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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

LiDAR (Light Detection And Ranging) systems are capable of providing 3D positional and spectral information (in the utilized spectrum range) of the mapped surface. Due to systematic errors in the system parameters and measurements, LiDAR systems require geometric calibration and radiometric correction of the intensity data in order to maximize the benefit from the collected positional and spectral information. This paper presents a practical approach for the geometric calibration of LiDAR systems and radiometric correction of collected intensity data while investigating their impact on the quality of the derived products. The proposed approach includes the use of a quasi-rigorous geometric calibration and the radar equation for the radiometric correction of intensity data. The proposed quasi-rigorous calibration procedure requires time-tagged point cloud and trajectory position data, which are available to most of the data users. The paper presents a methodology for evaluating the impact of the geometric calibration on the relative and absolute accuracy of the LiDAR point cloud. Furthermore, the impact of the geometric calibration and radiometric correction on land cover classification accuracy is investigated. The feasibility of the proposed methods and their impact on the derived products are demonstrated through experimental results using real data.

The importance of geometric calibration and radiometric correction of active remote sensing data has been emphasized for Japan Earth Resources Satellite-1 Synthetic Aperture Radar (JERS-1 SAR) [

The geometric calibration of LiDAR systems aims at estimating and removing all the systematic errors from the point cloud coordinates such that only random errors are left. Systematic errors in the LiDAR data are mainly caused by biases in the system parameters, e.g., biases in the mounting parameters relating the system components (lever arm and boresight angles) and biases in the measured ranges and mirror angles. The elimination/reduction of such systematic errors or their impact has been the focus of the LiDAR research community in the past few years. The existing approaches can be classified into two main categories: system-driven (calibration) and data-driven (strip adjustment) procedures. This categorization is mainly related to the nature of the utilized data and mathematical model. System-driven (calibration) procedures are based on the physical sensor model relating the system measurements/parameters to the ground coordinates of the LiDAR points. These procedures incorporate the system’s raw data or at least the trajectory and time-tagged point cloud for the estimation of biases in the system parameters with the help of the LiDAR point positioning equation. In this paper, the term “raw data” is used to denote all the quantities present in the LiDAR point positioning equation (

As already mentioned, rigorous geometric calibration procedures are based on the physical mathematical model (

Radiometric correction of the LiDAR data is a relatively new research area. It aims at converting the recorded intensity data into the spectral reflectance of an object. The radar equation has been proposed for radiometric correction of the LiDAR intensity data in [

Despite the previous research on geometric calibration of LiDAR systems and radiometric correction of the intensity data, the impact of these approaches on the quality of the derived products is still an open research area. This paper presents methods for geometric calibration and radiometric correction of airborne LiDAR data while evaluating their effect on the geo-positional accuracy and classification of the intensity data. The geometric calibration involves a quasi-rigorous procedure for the estimation of biases in the system parameters. The geo-positioning accuracy of the adjusted point cloud is assessed based on quantifying the degree of compatibility between LiDAR and control surfaces before and after the calibration process. The radiometric correction utilizes the radar equation to determine the spectral reflectance of objects. Then, land cover classification is conducted. Accuracy assessment with checkpoints acquired from an orthophoto is used to assess and compare the classification results from the original and the geometrically calibrated and radiometrically corrected datasets.

In this section, the proposed geometric calibration procedure for the estimation of biases in the system parameters is described. This method, which is denoted as “Quasi-Rigorous” since it only requires the time-tagged point cloud and trajectory position data, utilizes LiDAR data in overlapping strips. Biases in the system parameters are estimated by reducing discrepancies between conjugate surface elements in overlapping strips and control data, if available.

