NEW HORIZONTAL ACCURACY ASSESSMENT TOOLS AND TECHNIQUES FOR LIDAR DATA INTRODUCTION

Transcription

1 NEW HORIZONTAL ACCURACY ASSESSMENT TOOLS AND TECHNIQUES FOR LIDAR DATA John A. Ray Ohio Department of Transportation Columbus, Ohio Lewis Graham GeoCue Corporation Madison, Alabama ABSTRACT Much of the focus to date in LiDAR geometric data assessment has been on vertical accuracy. However, with the increasing use of LiDAR derived images such as LiDAR orthos and LiDAR synthetic stereo models, knowledge of the horizontal accuracy of LiDAR data is essential to project error modeling. We present new assessment tools and techniques for both assessing the horizontal geometric accuracy characteristics of LiDAR data and for effecting global data adjustments. We provide practical examples in highway project analysis as well as ideas for future work. Application of these and similar tools show great promise for deploying LiDAR as an engineering tool of choice. INTRODUCTION The Ohio Department of Transportation (ODOT) performs a portion of the remote sensing and digital mapping for the highway improvement projects delivered annually by ODOT in an effort to fulfill its mission. In 2004, ODOT implemented airborne LiDAR technology as a means to reduce the project development duration and improve the vertical accuracy of the spatial data generated from remote sensing operations. Within a few months, the processes and procedures established were capable of yielding spatial data with a vertical RMSE (Root Mean Square Error) approaching 0.10 feet on hard surfaces. The procedures were the result of research and testing of various flight operation parameters, GPS attributes, LiDAR control methodologies, and data processing techniques. Some of the procedures adopted were somewhat unique, such as adjustment of the LiDAR dataset to points collected using special targets as control (see Figure 1). Figure 1. LiDAR control target being positioned with GPS.

2 When delivering any geospatial product, it is important to be able to report the accuracy of the final product(s) being delivered, but it is critical that the accuracy be known when using the data for engineering purposes. In an effort to express the quality of the LiDAR derived data, ODOT began providing a Quality Control Report for each project. The Quality Control Report details the points used to adjust and determine the vertical accuracy of the LiDAR generated TIN (Triangulated Irregular Network). Reporting the vertical accuracy has also been beneficial in helping educate designers about the potential uses for LiDAR data. After gaining some additional experience, ODOT transitioned to using pavement surfaces for LiDAR control instead of the specially designed circular targets. The targets functioned well, but were labor intensive to install and required placement just prior to the flight, which became problematic. As ODOT s LiDAR process continued to evolve, multiple quality control procedures were implemented. One such quality control procedure is to overlay the planimetric features (including pavement markings) collected using photogrammetric methods on top of the LiDAR intensity image also known as a LiDAR ortho ( Figure 2). This simple check helps identify any potential problems with the GPS/INS system, scale factor and coordinate transformation issues. It soon became apparent that the visual agreement between the LiDAR data and the planimetric features were very good where a clear difference in the intensity of the LiDAR data was evident. Survey control points were also used to check for horizontal positioning. Note that the density of the LiDAR data (typically 12 pts. /m 2 ) certainly has a positive impact on the ability to identify features such as pavement markings. Figure 2. Survey control points on a LiDAR ortho. With ODOT s desire to report the horizontal accuracy in addition to the vertical accuracy of the final products being created, ODOT discussed the issue with The Ohio State University s Center for Mapping. The discussions resulted in The Ohio State University (OSU) proposing a research and development project entitled Airborne LiDAR Reflective Linear Feature Extraction for Strip Adjustment and Horizontal Accuracy Determination. The goal of this project was to automatically extract features within the vicinity of control points in order to determine the horizontal accuracy of the LiDAR dataset. Even though OSU is very good at research and development, they are not in the commercial software and support business. With the ultimate goal being the integration of the horizontal accuracy determination and adjustment capability into an existing workflow, the new functionality would ideally be placed in a product that was currently used within the process. After consideration of many factors, we discussed the idea with GeoCue Corporation of Madison Alabama. The GeoCue software was a logical place to house the new functionality for

