GENERALIZED PROCRUSTES ANALYSIS AND ITS APPLICATIONS IN PHOTOGRAMMETRY


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1 SWISS FEDERL INSIUE OF ECHNOLOGY Insttute of Geodes and Photograetr EHHoenggerberg, Zuerch GENERLIZED PROCRUSES NLYSIS ND IS PPLICIONS IN PHOOGRMMERY Prepared for: Praktku n Photograetre, Fernerkundung und GIS Presented to: Prof. rn W. GRUEN Prepared b: M. Devr KC June,
2 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca LE OF CONENS. INRODUCION. PROCRUSES NLYSIS: HEORY ND LGORIHMS 4.. Who s Procrustes? 4.. Orthogonal Procrustes nalss 4.. Etended Orthogonal Procrustes nalss (EOP) 6.4. Weghted Etended Orthogonal Procrustes nalss (WEOP) 9.5. Generaled Orthogonal Procrustes nalss (GP).6. heoretcal Precson for GP 5. PPLICIONS IN PHOOGRMMERY 6.. Eaple 7.. Eaple 8.. Eaple 9.4. Coparson of the two ethods 4. CONCLUSIONS REFERENCES
3 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca. INRODUCION Soe easureent sstes and ethods can produce drectl D coordnates of the relevant obect wth respect to a local coordnate sste. Dependng on the etenson and shape coplet of the obect, t a requre two or ore vewponts n order to cover the obect copletel. hese dfferent local coordnate sstes ust be cobned nto a coon sste. hs geoetrc transforaton process s known as regstraton. he fundaental proble of the regstraton process s estaton of the transforaton paraeters. In the contet of tradtonal leastsquares adustent, the lnearsaton and ntal approatons of the unknowns n the case of or ore densonal slart transforatons are needed due to nonlneart of the functonal odel. Procrustes analss theor s a set of atheatcal leastsquares tools to drectl estate and perfor sultaneous slart transforatons aong the odel pont coordnates atrces up to ther aal agreeent. It avods the defnton and soluton of the classcal noral equaton sstes. No pror nforaton s requested for the geoetrcal relatonshp estng aong the dfferent odel obects coponents. ths approach, the transforaton paraeters are coputed n a drect and effcent wa based on a selected set of correspondng pont coordnates (enat and Croslla, ). he ethod was eplaned and naed as Orthogonal Procrustes proble b Schoeneann (966) who s a scentst n the Quanttatve Pscholog area. In ths publcaton, Schoeneann gave the drect leastsquares soluton of the proble that s to transfor a gven atr nto a gven atr b an orthogonal transforaton atr n such a wa to ne the su of squares of the resdual atr E. he frst generalaton to the Schoeneann (966) orthogonal Procrustes proble was gven b Schoeneann and Carroll (97) when a least squares ethod for fttng a gven atr to another gven atr under choce of an unknown rotaton, an unknown translaton and an unknown scale factor was presented. hs ethod s often dentfed n statstcs and pschoetr as Etended Orthogonal Procrustes proble. fter Schoeneann (966), slar ethods were proposed n coputer vson and robotcs area (run et al., 987, and Horn et al., 988). he soluton of the Generaled Orthogonal Procrustes proble to a set of ore than two atrces was reported (Gower, 975, en erge, 977). Further generalaton n the stochastc odel s called Weghted Procrustes nalss, whch can be dfferent weghtng across coluns (Lsst et al., 976) or across rows (Koschat and Swane, 99) of a atr confguraton. n approach that can dfferentl weght the hoologous ponts coordnates was gven (Goodall, 99). ethod that can take nto account the stochastc propertes of the coordnate aes was gven b enat and Croslla (). Ipleentaton detals and two dfferent applcatons of Procrustes nalss n Geodetc Scences were gven b Croslla and enat (, ): photograetrc block adustent b ndependent odels, and regstraton of laser scanner pont clouds. he reader can also fnd a detaled surve of the Procrustes analss and ts soe possble applcatons n the Geodetc Scences n (Croslla, 999). he report s organed as follows. In the second secton, the atheatcal background and the algorthc aspects of the Procrustes analss s gven. In the thrd secton, two dfferent
