GLOBAL REGISTRATION OF NON STATIC 3D LIDAR POINT CLOUDS: SVD FACTORISATION AND ROBUST GPA METHODS

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1 GLOBAL REGISRAION OF NON SAIC 3D LIDAR POIN CLOUDS: SVD FACORISAION AND ROBUS GPA EHODS Fabo Croslla*, Alberto Benat Unversty of Udne, Dp d Georsorse e errtoro, Va Cotonfco 4, I-33 UDINE (Italy) fabo.croslla@unud.t, alberto.benat@unud.t KEY WORD: Regstraton, Non statc confguraton, LDAR, SVD factorsaton, Robust generalsed Procrustes analyss ABSRAC: he paper reports two analytcal methods capable to relably perform the smultaneous global regstraton of non statc 3D LDAR pont clouds, and nvestgates ther applcablty by analysng the results of some prelmnary numercal examples. he frst method, proposed by Xao (5), and Xao et al. (6), apply a drect SVD factorsaton to non statc 3D fully overlappng pont clouds charactersed by target ponts. he factorsaton s appled to a matrx, sequentally contanng by rows the coordnates of the correspondng targets present n the cloud scenes. Besdes the rgd transformaton parameters, a number of shape bases s determned for each pont cloud, whose lnear combnaton descrbes the dynamc component of the scenes. A lnear closed-form soluton s fnally obtaned, enforcng lnear constrants on orthonormalty of the rgd rotatons and on unqueness of the lnear bases. he second method analysed s the so called Robust Generalsed Procrustes Analyss, recently proposed by the authors. o overcome the lac of robustness of Generalsed Procrustes Analyss, a progressve sequence nspred to the forward search was developed. Startng from an ntal partal pont cloud confguraton satsfyng the LS prncple, the confguraton s updated, pont by pont, tll a sgnfcant varaton of the regstraton parameters occur. hs reveals the presence of non statonary ponts among the new elements ust nserted, that are therefore not ncluded n the regstraton process. Both methods are capable to correctly determne the regstraton parameters, when compared to the commonly appled two steps method, where the regstraton of deformable shapes s based by non - rgd deformaton components.. INRODUCION In some papers publshed a few years ago (e.g. Benat and Croslla, ), the authors proposed the Generalsed Procrustes Analyss to perform a hgh precson smultaneous regstraton of multple partally overlappng 3D pont clouds acqured wth terrestral laser scannng devces. he proposed technque requres for each pont cloud the matchng of a suffcent number of artfcal targets, eventually pre-sgnalsed on the obect surface to survey. Furthermore, the same authors have recently proposed (Benat, Croslla, Sepc, 6) an automatc regstraton technque that does not requre any manual matchng of the target ponts, but that nstead uses the morphologcal or the radometrc local varatons on the surveyed surface. he method, by studyng the dfferental propertes of the sampled pont surface, computes at frst the local values of the Gaussan curvature, then apples a topologcal research to defne for each pont cloud the correspondng zones charactersed by the same curvature values. By applyng an SVD algorthm, t s possble to automatcally solve a coarse regstraton followed by an Iteratve Closest Pont (ICP) global refnement. Both regstraton approaches can be correctly appled f the obect does not change ts shape durng the survey of the complete sequence of pont clouds. hat s, the regstraton problem conssts n the defnton of the correct smlarty transformaton parameters for each pont cloud. On the other hand, regstraton and modellng of dynamc pont cloud scenes s a promnent problem for robot navgaton, for reconstructon of deformable obects, and for montorng envronmental phenomena. he recovery of the resultng shapes can be regarded as a combnaton of rgd smlarty transformatons of the 3D pont clouds and unnown non - rgd deformatons. In the lterature (e.g. Dryden and arda, 999), the problems soluton s usually carred out n two consecutve steps. he frst step regsters the pont clouds by smlarty transformaton, consderng the deformable shapes as contamnated by Gaussan nose. he second step determnes the lnear deformable model of the regstered shapes by applyng Prncpal Component Analyss (PCA) to the regstraton resduals. Proceedng n ths way, the regstraton of deformable shapes s based