A MULTI-SCALE CURVE MATCHING TECHNIQUE FOR THE ASSESSMENT OF ROAD ALIGNMENTS USING GPS/INS DATA

Transcription

1 6th International Symposium on Mobile Mapping Tehnology, Presidente Prudente, São Paulo, Brazil, July 1-4, 009 A MULTI-SCALE CURVE MATCHING TECHNIQUE FOR THE ASSESSMENT OF ROAD ALIGNMENTS USING GPS/INS DATA J. Stratakos, V. Gikas *, K. Fragos NTUA, Shool of Rural and Surveying Engineering, National Tehnial University of Athens, 9 H. Polytehniou St., Zografos, Athens, 15780, Greee vgikas@entral.ntua.gr KEY WORDS: mobile mapping, road geometry, alignment, multi sale, urve mathing, GNSS/INS ABSTRACT: In this paper a methodology and an algorithm is presented for the automated extration of road alignments using GPS/INS navigational data. Centerline geometry assessment is derived in the form of traditional design elements. The ore of the proposed algorithm relies on the use of multi-sale orrelation analysis (similarity and affinity transformations) and fundamental urve mathing tehniques that were suitably adapted to adhere the nature of road alignment data. A mobile mapping vehile omprising a high auray multi-sensor navigational system was used to verify the orretness and level of automation of the algorithm as well as the feasibility of the assoiated software. 1. INTRODUCTION Reent advanes in land-based mobile mapping systems have resulted an enormous improvement in apturing high quality / low ost field data that an serve for the aurate extration of road geometry features. Suh systems mostly rely on vehile navigation (GNSS / INS) and optial sensor (image sequene / laser profiler) data that an be proessed to aquire road axis, lane line and ross setion geometry information [Toth and Grejner Brzezniska, 00; Wang et al, 007; Wang et al 008; Zhiyong et al 007]. Previous researh on road axis geometry extration mainly onentrates on semi-automated systems that model enterline geometry in the form of a polyline or a line of ontinuous urvature (e.g. splines) [Cefalo et al,007; Makanae, 004]. However, as for the road engineers, enterline geometry is muh more useful if presented in terms of traditional design elements, some researhers foused on algorithms to serve this need [Can and Kusu, 008; Choi and Sung, 007; Gikas and Daskalakis, 008; Karamanou, 009; Karamanou et al, 009]. These algorithms, notwithstanding an generally produe satisfatory results, they suffer lak of automation and in ertain ases (e.g. noisy data, peuliar geometry) they annot represent reliably the atual road geometry harateristis. The ore idea behind the existing algorithms relies on azimuth and urvature alulations of the enterline alignment. However, this approah an introdue exessive noise in the omputed parameters due to suessive derivation operations. This problem ould be partially mitigated if a low-pass filter is applied at every derivation step [Stratakos et al, 009]; howbeit, it an still affet the results as useful information is smeared out due to data filtering - espeially, in the lothoid setions. Obviously, in order to overome this problem an alternative methodology is required that would not depend on urvature omputations. The algorithm proposed in this paper is based on the use of a fundamental urve mathing method, known as Diret Curve Mathing (DCM) tehnique [Kohlmayr, 006; Wolfspon, 1990]. The method originates in the fields of applied mathematis, omputational geometry and omputer siene. Its priniple relies on RST (rotation, saling, translation) omputations between two sets of point data so that, the differene in the shape between the urves representing the two data sets omply with ertain mathing (minimization) riteria. In this study, the first data set orresponds to a subset of the vehile navigation data redued on the enterline alignment, whereas the seond one orresponds to the referene data of a geometri element (i.e. a irle ar, a lothoid or straight line) of predefined harateristis. In order to produe referene data for a roadway design element its geometri properties should be taken into aount. Thus, the similarity transformation is used to generate a luster of referene ar irles and lothoids from an initial pair of urves. In ontrast, other types of transition urves, suh as ubi parabolas, deem neessary the use of affinity transformation. In fat, the similarity and affinity transformations are sale transformations in one and two dimensions respetively. In the first ase, the initial and final urves have the same sale ratios in X and Y diretions, whereas in the affinity transformation apply different sales [Kohn, 1984]. The oneptual meaning of sale in these transformations as well as in the proedure of referene data preparation is very important throughout this work. Seeing that referene urves of different sales are used for the appliation of the DCM tehnique, the proposed methodology resides in the field of multi-sale analysis. Moreover, this analysis may not onsidered as a trivial one sine it ensembles multi-sale transformations with both uniform and non-uniform aspet ratios.. MULTI-SCALE CORRELATION As already stated, saling plays a fundamental role in the generation of a referene urve and by extension to the implementation of the DCM tehnique. The similarity transformation of a Eulidean spae is defined as a funtion f * Corresponding author

