Stepwise Decomposition of Full‐Waveform Data Based on Levenberg Marquardt

www.as‐se.org/ssms Studies in Surveying and Mapping Science (SSMS) Volume 2, 2014

Stepwise Decomposition of Full‐Waveform Data Based on Levenberg Marquardt Pengcheng Li*1, Qing Xu1, Pingyuan Cui2, Shuai Xing1, Chaozhen Lan1,2 1

Zhengzhou Institute of Surveying and Mapping, No. 66 Longhai Road, Zhengzhou, China

2

School of Aerospace Engineering, Beijing Institute of Technology, Beijing, China

*

lpclqq@163.com; xq@szdcec.com; cuipy@bit.edu.cn; xing972403@163.com; lan_cz@163.com

Abstract Compared with traditional airborne LiDAR, the advantage of full‐waveform LiDAR is that it digitalizes the full backscattered information. Waveform decomposition is the most important part of waveform data processing. A method of stepwise decomposition of full‐waveform data based on Levenberg Marquardt has been proposed, which employs stepwise decomposition and uses Levenberg Marquardt, which is a Gaussian function with non‐linear least squares fitting algorithm, to obtain precise fitting waveform. Experiment results with 4 different waveform data acquired by RIEGL airborne LiDAR have proved that this method is available and effective.

Keywords Full‐waveform LiDAR; Waveform Decomposition; Gaussian function; Levenberg Marquardt

Introduction In the last decade, airborne LiDAR (Light Detection and Ranging) played a very significant role in the fields of terrain surveying and mapping, forest monitoring and city modelling. It was accepted as an important supplement of traditional photogrammetry [1].

The technique of airborne LiDAR integrates Laser Measuring device, Global Positioning System (GPS) and Inertial Measuring Units (IMU) [2]. Laser measuring device determines the distance between emitting source and laser points; GPS gets the exact position of the shooting point and IMU measures the attitude parameters. By synchronized and harmonious cooperation of the above three parts, it can directly obtain the 3 dimensional coordinates of targets.

Although airborne LiDAR has positive effects on many fields, there still exist some problems. Nearly all processing algorithms about point clouds are based on the geometry information, inevitably there will be some errors when terrain surface is complex. And now most airborne LiDAR systems just record the peaks of wave using built‐in detection method and lost large number of information. For solving the above problems, a new‐type airborne LiDAR has appeared, which can digitize all backscattered information completely. It is called airborne full‐waveform LiDAR. Technique of full-waveform LiDAR The first full‐waveform systems were designed in the 1980s for bathymetric purposes. And the first truly operational topographic system, LVIS appeared in 1999 and demonstrated the value of recording the entire waveform for vegetation analysis. During that time, some experimental systems were also generated by NASA, and RIEGL developed the first commercial full‐waveform LiDAR system, RIEGL LMS‐Q560 till 2004 [3].

All full‐waveform systems contain a waveform digitizer which can record the entire waveform data by a certain interval. For example, Optech’s IWD‐2 (Intelligent Waveform Digitizer‐2) and Leica’s WDM65 are the important parts to do this.

The differences on data recording determine their different post processing methods. In airborne full‐waveform LiDAR data, there is not only geometry information, but also some physical information of targets, such as amplitude, pulse width and backscatter cross‐section etc. The data could be obtained by the most important step: waveform decomposition [4]. Users can get waveform parameters and discrete point clouds after decomposing the waveform, and improve precision of filtering results and DEM combined with such waveform parameters [5]. In addition, a more exact classification with these waveform features has been realized [6]. Finally, the building models could be reconstructed based on the classified building points, so as to realize the city modelling. It can be seen from the above that, the step of waveform decomposition guarantees the generation of high quality LiDAR products. The general processing flow of full‐waveform LiDAR data is shown in Fig. 1.

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FIG. 1 PROCESSING FLOW OF FULL‐WAVEFORM LIDAR DATA

FIG. 2 WORK FLOW OF METHOD

Stepwise Decomposition of Full-Waveform Data Based on Levenberg Marquardt Waveform decomposition means decomposing the waveform into a sum of components or echoes, so as to characterize the targets along the path of the laser beam. This approach could maximize the detection rate of peaks, generate a denser point clouds, and extend waveform processing capabilities. Generally, it is realized by waveform modelling and fitting. For example, Hofton gave a description of a non‐linear least–squares method which was used in LVIS waveform data; Persson developed a pulse detection method based on Expectation‐Maximization algorithm, and the EM algorithm was improved by Qi L and Hongchao M; and Hernández‐Marín proposed a fitting method using Reversible Jump Monte Carlo Markov Chain. For the non‐linear least–squares method, it may fail when encountering complex terrain or objects which are recorded by small feet airborne system. Therefore, we propose a stepwise decomposition method based on Levenberg Marquardt to solve this problem. The main idea of our approach is to detect peaks and estimate parameters preliminarily, then select Gaussian model and Levenberg Marquardt algorithm [7] to fit the original pulse, finally find the missing peaks again and again through the difference between modelled and original signals till no more peaks appear. The work flow of this approach is shown in Fig. 2.