by non - rgd deformaton components. It s therefore necessary to apply some procedures that mae possble to relably estmate the roto-translaton components, and the deformable shapes. he paper synthetcally descrbes two methods recently proposed n the lterature, and analyses the results obtaned for the regstraton of a 3D scene charactersed by statc and dynamc elements. he frst method, ntroduced by Xao (5), solves the combned problem of regstraton and dynamc shape modellng by a drect factorsaton of the ponts coordnate matrx, contanng by rows for each acqured scene the 3D sampled model pont coordnates. he method wors well when the dynamc obect shape can be descrbed by a lnear combnaton of a small number of shape bases, that, together wth the smlarty transformaton parameters for each cloud, are the unnown elements of the ont regstraton and shape modellng problem. he second method proposed (Croslla, Benat; 6) represents a robust soluton of the Generalsed Procrustes problem. he descrbed algorthm derves from the Robust Regresson Analyss based on the Iteratve Forward Search approach proposed by Atnson and Ran (), and Cerol and Ran. he procedure starts from a partal pont confguraton only contanng statonary ponts. At each teraton, the transformaton parameters are determned, and the ntal dataset s enlarged by one or more new ponts, tll a sgnfcant varaton of the transformaton parameters occur. At ths pont the method allows to dentfy n the varous confguratons the remanng non statonary ponts that represent the dynamc component of the scene.

2 . JOIN REGISRAION AND SHAPE ODELING BY SVD FACORISAION he method proposed by Xao (5) and Xao et al. (6) s based on the fact that the shape S of a deformng obect, or of a non statc scene at epoch ( = N), can be modelled as a lnear combnaton of shape bases B (= K). Each bass s a (D P) matrx, where D s the space coordnate dmenson and P s the number of the ponts. Accordng to these postons, we can wrte that: K S = l B = where l s a coeffcent to apply to the B bass n order to defne the shape S at epoch. Of course, every shape S can be measured from a dfferent pont of vew, (eventually) wth a dfferent scale, and the coordnates of each shape model may be defned wth respect to a dfferent coordnate system. herefore, the measured shape W can be consdered as a smlarty transformaton of the shape S, that s number of shape bases requred to descrbe the shape varaton model. In fact, f ran(w)=dk, than K=ran(W)/D. Furthermore, the SVD of W allows to obtan at frst a (DN DK) matrx %, and a (DK P) matrx B %. atrces and B, contanng the unnown terms of the problem reported n Equaton, can be determned by applyng a further unnown lnear correctve transformaton (DK DK) matrx G to the matrces % and B %, that s = G % n order to satsfy: % % W = GG B= B B= G B % (4) atrx G can be parttoned nto the followng K sub-matrces of sze (DK D): G = (G G K ) W R S t () = c + ' where c s a non zero scalar, R s a (D D) rotaton matrx, t s a translaton vector and s a unt vector. Combnng formula and (), matrx W can be expressed as ( µ L µ )( L ) W = R R t B B Now, consderng all the N measurement epochs, a (DN P) matrx W can be defned. Of course matrx W can be consdered as a result of the followng matrx expresson W = B+ t Where s a (DN DK) matrx Sub-matrces G (=...K) satsfy the followng property: % µ R G % = = G % N µ NRN Now, let Q = G G be a (DK DK) matrx. hen, from Eq. (5) t s possble to consder the followng general condton: Q % % = µ µ RR (, = N) (6) that has to be consdered n order to satsfy two fundamental constrants:. orthonormalty of the rotatons. unqueness of the shape bases. he frst constrant s satsfed by the followng condton: (5) µ R L µ R = µ N N µ R L NRN B s a (DK P) bass matrx B= ( B B ) (DN ) translaton vector ( N ) t L t. L and t s a he ont regstraton and shape modellng problem proposed by Xao et al. (6), consders at ths pont a drect factorzaton of matrx W. Before dong so, all the pont coordnates are translated nto a barycentral system, so to neglect the translaton components. From now on, let W be the coordnate matrx, where the coordnates of each sampled pont cloud are referred to the correspondng centrod. Next step s to proceed to the Sngular Value Decomposton (SVD) of W, so to obtan a factorzaton that can be wrtten as W = B %%. he ran of W s mn(dk, DN, P). Snce generally DN>DK and P>DK, the SVD of W maes possble to determne K, that s the Q % % ' = µ I (=..N) (7) ( dxd) As an example for D =3, snce Q s symmetrc, and due to the presence of the unnown term µ, for each submatrx % (=...N), condton (7) generates the followng system of lnear equatons m% Q m% ' m% Q m% ' = () () m% Q m% ' m% Q m% ' = () ' = (8) () () where m%, m%, (D DK) sub-matrx m% are the frst, second and thrd row of the %. Enforcng the orthonormalty constrants alone, s not enough n the case n whch a deformaton of the pont clouds occurs. It s