The proposed method will be explained in the following subsections. First, the mathematical model relating conjugate surface elements in overlapping LiDAR strips as well as overlapping LiDAR and control surfaces in the presence of systematic errors is derived. Based on the analysis of the derived mathematical model, remarks regarding the necessary flight and control configuration for LiDAR system calibration are outlined. The established mathematical model for the calibration procedure is derived based on point primitives (

The coordinates of the LiDAR points are the result of combining the derived measurements from each of its system components, as well as the mounting parameters relating such components. The relationship between the system measurements and parameters is embodied in the LiDAR point positioning equation [_{G} is derived through the summation of three vectors (_{o}_{G}_{ω,φ,κ}, _{Δω,Δφ,Δκ}, and _{SααSββ} (refer to _{o}_{G}_{o}_{ω,φ,κ} stands for the rotation matrix relating the ground and IMU coordinate systems—which is derived through the GPS/INS integration process. The term _{Δω,Δφ,Δκ}_{SααSββ} refers to the rotation matrix relating the laser unit and laser beam coordinate systems with _{α}_{β}

– _{G}

–

–

The LiDAR point positioning mathematical model presented in _{α}_{β}_{G}_{t}_{nf}_{b}_{n}_{G}

_{G}_{G}_{G}

Having two conjugate points in overlapping strips, which will be denoted by subscripts _{GA} (_{GB} (_{G}

The procedure for estimating the quantities (

For a LiDAR point mapped at time

Then, a straight line is fitted through the selected trajectory positions to come up with a local estimate of the trajectory. After defining the local trajectory, the necessary quantities can be estimated as follows:

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▪

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Analyzing the mathematical expressions for the discrepancies between bias/noise contaminated coordinates of conjugate points in overlapping strips—

Using only overlapping strips, one cannot estimate the vertical bias in the lever arm

The impact of the range bias

The impacts of biases in the boresight heading angle

Having opposite flight lines with 100% overlap is optimal for the recovery of the planimetric biases in the lever arm (

The impact of biases in the lever arm on the introduced discrepancies is independent of the flying height. However, the impact of the biases in the boresight angles increases with an increase in the flying height. Moreover, for a given flight height and relatively flat terrain (_{A} and z_{B}

Having two parallel flight lines with less than 100% overlap is necessary for the estimation of the bias in the boresight heading angle _{A}_{B}

To estimate biases in the vertical component of the lever arm

For reliable estimation of the bias in the mirror angle scale

Based on the above discussion, one can conclude that the optimal/minimal flight and control configuration for the estimation of the system parameters consists of three overlapping pairs and one vertical control point. More specifically, two strip pairs, which are captured from two flying heights in opposite directions with 100% overlap, and two parallel flight lines, which are flown in the same direction with the least overlap possible, are needed. As for the parameters to be estimated, due to high correlation, it is not recommended to simultaneously solve for the vertical bias in the lever arm

The mathematical model that has been developed so far is based on the availability of conjugate points in overlapping strips,

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–

–

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For LiDAR data, there is no point-to-point correspondence between overlapping strips or between a given strip and a control surface. Therefore, the abovementioned LSA solution cannot be used to come up with an estimate of the biases in the system parameters. Therefore, the following subsections will deal with the necessary modification to the stochastic model in

In this research, one of the strips, denoted by _{1}_{2}_{1}_{2}

For a given point-patch pair, we will assume that one of the vertices of the TIN patch in _{2}_{1}_{2}_{1}

The main objective for the development of the modified LSA is to deal with the model in ^{′}. Such a condition signifies that the modified weight matrix is not positive-definite (^{′−1} does not exist). Therefore, the modified variance-covariance matrix will be represented as follows:

Using the modified weight matrix, the LSA target function can be redefined as ^{′}

The last step is to estimate the a-posteriori variance factor
^{′} – refer to

In summary, from an implementation point of view, the LSA solution to the stochastic model in

So far, we established that by modifying the weight matrix to satisfy the condition in ^{′}. This can be established according the following procedure. First, one starts by defining a new coordinate system (

The weight matrix of the transformed coordinates in the

Finally, the modified weight matrix in the ^{′}_{XYZ}D

In summary, the proposed Quasi-rigorous calibration procedure proceeds as follows:

The correspondence between points in _{1}_{2}

For each conjugate point-patch pair from overlapping LiDAR strips, (e.g., patch vertex _{GB}(_{2}_{GA}(_{1}

When a control surface is used, it is represented by the original points (due to its sparse nature) and the LiDAR strips are represented by triangular patches. Then, for each conjugate point-patch pair, one can write the observation equations similar to those in

For each conjugate point-patch pair, the weight of the observed discrepancy vector _{GA}(_{GB}(

The established correspondences and the corresponding observations with their modified weight matrices are used to derive an estimate of the system biases. Then, the estimated system biases are used to reconstruct an adjusted point cloud using

Following the reconstruction of the LiDAR point cloud, the correspondence between point-patch pairs might change. Therefore, a new set of correspondences has to be established. The new correspondences are then utilized to derive a better estimate of the system biases.

