservable local peaks in the stretched raw waveforms, and high classification accuracy for differentiating herbaceous biomass levels. Index Terms—Deconvolution, Light Detection And Ranging (LiDAR), nonnegative least squares (NNLS), Richardson–Lucy (RL), simulation, waveform, Wiener filter (WF).
I. I NTRODUCTION
A
NEW generation of airborne laser scanners [1]–[4], namely, full-waveform Light Detection And Ranging (LiDAR) systems [5], [6], is emerging as a prominent tool to provide detailed 3-D structural measurements remotely. The physical principle of LiDAR remote sensing systems is based on the emission of short laser pulses from an airborne platform with a high temporal repetition rate, followed by the measurement of the return trip time from the Earth surface target and back to the sensor. Full-waveform LiDAR is able to record the entire signal of the backscattered laser pulse, followed by digital sampling with extremely high temporal
Manuscript received August 10, 2010; revised November 16, 2010; accepted December 15, 2010. Date of publication February 16, 2011; date of current version May 20, 2011. This work was supported by Ph.D. research funding provided by the Rochester Institute of Technology. J. Wu and J. A. N. van Aardt are with the Chester F. Carlson Center for Imaging Science, Rochester Institute of Technology, Rochester, NY 14623 USA (e-mail:
[email protected];
[email protected]). G. P. Asner is with the Department of Global Ecology, Carnegie Institution for Science, Stanford, CA 80309 USA (e-mail:
[email protected]). Digital Object Identifier 10.1109/TGRS.2010.2103080
resolution (e.g., 1 ns). This characteristic amplitude profile of the recorded reflections is called a waveform. Moreover, information associated with the illuminated object can potentially be extracted or decoded from the waveform, since the features of the waveform, e.g., shape, integrated area, intensity, etc., directly relate to the geometry and radiative properties of the illuminated surface [7]–[16]. For spatially distributed targets, the return signal associated with a small-footprint waveform is mainly a superposition of echoes from scatterers at different discriminable ranges (e.g., 1 ns/0.15 m vertical resolution). Those scatters that cannot be discriminated by the sensor due to resolution limitations in the vertical axis, e.g., 0.15 m discretization, could also affect the shape of waveform in terms of width, slope, height, etc. However, we assumed that this effect is relatively minor and focused on the discriminable target for our return signal modeling. The backscatter response can thus be expressed as the combination of a convolution in terms of system and environment contributions [5] Pr (t) =
N
D2 4πλ2 Ri4 i=1
(Pt ∗ ηsys )(t) ∗ system contribution
(ηatm ∗ σi )(t) Environment contribution
(1) where “∗” is the convolution product, Pr (t) is the received signal as a function of time (waveform), t is the travel time for the transmitted laser pulse, N is the number of targets which contribute to the received signal, D is the aperture diameter of the receiver optics, Pt (t) is the emitted signal, λ is the wavelength, R is the distance from the LiDAR system to the target, ηsys and ηatm represent the system transmission and atmospheric factors, respectively, and σ(t) is the cross section of the illuminated target. This approach to LiDAR radiative transfer modeling [17]–[20] effectively relates the outgoing (transmitted) LiDAR signal and the return signals, and it also takes into account the detector and target characteristics. However, the system contribution waveform is typically not a perfect delta function, instead having a certain width, e.g., a Gaussian function. As a consequence, the convolution of system and environment contributions, which reveals the true distribution of the scattering substance along the optical path, also called the cross section, could result in the decrease of effective temporal resolution of the LiDAR signal, i.e., generating a smeared response instead of a temporally well-defined response. Such loss of resolution theoretically can be recovered by using deconvolution techniques to reduce unwanted system contributions, resulting in only the true cross-sectional contribution
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Fig. 1.
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Workflow of LiDAR waveform simulation using DIRSIG.
to the signal. Different deconvolution approaches have been applied to solve the true cross section in the literature. Jutzi and Stilla [21], for example, proposed to use the Wiener filter (WF) [22] to estimate the surface response from the noisy received waveform by assuming that a plane surface is perpendicular to the pulse propagation direction and the surface is illuminated by an infinitesimal footprint. Nordin [23] also mentioned that more canopy and ground echoes can be detected when using a waveform deconvolved via the Richardson–Lucy (RL) algorithm [24]. Harsdorf and Reuter [25] presented a deconvolution comparison between a Fourier transform approach and the nonnegative least squares (NNLS) [26] and RL algorithms using single arbitrary simulated waveforms and concluded that RL performed best based on visual comparison of the deconvolution results. Roncat et al. [27] also presented an approach to retrieve the backscatter cross section in full-waveform LiDAR data using uniform B-splines. However, the authors essentially solved a decomposition problem instead of deconvolution, and the retrieved individual cross sections consisted of symmetric scatterers, which is not practical in the real world. Moreover, these existing results and conclusions are typically based on the observation of several deconvolution samples, rather than a quantitative comparison. The lack of quantitative comparisons is mainly due to our inability to accurately describe the true
cross sections from a realistic scene. It is evident that the selection of the optimal deconvolution approach for LiDAR waveform preprocessing is barely addressed in the literature. In this paper, we performed a quantitative comparison between the three most widely used deconvolution techniques in the LiDAR waveform processing literature, namely, RL, WF, and NNLS algorithms, by employing the Digital Imaging and Remote Sensing Image Generation (DIRSIG) LiDAR simulation model to generate a truth cross-sectional data set. The aim of this study is to contribute to a better understanding of the advantages and disadvantages of different deconvolution algorithms as applied to LiDAR waveform preprocessing, toward extracting more representative and accurate 3-D structural parameters from remotely sensed scenes. II. LiDAR WAVEFORM S IMULATION DIRSIG, developed by the Digital Imaging and Remote Sensing Laboratory at Rochester Institute of Technology, was used to simulate the realistic inaction between the outgoing laser pulse and vegetation. The DIRSIG model [28], [29] is designed to simulate returned fluxes for a scene as a function of time, using Monte Carlo [30] ray tracing techniques, and is based on outgoing laser pulses that are generated by a