3 therefore necessary to enforce also the second constrants that guarantee the unqueness of the bases. he problem can be solved analysng the ndependence propertes of the measured shape random samples. hat s, t s necessary to determne K measured shapes that contan ndependent deformable shapes. hs can be done by measurng the condton number for all the possble permutaton sets of (DK P) sub-matrces of W, and by choosng the set that mnmzes that value (Xao et al., 6). Smaller condton number means hgher ndependence. he deformable shapes contaned n the selected K measurement shapes are consdered as the unque bases. Snce scalng does not nfluence the ndependence of the shapes, the scalars µ are absorbed nto the bases, and then the chosen K measurements are smply the rotated bases. Denotng the K selected bass measurements as the frst K measurements n the barycentral coordnate matrx W, t follows that W = RB (=...K). he correspondng coeffcents are thus: µ = (= K) (9) µ = (, = K ; ) Accordng to Equatons (7), and (9) the unqueness of the bases s satsfed by the followng condtons Q % % ' = ( =.. ( dxd K ; =..N; ) (a) ) Q % % ' = I (==) (b) ( dxd) As n the prevous example for D=3, for each matrx product reported n (a), we can wrte the followng set of lnear equatons ' = () ' = () () () ' = () Whle, for each matrx product reported n (b) t follows the followng equatons ' = () ' = () () () ' = () ' = Systems (8), (), and () enlarged for all possble ndexes and mae possble to fnd an nconsstent system of lnear equatons n the unnown terms of the symmetrc matrx Q upper trangle that can be solved by least squares. Once Q s determned, to compute G, t s necessary to apply an SVD to matrx Q, snce Q = GG. hs decomposton allows to determne matrx G apart for an arbtrary (D D) orthonormal transformaton F, snce GFFG = Q. hs ambguty s due to the fact that matrces G (=...K) are ndependently estmated under dfferent coordnate systems (Xao et al., 6). herefore matrces G (=...K) have to be transformed under a unque reference system. Before dong so, t s necessary to determne for each the rotaton matrces R relatng to each scene. Rememberng that G % = µ R (=...N), snce R s orthonormal,.e. R =, than R =± G G. In ths way K sets of rotaton matrces R (=...N) are computed. Specfyng one of the sets as the reference one, an Ordnary Procrustes Analyss (OPA) s appled to all the other sets so to algn them to the selected one. he result furnshed by OPA maes also possble to transform G (= K) under a common coordnate system, and n ths way the searched transformaton matrx G s acheved. he coeffcents are then computed by (5), and the shape bases B are recovered by (4). In ths way the shape of a non statc scene at epoch can be fnally determned by. 3. ROBUS GENERALISED PROCRUSES ANALYSIS Generalsed Procrustes Analyss (GPA) s a well nown multvarate technque used to provde multple and smultaneous L.S. smlarty transformatons of data sets composed of P correspondng D-dm ponts, whose coordnates are referred to dfferent reference frames, and charactersed by measurement nose. he followng least squares obectve functon has to be satsfed: S= tr ( c + ) ( c + ) X R t X R t < ( c + ) ( c + ) X R t X R t = mn under the orthogonalty condton R R = I; where X X are data matrces of sze (P D), each one contanng the coordnates of the same set of P correspondng ponts defned n dfferent reference frames; s the (P ) auxlary untary vector; t, R and c are the unnowns (= ),.e. the (D ) th translaton vector, the (D D) th rotaton matrx, and the th sotropc scale factor, respectvely. he soluton of Equaton represents the GPA problem descrbed by Krstof and Wngersy (97), Gower (975), ten Berge (977), and Goodall (99). hs problem has an alternatve formulaton. Sad p X = c X R + t, the followng measures: p p p p p p = tr < < X X X X X X (4) p p p X H = tr X H X H (5)