2 from the D spae into itself that multiplies all distanes by the same sale fator r, so that for any set of points a and b it applies: d( f( a), f( b)) = r d( a, b) where, d(a,b) is the Eulidean distane between points a and b. Hene, in the ase of a irle element an initial ar irle is generated and subsequently ar irles of different radii are produed by applying the similarity transformation on the prototype feature. That is, for an ar irle with enter oordinates (x,y ) and radius R formulated by: ( x x) + ( y y) = R a similarity transformation of a sale fator r will transform its point oordinates (x,y) to (x,y ), where x and y are defined by equations: x = r x Substituting x and y oordinates in Eq. it results: In fat, Eq. 6 provides the parameters (urvature radius and enter oordinates) of the transformed irle for a sale fator r. Reversely, an ar irle of radius R an be generated from a base irle of radius R by multiplying its point oordinates by the sale fator r given by: (1) () (3) y = r y (4) x' y' ( x ) ( y ) R r + r = ( ) ( r x x') + ( r y y') = r R = R' (5) (6) well. The ubi parabola is suh a ase, ommonly used in railway engineering. This transition urve is defined by: y = K x 3 (11) where, K relates to the urve length and the radius of the adjaent ar irle. In this ase, as y oordinates relate to x oordinates the affinity transformation applies, so that: x = rx x y = ry y (1) (13) In order to simplify omputations, by setting rx=1 and substituting Eq. 1 and 13 to Eq. 11 we get: y' ( r K) x' K' x' 3 3 = y = Κ ry =, rx = 1 Κ (14) (15) Similarly, the multi-sale orrelation priniple an be applied on other types of transition urves not disussed here (e.g. the sinusoidal, the sine half-wavelength diminishing straight urve, the Bloss transition, the ubi transition and the bi-quadrati transition urve) [Bloss et al, 1979; Meek and Walton 1989; Meek and Walton, 199]. One a prototype of eah urve type is stored in a library file it beomes easy to alter its nominal design parameters by issuing the appropriate sale transformation with the minimum omputational effort. R r = (7) R Conlusively, by generating a pool of referene irles it is possible to determine whih of them would fit best the sample data. The same working priniple applies for the ase of lothoid transitions. The parametri equations of a lothoid of a parameter A are given by [Bass, 1984]: t π u x() t = A os( ) du 0 π u yt () = A sin( ) du From Eq. 8 and 9 it is evident that the lothoid parameter A is simply a onstant fator of du. Therefore, using Eq. 3 and 4 a similarity transformation will transform (x,y) oordinates to (x,y ) so that, the sale ratio between the initial and the transformed lothoids is: Figure 1 depits a set of ar irles and a set of lothoid urves of various urvature radii and lothoid parameters generated using a similarity transformation. In road / highway design, in addition to lothoids other transition urves an be used as (8) t (9) 0 A' r = (10) A Figure 1. A set of referene irles and lothoid transitions of varying design parameters produed using multisale orrelation analysis 3. THE DIRECT CURVE MATCHING ALGORITHM One a pool of referene urves has generated, urve mathing between sample (observed) and referene data is aomplished using the DCM tehnique. The riterion for the seletion of the best mathing referene urve is based upon the residual values obtained from omparing the sample data against a multiple of referene urves. In fat, the referene urve with minimum RMS value is said to math best the sample data, and onsequently its parameter value is assigned to the sample data set. In mathematial terms, the RMS value of the residuals between the referene y-oordinates YR and the sample y- oordinates Y S, Er is given by: 1 Er = Y Y n ( Ri Si) (16) n i = 1