3 several reasons. One of the primary reasons was the existing use of the GeoCue software as a LiDAR processing and management tool. The GeoCue software was also used by many other LiDAR data providers which would then create an opportunity for reporting the horizontal accuracy component on an industry-wide basis. When GeoCue was approached regarding the issue, they expressed an interest in developing the functionality within their suite of software products. A project soon followed to create a module known as FMAS (Feature Matching and Adjustment System). While the primary objective of the GeoCue effort was to integrate the OSU developed algorithms seamlessly into their software suite, GeoCue developed other interactive techniques for measuring and adjusting data based on measuring point and linear features, thereby increasing system functionality and the end users options. Of particular utility, and the subject of this paper, is a novel algorithm developed by GeoCue that allows measurement and correction of data on non-conjugate line segments. LINEAR CONTROL GeoCue Corporation has subsequently developed a system to measure control points and associated conjugate points in orthos and LIDAR Orthos. This system (The Feature Matching and Adjustment System CuePac, or the FMAS CuePac) is intended for assessing the horizontal and vertical accuracy of LIDAR data and the horizontal accuracy of orthos. The system can also compute a 6 parameter linear transform in the horizontal components and a simple shift in Z. The user can optionally apply these corrections to a LIDAR data set using the distributed processing system of the GeoCue software system (GeoCue). The system is primarily designed to use externally provided control data in the form of FMAS Control Features. FMAS supports: Points 2 point line segments Polylines Simple polygons Hierarchical combinations of the above (e.g. traffic stop bar) As we tested the initial version of the system, we realized a particularly valuable workflow of comparing LIDAR data horizontal position by using a known control orthophoto. Basically the flow is as shown in Figure 3 1. During this investigation we recognized the value in developing an algorithm to solve for an affine correction using only parallel line segments (conjugate lines). 1 Of course, one could argue that the wrong elevations have been used in generating the orthos since the LIDAR data are assumed off in X, Y. This is a phenomenon that will be quantified in a future paper.

4 Figure 3. Adjusting LIDAR with metric orthophotos. Conjugate Lines When implementing the flow of Figure 3 we quickly discovered that it is very difficult to accurately digitize point features (such as the end of a paint stripe) in LIDAR intensity images. What looked to be more fruitful was digitizing line segments. However, accurately digitizing the start and end points of the conjugates of the line segments in LIDAR orthos proved unreliable. This is primarily due to the resolution difference between these sources and the fact that often the best control is a segment of a line such as an edge-of-pavement mark. An example of this difficulty is illustrated in Figure 4. On the left is a 1.5 m LIDAR ortho derived from 2 m post spaced LIDAR data. On the right is a USGS 0.33 m natural color ortho. Notice that linear features are clearly visible in the LIDAR ortho. However, precisely delineating the start or end point of a line segment is not possible due to the lower resolution of the LIDAR ortho. Figure 4. LIDAR Ortho (left) compared to a natural color Ortho (right). What did look promising was to digitize a control line segment such that the segments were on the same linear feature but not necessarily conjugate (see Figure 5). Notice that we have digitized a segment along the same linear

5 feature (the white stripe down the west side of a runway) but these segments are not conjugates since they represent different regions of the conjugate line 2. Figure 5. Control segment (vertical orange on right) and Conjugate (green line on left). Note that it is important to realize that the control and conjugate lines are conjugate but not the segments. We then faced the task of developing algorithms to use these semi-conjugate line segments in our point driven system. Thus the statement of the problem is: given a control segment and a conjugate line on which the conjugate segment is located (but whose endpoints are currently unknown), find an algorithm to determine the end points of the conjugate system. Considering the Segments Consider the example of Figure 6. Here we have 4 control segments and 4 target segments. The target segments represent segments of linear features where the linear features are the conjugate features of Control lines. Since these target segments are digitized along conjugate lines but not in conjugate locations, we will refer to these collections as Control Segments (denoted by C ) and Related Segments (denoted by R ). The blue vector V represents the average relative displacement between the control data set and the conjugate data set. C 2 C 1 R 2 R 1 C 3 C 4 R 3 V R 4 Figure 6. Example of Control Segments and Related Segments. 2 Recall that a line is defined by two points and extends to infinity in both directions whereas a line segment terminates at each end by the defining points.

