GENERALIZED PROCRUSTES ANALYSIS AND ITS APPLICATIONS IN PHOTOGRAMMETRY


 Benedict Johns
 2 years ago
 Views:
Transcription
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
Least Squares Fitting of Data
Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 19982016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng
More informationElastic Systems for Static Balancing of Robot Arms
. th World ongress n Mechans and Machne Scence, Guanajuato, Méco, 9 June, 0 _ lastc Sstes for Statc alancng of Robot rs I.Sonescu L. uptu Lucana Ionta I.Ion M. ne Poltehnca Unverst Poltehnca Unverst Poltehnca
More informationAn Enhanced KAnonymity Model against Homogeneity Attack
JOURNAL OF SOFTWARE, VOL. 6, NO. 10, OCTOBER 011 1945 An Enhanced KAnont Model aganst Hoogenet Attack Qan Wang College of Coputer Scence of Chongqng Unverst, Chongqng, Chna Eal: wangqan@cqu.edu.cn Zhwe
More information21 Vectors: The Cross Product & Torque
21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rghthand rule for the cross product of two vectors dscussed n ths chapter or the rghthand rule for somethng curl
More informationELE427  Testing Linear Sensors. Linear Regression, Accuracy, and Resolution.
ELE47  Testng Lnear Sensors Lnear Regresson, Accurac, and Resoluton. Introducton: In the frst three la eperents we wll e concerned wth the characterstcs of lnear sensors. The asc functon of these sensors
More informationPoint cloud to point cloud rigid transformations. Minimizing Rigid Registration Errors
Pont cloud to pont cloud rgd transformatons Russell Taylor 600.445 1 600.445 Fall 000014 Copyrght R. H. Taylor Mnmzng Rgd Regstraton Errors Typcally, gven a set of ponts {a } n one coordnate system and
More informationSimple Correspondence Analysis: A Bibliographic Review
nternatonal Statstcal Revew (2004), 72, 2, 257 284, Prnted n The Netherlands c nternatonal Statstcal nsttute Sple Correspondence Analyss: A Bblographc Revew Erc. Beh School of Quanttatve ethods and atheatcal
More informationRecurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
More informationA Fuzzy Optimization Framework for COTS Products Selection of Modular Software Systems
Internatonal Journal of Fuy Systes, Vol. 5, No., June 0 9 A Fuy Optaton Fraework for COTS Products Selecton of Modular Software Systes Pankaj Gupta, Hoang Pha, Mukesh Kuar Mehlawat, and Shlp Vera Abstract
More informationA Statistical Model for Detecting Abnormality in StaticPriority Scheduling Networks with Differentiated Services
A Statstcal odel for Detectng Abnoralty n StatcProrty Schedulng Networks wth Dfferentated Servces ng L 1 and We Zhao 1 School of Inforaton Scence & Technology, East Chna Noral Unversty, Shangha 0006,
More informationQUANTUM MECHANICS, BRAS AND KETS
PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented
More information8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
More informationQuality of Service Analysis and Control for Wireless Sensor Networks
Qualty of ervce Analyss and Control for Wreless ensor Networs Jaes Kay and Jeff Frol Unversty of Veront ay@uv.edu, frol@eba.uv.edu Abstract hs paper nvestgates wreless sensor networ spatal resoluton as
More informationHow Much to Bet on Video Poker
How Much to Bet on Vdeo Poker Trstan Barnett A queston that arses whenever a gae s favorable to the player s how uch to wager on each event? Whle conservatve play (or nu bet nzes large fluctuatons, t lacks
More informationDescription of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t
Indeternate Analyss Force Method The force (flexblty) ethod expresses the relatonshps between dsplaceents and forces that exst n a structure. Prary objectve of the force ethod s to deterne the chosen set
More informationPSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 12
14 The Chsquared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed
More informationII. THE QUALITY AND REGULATION OF THE DISTRIBUTION COMPANIES I. INTRODUCTION
Fronter Methodology to fx Qualty goals n Electrcal Energy Dstrbuton Copanes R. Rarez 1, A. Sudrà 2, A. Super 3, J.Bergas 4, R.Vllafáfla 5 12 345  CITCEA  UPC UPC., Unversdad Poltécnca de Cataluña,
More informationx f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60
BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true
More informationBANDWIDTH ALLOCATION AND PRICING PROBLEM FOR A DUOPOLY MARKET
Yugoslav Journal of Operatons Research (0), Nuber, 6578 DOI: 0.98/YJOR0065Y BANDWIDTH ALLOCATION AND PRICING PROBLEM FOR A DUOPOLY MARKET PengSheng YOU Graduate Insttute of Marketng and Logstcs/Transportaton,
More informationStochastic Models of Load Balancing and Scheduling in Cloud Computing Clusters
01 Proceedngs IEEE INFOCOM Stochastc Models of Load Balancng and Schedulng n Cloud Coputng Clusters Sva heja Magulur and R. Srkant Departent of ECE and CSL Unversty of Illnos at UrbanaChapagn sva.theja@gal.co;
More informationAn Electricity Trade Model for Microgrid Communities in Smart Grid
An Electrcty Trade Model for Mcrogrd Countes n Sart Grd Tansong Cu, Yanzh Wang, Shahn Nazaran and Massoud Pedra Unversty of Southern Calforna Departent of Electrcal Engneerng Los Angeles, CA, USA {tcu,
More information+ + +   This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
More informationThe Mathematical Derivation of Least Squares
Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the ageold queston: When the hell
More informationCapacity Planning for Virtualized Servers
Capacty Plannng for Vrtualzed Servers Martn Bchler, Thoas Setzer, Benjan Spetkap Departent of Inforatcs, TU München 85748 Garchng/Munch, Gerany (bchler setzer benjan.spetkap)@n.tu.de Abstract Today's data
More informationGPS Receiver Autonomous Integrity Monitoring Algorithm Based on Improved Particle Filter