4 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca applcatons of the Procrustes analss n photograetr are presented. hs secton also copares the Procrustes nalss and the conventonal LeastSquares soluton wth respect to accurac, coputatonal cost, and operator handlng. Dscusson and concluson are gven n the fourth secton.. PROCRUSES NLYSIS: HEORY ND LGORIHMS.. Who s Procrustes? he nae of the ethod coes fro Greek Mtholog (Fgure ). Procrustes, or "one who stretches," (also known as Prokrustes or Daastes) was a robber n the th of heseus. He preed on travelers along the road to thens. He offered hs vcts hosptalt on a agcal bed that would ft an guest. He then ether stretched the guests or cut off ther lbs to ake the ft perfectl nto the bed. heseus, travellng to thens to cla hs nhertance, encountered the thef. he hero cut off the evldoer's head to ake h ft nto the bed n whch an "guests" had ded (Greek Mtholog Reference). Fgure : Procrustes n Greek Mtholog (Procrustes ccoodates ob aat).. Orthogonal Procrustes nalss Orthogonal Procrustes proble (Schoeneann, 966) s the least squares soluton of the proble that s the transforaton of a gven atr nto a gven atr b an orthogonal transforaton atr n such a wa to ne the su of squares of the resdual atr E . Matrces and are (p k) densonal, n whch contan p correspondng ponts n the kdensonal space. Least squares soluton ust satsf the followng condton tr { E E} tr{ ( ) ( ) } n () he proble also has another condton, whch s the orthogonal transforaton atr, I () oth of the condtons can be cobned n a Lagrangean functon, { E E} + tr{ L( I) } F tr () 4
5 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca {( ) ( ) } + tr L ( I) { } { + } + tr{ L ( I) } F tr (4) F tr (5) where L s a atr of Lagrangean ultplers, and tr{ } stands for trace of the atr. he dervaton of ths functon wth respect to unknown atr ust be set to ero. F + L ( + L ) (6) where ( ) and (L+L ) are setrc atrces. Let us ultpl equaton (6) on the left sde b, L + L + (7) ( L + L ) ( ) ( ) ( ) L + L (8) Snce ( ) s setrc, ( ) ust also be setrc. Rend that (L+L ) s also setrc. herefore, the followng condton ust be satsfed. ( ) ( ) (9) Multplng Equaton (9) on the left sde b, ( ) ( ) and on the rght sde b () ( ) ( ) () Fnall, we have the followng equaton usng Equatons () and (), ( )( ) ( ) ( ) () Matrces [( )( ) ] and [( ) ( )] are setrc. oth of the have sae egenvalues. svd ( )( ) svd ( ) ( ) () where svd{ } stands for Sngular Value Decoposton, nael EckartYoung Decoposton. he result s, VD s V WDsW (4) 5
6 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca hs eans that, V W (5) Fnall, we can solve the unknown orthogonal transforaton atr. V W (6).. Etended Orthogonal Procrustes nalss (EOP) he frst generalaton to the Schoeneann (966) orthogonal Procrustes proble was gven b Schoeneann and Carroll (97) when a least squares ethod for fttng a gven atr to another gven atr under choce of an unknown rotaton, an unknown translaton t, and an unknown scale factor c was presented. hs ethod s often dentfed n statstcs and pschoetr as Etended Orthogonal Procrustes proble. he functonal odel s the followng E c + t (7) where [... ] s ( p) unt vector, atrces and are (p k) correspondng pont atrces as entoned before, s (k k) orthogonal rotaton atr, t s (k ) translaton vector, and c s scale factor. In order to obtan the least squares estaton of the unknowns (, t, c) let us wrte the Lagrangean functon where { E E} + tr{ L( I) } F tr (8) ( c + t ) ( c + t ) + tr{ L ( I) } F tr (9) { E E} tr{ } + c tr{ } + p t t c tr{ } tr{ t } c tr{ t } tr + and p s a scalar, nael nuber of rows of the data atrces. he dervatons of the Lagrangean functon wth respect to unknowns ust be set to ero n order to obtan a least squares estaton, F c c + c t + L ( + L ) () F p t t F c tr c + c { } tr{ } + tr{ t } () () he translaton vector fro Equaton () s that 6
7 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca 7 ( ) p c t () In Equaton (), ( ) and (L+L ) are setrc atrces. Let us ultpl Equaton () on the left sde b ( ) L L t c c c (4) ( ) ( ) ( ) L L c t c c L L + + (5) Snce ( ) s setrc, [ c t ] ust also be setrc. Rend that (L+L ) s also setrc. s. t (6) ccordng to Equaton (), Equaton (6) can be wrtten as ( ). s c p (7) s. p c p + (8) Snce p s setrc, the rest of the equaton ust also be setrc, s. p (9) s. p () s. p I () Let us sa, p I S () where atr S s (k k) densonal. In order to satsf Equaton (), the followng condton ust be satsfed. Note that transpose of the atr s equal to tself, f the atr s setrc.