6 We will use the notation defined in Table 1. Table 1. Notation Symbol C i R i c i,k r i,k G i g i,k L i m i Definition The i th control segment The i th related segment The k th vertex (node or point) on the i th control segment The k th vertex (node or point) on the i th related segment The i th conjugate segment The k th vertex (node or point) on the i th conjugate segment The i th line (as opposed to segment) The slope of the i th segment (or line) An approximation or estimate of x It is very important to appreciate that the user, in general, has not attempted to digitize conjugate segments in the target. The only requirement is that the user digitizes a segment on the same linear feature as was used to digitize the control segment. This means that R i is not the conjugate segment of C i. Thus we know that in the case of no error between the control image and the target image,. C i and R i will both lie on the same line but they will not be (in general) coincident. This is why we say that R i is related to C i as opposed to being the conjugate. An illustration of our example, post adjustment, is depicted in Figure 7. Note that even after a fairly good adjustment (as indicated by the diminished length of the displacement vector, V), the related segments are not coincident with the control segments. This is due to the fact that the related segments were digitized on lines conjugate to the lines defined by the control segments but not in conjugate locations along these lines. C 1 R 1 R 2 C 2 C 3 C 4 R 3 V R 4 Figure 7. The appearance of adjusted data without conjugate segments. The Mathematical Approach We know from the specification of the problem that a control segment, C i, defines a line L i and that, with no error, the related segment in the data to be adjusted (R i ) should be coincident with L i. Thus we could use a linear constraint to find adjustment parameters. However, a direct linear constraint will not solve for a scale difference. This makes intuitive sense because scaling something of infinite extent and zero width (a line) yields the same line! Our approach has been to attempt to find the true conjugates of the control segments and then apply standard point adjustment algorithms to these data.

7 Consider a magnified view of the example data. If we can approximate the maximum displacement V m, we know each conjugate point g i,j must be within a distance V m of its control point c i,j (we refer to this as the radius of ambiguity) 3. Furthermore, g i,j must be on the line defined by R i (within digitizing error). This is illustrated in Figure 8. The conjugate points must lie on the orange lines in this figure. With this approximation of the locations of the conjugate points, we will now develop an algorithm to refine he estimates. The approach that we will take is to use orthogonal independence to separately estimate the x and y components of each conjugate point. r = V m C 1,2 g 1, 2 lies somewhere on this segment C 1,1 V m Figure 8. Radius of ambiguity of conjugate points. Consider horizontal and vertical segments as shown in Figure 9. Here we have depicted the radius of ambiguity of each conjugate point and sketched the possible location of each in orange. Now consider the special circumstances that exist for these lines. 3 There is a bit of a subtle point here. If conjugate points are collected by human interaction or an automated technique such as correlation, one cannot make this statement (due to outlier collection errors). However here we are discussing the true and as of yet uncollected conjugate points and thus we can quantify a radius of ambiguity.

8 C 2 R 2 V C 3 Radius of ambiguity R 3 Figure 9. The special case of horizontal and vertical segments. We know, for a horizontal control line, that the y component of the conjugate point is given by the y value of the conjugate line (which is a constant since the line is horizontal). Similarly, the x component of the conjugate of a vertical line is given by the x value of the conjugate vertical line. Thus you will note that by selecting horizontal lines we can remove (to within the error of digitizing) the ambiguity in the y coordinate and by selecting vertical lines we remove the ambiguity in the x coordinate. This leads us to use the slope of the line to motivate the following arguments: (1) (2) where, are the orthogonal components of the standard deviation 4. Note that we are assuming that the slope of the control line is approximately the same as the slope of the conjugate line. In other words horizontal lines provide strength to the y component of the fit and vertical lines to the horizontal fit. Conversely, a horizontal line provides no information about the x component of a conjugate point ) and a vertical line lends no information to the y component of a conjugate point ). An inspection of Figure 9 lends intuitive credence to this. First Pass Conjugate Approximation The previous observations provide the basis for an algorithm that will allow us to approximate the conjugate points (the g i,k ). Our first approximations for the conjugate points are the points defined by the intersection of a line perpendicular to the control point C i,k and intersecting R i. This is illustrated in Figure 10. We denote this first set of approximated conjugate points as where i is the i th related line and j is 1 or 2 representing the end point of the approximated conjugate line segment. Note that these approximations can be as far as V m /2 from their correction locations and thus will require a refinement. 4 One could argue that this should be more properly the sample standard deviation but another subtly occurs here. This will be addressed in a future paper. We do use sample standard deviation in the least squares adjustments.