4 are perfectly equvalent (e.g. Borg and Groenen, 997), where H s the unnown centrod. herefore Eq. (5), nstead of Eq. (4), can be mnmsed so to determne the unnowns {c, R, t} (= ) that mae t possble to teratvely compute the fnal p X (= ). p atrx Hˆ = X represents the LS estmate of H. Note that = H+ E = X, where vec ( E ): N, Σ =σ ( Q Q ) and σ p { } n has a factored structure. In the current algorthm mplementaton of the Robust Generalsed Procrustes problem soluton, the procedure starts from a partal pont confguraton contanng only statonary data. At each teraton, the ntal dataset s enlarged by one or more ponts, tll a sgnfcant varaton of the transformaton parameters occurs. In order to defne the ntal confguraton subset X of X,.e. the one contanng statonary data, t s necessary to compute the LS estmate of the correspondng centrod H, and consequently determne the smlarty transformaton parameters for all the = data sub-matrces X : H X (6) ˆ = c R + t = where H ˆ corresponds to the LS estmate of the unnown H. hs procedure s repeated for every = P possble S confguraton subset X, where S s the number of ponts formng the subset. Now, the global pseudo-centrod s computed by applyng the transformaton parameters, relatve to the -th data submatrx X, to the full correspondng X, obtanng X P() : P() ( c ) = + = = = H % X R t X (7) o defne the ntal subset X contanng statonary ponts, the least medan of squares (LS) prncple s appled (Rousseauw, 984). As well nown, ths regresson method can normally reach a brea down pont as hgh as 5%: among all the possble confguraton subsets X, the one satsfyng the followng LS condton s chosen as the ntal one: P () P () ( ) = meddag X H % X H % = mn (8) hs ntal subset s then enlarged onng up the pont for whch: P () P () ( ) = dag X H % X H % = mn (9) selected from the remanng (P S) ponts of the confguraton, not belongng to the ntal subset. he LS estmate of the enlarged partal centrod H (+), and the S-transformaton parameters for the sub-matrces X (+), are computed agan as: H X X () ( + ) ( + ) ( + ) ( + ) ( + ),P ˆ + = c R + t = = = Now, Procrustes statstcs (Sbson, 979; Langron and Collns, 985) s appled to verfy whether a sgnfcant varaton of the S-transformaton parameters occurs by enlargng the orgnal selected data subset. o ths am, the total dstance between the partal centrod H ˆ,P ( + ) and the sub-matrces X, obtaned by applyng to the orgnal X the S-transformaton parameters relatng to the (+) dataset, s computed:,p [ ( + ) ],P[ ˆ + ] tr ˆ G = ( X H ) ( X H ) () = he followng dstances are also computed:,p [ ( + ) ],P[ ˆ + ] tr ˆ Gt = ( X + dt H ) ( X + dt H ) (a) =,P [ ( + ) ],P[ ˆ + ] tr ˆ GtR = ( X dr + dt H ) ( X dr + dt H ) = (b),p [ ],P tr c ˆ c G = d X + dr + dt H d X + dr + dt Hˆ [ ] trc = (c) after havng taen care of the fact that the translaton components relatng to the (+) subset must be prevously reduced by the dfference between the centrods of H ˆ ( +) and H ˆ. hese dstances are resdual dstances after a Procrustes transformaton. In partcular G t s the resdual dstance after a translaton, G tr s the resdual dstance after a translaton and a rotaton, and G trc s the resdual dstance after a translaton, a rotaton, and a scalng. Assumng a proper frst nd error α, and the proper degrees of freedom df and df, the reecton of the null hypothess for the followng tests (Langron and Collns 985): G G G G G G ; ; GtRc GtRc GtRc t t tr tr trc > F α, df, df ndcates a sgnfcant varaton of some or of all the transformaton parameters at ths step, due to the possble enterng nto the X (+) datasets of non statonary data. If the null hypothess for all the tests s accepted nstead, the teratve process contnues wth the nserton of a further new pont X (+), satsfyng Equaton 9 wthn the remanng ones of the dataset. 4. ALGORIH IPLEENAION AND ESING he SVD factorsaton, and the Robust GPA methods were mplemented n atlab, n order to test ther capablty to correctly regster models by usng both statc and non-statc tepont confguratons. he experments, related to smulated envronments, let us to ntroduce varably modulated measurement nose n the te-pont coordnates.