8 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca S S () Multplng Equaton () on the left sde b, S S (4) and on the rght sde b S S (5) Fnall, we have the followng equaton usng Equatons (4) and (5), SS S S (6) Matrces [SS ] and [S S] are setrc. oth of the have sae egenvalues. svd { SS } svd{ S S} (7) where svd{ } stands for Sngular Value Decoposton, nael EckartYoung Decoposton. he result s, VD s V WDsW (8) where atrces V and W are orthonoral egenvector atrces, and D s s the dagonal egenvalue atr. ccordng to Equaton (8), V W (9) Fnall, we can solve the unknown orthogonal transforaton atr. VW (4) In the calculaton phase, one should take nto account the followng equaton. ccordng to (Schoeneann and Carroll, 97) svd p { S } svd I VDW D Ds, (4) In order to solve the scale factor c, let us substtute Equaton () n Equaton () c tr I p tr I p (4) Fnall, translaton vector t can be solved fro Equaton () ( c) p t (4) 8
9 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca.4. Weghted Etended Orthogonal Procrustes nalss (WEOP) WEOP can drectl calculate the leastsquares estaton of the slart transforaton paraeters between two dfferentl weghted odel pont atrces. hs a s acheved when the followng condtons are satsfed (Goodall, 99). tr ( c + t ) W ( c + t ) W n P K (44) I (orthogonalt condton) (45) where atrces and are (p k) odel pont atrces, whch contan the coordnates of p ponts n R k space. Matrces W P (p p) and W K (k k) are optonal weghtng atrces of the p ponts and k coponents, respectvel. Model ponts atr s transfored nto bestft of the odel ponts atr, b the unknown transforaton paraeters, nael orthogonal rotaton atr (k k), translaton vector (k ) t, and scale factor c. he vector s (p ) unt vector. t the frst attept, let us assue that W K I, and let us rearrange Equaton (44) n order to obtan a slar epresson as n Equaton (9). For the sake of ths a, atr W P can be decoposed nto lower and upper trangle atrces b Cholesk Decoposton. So that Fnall, WP Q Q (Cholesk Decoposton) (46) tr tr ( c + t ) Q ( c + t ) I n Q (47) {( c Q + t Q Q )( cq + Q t Q) } n tr (48) ( c + Q t Q) ( cq + Q t Q) n Q (49) substtutng w Q., w Q., and w Q. tr ( c + t ) ( c + t ) n w w w w w w (5) Equaton (5) s the sae epresson as n Equaton (9). herefore, ths proble can be solved b the sae ethod as n Etended Orthogonal Procrustes (EOP) analss. Perforng the Sngular Value Decoposton of atr product: svd w w w I w VDW w (5) w 9
10 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca where V and W are orthonoral egenvector atrces, and D s the dagonal egenvalue atr. Note that the denson of svd{ } part s (k k). he unknowns can be found as entoned before. V W (5) w w w w c tr w I w tr w I w (5) w w w w t ( w c w) (54) w w w n teratve soluton ethod for the case of WK I was gven b Koschat and Swane (99). lso, a drect soluton ethod that can take nto account the stochastc propertes of the coordnate aes n the case of Generaled Orthogonal Procrustes nalss (GP) was gven b enat and Croslla (). hs ethod wll be eplaned n the followng secton..5. Generaled Orthogonal Procrustes nalss (GP) Generaled Procrustes nalss s a wellknown technque that provdes leastsquares correspondence of ore than two odel ponts atrces (Gower, 975, en erge, 977, Goodall, 99, Drden and Marda, 