3 Also, the best fitting referene urve to the sample data exhibits a minimum RMS value given by: Er( r y ) = min (17) Note that in Eq.17 only the r y sale fator appears. This is beause r x is set either equal to r y or equal to unity depending on the type of transformation used. Eq. 17 an be solved for r y using well known minimization algorithms from numerial analysis. The speifi method applied in this study is desribed as follows: Firstly, the Er parameter is alulated from fitting a straight line to the sample data. This error value is named Ers. As a straight line orresponds to an infinitely large radius the orresponding urvature value (Ps) equals zero. Then, the Er value is omputed for the minimum parameter value allowed for the feature type examined based on road design riteria. Suh design riteria limits refer to the minimum ar radius or transition parameter and is named P. The resulting Er value is named Er. Consequently, the algorithm enters a loop in whih the Ers and Er values are alulated repeatedly using urvature values omputed from the following equations: + P Ers Ps = 0.75 Ers + Er + P Ers P = 1.5 Ers + Er (18) (19) At every iteration step the error values Ers and Er redue mutually. The proedure stops when the differene between Ps and P values is under a preset low limit or when the last five results of Ers and Er do not differ in their integer parts. In ases where the enterline positions suffer from an inreased level of noise this aperture limit may be loosened aordingly. This algorithm onverges fast and provides results of an adequate preision even in the ase of exessive noise in the Figure. Multi-sale diret urve mathing sequene position data. The proportional fators in Eq.18 and 19 are used to adjust the rate of onvergene, that will have a positive impat on algorithm stability. More speifially, notwithstanding values 0.75 and 1.5 used in Eq. 18 and 19 were hosen experimentally, they were proved to produe statistially best results. As an example, Figure depits the operating priniple of the algorithm in the ase of an ar irle. In this plot, for pratial reasons, the ar urvature is displaed on the x-axis instead of its radius value. The algorithm starts with the alulation of Er and Ers errors for the maximum urvature value allowed (lowest irle radius permitted by the design standards) and a straight line (zero urvature). Then, new estimates of Ps and P values are omputed using Eq. 18 and 19 and aordingly a pair of sale fators are obtained using Eq. 7 to produe referene irles of a different radius. Consequently, the updated referene ar irles are orrelated with the sample data and a new pair of Er and Ers values is omputed using Eq. 16. The proess is repeated until Ps and P values oinide and onverge to a minimum Er value. 4. THE CENTRELINE MODELING ALGORITHM In this study, the proposed enterline geometry extration algorithm integrates multi-sale orrelation analysis and the DCM tehnique. As a first step, the navigation data obtained from the mobile mapping system in forward and reverse diretions are proessed to produe the enterline alignment in the form of individual data points. Various algorithms exist for this purpose; for example see Gikas and Daskalakis (008). Then, the resultant data file is sampled so that a small portion of the data is loaded for proessing at a time. This is ahieved using predefined sample length settings. Data proessing is performed sequentially, in a number of omputational steps shown in the flowhart diagram of Figure 3. In this proess is assumed that sample length it remains onstant over time. In fat, a running window of onstant width is defined so that at every proessing step the last point in the rolling window is replaed with a new point at the front. In order to failitate omparison between the sample and referene data, eah time the sample data set is updated, a rotation and an offset operation is applied to transform sample data from their initial datum to the oordinate system of the referene urve. A subroutine is also used to identify the turn diretion (left of right wise) of the urve for later use. At this stage of the proess andidate referene urves are produed using prototype urves of predefined design parameters (typially, R=500 m and A=100 m). One both sample and referene data sets have been prepared, the multi-sale orrelation and the DCM algorithm are being applied in the data. More speifially, this omputational stage implements three lones of the same mathing tehnique. The first one returns the radius and the minimum Er value that orrespond to the best mathing ar irle in the sample data. The seond and third lones return the lothoid parameter and aompanying Er value for a lothoid-tangent and a tangentlothoid sequenes respetively. In onlusion, the algorithm returns three Er error values. These orrespond to the best mathing straight line, ar irle, and lothoid transition. Finally, the sample data are assigned to the design element type with minimum Er error value.