Such a procedure is repeated until the corrections to the estimated calibration parameters are almost zero.

One should note that each pair of pseudo-conjugate points provides three observations of the form in _{1}_{2}

The impact of the geometric calibration on the relative accuracy of the LiDAR point cloud can be assessed by checking the degree of compatibility between conjugate surface elements in overlapping strips before and after reconstructing the point cloud using the estimated biases. In this work, the compatibility will be evaluated qualitatively and quantitatively. The qualitative evaluation will be performed by visual inspection of profiles generated using the original and adjusted point cloud to check any improvements in the quality of fit between overlapping strips. The quantitative assessment, on the other hand, will be performed by computing the necessary 3D transformation parameters for the co-alignment of overlapping strips before and after the calibration procedure. For the computation of the 3D transformation parameters, the proposed Iterative Closest Patch (ICPatch) procedure in [

The impact of the geometric calibration on the absolute accuracy can be evaluated by quantifying the degree of compatibility between LiDAR and control surfaces before and after the calibration process. From an implementation point of view, such a procedure would not always be feasible due to the control surface requirement. In this work, linear features extracted from the LiDAR data before and after the calibration process are used for the geo-referencing of an image block covering the same area. The absolute accuracy of the derived ground coordinates from the geo-referenced image block is evaluated using a check point analysis. The adopted methodologies for linear features extraction and their utilization for the photogrammetric geo-referencing are detailed in [

The physical properties of the laser energy are considered with respect to the sensor configuration and environmental parameters using the radar equation, which is proposed to model the power of the received signal [

In this equation, the received signal power _{r}_{t}_{r}_{t}_{sys}_{atm}

In the above equation, Ω is the scattering solid angle, _{s}_{s}

The radiometric correction aims at converting the intensity value _{s}_{r}_{r}_{r}_{t}_{sys}_{t}_{atm}

Land cover classification is conducted to evaluate the impact of the geometric calibration and the radiometric correction of the LiDAR data on the final data products. Two datasets are prepared from the original and modified LiDAR data after geometric calibration and radiometric correction. Each set of data includes the interpolated digital surface model and the interpolated intensity. Four land cover classes (tree, grass, soil and built-up area) are identified in both datasets. Using the same training sites, Maximum likelihood classification is conducted on both datasets and finally the classification results are evaluated by conducting an accuracy assessment using check points, which can be generated from an orthorectified aerial photo. The overall accuracy and kappa coefficient are used to evaluate the classification results.

A real LiDAR dataset was acquired to test the feasibility of the proposed methods. The study area covers the British Columbia Institute of Technology (BCIT) located at Burnaby, British Columbia, Canada (122°59′W, 49°15′N). The area contains buildings and parking lots connected by sidewalks and paved road segments. Individual shrubs and open spaces covered by grass can also be found in the surveyed area. The LiDAR mission was conducted on July 17, 2009 from 14:37 to 15:15 local time. The day of the mission was a sunny day with a temperature of 29.8 °C. The visibility and the pressure were 48.3 km and 101.81 kPa, respectively, as delivered by the National Climate Data and Information Archive from Environment Canada. The LiDAR sensor used was a Leica ALS50 operating at a 1.064 μm wavelength with 0.33 mrad beam divergence. The captured LiDAR data consists of six strips that cover a 1 km by 2 km area. The configuration of the flight lines is shown in _{1} (1150 m) is 1.5 points/m^{2} while for the flying height H_{2} (540 m) the average point density is 3.7 points/m^{2}. In the surveyed area, thirty-seven control points were established by a GPS survey. These control points were used for the geometric calibration and check point analysis to evaluate the absolute accuracy of the adjusted point cloud.

In addition to testing the feasibility of the proposed geometric calibration procedure, we would like to investigate whether the calibration results are significantly different when using more overlapping strip-pairs than the minimum recommended configuration as discussed in Section 2.2.