998, org and Groenen, 997). It satsfes the followng least squares obectve functon: tr + [( c + t ) ( c + t )] [( c + t ) ( c + t )] n (55) where,,, are odel ponts atrces, whch contan the sae set of p ponts n k densonal dfferent coordnate sstes. ccordng to Goodall (99), there s a atr Z, also naed consensus atr, n whch contans the true coordnates of the p ponts defned n a ean and coon coordnate sste (Fgure ). he soluton of the proble can be thought as the search of the unknown optal atr Z. Z + E ˆ c + t,,, (56) vec { } ( E ) ~ N Σ σ ( Q Q ), (57) P K where E s the rando error atr n noral dstrbuton, Σ s the covarance atr, Q P s the cofactor atr of the p ponts, Q K s the cofactor atr of the k coordnates of each pont, stands for the Kronecker product, and σ s the varance factor. Let Σ σ I Least squares estaton of unknown transforaton paraeters, c, and t (,,,) ust satsf the followng obectve functon, as entoned before n Equaton (55),
11 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca c t Â c t Â Z Â c t Fgure : GP concept (Croslla and enat, ) + ˆ ˆ n (58) Let us defne a atr C that s geoetrcal centrod of the transfored atrces, as follows: C ˆ (59) he followng two obectve functons ( ˆ ˆ ) ( ˆ ˆ ) ˆ ˆ tr + + 6) ( ˆ C) ( ˆ C) ˆ C tr (6) are equvalent (Krstof and Wngersk, 97, org and Groenen, 997). herefore, Generaled Orthogonal Procrustes proble can also be solved nng Equaton (6) nstead of Equaton (6). Fro a coputatonal pont of vew, ths soluton ethod s spler than the other one. Note that both of the solutons are teratve and equvalent, but two dfferent was. In the followng, onl the soluton ethod that poses the nu condton n Equaton (6) wll be epresses n detaled. he other soluton that poses the nu condton n Equaton (6) was proposed b Gower (975), and proved b en erge (977).
12 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca he soluton of the GP proble can be acheved usng the followng nu condton ( ˆ C) ( ˆ C) n ˆ C tr (6) n a teratve coputaton schee of centrod C n a such a wa: Intale: Defne the ntal centrod C Iterate: Drect soluton of slart transforaton paraeters of each odel ponts atr wth respect to the centrod C b eans of Weghted Etended Orthogonal Procrustes (WEOP) soluton fter the calculaton of each atr Â s carred out, teratve updatng of the centrod C accordng to Equaton (59) Untl: Global convergence,.e. stablaton of the centrod C he fnal soluton for the centrod C shows the fnal coordnates of p ponts n the aal agreeent wth respect to least squares obectve functon. Unknown slart transforaton paraeters (, c, and t ) can also be deterned b eans of WEOP calculaton of each odel ponts atr to the centrod C. he centrod C corresponds the least squares estaton Ẑ of the true value Z. he proof of ths defnton was gven b Croslla and enat (). C Zˆ ˆ (6) In the followng parts of ths secton, dfferent optonal weghtng strateges wll be epressed. For further detals and proof of the stateents, author refers Croslla and enat (), enat and Croslla (). Case : Let us consder the followng case, { }, Q I ( ) ~ N Σ σ ( Q Q ) vec E (64), P K K where QP I, but dagonal,.e. each row of Â has dfferent dsperson wth respect to the true value Z. ut Q P reans constant when varng,,,. In ths case, centrod C s sae as n Equaton (59) or (6).
