Matching of 2D Laser Signatures based on Spatial and Spectral Analysis

Transcription

1 Matchng of 2D Laser Sgnatures based on Spatal and Spectral Analyss A. Aboshosha, H. Tamm and A. Zell [aboshosha, tamm, Rechnerarchtektur Abt., WSI, Unverstät Tübngen Sand, D-7276, Tübngen, Germany. Abstract. In ths artcle we present a comparatve study of three approaches for laser sgnature matchng n ndoor terrans. The frst approach analyses the sgnature usng spatal doman analyss, the second one reles on spectral doman analyss, and the thrd approach nvestgates the sgnature wth a spatal-spectral analyss usng wavelets. The man objectve of ths comparson s to explore the compresson and dentfcaton capabltes of these approaches. Furthermore we study the possblty of pattern matchng under some lmted varatons of the dynamc envronments e.g. phase shft and scalng. Generally, matchng and analyss of 2D laser scans underles self localsaton algorthms for moble autonomous robots. I. Introducton Self-localzaton has been referred to as the most fundamental capablty of an autonomous robot. Hence, t s consdered as a key subject n robotcs. Estmaton of a moble robot pose s the process of deducng the locaton relatve to ts envronment from ts sensor data. 2D laser sgnatures are hghly precse compared wth sonar []. Therefore, a varety of localzaton algorthms rely on laser sgnature matchng and analyss to estmate the pose of robots, see fgure (). A crtcal problem n localsaton s the processng of a huge amount of data needed to model the envronment and, consequently, large computatonal effort needed by the robot to match poses. Ths problem s exaggerated by real-tme condtons. To overcome ths problem we employ feature extracton/compresson technques. Then, we apply the comparson on the features rather than on the raw data. Referrng to the relevant research work, Lu and Mlos proposed a soluton for laser scan matchng of ponts and tangents usng least squares [6]. Mota and Rbero employed the maxmum lkelhood algorthm to match 2D laser scans n 3D reconstructed models []. Bengtsson and Baerveldt presented a scan matchng algorthm, IDC-S, Iteratve Dual Correspondence Sector, whch deals wth changes n the envronment by dvdng the scans n several sectors, whch are matched separately [4]. Gutmann and Schlegel [8] combned the approach wth the pont to lne matchng of Cox [7, 9]. Both are teratve methods,.e. they need a relatvely large amount of processng tme. Therefore, they are used as offlne algorthms after all dstance data was already acqured. In the method of Weß et. al., hstograms are used as a base of scan ponts matchng [6]. Röfer (Bremen Autonomous Wheelchar) extended the method of Weß et. al. to get faster matchng []. Concernng the actvtes of the laboratory for autonomous robots at the Unversty of Tübngen, several pattern matchng algorthms have been nvestgated for robot self-localzaton. Feyrer and Zell employed the laser sgnature of human legs to underle pursung persons [4]. Mojaev and Zell [] ncorporated scan ponts nto occupancy maps by whch these local grds were matched to generate a global map. Aboshosha and Zell employed the spectral doman analyss to match laser and geomagnetc sgnatures [3]. Another technque has been ntroduced by Bber and Straßer to match 2D laser scans relyng on the normal dstrbuton transform []. In ths paper we concentrate on the comparson of three approaches. The frst approach s the spatal doman analyss usng Eucldean dstance (ED) and the cross correlaton algorthm (CCA). The second one s the spectral doman analyss usng dscrete cosne transform (DCT). The thrd one s an ntegraton of both domans under the Haar wavelet transform (HWT).

2 The remander part of ths paper s organzed as follows; secton (II) presents the use of spatal doman analyss to match laser sgnatures, n general laser data seres. Secton (III) llustrates applyng spectral doman analyss to extract compressed features of sgnatures. Secton (IV) focuses on the mplementaton of the Haar wavelet transform to compress and compare data seres. Secton (V) comprses the expermental results of pose trackng usng the llustrated algorthms under rotaton and lmted translaton. Secton (VI), demonstrates employng laser sgnatures (laser-mlestones) n mprovng the vector mappng paradgm (VMP). Fnally, secton (VII) s the concluson. II. Spatal Analyss Spatal doman analyss s an effcent method for data seres matchng. Ths method can be mplemented by deducng the Eucldean dfference of two seres: x and y (equ.: ) or the cross correlaton of both of them (equ.: 2). ED( x, y) = ( x ) 2 y () = The correlaton between two seres (cross Fgure. Deducton of robot poses usng laser sgnatures. correlaton) s a standard approach to feature detecton as well as a component of more sophstcated technques. It s well known that cross correlaton can be effcently mplemented n the transform doman, the cross correlaton s preferred for feature matchng applcatons that do not have a smple frequency expresson. Cross correlaton s a standard method for estmatng the degree to whch two seres are correlated. Consder two seres x and y where =,2,...,. The cross correlaton ρ at delay d (sgnature phase shft) s defned as: ρ ( xy, ) d ( x m )( y m ) x d y = = 2 2 ( x mx) ( y d my) = = = cov( x and y) Var( x) Var( y) (2) Where m x and m y are the means of the correspondng seres. If the above s computed for all delays d =,2,..., (phase shft) then t results n a cross correlaton seres of twce the length as the orgnal seres. The correlaton coeffcent s a normalzed measure of the degree of correlaton between two seres, and the normalzaton s such that ρ always les wthn the range < ρ <. In fgure (2) the best match s the global mnmum of the Eucldean dstance, whle the best match as shown n fgure (3) s the global maxmum of the cross correlaton. 2 ED Rotaton (Degree) Correlaton CCA Fgure 2. Eucldean Dstance w.r.t. rotaton. Fgure 3. Correlaton w.r.t. rotaton.