As already mentioned, for the estimation of the biases in the vertical component of the lever arm (

To evaluate the impact of the geometric calibration on the relative accuracy, the compatibility of overlapping strips before and after the calibration procedure (using the different experiment scenarios) is assessed. The compatibility of the point cloud is evaluated qualitatively and quantitatively. The qualitative evaluation is performed by visual inspection of profiles generated using the original and adjusted point cloud to check any improvements in the quality of fit between overlapping strips. The improvement in the strips compatibility is illustrated in _{T}_{T}

The qualitative and quantitative evaluations demonstrate compatible results from the different investigated scenarios. One should finally note that the range bias, which was removed in the experiments using control information, does not lead to significant discrepancies among conjugate surface elements in overlapping strips. Therefore, we cannot evaluate the introduced improvement when adding the range bias in the calibration process by checking the compatibility among overlapping strip-pairs. Such an analysis is done next in the absolute accuracy verification.

For all investigated scenarios, significant improvement in the planimetric accuracy can be observed (refer to the highlighted cells). This can be explained by the fact that the main detected bias in the studied dataset is in the boresight roll and pitch angles, which mostly affect the horizontal accuracy. One can also note that when reducing the number of utilized overlapping strip-pairs (experiment IV), the results are not negatively affected. For the experiments using control information (experiments II, III, and IV), where we solved for the range bias, we can observe that the bias value in the vertical direction has been significantly reduced (refer to the circled values in

To assess the impact of radiometric correction, the variance-to-mean ratio of the intensity values for the whole dataset are plotted against the surface slope before and after the radiometric correction (

To further assess the impact of the radiometric correction, the intensity values before and after radiometric correction are compared for different land cover classes that have been identified with the help of an overlapping orthophoto of the study area. This comparison is conducted to check whether the proposed procedure would have any impact on the homogeneity and separability of the investigated classes.

Using more than 1,000 check points that have been identified with the help of an orthophoto over the study area, we compared the four-land cover classification results (Tree, Built-up Area, Grass, and Soil) with the manual classification of the check points.

The overall classification accuracy using the intensity from the original LiDAR data is about 63.0%. After geometric calibration and radiometric correction, the overall accuracy increased to 70.5%. The kappa coefficient has also increased from 0.442 to 0.558. Considering each individual land cover class, the kappa coefficient of the tree and built-up features are found to be always lower than 0.5 using the original LiDAR dataset. After geometric calibration and radiometric correction, the kappa coefficient of the tree class increases from 0.464 to 0.725 and the kappa coefficient of the built-up class increases from 0.486 to 0.593. This improvement in the classification results is due to the high separability of the intensity values between different class features after the radiometric correction. A slight improvement of the kappa coefficient of the grass class can also be detected after the geometric calibration and radiometric correction. It can be also observed that the proposed process helps in reducing the confusion between grass and tree classes in LiDAR data classification.

One can note a significant impact of the geometric calibration and radiometric correction in the tree class areas. In

In this research, methodologies for the geometric calibration and radiometric correction of the LiDAR system and collected data have been presented. The introduced geometric calibration procedure is denoted as the Quasi-rigorous due the fact that few reasonable assumptions are made for its development. This method only assumes that we are dealing with an almost vertical LiDAR system, which is quite realistic for flight missions with a steady platform. To conduct such a calibration, we require time-tagged point cloud and trajectory position data. In contrast to the position and orientation information requirement for each pulse in the rigorous calibration, the Quasi-rigorous procedure only requires a sample of the trajectory positions at a much lower rate. Access to this type of data is not a concern. Since this calibration procedure derives approximations of some of the system raw measurements, the proposed procedure can provide as a by-product the necessary information for the radiometric correction of the LiDAR intensity data when system raw measurements are not available (

Future work will focus on more testing using real datasets from operational systems. Also, the quasi-rigorous geometric calibration will be extended to include the attitude information in the calibration process. Furthermore, an automated procedure for the identification of useful areas within the data for reliable and faster estimation of the parameters will be implemented in the geometric calibration process. In addition, radiometric correction and land cover classification will be investigated by using full-waveform LiDAR data as it provides additional information (such as the transmitted laser pulse, the echo width, the cross section of the echo, etc.) comparing to the traditional multi-return LiDAR data. It is expected that the full-waveform LiDAR data will improve the point cloud density and the dimensionality of feature space leading to better classification and segmentation.