13 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca Case : Let us treat a ore general schee, ( ) ( ) { } I Q, Q Q Σ E σ K K P, N ~ vec (65) where I Q P, but dagonal,.e. each row of Â has dfferent dsperson wth respect to the true value Z and the dsperson vares for each odel ponts atr,,,. In ths case, the centrod C s defned as follow: P ˆ Q P, P P C (66) lso n ths case, centrod C corresponds to the classcal least squares estaton Ẑ of the true value Z. Note that the posed least squares obectve functon s ( ) ( ) n ˆ ˆ tr Z P Z (67) Case : In real applcatons (for eaple block adustent b ndependent odels), all of the p ponts could not be vsble n all of the odel ponts atrces,,,. In order to handle the ssng pont case, Coandeur (99) proposed a ethod based on assocaton to ever atr a dagonal bnar (p p) atr M, n whch the dagonal eleents are or, accordng to estence or absence of the pont n the th odel (Fgure ). hs soluton can be consdered as ero weghts for the ssng ponts M M M Fgure : Incoplete (p ) odel ponts atrces and resultng M (p p) oolean dagonal atrces (adapted fro enat and Croslla, ).
14 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca Least squares obectve functon and centrod C are as follows n the ssng pont case: or where or tr [( c + t ) C] M [( c + t ) C] n (68) ( ˆ C) M ( ˆ C) n tr (69) ( + ) C M M c t (7) C M MÂ (7) In order to obtan a ore general schee, one should consder the cobned weghted/ssng pont soluton. he weght atr P and the bnar atr M can be cobned n a product atr, as follow: P D M P P M, P Q (7) Note that D s also dagonal. In ths case, the correspondng least squares obectve functon wll be tr [( c + t ) C] P M [( c + t ) C] n where the centrod C becoes (7) ( + ) C PM PM c t (74) Case 4: he eplaned stochastc approaches up to ths secton deal dfferent weghtng strateges aong the odel ponts, not aong the coordnate coponents. In order to account for the dfferent accurac of the tepont coordnate coponents, enat and Croslla () proposed an ansotropc error condton. 4
15 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca { }, Q I and Q I ( E ) ~ N, Σ σ ( Q Q ) vec (75) P K P K where Q P and Q K are dagonal cofactor atrces. hen, weght atrces and P P Q (76) K K Q (77) he product atr for the weghted/ssng pont soluton s sae as the prevous defnton, D M P P M (78) where M s the bnar (oolean) atr. he correspondng least squares obectve functon wll be ( ˆ C) D ( ˆ C) K n tr (79) where the centrod C s vec( C ) ( ) K D K Dvec ˆ (8) where centrod C corresponds to the classcal least squares estaton Ẑ of the true value Z. Note that [ K D ] and vec( ˆ ) atrces are (kp kp) and (kp ) densonal, respectvel. For further detals and the proof of the defnton, author refers enat and Croslla ()..6. heoretcal Precson for GP Croslla and enat () gave the forulaton of the a posteror covarance atr of the coordnates of each pont as follow: S [ ] ( ˆ ) [ ] ( ˆ ) k k C C k [ k ] (8) where k s the nuber of densons, s the nuber of estence of the th pont n the ˆ C s the kdensonal row vector for the th pont n the all odels, and atr ( ) [ k ] th odel ponts atr. he offdagonal eleents of the S atr show the algebrac correlaton aong the coordnate aes for the th pont, not phscal correlaton. 5
16 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca. PPLICIONS IN PHOOGRMMERY s entoned before n Secton (.5), the frst step of Generaled Orthogonal Procrustes nalss (GP) s defnton of the ntal centrod C. One should defne one of the odels as fed, and sequentall lnk the others b eans of WEOP algorth. Instead of sequentall regsterng pars of sngle odels, enat and Croslla () proposed the orentaton of each odel wth respect to the topologcal unon of all the prevousl orented odels. hs process s shown scheatcall n Fgure (4). 