Model Selection Model selection is the foundation of waveform decomposition, a not suitable model should have side effect on the

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fitting result. Generally, a single function is always used to model all echoes of the waveforms. A waveform y could be decomposed into a sum of n components: ∑

y

x

ϕ

f x

b (1) ∑ ϕ

Where, f is waveform model, ϕ is the echo model whose parameters are θ f distributed points, y y ,…,N is the sampled waveform and is the noise.

, x

,…,N

are well

Most methods regard the echo waveform as the sum of several Gaussian functions, and Wagner (2006) [8] also pointed out that more than 98% of the waveforms with RIEGL system could be fitted with a sum of Gaussian functions. Therefore, we selected the Gaussian function as the model to fit, the analytical expression of the Gaussian function is: ϕ x

μ

A exp

(2)

Where, A is pulse amplitude, σ is pulse width and μ is pulse distance, so θ

A , w , μ .

Fitting Parameters Optimization During waveform decomposition process, the step of fitting is very important. We regard this problem as a non‐linear least‐squares problem to solve [9], therefore, the algorithm of Levenberg Marquardt was used here. This algorithm was proposed by Levenberg in 1944, and was improved by Marquardt in 1963 [7]. And its principle is described below: Suppose f x , p was a function with parameters p , p , … , p , and the observation data was x , y , i 1,2, … , n.

Firstly, the parameters were initially set asp p , p , … , p , the functionf x , p was expanded in Tailor at p and the target function was set as: Q

∑

y

,

∑

f x ,p

p

λ∑

p

p

p

(3)

Where, λ is damping coefficient, and it is Gaussian‐Newton when λ is 0. The first derivative on Q is set as 0 to make Q smallest: Q

0

2∑

y

,

∑

f x ,p

p

,

p

2λ p

p

,k

1,2, … m (4)

And the solution is obtained: p

p p

p

p

p a a

p

a

λ a

λ a

a

…

a

… … … a

a

a a

(5)

a

λ

Suppose J x, p is Jacobian matrix of f x, p , H x, p is the Hesse matrix, then formula (5) can be simplified as: p

p

H x, p

λE

JT x, p

y

f x, p

(6)

This is the iteration formula of LM algorithm. During the calculating process, the iteration should be stopped when is large, set the obtainedp as new the absolute value of difference between p and p is small. When p p is convergent or the iteration times reach the limit. p to calculate it again, and repeat it till p p Experiment Tests Data The airborne full‐waveform data in urban area was selected here, which was obtained by RIEGL and supplied by Martin Isenburg. The format of each manufacturer’s full‐waveform data was different, for example, RIEGL’s data

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contains two file: all the pulse data were recorded in LWF file and the property of each pulse was recorded in LGC file [10] .

We chose 4 different types of pulses to testify the feasibility of our algorithm, and they were ground, building roofs, building edges and vegetation.

Results and Analysis From Fig. 4 to Fig. 7 are the decomposing results of ground, building roof, building edge and vegetation. The black curve in (a) is original pulse, while the blue one is the pulse without outliers; and the red curve in (b) stands for the modelled pulse.

Before decomposition process, the noises should be eliminated as they have side effect on detecting peaks. From Fig. 4(a) to Fig. 7(a), there are several fluctuations in the pulse, some are peaks generated by object echoes, and others are system noise. The level of coincidence between modelled pulse and pulse without outliers is shown from Fig 4(b) to Fig 7(b), which proves the reliability of decomposition results.

The statistics of peaks and RMSE between modelled pulse and pulse without outliers are shown in Table 1.

Theoretically, there should be only one peak in the returning pulse of ground and building roof; and two peaks in the returning pulse of building edge; several peaks in the returning pulse of vegetation as laser’s strong penetration through it. And the above theories are proved by the statistics in Table 1. In addition, the RMSEs between modelled pulse and pulse without outliers of the 4 different objects are all less than 1, which prove that the fitting method is effective. Therefore, the stepwise decomposition method of full‐waveform data based on Levenberg Marquardt we proposed is practicable and valid.

(a) (b) FIG. 4 RESULT OF GROUND PULSE

(a) (b) FIG. 5 RESULT OF BUILDING ROOF PULSE

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(a) (b) FIG. 6 RESULT OF BUILDING EDGE PULSE

(a) (b) FIG. 7 RESULT OF VEGETATION PULSE TABLE 1 STATISTICS OF PEAKS AND RMSE

peaks RMSE

Ground 1 0.485772

Building roof 1 0.835702

Building edge 2 0.874325

Vegetation 5 0.466112

Conclusion Compared with traditional airborne LiDAR, waveform decomposition is the most significant step in full‐waveform data processing. In this paper, a stepwise decomposition method of full‐waveform data based on Levenberg Marquardt is proposed here. And we choose 4 different types of pulses to testify availability of our method.

The experimental results have proved that our method is valuable and practicable.

ACKNOWLEDGEMENTS

This work was supported by National Natural Science Foundation Project (41371436).

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