3 III. Spectral Analyss of Sgnatures The DCT transforms a sgnal from a spatal representaton nto a frequency representaton. Lower frequences contrbute more to a sgnal than hgher frequences, so f we transform a sgnature nto ts frequency components and throw away data about hgher frequences we can reduce the amount of data needed to descrbe those sgnatures wthout sacrfcng too much sgnature qualty. The DCT transform can be computed as follows; π (2 j )( ) y =Λ xj cos, =,..., F, = Λ= (3) j= F n where, x s the data seres, at tme t, x, =,...,, F s the o. of DCT coeffcents, y are the DCT coeffcents and s the sgnature length. For all frequences (F=), vares from to, there s no compresson. To compress the sgnature (by omttng hgh frequences), vares from to fl (F=fl), where fl s called the frequency lmter or the number of DCT coeffcents, see fgures (4 and ). The nverse DCT can restore the orgnal sgnature usng a lmted number of DCT coeffcents, see fgures (4, and 6). F π (2 )( j ) x = Λ y cos, =,..., and j =,..., F (4) 2 j j j= Fgure 4. Orgnal sgnature. Fgure. Low DCT coeffcents. Fgure 6. Restored sgnature. Comparng DCT coeffcents vectors nstead of raw sgnatures s faster than the tradtonal matchng technques due to ts compresson capablty. Hence, the elapsed tme requred to match sgnatures wll be reduced. - Scalng property: ω If f () t F( ω ) then f ( at) F, see fgure (7). The value a > compresses the tme axs and a a expands the frequency axs, ω s the frequency, the value a < expands the tme axs and compresses the frequency axs, t s the tme and a, to are constants. Hence, the spectral analyss can counteract the nfluence of scalng, see fgure (7) Fgure 7. Sgnature scalng and ts correspondng DCT coeffcents.

4 2- Phase shft property: j If ( ) ( ω) then ( ) ω to f t F f t to e F( ω). Fgure (8) shows the change of DCT characterstcs due to rotaton that ndcate the capablty of the spectral analyss to tolerate the rotaton for a certan lmt Fgure 8. Sgnature phase shft and ts correspondng DCT coeffcents. Fgures (9 and ) show the robustness of the DCT wth respect to sgnature phase shft and translaton, respectvely. On the left we show how we can fnd the best match and headng of the robot at the global mnmum although the robot s one meter apart from ts home locaton. The rght graph shows the change n match error due to translaton Best suted pose Headng (degree) Dsplacement (m) Fgure 9. DCT matchng error w.r.t. rotaton. Fgure. DCT matchng error due to dsplacement. IV. Haar Wavelet Transform (HWT) Among many wavelet algorthms ncludng Daubeches wavelets, Mexcan Hat wavelets and Morlet wavelets, the Haar Wavelets are especally popular due to ther smplcty and lmted support. The HWT enable applyng our approach n onlne mode on onboard computers of moble robots. We want to have a decomposton that s fast to compute and requres lttle storage for each sequence. The Haar wavelet s chosen for the followng reasons: () t allows good approxmaton wth a subset of coeffcents, (2) t can be computed quckly and easly, requrng lnear tme n the length of the sequence and smple codng, and (3) t preserves Eucldean dstance. The most nterestng dssmlarty between DCT and HWT transforms s that ndvdual wavelet functons are localzed n space, whle DCT functons are not. Ths localzaton feature, along wth wavelets' localzaton of frequency, makes many functons and operators usng wavelets "sparse" when transformed nto the wavelet doman. Ths sparseness, n turn, results n a number of useful applcatons such as data compresson, detectng features n patterns, and removng nose from data seres. One way to see the spatal-spectral resoluton dfferences between the DCT and the HWT transforms s to look at the bass functon coverage of the spatal-spectral doman. The Haar wavelet uses a rectangular wndow to sample the data seres. The frst pass over the tme seres uses a wndow wdth of two. The wndow wdth s doubled at each step untl the wndow encompasses the entre data seres.