5 enttes (buldngs, roofs, walls, roads), and 4 relate to movng obects (cars). Statc te-ponts are evdenced by red damond symbols, non statc ones by blue crcles. Fgure : Global regstraton by Robust GPA Fgure shows the result of the global algnment of the four models of Fgure, performed by Robust GPA. he method dentfes all the non-statc te-ponts, and treats them as outlers: the global regstraton s then acheved by way of the largest statc te-pont subset, common to all the models Fgure 3: e ponts dstrbuton after Robust GPA regstraton: statc ones (red damonds) appear precsely overlapped. Fgure 3 shows the te-pont dstrbuton after the regstraton: non-statonary ponts are automatcally detected, and outlned by blue crcle symbols, whle statc ones are mared by overlappng red damonds. 4 3 Fgure a,b,c,d: Smulated models of an urban envronment, and statc and non-statc te-ponts for model regstraton. As example, we report one of these tests. Fgures a to d depct four models (or scenes) of one reconstructed urban envronment, n dfferent reference systems (or poses). Of the te-ponts employed for the model algnment, 8 dentfy statc Fgure 4: e ponts dstrbuton after Ordnary GPA regstraton: statc te-ponts are not overlapped, and the hgh value of the resduals represents a dstorted reconstructon

6 Fgure 4 represents, for comparson, the results obtaned performng the regstraton by an ordnary GPA: non-statc teponts heavly contamnate the regstraton accuracy by a quantty proportonal to ther number, dsplacement length and relatve poston (leverage effect). Several experments were performed varyng te-ponts locaton, number, and accuracy. A detaled analyss of the numercal experments wll be presented n a future wor. 5. CONCLUDING REARKS Our nvestgatons concernng the regstraton methods for non statc models are stll under development, nevertheless some clear consderatons regardng the methods dscussed here can be expressed. As commonly nown, ordnary GPA, although very effcent, may fal n achevng an acceptable regstraton accuracy due to the presence of outlers or non-statonary data. On the contrary, the SVD factorsaton method (Xao et al., 6) reported n the paper, does not exclude, but s capable to employ the non-statonary ponts for a correct regstraton process. oreover t furnshes the geometrc bases to reconstruct the deformable shapes. But ths method, although robust aganst measurement nose, ntroduces a restrctve operatve condton: the shape deformatons, n whole, must span all the model space dmensons. As mentoned n Secton, the shape of a deformable obect can be regarded as a lnear combnaton of a selected number of shape bases. When at least three ponts smultaneously move along three dfferent fxed drectons n the 3D space, ther traectores form a deformaton bass of ran 3. If two ponts move along fxed drectons wthn a D plane, ther traectores form a ran- shape bass. If fnally one pont moves along a fxed drecton, ts traectory forms a ran- bass. Non-degenerate bases of a 3D non rgd shape are characterzed by a full ran 3 and, accordng to what reported n Secton, a closed form soluton exsts enforcng lnear rotaton, and bass constrants. Degenerate deformatons often occur n practce,.e. some bases are of ran or. Relatng to the reported example, cars movng ndependently on a straght plane road refer to ran- deformaton of the scene. Cars movng along two dfferently orented straght