4 [ ] fed ~ [ ] ~ ~ [ ] etc. Fgure 4: Intal regstraton (adapted fro enat and Croslla, ) he approated shape of the whole obect obtaned n ths wa provdes an ntal value for the centrod C. If the proble nclude the datu defnton, e.g. n the case of block adustent b ndependent odels, a fnal WEOP s also needed to transfor the whole obect nto the datu usng ground control ponts. In fact Generaled Orthogonal Procrustes nalss (GP) s a free soluton, snce the consensus atr Z s n an orentatonpostonscale n the kdensonal space. In other words, t does not nvolve the datuconstrants, e.g. ground control ponts. One of the ost possble photograetrc applcatons of the GP s block adustent b ndependent odels, whch needs datu defnton. n adaptaton of GP ethod nto block adustent b ndependent odels proble was gven b Croslla and enat (). t each teraton, all odels Â of the block, one at a te, are rotated, translated and scaled to locall ft the teporar centrod C b usng the WEOP and the coon te ponts estng between Â and C. he centrod s coputed fro two sets of te and control pont coordnates together, all n the ground coordnate sste. he control ponts, possbl wth dfferent weghts, pla the role of constrants n the centrod coputaton. he produce the sae effect as pseudoobservaton equatons of the control pont coordnates n the conventonal soluton of the block adustent. Durng the adustent, the centrod s not constant, but changes at each teraton because the te pont coordnates are constantl recoputed and updated, whle the control pont coordnates are kept fed. s soon as a odel Â s rotated, translated and scaled and ts new coordnates stored, these changes are edatel appled and the centrod confguraton s updated. he process ends when the centrod confguraton varatons between two subsequent teratons are saller than a predefned threshold. hs event eans that the least squares ft aong the odels has been obtaned (Croslla and enat, ). 6
17 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca In the followng parts of ths secton, dfferent eaples wll be gven n order to copare the Procrustes ethod wth the conventonal leastsquares adustent. ll of the eaples were perfored on a PC that has the followng specfcatons: Wndows Professonal OS, Intel Pentu III 45 Mh CPU, and 8 M RM... Eaple t the frst attept, conventonal leastsquares adustent for slart transforaton and WEOP were copared accordng to ther coputatonal epense. he proble s the leastsquares estaton of the slart transforaton paraeters between two odel pont atrces, as entoned before n Secton (.4). snthetc odel ponts atr, n whch are ponts n densonal space, and ts transfored counterpart was generated. ddtonall, the coordnate values of the atr were dsturbed b the rando error e that s n the followng dstrbuton, { µ, σ 5 } e ~ N ± he coputaton tes were gven n able (). s entoned before, Weghted Etended Orthogonal Procrustes (WEOP) soluton s a drect soluton as opposed to the conventonal leastsquares soluton. he ntal approatons of the unknowns were calculated usng a closedfor soluton proposed b Dewtt (996), snce the functonal odel of the conventonal leastsquares soluton for ths proble s not lnear. (8) Iteratons Coputaton te (sec.) Leastsquares adustent.9 WEOP . able : Conventonal leastsquares soluton versus WEOP. Of course, sae results for the unknown transforaton paraeters, c, t were obtaned n both solutons, n spte of two dfferent soluton was. In WEOP soluton, the core of the coputaton s Sngular Value Decoposton of the (k k) atr, n ths eaple t s ( ). he used soluton strateg n conventonal leastsquares soluton s wellknown ethod,.e. noral atr parttonng b eans of groups of the unknown, Cholesk decoposton, and backsubsttuton. Note that the coordnates of the control ponts were treated as stochastc quanttes wth proper weghts. v v L C t + l ; P (8) I l L C ; P L C where t and are unknown vectors of absolute orentaton paraeters and obect space coordnates, respectvel. herefore, the denson of the noral equatons atr n ths eaple was [ (7 + pk) (7 + pk) ], nael (7 7). 7