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well-defined source system. The advantage of using a LiDAR waveform simulation approach is that we can arbitrarily change LiDAR system settings, e.g., pulse width, beam divergence (footprint size), wavelength, etc., thereby providing the flexibility to characterize the object structure for a variety of scenarios. On the other hand, the structural parameters of the virtual object, e.g., tree, grass, etc., are known exactly, thus enabling us to link tree height, crown shape, volume, biomass, leaf area, etc., to the simulated waveform. Fig. 1 shows the workflow of the LiDAR waveform simulation for a tree using DIRSIG. A 3-D virtual deciduous tree was created as input to the DIRSIG LiDAR simulation by using the tree generation software Arbaro [31]. Materials (leaves, branches, and ground) were mapped to each facet of this 3-D model, and valid emissivity and extinction coefficients, which are based on the measurement of actual vegetation, were assigned to each material to simulate the absorption, reflection, and transmission processes for each pulse and the vegetation it interacts with. An operationally viable waveform LiDAR platform was set up in the DIRSIG environment as per the system configuration lists (Fig. 1): The goal was to match our virtual system with the commercially available small-footprint waveform LiDAR systems, e.g., Optech ALTM and the Leica LMS series. A varying outgoing pulse width (2/8/16 ns) was used in order to test the effect of pulse width on deconvolution results. The selection of pulse width was motivated by an outgoing pulse width of 16 ns, as implemented in the operational LiDAR waveform data collected by Carnegie Airborne Observatory [32] using Optech ALTM 3100 system. This operational selection is based on the need for the laser pulse to have enough energy to penetrate dense canopy in all woody or forested environments. A 2 ns outgoing pulse width was used to generate the approximated or truth data set. The standard setting for the ALTM 3100 system is 8 ns, and it was used as an intermediate pulse width between 2 and 16 ns. Therefore, the outgoing pulse width setting of our simulation is congruent with an applicable operational system, so that the results can guide the waveform preprocessing that will be applied to the real data. We also restricted the outgoing laser pulse to only nadir view to reduce undue variation in target backscatter and attenuation due to longer atmospheric path length. The transmitted pulses in operational sensors typically are asymmetric in shape, i.e., they have a slightly longer tail in the trailing edge versus the leading edge. However, the shape of the outgoing pulse in the simulation was assumed to approximate a Gaussian distribution based on our observation of the actual outgoing pulse from the Carnegie Airborne Observatory [32] (Fig. 2) and for the following reasons: First, as can be observed in Fig. 2, the shape of the actual pulses closely approximates a “Gaussian” distribution, and the observed asymmetry is minimal. Second, the shape of the outgoing pulse could vary in terms of the slope and intensity in reality. We used a Gaussian approximation in order to keep maintain consistency in the shape of the outgoing pulse across all the waveforms for our simulation. Third, since we conducted a relative comparison between three deconvolution algorithms in this study, the shape of the outgoing pulse is assumed to have no absolute impact on the comparison results. In this paper, the plot of the tree was divided into a 40 × 40 pixel grid with a footprint size that is equal to 0.5 m,
Fig. 2. Illustration of the real outgoing pulses used by Carnegie Airborne Observatory system (ALTM 3100).
while the waveforms were sampled in 225 time bins for each pixel after implementing the simulation. The “x-axis” for each waveform corresponds to the time bins, which can be converted to the height above ground, starting from 24.995 m to −8.605 m at increments of 0.15 m. This range was arbitrarily set based on the size of the tree and considering temporally delayed returns due to the multiple scattering of photons [33]. The negative value was used so that multiple scatters could also be included in the simulated waveform, e.g., the minor peak after ground response. The “y-axis” of the output waveform represented the number of photons detected for that pixel at different heights or time bins, which directly relates to the intensity of the waveform signal in the real waveform LiDAR system. Fig. 1 also shows a typical waveform generated from the simulated tree canopy. It basically consists of three parts: the canopy (where most of the energy is reflected), the base of the tree (trunk without branches), and the ground response. In some situations, the ground response may not be recorded since there are not enough energy transmitted by branches and leaves to reach the ground. Postground response, as mentioned before, may be observed due to multiple scattering of photons and delayed signal travel time. III. S IGNAL D ECONVOLUTION The LiDAR signal is typically smeared, and the effective spatial resolution decreased due to a series of convolutions shown in the mathematic expression (1) of the LiDAR waveform model. The goal is to recover the cross section σ(t) of the illuminated target, which corresponds to the true distribution of optically active substances along the ray path of the LiDAR pulse. We can first simplify (1) to solve this deconvolution problem, by assuming the following: 1) only a single target along the laser path so that no decomposition involved (N = 1); 2) neglecting the atmospheric factors; and 3) removing the constant terms, since it will not affect the shape of the waveform for a cross section. Finally, we derived the received LiDAR signal P (t) (Fig. 3), described by the convolution integral +∞ R(t −t)Pδ (t )dt +N (t) = (R ∗ Pδ )(t)+N (t) (2) P (t) = −∞
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that the noise and the signal are statistically independent and results in the Wiener filter, constructed in the frequency domain P¯ (f ) 2 (5) W (f ) = ¯ (f ) 2 P¯ (f ) 2 + N
Fig. 3. Illustration of the (a) system contribution R(t) and (b) received LiDAR signal P (t).