5 In order to study the laser sgnature usng HWT, we expose the sgnal to a recursve (mult-resoluton) transform. Each tme, we extract a set of coeffcents, whch deduce the data varaton found n the sgnal at a gven sub-band [2]. Fgures (, 2, 3 and 4) show a laser scan, the correspondng detal and approxmaton coeffcents, and the reconstructed sgnature w.r.t the orgnal one at wavelet decomposton level 3. The sgnal s entaled wth the approxmate coeffcents found wth the hghest sub-band. In order to perform compresson n the wavelet doman, a gven number of the lower subbands s beng neglected when defnng our sgnature Fgure. Orgnal laser sgnature. Fgure 2. Approxmaton coeffcents at wavelet decomposton levels Fgure 3. Reconstructed laser sgnature w.r.t. the orgnal one at wavelet decomposton levels 3. Fgure 4. Detal coeffcents at wavelet decomposton levels 3. V. Expermental Results To compare the presented approaches, we have performed some experments usng a Sck LMS 2 laser scanner mounted on our B2-RWI robot platform (Coln) n the laboratory for autonomous moble robots. The laser sgnature of an arbtrary landmark has been regstered and laser scan samples from four nearby locatons have been stored. The stored samples comprse both translaton and rotaton. Then we apply HWT and DCT compresson on the scans usng equal compresson factor. Fnally we used the selected landmark and searched for t among all samples. To compare the results farly we use mean square error n order to compare the spatal, DCT, and HWT sgnatures wth the correspondng ones. Fgures () shows the expermental results. It s worth mentonng that we use a compresson factor of a hgh rate for DCT and HWT wthout sacrfcng the nature of the sgnature. These experments show that both DCT and HWT are not superor to each other as ther sgnatures are stll capable of embracng the nformaton even wth hgh compresson rates. Fgure () clarfes that both DCT and spatal technques comprse a unque global (reference pose) meanwhle they ncorporated multple local mnma. Therefore, both DCT and HWT are consdered partally nvarant to translaton and rotaton. VI. Laser-Mlestones for VMP Relyng on the precedng analyss, the laser-landmarks can be precsely recognzed n spatal, spectral or wavelet domans. We beneft from ths dea n markng specfc postons (nodes), locatng on the mappng vector, as unque landmarks that label these postons, see fgure (6). The set of these nodes and vectors constructs an accurate reference model of the workspace. Ths model s employed to correct the odometrc errors assocated wth the VMP []. Laser-landmarks, locatng on the mappng

6 vector, are consdered as mlestones that descrbe the metrc straghtforward translaton from the start pont toward the goal. Several mappng algorthms have been nvestgated to model the robot s workng envronment. The most wdely used methods are the occupancy grd, the topologcal graph and maps ntegratng both. Each of these methods has ts characterstcs, advantages and dsadvantages. Throughout ths work the VMP has been used. Ths technque s sutable for robots equpped wth a laser scanner. The VMP s orgnally based on the occupancy grd wth a reduced map sze. The man dea of the vector mappng s to determne an empty space as a regon of cells between the laser scanner (robot s poston) and the detected obstacles. In other words, these empty cells wll be represented by end ponts of the laser rays (vectors) and the poston of the laser scanner. The measurements of the VMP are gven n a polar coordnate system whose orgn s the poston of the scanner (robot), whle the end of the vectors s the obstacle boundares. Ths map reduces both computer power and memory requrements. In ths method, t s requred to record just the start (once per scan) and the end of the vectors so the regon n between s consdered free space that wll preserve the Fgure. Pose trackng from 4 nearby locatons under nfluence of rotaton relyng on CCA, ED, DCT and HWT algorthms. consstency of the maps. The laser scanner sends out 8 laser rays per scan wth angular resoluton. The resoluton of the VMP can be controlled usng ether hardware or software technques. The VMP maps can be easly converted to the tradtonal forms. As example, the occupancy grd can be generated by markng all cells ncluded n the vector regon by null (empty) and the remander cells by one (occuped). Also the topologcal graph and ntegrated form can be estmated through the occupancy grd. Based on vector mappng the contour-graph can be generated. Ths graph ncludes the contours of the exstng obstacles. Fgure (6) descrbes usng laser mlestones to deduce the metrc straghtforward translaton from the start pont. They are denoted from to, whle the frst and the last ponts ( and ) are used to assembly the whole map-segments. VII. Concluson The major contrbutons of ths artcle arse from the formulaton of new matchng approaches, usng the spatal and spectral domans to the modellng and dentfcaton of laser sgnatures, whch provde mproved computatonal effcency n the postonng technques. By manpulatng the manner n whch feature nformaton of laser sgnatures s ncorporated nto the model, t can be shown that sgnfcant mprovements n the performance of the algorthm can be realsed. Moreover, the smplcty and the effcency of dynamc pose trackng technques succeeded to mprove the robot pose estmaton process. The expermental work shows that usng ether DCT or HWT can compress the laser scans up to 9% wthout sacrfcng too much the features of the sgnature. Acknowledgment The frst and the second authors would lke to acknowledge the fnancal support by the German Academc Exchange Servce (DAAD) of ther PhD scholarshps at the Unversty of Tübngen.