9 c 1,1 C 1 R 1 c 3,1 c 1,2 C 3 R 3 c 3,2 Figure 10. First approximation to conjugate points. Second Pass Refinement We next need to quantify, for each of these approximated conjugate points. We do this as follows: Let be the unsigned angle between the ith line defined by Ri and the horizontal. Thus. We now want to compute weights for the x and y components of the conjugate points such that (3) since we are going to solve for the initial refinement in a weighted, separable least squares adjustment. Our base model 5 uses: (4) (5) We chose this because the sine and cosine are obviously the slope components with the advantage of not blowing up at the extremes of horizontal and vertical. We needed a much lower weight on diagonal lines and the reciprocal relationship between W(x) and W(y). We used the 4 th power (rather than 2 nd ) because we wanted a rapid off-axis dive in the weight. The weights as a function of are plotted in Figure Weight equations are a topic for further research.

10 1 W x ( m) W y ( m) m Figure 11. Independent coordinate least square weights. We next computed a 6 parameter affine transformation: using a weighted least squares adjustment. The least squares algorithm selected must support independent x and y weights since this is the trick of the first pass adjustment. Note that true conjugate points can be mixed in with this phase of the adjustment (if available) and, indeed, they add strength to the solution. These normal conjugate points should have weights inversely proportionate to their sample standard deviations (if known), normalized to the same weight range as the pseudo conjugate points ( ). If the deviations are unknown, use W = 1.0. The result of the least square solution will be a transform that represents the true adjustment between the control system and the target system. Thus this adjustment can now be used to construct conjugate points for the related segments. If is the six parameter transform that projects the c 1,1 C 2 conjugate space (G) onto the control space (C), then (since we are using a six C 1 R 2 parameter transform you will need to augment the T matrix) projects C onto G. R 1 C 3 We now make a second pass adjustment of the approximated conjugate C 4 R 3 points by the following procedure. Project all of the control points through yielding a result similar to R 4 Figure 12 where we denote the back-projected control V points as green diamonds. Figure 12. Control points (block) projected into conjugate space (green diamonds). (6) (7)

11 Our projected conjugate points ( ) will vary somewhat from the digitized Related lines due to the general measurement error, the fact that the error is not really exactly modeled by a six parameter transform and the distribution of the original pseudo conjugate points. However, our second pass conjugate point g i,k must be on R i (again, to within our digitizing error) since this is the original assumption of collecting conjugate lines. Thus we perform a second pass adjustment by the following procedure. Construct a line through the points c i,j,. Find where this line intersects R i. Make this the new (and final for the purposes of this paper) conjugate point g i,j. This is illustrated, greatly magnified, in Figure 13. c 1,1 g 1,1 C 1 R 1 c 1,2 g 1,2 Figure 13. Constructing refined conjugate points. These new conjugate points are now considered the true conjugates. Final Adjustment The newly computed g i,j are now treated as the true conjugate points to the original control segment end points c i,j. The weights of the deviations of these computed conjugate points are now set such that. These control, conjugate pairs are now used in a second pass, equally weighted least squares adjustment with all other ordinary measurement pairs added to the data set. This yields the final transform T. Any major changes in T between the intermediate adjustment and the final adjustment are indicative of a weak original geometry. In the example of adjusting a LiDAR data set to known linear control, the final process would be to apply the resulting transform T to the each and every LiDAR data point. We do this using a distributed processing function within the GeoCue Enterprise system, allowing the LiDAR files to be processed in parallel. CONCLUSIONS The Ohio Department of Transportation has begun using the horizontal accuracy determination and correction functionality contained in the GeoCue LiDAR FMAS CuePac module. To date, five projects have been analyzed with the software. The current process involves running the algorithms after the strip alignment task has been performed. The control points are loaded and the software automatically drives the view to the next control point in the list for assessment. The features within the LiDAR data are manually digitized in order to facilitate comparison to the control points. While manually digitizing the LiDAR features does create some error, careful selection of features with adequate intensity delineation minimizes the magnitude of these errors. Additionally, as mentioned earlier, avoiding the endpoints of features such as stop bars where LiDAR data may not have sufficiently defined the feature, is recommended. The existing FMAS CuePac is an intuitive and user friendly tool capable of comparing control points and LiDAR intensity features. The software provides the necessary statistics to analyze and adjust the data to remove

12 any global bias. The horizontal accuracy can now be easily quantified. Specifying the horizontal as well as the vertical accuracy of LiDAR data should quickly become the standard. REFERENCES De Berg,M., Van Krevald, M., Overmars, M., Schwarzkopf, O., Computational Geometry, Springer-Verlag, Berlin. Bevington, P., Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, New York. Press, W., et al., Numerical Recipes in C, Cambridge University Press, New York.