where R(t) is the system contribution term which is equal to the convolution of the outgoing waveform or transmitted pulse (generally provided by the commercial LiDAR system with waveform digitizing capabilities) and system impulse response (Fig. 3) and can be estimated from the return from flat ground (Lambertian surface), Pδ (t) is the target cross section, and N (t) is the additive noise term. We compared three deconvolution algorithms for the purpose of this research: RL, WF, and NNLS. These three deconvolution approaches have been widely used in the literature due to their stability, efficiency, and ease of implementation. However, depending on the application, each of the approaches has its own advantages and limitations. For example, RL and NNLS are based on an iterative solution, which typically requires a longer computational time but results in more accurate outputs; WF, on the other hand, has fast implementation time but is not able to overcome the negative value and usually results in a “ringing effect” in the deconvolved signal [23]. The next section provides an overview of each of these approaches.
¯ (f ) (noise signal N (t) in the frequency domain) can be where N estimated from the background noise and P¯ (f ) is estimated by low-pass filtering of the received signal P (t) in the frequency domain. The final estimation of the Pδ (t) term (target cross section) is described by the following, followed by an inverse Fourier transformation to the time domain: P (f ) · W (f ) P¯δ (f ) = R(f )
(6)
so that the sum of the square error
2
P¯δ (t) − Pδ (t) 2 = P¯δ (t) − Pδ (t) = min .
(7)
t
C. NNLS Algorithm The classic form of NNLS problem can be expressed as follows: Given a matrix A ∈ Rm×n and the set of observed values given by b ∈ Rm , find a nonnegative vector x ∈ Rn to minimize the function f (x) = 1/2Ax − b2 , i.e., 1 Ax − b2 x 2 subject to x ≥ 0. min f (x) =
(8)
We can thus express the deconvolution problem with respect to Pδ (t) in the form of minimizing the sum of the square error (R∗Pδ )(t)−P (t)2 = N (t)2 = min, Pδ (t) ≥ 0
∀t.
(9)
A. RL Algorithm The RL algorithm is an iterative algorithm originally developed for astronomical image restoration [24]. It is derived directly from the Bayes theorem. The RL algorithm can also be used for deconvolution when we regard a LiDAR waveform profile as an image with the dimension 1 × N. The ith iteration solution can be calculated by Pδi+1 (t) = Pδi (t) ·
P (t)
R(t) ∗ R ∗ Pδi (t)
.
(3)
The residual of each iteration is computed as
ri (t) = P (t) − R ∗ Pδi (t).
(4)
The residual will converge as the iteration progresses. The user can terminate the iteration, either by selecting a specific residual threshold or by setting a constant iteration number.
B. WF The WF approach previously has been used by researchers for the deconvolution of LiDAR waveforms [21]. It assumes
The solution Pδ (t) can be calculated iteratively as the finite convergence of the error without any prior information about Pδ (t) and N (t) according to Lawson and Hanson’s algorithm. More details about the steps of iterative solution can be found in [26]. IV. M ETHODOLOGY In this section, we focus on the methodology developed for comparing the different deconvolution algorithms and quantifying the results. In order to evaluate the potential of deconvolution on waveform processing for vegetation applications in particular, we tested the algorithms in context of two vegetation structural assessments: 1) ability to recover the true cross-sectional profile of an illuminated object, based on the waveform simulation of a virtual 3-D tree model, and 2) ability to differentiate herbaceous biomass based on the waveform simulation of virtual grass patches. A. Recovering the True Cross Section of a Vegetation Target One of the basic goals of deconvolution of LiDAR waveforms is to remove the unwanted system contribution and extract the true cross-sectional profile of the illuminated object. However, this true cross section is typically impossible or difficult to measure directly for real targets such as trees, grasses,
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TABLE I S PECIFICATIONS OF THE 3-D V IRTUAL T REES U SED FOR WAVEFORM S IMULATION
and buildings. We circumvented this problem by operating in the DIRSIG simulation environment. We simulated a nearperfect outgoing pulse with extremely narrow pulse width. In this case, most of the backscattered response contained in the return signal will result from the target itself, thus approximating the true cross section. We set the width of outgoing pulse to 2 ns, given that the sampling rate of the waveform is 1 ns. An outgoing pulse width < 2 ns might result in artifacts after deconvolution. We incrementally increased the width of the outgoing pulse, from 2 to 8 to 16 ns to simulate the setting of real small-footprint LiDAR sensors, e.g., the Optech ALTM 3100 operated by the Carnegie Airborne Observatory [32]. This was followed by the application of the three deconvolution algorithms to the simulated return signal and comparison with the true cross-sectional data, which are generated from the 2 ns outgoing pulse. Two waveform data sets, namely, the 2 ns true response and the deconvolved comparison, should be very similar to each other in terms of the shape if the deconvolution worked properly. We simulated the complexity and diversity of natural trees by generating six different virtual 3-D trees at a fine scale, whose specifications and rendered images are listed and shown in Table I and Fig. 4, respectively. Each tree plot consisted of branches, leaves, and ground associated with their respective valid emissivity and extinction coefficients [29]. As mentioned in Section II, the plot for each tree was divided into a 40 × 40 pixel grid with a waveform footprint size that is equal to 0.5 m, which resulted in a waveform with 225 time bins (from 24.995 to −8.605 m at an increment of 0.15 m) for each pixel after implementing the simulation. Finally, three sets of simulated waveforms were generated for outgoing pulse widths of 2, 8, and 16 ns for each tree plot. Three metrics were used to assess the performance of the respective deconvolution algorithms in terms of recovering the cross section. 1) Root mean square error (RMSE) value between the truth and deconvolved waveforms 2 P˜δ,2 (t) − Pδ,w (t) n m RMSE = m×n