18 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca.. Eaple In the second eaple, a real data set, whch conssted 5 odel ponts atrces obtaned fro a closerange laser scanner devce, was used. he data set ncludes totall te ponts n the densonal space, also n unt of eter. he epected a pror precson of the coordnate observatons s σ ± along the coordnate aes. he a s to cobne all odels nto a coon coordnate sste n order to obtan the whole obect boundar. wo dfferent ethods were eploed n order to acheve the soluton; block adustent b ndependent odels as conventonal leastsquares soluton, and/versus Generaled Orthogonal Procrustes ethod (GP). able () shows the result. lock adustent b ndependent odels Generaled Orthogonal Procrustes (GP) Iteratons Coputaton σ (.) tes (sec.) able : lock adustent b ndependent odels versus Generaled Orthogonal Procrustes (GP). In able () σ value of GP ethod was calculated accordng to the devatons of the transfored coordnates fro the fnal centrod C. In block adustent b ndependent odels ethod, sae soluton strateg entoned n Secton (.) was eploed. hree of the te ponts were nvolved as control ponts n order to defne the datu usng the sae functonal odel n Equaton (8). In contrar, Generaled Orthogonal Procrustes (GP) soluton s copletel free soluton. In other word, t does not nvolve an obect space constrant. hs crcustance s also the reason of slght dfference between the two σ values. One of the ost portant advantage of the GP ethod aganst to block adustent b ndependent odels ethod s ts drastcall less eor requreent. Requred basc eor ses for ths eaple are gven n the followng part. Note that the varables are double precson, e.g. 8 btes. For block adustent b ndependent odels ethod: For N : (u u ) 5. (7 7) 45 varables For N : ( u) (p k) (5.7) (. ) 5 varables For N : p k. varables otall : 6 tes For Generaled Orthogonal Procrustes (GP) ethod: For unknowns of each odel : u varables For centrod C : p k. varables otall : 5 tes 8
19 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca where N, N, and N are the parttoned subparts of the noral equatons atr, s nuber of the odels, p s nuber of ponts, k s nuber of densons, and u s nuber of unknown transforaton paraeters for a odel... Eaple In the last eaple, a snthetc data set, whch conssted 9 odel ponts atrces, s used. he data set ncludes totall te ponts, n whch of the are control ponts, n the  densonal space. he data set was slghtl dsturbed b the followng rando error e: {, σ ±. } e ~ N µ (84) untless able () shows the calculaton nforaton of the two ethods,.e. block adustent b ndependent odels as conventonal leastsquares soluton, and/versus Generaled Orthogonal Procrustes ethod (GP). In both ethods, control ponts were eploed as datudefntons. In GP ethod, the control ponts were treated as n the ethod, whch adapts the GP ethod to block adustent b ndependent odels (Croslla and enat, ), as epressed n Secton (). Iteratons Coputaton tes (sec.) σx (untless) σy (untless) σz (untless) lock adustent b ndependent odels Generaled Orthogonal Procrustes (GP) able : lock adustent b ndependent odels versus Generaled Orthogonal Procrustes (GP). s entoned before n Secton (.), the ost coputatonall epensve part of the Procrustes ethod s Sngular Value Decoposton of the (k k) atr, n ths eaple t s ( ). he used soluton strateg n conventonal leastsquares soluton s wellknown ethod,.e. noral atr parttonng b eans of groups of the unknown, Cholesk decoposton, and backsubsttuton. Note that the coordnates of the control ponts were treated as stochastc quanttes wth proper weghts. In the case of datudefnton, ver slow convergence behavor of the Generaled Orthogonal Procrustes (GP) ethod copared to conventonal block adustent soluton can be shown fro able (). 9
20 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca.4. Coparson of the two ethods he Procrustes analss s a lnear leastsquares soluton to copute the slart transforaton paraeters aong the ( ) odel ponts atrces n k densonal space. Snce ts functonal odel s lnear, t does not need ntal approatons for the unknown slart transforaton paraeters. ut n the case of conventonal leastsquares adustent, the lnearsaton and ntal approatons of the unknowns n the case of or ore densonal slart transforatons are needed due to nonlneart of the functonal odel. In the lterature, there are an closedfor solutons to calculate the ntal approatons for the unknown slart transforaton paraeters (hopson, 959, Schut 96, Oswal, alasubraanan, 968, Dewtt, 996). he Procrustes analss does not has a restrcton on the