This research work is supported by a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada (NSERC) and the GEOIDE Canadian Network of Excellence, Strategic Investment Initiative (SII) project SII P-IV # 72. The authors would like to thank McElhanney Consulting Services Ltd, BC, Canada for providing the real LiDAR and image datasets. Also, the authors are indebted to Dan Tresa, McElhanney Consulting Services Ltd, for the valuable feedback.

1. Model

2. LSA Target Function ϕ(

Since ^{′}

Expanding

3. Solution Vector

The solution vector

4. Variance-covariance matrix of the solution vector ∑{

Using the law of error propagation, the variance-covariance matrix of the solution vector ∑{

Since for a pseudo inverse, ^{′}^{′+}^{′} = ^{′}[

5. A-posteriori variance factor

The a-posteriori variance factor

Since ^{′}

Expanding

Given that the trace of a scalar equals to the scalar, i.e., tr(S) = S and that the trace operation is commutative, i.e., tr(AB) = tr(BA) [

Based on the properties that tr(A) + tr(B) = tr(A+B) and that _{n} is an

The term ^{T}

Expanding

Substituting

Given that ^{′}

Based on the property that

Given that ^{′}^{′+}) = ^{′}^{′+}^{′} = ^{′},

Finally, we can get the expression for the a-posteriori variance factor

Coordinate systems and involved quantities in the LiDAR point positioning equation.

(

Flight and control configuration of the LiDAR dataset.

Profiles along the

Variance-to-mean ratio of the intensity data (before and after radiometric correction) for different slopes.

Comparison of classification results of original and the geometrically calibrated and radiometerically corrected LiDAR dataset.

Experiments description (used overlapping strip-pairs and number of control points).

I | 1&2, 3&4, 4&5, 5&6 | 0 |

II | 1&2, 3&4, 4&5, 5&6 | 37 |

III | 1&2, 3&4, 4&5, 5&6 | 1 |

IV | 1&2, 4&5, 5&6 | 37 |

Estimated biases in the system parameters for the different experiments.

I | 0.01 | 0.00 | −29.5 | −88.7 | 3.0 | - | 0.0002656 |

II | 0.00 | 0.00 | −29.0 | −91.0 | 3.6 | 0.118 | 0.00010377 |

III | 0.01 | 0.00 | −29.5 | −89.6 | 3.5 | 0.142 | 0.00009348 |

IV | 0.02 | 0.05 | −38.8 | −90.7 | −32.1 | 0.115 | 0.00003607 |

Discrepancies between overlapping strips before and after applying the calibration parameters estimated using the different scenarios.

RMSE analysis of the photogrammetric check points using extracted control linear features from the LiDAR data before and after the calibration procedure.

Mean and standard deviation of the intensity data for different land cover classes before and after radiometric correction.

Built-Up Area | 13.9 ± 4.1 | 10.6 ± 3.3 |

Grassland | 40.6 ± 9.5 | 32.7 ± 5.7 |

Soil | 24.7 ± 5.1 | 20.5 ± 2.7 |

Tree | 21.4 ± 9.3 | 82.7 ± 44.2 |

Confusion matrix of the classification results using original LiDAR dataset.

Tree | 130 | 40 | 20 | 18 | 208 | 0.464 |

Built-up | 119 | 386 | 12 | 16 | 533 | 0.486 |

Grass | 34 | 26 | 59 | 43 | 162 | 0.294 |

Soil | 20 | 16 | 10 | 61 | 107 | 0.502 |

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Total | 303 | 468 | 101 | 138 | 1010 | |

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Overall Accuracy = 63.0% | Average Kappa Coefficient (KC) = 0.442 |

Confusion matrix of the classification result using geometrically calibrated and radiometrically corrected LiDAR dataset.

Tree | 153 | 23 | 12 | 4 | 192 | 0.725 |

Built-up | 62 | 399 | 23 | 23 | 507 | 0.593 |

Grass | 37 | 35 | 94 | 26 | 192 | 0.402 |

Soil | 11 | 24 | 19 | 65 | 119 | 0.486 |

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Total | 263 | 481 | 148 | 118 | 1010 | |

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Overall Accuracy = 70.4% | Average Kappa Coefficient (KC) = 0.558 |