plane roads refer to a ran- deformaton of the scene. Fnally, cars movng on two dfferently orented straght and slope roads refer to ran-3 non-degenerate deformaton of the scene. he soluton of degenerate deformatons could requre further and computatonally heavy constrants, or may not exst (Xao and Kanade, 4). Robust GPA overcomes the drawbacs due to nsuffcent ran deformatons provdng a correct model regstraton. If the number of non-statc te-ponts s less than the LS breadown lmt of 5%, and the number of the statc te-ponts s at least equal to the model space dmensons, Robust GPA can represent a vald complement, or a valuable alternatve, to SVD factorsaton for the deformable shape regstraton, and for the relatve non-statonary components detecton. REFERENCES LDAR close range mages, Internatonal Archves of Photogrammetry, Remote Sensng and Spatal Informaton Scences, 5-7 september 6, Dresden, Germany. Borg, I., Groenen, P.J.F., 997. odern multdmensonal scalng: heory and applcatons. New Yor, Sprnger. Cerol, A., Ran,., 3. Robust ethods for the Analyss of Spatally Autocorrelated Data, Statstcal ethods & Applcatons,, pp Croslla, F.; Benat, A. 6. A forward search method for robust generalsed Procrustes analyss, Data Analyss, Classfcaton and the Forward Search, S. Zan, A. Cerol,. Ran and. Vch (Eds), Sprnger-Verlag, pp Dryden, I.L.; arda, K.V Statstcal Shape Analyss, J. Wley, Chchester, pp Goodall, C., 99. Procrustes methods n the statstcal analyss of shape, Journal Royal Stat. Soc.. Part B 53,, pp Gower, J. C., 975. Generalzed Procrustes analyss, Psychometra, 4, pp Langron, S. P.; Collns, A. J.; 985. Perturbaton theory for Generalzed Procrustes Analyss. Journal Royal Statstcal Socety, 47(), pp Krstof, W.; Wngersy, B., 97. Generalzaton of the orthogonal Procrustes rotaton procedure to more than two matrces. Proc. of the 79-th Annual Conv. of the Amercan Psychologcal Ass., 6, pp Rousseeuw, P. J., 984. Least edan of Squares Regresson, J. of the Amercan Statstcal Assocaton, 79(388), pp Sbson, R., 979. Studes n the Robustness of ultdmensonal Scalng: Perturbatonal Analyss of Classcal Scalng, Journ. R. Stats. Soc., B, 4, pp ten Berge, J.. F., 977. Orthogonal Procrustes rotaton for two or more matrces. Psychometra, 4(), pp Xao, J., 5. Reconstructon, Regstraton and odelng of Deformable Obect Shapes, PhD thess, ech. eport CU-RI- R-5-, Robotcs Insttute, Carnege ellon Un., Pttsburgh Xao, J.; Georgescu, B.; Zhou, X.; Comancu, D.; Kanade. 6. Smultaneous Regstraton and odelng of Deformable Shapes, 6 IEEE Computer Socety Conference on Computer Vson and Pattern Recognton,, pp Xao, J.; Kanade,., 4. Non-Rgd Shape and oton Recovery: Degenerate Deformatons. IEEE Internatonal Conference on Computer Vson and Pattern Recognton (CVPR), pp ACKNOWLEDGENS hs wor was carred out wthn the research actvtes supported by the INERREG IIIA Italy-Slovena 3-6 proect "Cadastral map updatng and regonal techncal map ntegraton for the Geographcal Informaton Systems of the regonal agences by testng advanced and nnovatve survey technques" Atnson, A.C., Ran.,. Robust Dagnostc Regresson Analyss, Sprnger, N.Y. Benat, A., Croslla, F.,. Generalzed Procrustes Analyss for sze and shape 3D obect reconstructon. Gruen, Kahmen (Eds), Optcal 3D easurement echnques, pp , Venna, Austra. Benat, A., Croslla, F., Sepc, F., 6. Automatc morphologcal pre-algnment and global hybrd regstraton of