(10)
where P˜δ,2 (t) is the true cross section approximated by the direct simulation results using an outgoing pulse width that is equal to 2 ns. Pδ,w (t) corresponds to the deconvolved waveform using outgoing pulse widths that are equal to w = 8 and 16 ns. m and n are the number of bands for each waveform and the total number of waveforms for the plot (e.g., m = 225 and n = 1600 in this paper), respectively. 2) We furthermore evaluated the sensitivity to local peaks of a target waveform detection by determining where the sign of the first derivative of the waveform changes for different deconvolution approaches, which is defined as Sensitivity =
# of true detections . total # of true peaks
(11)
True detection is defined by the time bin index of a detected local peak from the deconvolved waveforms that agrees with a true peak, which is extracted from the 2 ns waveform simulation for each of the six trees. 3) Another important metric, called false discovery rate, was also extracted and is defined as F alse discovery rate =
# of f alse detection total # of detected peaks
(12)
where false detection is the opposite of the true detections, described earlier. B. Differentiating Herbaceous Biomass As the laser pulse interacts with the layer of grass above ground, the waveform of the return signal is typically distorted, e.g., extended width, peak shift, etc., when compared with the signal reflected of flat bare ground. This is attributed to the fact that the signal scattered by the above-ground grass is temporally too close to the ground response and thus beyond the temporal or vertical resolution of operational LiDAR waveform systems (e.g., 0.15 m). However, the level of distortion theoretically relates to the amount of herbaceous biomass [33]. We used a statistic-based algorithm to extract features from these distorted waveforms for differentiating the biomass quantitatively and explored the significance of deconvolution on this use case.
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Fig. 4.
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Simulated 3-D trees used for waveform simulation and deconvolution assessment.
Fig. 6. Three-dimensional grass patch (the different herbaceous biomass levels were simulated by scaling the relative size of the grass facet while keeping the patch size 10 m × 10 m constant).
Fig. 5.
Workflow of herbaceous biomass classification algorithm.
Fig. 5 shows the workflow of the herbaceous biomass classification algorithm using the simulation data. First, five grass patches with the same size (10 × 10 m2 ) but with different herbaceous biomass were created using “Arbaro” (Fig. 6). The herbaceous biomass is modified by scaling the relative size of each grass facet on a per-patch basis. The scale factor ratios for these five patches were 0.2, 0.4, 0.6, 0.8, and 1. For example, the heights of these grass patches are 0.2, 0.4, 0.6, 0.8, and 1 m, respectively. Since the herbaceous biomass is equal to the
product of grass volume and density and the density is the same for all the patches, their herbaceous biomass ratios will be 0.2, 0.4, 0.6, 0.8, and 1, respectively. We generated 2000 waveforms in total with 400 for each patch (20 × 20 pixel with a 0.5 m footprint) using the DIRSIG simulation platform. The noise generated by uniformly distributed pseudorandom numbers was added to each waveform in order to simulate the real LiDAR signal-to-noise ratios, as well as for testing the robustness of deconvolution algorithms against noise. The amplitude of the noise (Fig. 7) was estimated by averaging the differences between the noisy and the smoothed waveform data collected by Carnegie Airborne Observatory systems [32]. This was followed by the application of deconvolution algorithms (RL, WF, and NNLS) on the simulated data. Principal component analysis (PCA) [34] was applied on all the simulated waveform data “G,” as described hereinafter to identify the vector that contributed most to signal variances. This was done to extract the uncorrelated feature associated with different herbaceous biomass levels that form these waveforms, represented in this case by the projection along the first principal
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Fig. 7. (Top) Real raw waveform data collected by Carnegie Airborne Observatory systems and (bottom) the smoothed waveform by preprocessing.
axis (13). We can thus retrieve feature that best explains the biomass variance by employing the PCA Y = aT 1G
(13)
where G is the m × n matrix, in which m is the number of bands for the waveform and n is the total number of waveforms, and a1 represents the first eigenvector associated with the largest eigenvalue of the covariance matrix of G. Next, we evaluated five different partitioning regions Ri (i = 1, 2, . . . , 5) to classify the different biomass levels by minimizing the average risk (14) associated with misclassification defined as a feature vector y that belongs to class ωk and that lies in Ri , i = k 5 5 r= P (y|ωk ) P (ωk ) dy (14) i=1 R
i
k=1
where P (ωk ) is the a priori probability for class k and P (y|ωk ) is the class-conditional probability density function that describes the distribution of the feature vectors in each of the classes. The partitioning regions Ri can be solved according to the Bayesian classifier [35], which is optimal with respect to minimizing the average risk. It can be expressed as follows: Assign y to Ri if P (y|ωi )P (ωi ) > P (y|ωk ) P (ωk )
∀k = i.