4 algorithm. In addition, the road setion was surveyed using onventional geodeti tehniques of a higher auray. Therefore, a reliability analysis between modeled and true geometry would be possible to aomplish. Figure 5. Computed Er (RMS) errors for the straight line, irle and lothoid model for the entire alignment. 5. Assessment of the Results and Disussion Figure 3. Flowhart of the proposed algorithm The algorithm is exeuted repeatedly so that an element type is identified for every subset of data that orresponds to the urrent loation of the running window. Finally, the parameter value of a enterline geometry element is omputed from averaging all individual parameter values of the same element type obtained from subsequent loop operations. Notably, for every sample data update the whole orrelation set is shifted by a single trak point. This attribute is onsidered important as it failitates to loate preisely the touh points (harateristi points) between tangent and spiral segments. Figure 4. Centreline alignment of the study region. It onsists of three sets of spiral-irle-spiral with straight lines in between 5. IMPLEMENTATION AND TESTING OF THE METHOD Data proessing assumed a rolling window of a 0-point length size, whih orresponds to approximately 1 m travel distane. Similarly, the length and point density of the referene data sets were adjusted aordingly. Figure 5 shows the Er (RMS) error values omputed for every design element in the alignment. In this plot in order to maintain the lower part of the diagram visible a logarithmi sale is used. As expeted, from this diagram it is evident that the straight line model math best all linear segments in the alignment. In these loations, the lothoid model results in higher error values, whereas the ar irle fit leads to extremely high error values. Clearly, in the linear segment region, neither ar irles nor lothoids an provide a better mathing due to onstraints applied in both the ar radius and the transition parameter adopted during referene urve generation (R max =5000 m, A max =500 m). Also, the moderate error values observed for the lothoid model (ompared to the irle one) are due to the varying urvature values of spiral urves. Similar onlusions are drawn from modeling the lothoid transition setions. Here, the lothoid model results in minimum Er values indiating the best mathing between observed and referene data sets. Small disrepanies between the rising and the falling edges in the lothoid-ar-lothoid region they possibly reveal small variations in the vehile speed resulting in a variable point density between the sample and the referene data sets. Finally, as shown in the same diagram, the ar irle regions are best approximated by the irle model. 5.1 Mobile Mapping Vehile and Data Desription To verify the orretness and the feasibility of the proposed algorithm software has been written to integrate the omplete proedure disussed in Setion 4.0. The software was written in the Pasal objet oriented programming language. In this paper the results of proessing a road alignment of nearly 1300 m long are presented (Figure 4). This setion features a multitude of road design elements of varying design parameters. Field data were aquired using a mobile mapping vehile bearing a high preision GNSS/INS/DMI navigation system, whereas raw data were proessed through a Kalman filter Figure 6. Enlargement of the lower part of Figure 5

5 Also, the algorithm was built to proess small subsets of navigational data in a sequential manner, and therefore, an be easily modified to operate in near real time mode. Finally, due to its simpliity, the algorithm is fast enough to ope with high frequeny kinemati data. A onsideration for further improvement of the method inludes the use of a non-onstant sample / referene urve length that is expeted to fully exploit the spatial harateristis of navigation data. Figure 7. Parameter value versus hainage omputed for road alignment examined in this study Figure 6 presents an enlargement of the lower part of Figure 5. The most important thing to note from this diagram is that the minimum value of the mathing error Er, is small and remains under 0.03 m. Obviously, suh small error values primarily ought to the high quality navigation data. Also, it should be pointed out that every Er value relates to a limited data volume (i.e., as muh as it orresponds to the sample window size); and therefore, Er errors are omputed against a very short travel path. Figure 7 shows summary results obtained from proessing the enterline data examined in this study. More speifially, this histogram presents parameter value against hainage, while the