7 Fgure 6. Descrpton of metrc translaton usng laser mlestones. References: [] A. Aboshosha and A. Zell, Robust Mappng and Path Plannng for Indoor Robots based on Sensor Integraton of Sonar and a 2D Laser Range Fnder, In: Proc. IES 23 Internatonal Conference, March 4-6, 23, Assut-Luxor, Egypt. [2] A. Aboshosha and A. Zell, An Introducton to Robot Dstrbuted Sensors, Techncal Report, o. WSI-23-3, Faculty of Informatcs, Unversty of Tübngen, Germany, ov. 23. [3] A. Aboshosha and A. Zell, Dsambguatng Robot Postonng Usng Laser and Geomagnetc Sgnatures, In: proceedngs of IAS-8, Amsterdam, ederlands, March, 24. [4] S. Feyrer, A. Zell, Robust Real-Tme Pursut of Persons wth a Moble Robot Usng Multsensor Fuson, 6th Internatonal Conference on Intellgent Autonomous Systems (IAS-6), Vence, Italy, pp. 7-7, 2. [] A. Mojaev, A. Zell, Onlne-Postonskorrektur für moble Roboter durch Korrelaton lokaler Gtterkarten. In: H. Wörn., R. Dllmann, D. Henrch, (Eds.): Autonome Moble Systeme. Informatk aktuell. Sprnger , 998. [6] F. Lu, E. Mlos, Robot Pose Estmaton n Unknown Envronments by Matchng 2D Range Scans. In: Journal of Intellgent and Robotc Systems 8: , 997. [7] I. Cox, Blanche - an Experment n Gudance and avgaton of an Autonomous Robot Vehcle. In: IEEE Transactons on Robotcs and Automaton 7: , 99. [8] J. S. Gutmann and C. Schlegel (996). AMOS: Comparson of Scan Matchng Approaches for Self-Localzaton n Indoor Envronments. In: st Euromcro Workshop on Advanced Moble Robots (Eurobot-96). [9] J. Leonard, H. F. Durrant-Whyte and I. J. Cox, Dynamc Map Buldng for an Autonomous Moble Robot. In: Proc. IEEE Int. Workshop on Intellgent Robots and Systems. 89-9, 99. [] T. Röfer, Usng Hstogram Correlaton to Create Consstent Laser Scan Maps. In: Proceedngs of the IEEE Internatonal Conference on Robotcs Systems (IROS-22). EPFL, Lausanne, Swtzerland , 22. [] P. Bber and W. Straßer, The ormal Dstrbutons Transform: A ew Approach to Laser Scan Matchng, In: Proceedngs of the 23 IEEE/RSJ Intl. Conference on Intellgent Robots and Systems Las Vegas, U.S.A., October 23. [2] Percval and Walden, Wavelet Methods for Tme Seres Analyss, Cambrdge Un. Press, 2. [3] E.J. Keogh and M.J. Pazzan, An Indexng Scheme for Fast Smlarty Search n Large Tme Seres Databases', In: Proc. of Conf. on Scentfc and Statstcal Database Management, 999. [4] O. Bengtsson and A-J. Baerveldt, Localzaton n Changng Envronments by Matchng Laser Range Scans, In: Proc. EURobot 99, Zürch, Schwez, September 999. [] J. G. Mota, M. I. Rbero, Localsaton of a Moble Robot usng a Laser Scanner on Reconstructed 3D Models, In: Proceedngs of the 3rd Portuguese Conference on Automatc Control, COTROLO 98, Combra, Portugal, pp , September 998. [6] G. Weß, C. Wetzler and E. von Puttkamer, Keepng Track of Poston and Orentaton of Movng Indoor Systems by Correlaton of Range-Fnder Scans. In: Proc. Int. Conf. on Intellgent Robots and Systems (IROS-94). Munch, Germany. 9-6, 994.