(15)
Finally, the classification accuracy for deconvolved waveforms from each herbaceous biomass level was assessed using a confusion matrix and by computing the mean of the diagonal values (expressed as the percent classified correctly in each entry). V. R ESULTS AND D ISCUSSION A. Recovery of the True Target Cross Section The deconvolved waveforms were compared with the truth data, as approximated by the direct simulation results using an outgoing pulse width that is equal to 2 ns, in order to assess
Fig. 8. Illustrations of the deconvolved waveforms for an outgoing pulse width of 8 ns. (Top) RL. (Middle) WF. (Bottom) NNLS. (Solid line) True cross section. (Bold dashed line) Deconvolved waveform. (Solid line with circle) Raw waveform with 8 ns pulse width.
deconvolution in terms of ability to recover the cross section of a tree plot. Fig. 8 shows a sample waveform from one of the tree plots, as deconvolved using the RL, WF, and NNLS algorithms. It was observed that, after deconvolution, the width of the local waveform components decreased and more local peaks were revealed, as we expected when removing the system contribution. The performance of the RL and WF approaches is close to that of the deconvolved waveforms, as can be seen when comparing them to the true cross section. However, it was observed that closely spaced adjacent local peaks could not be distinguished by either RL or WF, which could be due to the resolution of the outgoing pulse (8 and 16 ns). We also observed a documented drawback of the WF, called the “ringing effect” [23] caused by the loss of high-frequency components during the deconvolution processing. The sum of the remaining low-frequency components will introduce a wavelike artifact (a series of rings). This often resulted in extra minor peaks around the major local peaks. Although most of these minor peaks can be removed via filtering, e.g., thresholds, low-pass filtering, etc., it will be difficult to remove them completely. Results for the NNLS approach proved noisy, given the presence of multiple high-frequency peaks. We concluded that NNLS might be more sensitive to finding close peak neighbors when compared with RL and WF, but the results are not conclusive (Fig. 8). We also voxelized each waveform by converting every time bin into the “XY Z” coordinate in 3-D space and coding the intensity as gray scale. This was done to provide a comprehensive comparison in terms of visualization for all the
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Fig. 10. Results of the RMSE comparison for outgoing pulse widths of (top) 8 and (bottom) 16 ns.
Fig. 9. Three-dimensional representation of the LiDAR waveforms for tree 6 and all deconvolution approaches at 8 and 16 ns outgoing pulse widths.
waveforms. Fig. 9 shows the 3-D representations of the voxelized waveforms for tree 6. It is evident that the deconvolved 3-D waveform representations of the truth data agreed well with the real tree geometry, which is indicative of the potential of LiDAR waveforms for 3-D tree reconstruction. It was also observed that, by changing the width of the outgoing waveform, the 3-D representations of the raw waveforms exhibited increasing levels of blur, such that the ground became “thicker” and the details inside the canopy were lost. However, after applying the deconvolution, the temporal (vertical) resolution was recovered, as the figure shows. The performances of RL and WF were similar when using 8 ns as the outgoing pulse width, but RL stood out in the case of the larger outgoing pulse width (16 ns) when one considers that the ground section associated with WF is thicker and redundant local peaks could be found around the crown and below the ground level. The result for NNLS was obviously not satisfactory relative to the other two approaches, even when based on visual inspection. Fig. 10 shows the overall quantitative comparison of RMSE (10) between the truth and deconvolved waveforms for RL, WF, and NNLS at different outgoing pulse widths (8 and 16 ns). This figure again demonstrates that RL was superior to the other two approaches. The results also show that the selection of the width for the outgoing pulse could significantly affect the deconvolution outcome.
Fig. 11. Point clouds extracted from the local peaks of the waveforms (Tree 5). The intensity of point clouds is coded as gray scale. (a) True cross section (peak-to-discrete return assignment). (b) Raw waveforms (width: 8 ns) (peak-to-discrete return assignment). (c) Correct peak detection (peak-todiscrete return assignment). (d) Deconvolved waveforms (RL based) (peak-todiscrete return assignment).
The assessment of deconvolution accuracy using RMSE was expanded by evaluating if the location of peaks in each deconvolved waveform corresponded to those found in the truth data
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TABLE II S TATISTICS OF THE P EAK D ETECTION R ESULTS (A: T OTAL N UMBER OF D ETECTED P EAKS , T: N UMBER OF P EAKS T HAT M ATCH THE T RUTH WAVEFORM ; A LL THE VALUES A RE N ORMALIZED BY C OMPUTING THE R ATIO TO THE N UMBER OF P EAKS AT 2 ns, AS THE P ERCENTAGE S HOWS )
waveform (nondeconvolved; 2 ns outgoing pulse width). Fig. 11 shows the point clouds associated with the extracted peaks from various waveforms, similar to the output of discrete return LiDAR systems. Fig. 11(a) shows the points from the truth data of tree 5. Points located below the ground level typically correspond to the minor peaks after the ground response, which are attributed to the multiple scattering of the photons inside the crown. Fig. 11(b) shows the result using the raw simulated waveforms prior to deconvolution; it is evident that the density of the points decreased due to the loss of temporal (vertical) resolution caused by the system and noise contributions. To a large extent, deconvolution can reveal peaks hidden by this unwanted system interference; however, it could also introduce artifacts in the form of nonexisting returns due to the noise, as Fig. 11(c) shows. We further compared these detected points with those from the truth waveforms and came up with the correct peak detection, shown in Fig. 11(d). The overall statistics are listed in Table II. The quantitative results indicate the following: 1) deconvolution (RL, WF, and NNLS) dramatically increased the density of the point clouds, compared with the raw data; 2) as we increase the width of the outgoing pulse, the effect of deconvolution is negated; and 3) the fact that only about half of the detected peaks after deconvolution (RL, WF, and NNLS) matched the ground truth data suggests that no deconvolution algorithm can fully recover the full temporal response for these waveforms. This can be observed in Fig. 8, where, for certain close peak neighbors, the deconvolution approaches were unable to uncover both peaks, but resulted in a peak detection in-between. This is arguably still useful for vegetation applications, but it effectively represents a false
Fig. 12.