type of the design element is denoted by olor. As Built Measured % Differenes Type Length R / A Length R / A Length R / A Line C lothoid Curve Clothoid Line C lothoid Curve Clothoid Line C lothoid Curve Clothoid Line Table 1. Comparison of omputed versus as-built design parameters In order to examine the behavior of the algorithm in terms of reliability, a ross-omparison between the omputed against as-built alignment is attempted. Table 1 ontains summary results for the entire road setion. As it an be learly seen in this table all differenes observed stay well under 10% - representing an exeptionally good argument that the model is orret. Also, from the same plot it is evident that the differenes in design element length present higher values ompared to element parameter values. This observation relates diretly to the size of the sample window seleted; however, they are still small. 6. CONCLUSIONS In this paper, a new methodology is presented that deals with the problem of enterline geometry extration from navigation data in terms of traditional design elements. In ontrast with previous methods, the proposed algorithm relies on multi-sale orrelation analysis and urve mathing tehniques; hene, is simple to apply as well as it proved to be omputationally effiient. Provided that the algorithm is fed with high quality navigation data, then it an operated in an automati manner. REFERENCES Baass, K.G., The use of Clothoid templates in highway design, Transportation Forum, 1, p Bloss, W.L., Sham, L.J., Vinter, V., Interation-Indued Transition at Low Densities in Silion Inversion Layer, Phys. Rev. Lett., 43, pp Can, E. and Kusu, S., 008. Investigate of Transition Curves with Lateral Change of Aeleration for Highways Horizontal Alignment, FIG Working Week, Stoholm, Sweeden, June Cefalo, R., Grandi, G., Roberti, R., and Sluga, T., 007. Extration of Road Geometri Parameters from High Resolution Images Validated by Geodeti Tehniques, 4th Int. SIIV Congress, Palermo, Italy, Sept Zhiyong Chen, Qingwu Hu, Sheng Guo, Jianfeng Yuan 007. Appliation of L-MMS in Railroad Clearane Detetion, 5 th Int. Symposium on Mobile Mapping Tehnology, Padova, Italy. Choi, S. and Sung, J., 007. Data Generalization Algorithm for the Extration of Road Horizontal Alignment Design Using the GPS/INS Data in Letures in Computer Siene, Advanes in Hybrid Information Tehnology, Springer Berlin / Heidelberg, Vol. 4413, pp51-6. Gikas, V. and Daskalakis, S., 008. Determining Rail Trak Axis Geometry Using Satellite and Terrestrial Geodeti Data, Survey Review, 40, pp Karamanou, A., 009. Combination of Satellite and Inertial Geodeti Positioning Methods in Digital Survey of Road Nets, Phd Thesis in preparation, Shool of Rural and Surveying Engineering, National Tehnial University of Athens. Karamanou, A., Papazisi, K, Paradissis, D., Psarianos, B., 009. Preise Estimation of Road Horizontal and Vertial Geometri Features Using Mobile Mapping Tehniques, 6 th Int. Symposium on Mobile Mapping Tehnology, Sao Paulo, Brazil. Kohn, W., A Similarity Transformation for Numerial Integration of Pieewise-linear Seond-order Differential Equations Without Damping, SIMULATION, 43, 4, pp Kohlmayr, G.F., 006. Implementation of Diret Curve Mathing Methods for Transient Matrix Heat Transfer Testing, Journal of Applied Sientifi Researh, 4, 1, pp Makanae, K., 004. Highway Sequene Editor based on the Length-based Highway Produt Model. International Conferene on Computing in Civil and Building Engineering, Weimar, Bauhaus-Universität, ICCCBE.

6 Meek, D.S. and Walton, D.J., 199. Clothoid Spline Transition Spirals, Mathematis of Computation, 59(199), pp Meek, D.S. and Walton, D.J., The use of Cornu Spirals in Drawing Planar Curves of Controlled Curvature, Journal of Computational and Applied Mathematis, 5, pp Stratakos, J., Gikas, V., Fragos, K A Rigorous Algorithm for Automati Centerline Geometry Extration Using Multi-Sensor Navigational Data, 9 th Optial 3D Measurement Tehniques Conferene, Vienna, July 1-3. Toth, C.K., Grejner Brzezinska, D.A., 00. Near Real Time Road Centerline Extration, ISPRS. Photogrammet. Comput. Vision, Graz, Austria, pp Wang, Ch., Hassan, T., Lavigne, M., El-Sheimy, N., 007. ARVEE: Automati Road Geometry Extration System for Mobile Mapping, 5 th Int. Symposium on Mobile Mapping Tehnology, Padova, Italy. Wang, C., Hassan, T., El-Sheimy, N., Lavigne, M., 008. Automati Road Vetor Extration for Mobile Mapping Systems, The Int. Arhives of the Photogrammetry, Remote Sensing and Spatial Information Sienes, Vol. XXXVII. Part B3b,Beijing, China, pp Wolfson, H.J., On Curve Mathing, IEEE Transations on Pattern Analysis and Mahine Intelligene, 1, pp