nuber of k densons n the space of the data set. Its generc and fleble functonal odel can easl handle the k ( k > ) densonal slart transforaton probles wthout an arrangeent on the atheatcal odel. In photograetr area, we are ver falar to k ( k,) densonal slart transforatons. In the case of k > densonal slart transforaton probles, the functonal odel of the conventonal least squares adustent ust be etended/rearranged accordng to the nuber of densons of the data set. he Generaled Orthogonal Procrustes analss (GP) s a free soluton, n other words, t does not nvolve the control nforaton to defne the datu, ecept the adaptaton to block adustent b ndependent odels (Croslla and enat, ). hs confguraton can also be acheved n the conventonal leastsquares adustent b eans of nner constrants, or soetes referred as free net adustent. For the te beng, no work has been reported on the ost general stochastc odel, nael estence of correlaton aong the all easureents, for Procrustes analss. Fro the atheatcal pont of vew, conventonal least squares adustent has ver powerful atheatcal (functonal + stochastc) odel, whch can handle an phscall real stuatons, e.g. unknowns as stochastc quanttes, constrants aong the easureents and aong the unknowns, correlated easureents, etc In the Procrustes analss, the ost coputatonall epensve part of the calculaton s Sngular Value Decoposton of the (k k) atr, where k s the nuber of densons of the data set. ut t s relatvel slow convergence behavor akes ts coputaton speed equal wth copared to conventonal leastsquares adustent. Fro the software pleentaton pont of vew, the Procrustes analss needs drastcall less eor requreent than the conventonal leastsquares adustent, as eplaned b a sple eaple n Secton (..). he Procrustes ethod does not has an relablt crteron n order to detect and locale the blunders, although ths feature s vtal for the real applcatons, n whch easureents ght nclude the blunders. he conventonal leastsquares adustent has an powerful tools n order to locale and elnate the blunders, e.g. DataSnoopng and Robust ethods.
21 Generaled Procrustes nalss and ts pplcatons n Photograetr Devr kca 4. CONCLUSIONS he Procrustes analss s a leastsquares ethod to estate the unknown slart transforaton paraeters aong two or ore than two odel ponts atrces up to ther aal agreeent. ecause the estaton odel s lnear, t does not requre the ntal approatons of the unknowns. In geodetc scences, we are ver falar to solve the k, densonal slart transforatons b eans of conventonal leastsquares adustent. In fact, these two dfferent ethods offer two dfferent was to acheve the sae soluton. In ths report, a surve on Procrustes analss, ts theor, algorths, and related works has been gven. lso, ts applcatons n photograetr has been addressed. he prevous secton (.4.) gves a coparson between the Procrustes analss and the conventonal least squares adustent. he ost portant dsadvantage of the Procrustes ethod s lack of relablt crteron n order to detect and locale the blunders, whch ght be ncluded b the data set. Wthout such a tool, the results that produced b the Procrustes ethod can be wrong n the case of estence of blunders n the data set. CKNOWLEDGEMEN I would lke to thank Fabo Reondno of the Insttute of Geodes and Photograetr, EH Zuerch, for gvng e hs Sngular Value Decoposton progra. NOE In order to perfor the eperental part of ths seester Praktku, two progras,.e. block adustent b ndependent odels and the generaled Procrustes analss (GP), were developed as NSII C++ classes b the author, and are avalable n the nternal Web area of Char of Photograetr and Reote Sensng. REFERENCES run, K., Huang,., and losten, S., 987. Leastsquares fttng of two D pont sets. IEEE ransactons on Pattern nalss and Machne Intellgence, 9(5), pp enat,., Croslla, F.,. generaled factored stochastc odel for the optal global regstraton of LIDR range ages. IPRS, XXXIV (), pp enat,., Croslla, F.,. Generaled Procrustes analss for se and shape D obect reconstructons. In: Gruen,., Kahen, H. (Eds.), Optcal D Measureents echnques V, Venna, pp
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