False discovery rate versus sensitivity.
detection, even if this “false” detection is representative of target interaction on either side. The quantitative analysis was concluded by assessing two final metrics: sensitivity and false discovery rate, based on (11) and (12), respectively, and plotted in Fig. 12. We can see that all of the deconvolution algorithms enhanced the detection sensitivity when compared with the raw data.
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Fig. 13. Simulated waveforms from grass patches. (Top) Before deconvolution. (Bottom) After deconvolution. Herbaceous biomass ratios from patches 1 to 5 were 0.23 : 0.43 : 0.63 : 0.83 : 1. The plots are based on RL deconvolution algorithms with an outgoing pulse width that is equal to 8 ns.
NNLS performed the best in terms of sensitivity among these three deconvolution algorithms. However, this increased sensitivity came at the cost of a higher false discovery rate for the NNLS algorithm. For example, NNLS resulted in more peak detections, which increases the probability of detecting a true response for a given time bin. On the other hand, this boost in point density resulted in more false detections due to the noise or the application of the algorithm itself. Therefore, if minimization of the false detection is the most important consideration, the RL algorithm still appears to be the best choice. Hence, based on these quantitative tests (classification accuracy, RMSE, and peak detection) of three deconvolution algorithms, we concluded that RL is superior to others. This corroborated the findings of Harsdorf that the 1-D RL algorithm leads to the best results as per a visual comparison between arbitrarily designed pre-deconvolved and post-deconvolved waveforms [25]. However, our approach provides a more comprehensive comparison, both in the direct 3-D recovery accuracy for the truth waveform and the application of the deconvolved waveforms (biomass classification). This was underscored by statistic-based metrics toward measurement of the quality of the deconvolution algorithms applied to LiDAR
Fig. 14. (Top) Plot of the normalized cumulative sum of the eigenvalues for the simulated waveforms from grass patches. (Bottom) Eigenvectors associated with the four largest eigenvalues in descending order.
waveforms, thereby quantifying the advantages and disadvantages of the different algorithms. B. Classification Fig. 13 shows one of the simulated waveform pairs from the grass patches with five different herbaceous biomass levels, where patch 1 represents the lowest and patch 5 represents the highest biomass. These waveforms have a wider distribution and are more spread out prior to deconvolution, particularly in the leading edge (left) area. The peaks also shift to the left, or closer to the sensor, as the biomass increases. This can be explained by considering that, before the laser pulse hits the ground, it interacted with the above-ground grass, which backscattered part of the energy ahead of the energy closer to the ground. This led to an increase of width for the return signal, which became even more obvious after applying the deconvolution, shown in Fig. 13 (right). In order to better explain the classification algorithm, we can consider the results of post-deconvolution waveforms, using RL as an example. Fig. 14 shows the statistic-based feature
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Fig. 16. Comparison of the classification accuracies for no deconvolution and the three deconvolution algorithms in question.
Fig. 15. (Top) Projection on the first eigenvector versus projection on the second eigenvector. (Bottom) Histogram of the projection on the first eigenvector. TABLE III I LLUSTRATION OF C ONFUSION M ATRIX U SED FOR A SSESSING THE ACCURACY OF H ERBACEOUS B IOMASS C LASSIFICATION (BASED ON RL)
extraction using PCA. It was observed that more than 99% of the variance of the deconvolved waveforms can be explained by the first four eigenvectors associated with the largest eigenvalues in descending order, in which the first eigenvector contributed approximately 82% of the variance. If we evaluate the shape of the first eigenvector shown in solid line in Fig. 14 (bottom), it is clear that two local peaks with negative and positive values, respectively, can be related to the grass and ground scene components. The reverse signs of the intensity also suggest the tradeoff of energy contribution to the original waveform between the grass layer and ground. For example,
the more positive energy from the ground contributed to the original waveform, the more energy associated with the grass component will be subtracted and vice versa. This observation agrees with the plot in Fig. 13 that those waveforms from low biomass levels typically exhibit high intensity at the ground component and relatively less energy from the leading (left) edge, which corresponds to the energy contribution from grass; moreover, these simulation results corroborate previous research about the existence of a correlation between waveform shape metrics and the presence of varying levels of herbaceous biomass in the real world [33]. The second, third, and fourth eigenvectors were used to explain the slight peak shift across different biomass levels observed in Fig. 13, since the peaks of these eigenvectors are obviously spread out [Fig. 14 (bottom)]. Fig. 15 (top) shows the scatter plot of the projection on the first versus second eigenvectors. We can see that the points are separable in terms of different grass biomass levels along the x-axis, which corresponds to the projection on the first eigenvector. This observation agrees with the most significant contribution, i.e., grass structure, from the first eigenvector. It also shows that points associated with large herbaceous biomass (“o” and “+”) are basically located in the negative region along the x-axis. This suggests that the negative local peak, which was assumed to correspond to the grass layer in the first eigenvector, contributed positively to the final return signal. The inverse is true for the second ground-related peak. This effectively resulted in a shift of the return waveform to the left and with a larger width. It was also observed that the low biomass patches (“×” and “Δ”) were relatively similar, which was attributed to the herbaceous biomass not being linearly separated in the design of the grass patches, e.g., the ratios were 0.23 : 0.43 : 0.63 : 0.83 : 1. We assumed that the distribution of the projection values follows a normal distribution, as shown by the histogram fitting using a Gaussian curve [Fig. 15 (bottom)]. This was done in order to identify the thresholds for classifying the grass patches. The partitioning regions were computed by (15). Finally, the classification accuracy was assessed using a confusion matrix (see Table III as an example); the results are shown in Fig. 16. Four conclusions can be drawn from Fig. 16. 1) Deconvolution (RL, WF, and NNLS) improved the classification accuracy when compared with the results using
WU et al.: COMPARISON OF SIGNAL DECONVOLUTION ALGORITHMS BASED ON LiDAR WAVEFORM SIMULATION
the raw data without deconvolution, while the widths of the outgoing pulse were set to 8 and 16 ns. 2) RL stood out in terms of accuracy when compared to WF and NNLS. 3) The width of the outgoing pulse affected the classification results in that large widths negated the effect of deconvolution processing. 4) When the width of the outgoing pulse was set to 2 ns, the classification accuracy, based on waveforms without deconvolution, was better than the accuracies for deconvolved waveforms. This corroborated our assumption, stated in Section IV, that we can use the simulation results for outgoing pulses with a narrow width to approximate the true target response as reference for comparing different deconvolution algorithms. VI. C ONCLUSION The question of deconvolution algorithm choice, as a preprocessing step to waveform LiDAR usage, has remained unanswered. Previous attempts evaluated deconvolution approaches visually, without injecting quantitative assessments into studies [21], [23], [25], [27]. We have successfully developed a methodology based on four statistic-based quantitative metrics, namely, classification accuracy, RMSE, sensitivity, and false discovery rate, to compare three widely used deconvolution algorithms: RL, WF, and NNLS. This was done by taking advantage of high-fidelity waveform LiDAR simulations as our validation data. The results showed superior performance for the RL algorithm in terms of small RMSE between the deconvolved and truth waveforms and low false discovery rate for the recovery of the true 3-D tree cross section as one use case, and high classification accuracy for differentiating the herbaceous biomass levels as the second validation case. These results provide a quantifiable basis for the selection of the deconvolution approach in the waveform LiDAR processing chain. We have also demonstrated the potential of waveform LiDAR particularly for vegetation applications in terms of savanna woody and herbaceous biomass estimation. A PCAbased algorithm has been developed to extract features from the waveforms and relate these to herbaceous biomass levels. This could potentially provide a more efficient remote-sensingbased vegetation biomass assessment approach, particularly at senescent growth stages, when compared with traditionally expensive and time-consuming field data collection. This paper has also shown that the width of the outgoing waveform pulse has a major impact on waveform processing in that it directly affects the deconvolution results and our ability to extract finescale structural vegetation features. This could benefit LiDAR users and system engineers in terms of optimizing the system configuration for their specific application. Finally, the results hint at the possibility of 3-D reconstruction of tree structure by voxelizing the LiDAR waveform in 3-D space, which could prove useful for woody and foliar biomass estimation, crown volume assessment, etc. We plan to further explore how the tree (target) geometry relates to the variation in LiDAR return waveforms by using simulation data and developing algorithms to separate branches and leaf components inside the canopy. Our approaches will be applied to real waveform LiDAR data from the Carnegie Airborne Observatory
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[32] to develop a standardized waveform LiDAR processing chain and improve structural assessment in savanna and forest environments. ACKNOWLEDGMENT The authors are grateful for Ph.D. research funding provided by the Rochester Institute of Technology (RIT). They would also like to thank Dr. H. Rhody (RIT), Mr. D. Knapp, and Mr. A. Balaji (Carnegie Institution for Science) for their inputs during waveform LiDAR preprocessing discussions. R EFERENCES [1] G. Vosselman and H. Maas, Airborne and Terrestrial Laser Scanning. Boca Raton, FL: CRC Press, Mar. 2010. [2] G. Heritage and A. Large, Laser Scanning for the Environmental Sciences. Chichester, U.K.: Wiley-Blackwell, May 2009. [3] J. Shan and C. Toth, Topographic Laser Ranging and Scanning Principles and Processing. Boca Raton, FL: CRC Press, 2009. [4] A. Wehr and U. Lohr, “Airborne laser scanning—An introduction and overview,” ISPRS J. Photogramm. Remote Sens., vol. 54, no. 2/3, pp. 68–82, Jul. 1999. [5] C. Mallet and F. Bretar, “Full-waveform topographic lidar: State-of-theart,” ISPRS J. Photogramm. Remote Sens., vol. 64, no. 1, pp. 1–16, Jan. 2009. [6] C. Hug, A. Ullrich, and A. Grimm, “Litemapper-5600: A waveformdigitizing LIDAR terrain and vegetation mapping system,” Int. Archives Photogramm. Remote Sens. Spatial Inf. Sci., vol. XXXVI-8/W2, pp. 24–29, 2004. [7] M. Nilsson, “Estimation of tree heights and stand volume using an airborne LIDAR system,” Remote Sens. Environ., vol. 56, no. 1, pp. 1–7, Apr. 1996. [8] M. A. Lefsky, W. B. Cohen, S. A. Acker, G. G. Parker, T. A. Spies, and D. Harding, “LIDAR remote sensing of the canopy structure and biophysical properties of Douglas-fir and western hemlock forests,” Remote Sens. Environ., vol. 70, no. 3, pp. 339–361, Dec. 1999. [9] J. Drake, R. Dubayah, D. Clark, R. Knox, J. Blair, M. Hofton, R. Chazdon, J. Weishampel, and S. Prince, “Estimation of tropical forest structural characteristics using large-footprint LIDAR,” Remote Sens. Environ., vol. 79, no. 2, pp. 305–319, Feb. 2002. [10] H. Andersen, R. McGaughey, and S. Reutebuch, “Estimating forest canopy fuel parameters using LIDAR data,” Remote Sens. Environ., vol. 94, no. 4, pp. 441–449, Feb. 2005. [11] J. Anderson, M. E. Martin, M.-L. Smith, R. O. Dubayah, M. A. Hofton, P. Hyde, B. E. Peterson, J. B. Blair, and R. G. Knox, “The use of waveform LIDAR to measure northern temperate mixed conifer and deciduous forest structure in New Hampshire,” Remote Sens. Environ., vol. 105, no. 3, pp. 248–261, Dec. 2006. [12] A. Farid, D. C. Goodrich, R. Bryant, and S. Sorooshian, “Using airborne lidar to predict Leaf Area Index in cottonwood trees and refine riparian water-use estimates,” J. Arid Environ., vol. 72, no. 1, pp. 1–15, Jan. 2008. [13] J. A. B. Rosette, P. R. J. North, and J. C. Suarez, “Vegetation height estimates for a mixed temperate forest using satellite laser altimetry,” Int. J. Remote Sens., vol. 29, no. 5, pp. 1475–1493, Mar. 2008. [14] D. Bhattacharya, S. Pillai, and A. Antoniou, “Waveform classification and information extraction from LIDAR data by neural networks,” IEEE Trans. Geosci. Remote Sens., vol. 35, no. 3, pp. 699–707, May 1997. [15] M. A. Hofton, J. B. Minster, and J. B. Blair, “Decomposition of laser altimeter waveforms,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 4, pp. 1989–1996, Jul. 2000. [16] S. Pe’eri and W. Philpot, “Increasing the existence of very shallow-water LIDAR measurements using the red-channel waveforms,” IEEE Trans. Geosci. Remote Sens., vol. 45, no. 5, pp. 1217–1223, May 2007. [17] W. Wagner, A. Ullrich, V. Ducic, T. Melzer, and N. Studnicka, “Gaussian decomposition and calibration of a novel small-footprint full-waveform digitizing airborne laser scanner,” ISPRS J. Photogramm. Remote Sens., vol. 60, no. 2, pp. 100–112, Apr. 2006. [18] G. Sun and K. J. Ranson, “Modeling LIDAR returns from forest canopies,” IEEE Trans. Geosci. Remote Sens., vol. 38, no. 6, pp. 2617– 2626, Nov. 2000.
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Jiaying Wu (M’09) received the B.S. degree in optical information science from the University of Shanghai for Science and Technology, Shanghai, China, in 2006. He is currently working toward the Ph.D. degree with the Chester F. Carlson Center for Imaging Science, Rochester Institute of Technology (RIT), Rochester, NY. His Doctoral research involves studies of waveform LiDAR signal and image processing. From 2007 to 2008, he was a Research Assistant with the Real-Time Vision and Image Processing Laboratory, RIT. During this time, his research focused on the development of medial augmented reality algorithm based on 2-D/3-D image fusion to facilitate image-guided surgery. In 2008 summer, he was a Research Intern with Sharp Research Laboratories of America, Camas, WA. The research topic involved LCD display color modeling, and his work was published in SPIE Electronic Imaging in 2009. He also spent a summer as an Image System Engineer with Microsoft Corporation, Redmond, WA, in 2010, where he worked on video compression and quality assessment algorithms. His research interests include digital image processing, remote sensing, computer vision, and color science.
J. A. N. van Aardt received the B.Sc. and Hons. degrees in forestry from the University of Stellenbosch, Western Cape, South Africa, in 1996 and 1998, respectively, the M.S. and Ph.D. degrees in forestry–remote sensing from the Virginia Polytechnic Institute and State University, Blacksburg, VA, in 2000 and 2004, respectively. He is currently an Associate Professor within the Chester F. Carlson Center for Imaging Science, Rochester Institute of Technology, Rochester, NY. He is a member of the Digital Imaging and Remote Sensing (DIRS) group within the Center. This group focuses on a complete system approach to remote sensing applications based on system integration, processing workflow, and applied algorithm development. Before joining the Faculty at Rochester Institute of Technology, he worked in the academic (Katholieke Universiteit Leuven, Leuven, Belgium) and private (Council for Scientific and Industrial Research, South Africa) sectors. His research interests include the application of imaging spectroscopy and light detection and ranging for spectral–structural characterization of natural systems (remote sensing of natural resources).
Gregory P. Asner received the B.S. degree in civil and environmental engineering in 1991, the M.A. degree in geography in 1995, and the Ph.D. degree focusing on biogeochemistry and remote sensing in 1997, all from the University of Colorado, Denver. He is currently a Faculty Member of the Department of Global Ecology, Carnegie Institution for Science, Stanford, CA. He also holds a faculty position in the Department of Environmental Earth System Sciences, Stanford University, Stanford, CA. His scientific research centers on how human activities alter the composition and functioning of ecosystems at regional scales. He combines field work, airborne and satellite mapping, and computer simulation modeling to understand the response of ecosystems to land use and climate change. His most recent works include satellite monitoring of selective logging and forest disturbance throughout the Amazon Basin, invasive species and biodiversity in Hawaii rain forests, and El Nino effects on tropical forest carbon dynamics. His remote sensing efforts focus on the use of new technologies for studies of ecosystem structure, chemistry, and biodiversity in the context of conservation, management, and policy development. He directs the Carnegie Airborne Observatory, which is a new airborne laser and hyperspectral remote sensing platform designed for regional assessments of the carbon, water, and biodiversity services provided by ecosystems to society.