2066 JOURL OF COMPUTERS, VOL. 9, O. 9, SEPTEMBER 204 GPS Recever utonoous Integrt Montorng lgorth Based on Iproved Partcle Flter Ershen Wang School of Electronc and Inforaton Engneerng, Shenang erospace
More informationv a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
More informationWhat is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
More informationCausal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes causeandeffect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
More informationBasic Queueing Theory M/M/* Queues. Introduction
Basc Queueng Theory M/M/* Queues These sldes are created by Dr. Yh Huang of George Mason Unversty. Students regstered n Dr. Huang's courses at GMU can ake a sngle achnereadable copy and prnt a sngle copy
More informationAn Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
More informationStochastic Models of Load Balancing and Scheduling in Cloud Computing Clusters
Stochastc Models of Load Balancng and Schedulng n Cloud Coputng Clusters Sva Theja Magulur and R. Srkant Departent of ECE and CSL Unversty of Illnos at UrbanaChapagn sva.theja@gal.co; rsrkant@llnos.edu
More informationSupport Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
More informationFace Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching)
Face Recognton Problem Face Verfcaton Problem Face Verfcaton (1:1 matchng) Querymage face query Face Recognton (1:N matchng) database Applcaton: Access Control www.vsage.com www.vsoncs.com Bometrc Authentcaton
More informationStochastic Models of Load Balancing and Scheduling in Cloud Computing Clusters
Stochastc Models of Load Balancng and Schedulng n Cloud Coputng Clusters Sva Theja Magulur and R. Srkant Departent of ECE and CSL Unversty of Illnos at UrbanaChapagn sva.theja@gal.co; rsrkant@llnos.edu
More informationbenefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
More informationInventory Control in a MultiSupplier System
3th Intl Workng Senar on Producton Econocs (WSPE), Igls, Autrche, pp.56 Inventory Control n a MultSuppler Syste Yasen Arda and JeanClaude Hennet LAASCRS, 7 Avenue du Colonel Roche, 3077 Toulouse Cedex
More informationTwoPhase Traceback of DDoS Attacks with Overlay Network
4th Internatonal Conference on Sensors, Measureent and Intellgent Materals (ICSMIM 205) TwoPhase Traceback of DDoS Attacks wth Overlay Network Zahong Zhou, a, Jang Wang2, b and X Chen3, c 2 School of
More informationMaximizing profit using recommender systems
Maxzng proft usng recoender systes Aparna Das Brown Unversty rovdence, RI aparna@cs.brown.edu Clare Matheu Brown Unversty rovdence, RI clare@cs.brown.edu Danel Rcketts Brown Unversty rovdence, RI danel.bore.rcketts@gal.co
More informationPERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
More informationwhere the coordinates are related to those in the old frame as follows.
Chapter 2  Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of noncoplanar vectors Scalar product
More informationLOOP ANALYSIS. The second systematic technique to determine all currents and voltages in a circuit
LOOP ANALYSS The second systematic technique to determine all currents and voltages in a circuit T S DUAL TO NODE ANALYSS  T FRST DETERMNES ALL CURRENTS N A CRCUT AND THEN T USES OHM S LAW TO COMPUTE
More informationThe Development of Web Log Mining Based on ImproveKMeans Clustering Analysis
The Development of Web Log Mnng Based on ImproveKMeans Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
More informationGanesh Subramaniam. American Solutions Inc., 100 Commerce Dr Suite # 103, Newark, DE 19713, USA
238 Int. J. Sulaton and Process Modellng, Vol. 3, No. 4, 2007 Sulatonbased optsaton for ateral dspatchng n VendorManaged Inventory systes Ganesh Subraana Aercan Solutons Inc., 100 Coerce Dr Sute # 103,
More informationChapter 6 Balancing of Rotating Masses
Chapter 6 Balancng of otatng Masses All rotors have soe eccentrct. Eccentrct s present when geoetrcal center of the rotor and the ass center do not concde along ther length (gure ). Eaples of rotors are
More informationPortfolio Loss Distribution
Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets holdtomaturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment
More informationErrorPropagation.nb 1. Error Propagation
ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then
More information6. EIGENVALUES AND EIGENVECTORS 3 = 3 2
EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a nonzero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :
More informationCommunication Networks II Contents
8 / 1  Communcaton Networs II (Görg)  www.comnets.unbremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP
More informationLeast 1Norm SVMs: a New SVM Variant between Standard and LSSVMs
ESANN proceedngs, European Smposum on Artfcal Neural Networks  Computatonal Intellgence and Machne Learnng. Bruges (Belgum), 83 Aprl, dsde publ., ISBN 9337. Least Norm SVMs: a New SVM Varant between
More informationConversion between the vector and raster data structures using Fuzzy Geographical Entities
Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,
More informationA Novel Dynamic RoleBased Access Control Scheme in User Hierarchy
Journal of Coputatonal Inforaton Systes 6:7(200) 24232430 Avalable at http://www.jofcs.co A Novel Dynac RoleBased Access Control Schee n User Herarchy Xuxa TIAN, Zhongqn BI, Janpng XU, Dang LIU School
More information8 Algorithm for Binary Searching in Trees
8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the
More informationScan Detection in HighSpeed Networks Based on Optimal Dynamic Bit Sharing
Scan Detecton n HghSpeed Networks Based on Optal Dynac Bt Sharng Tao L Shgang Chen Wen Luo Mng Zhang Departent of Coputer & Inforaton Scence & Engneerng, Unversty of Florda Abstract Scan detecton s one
More informationA DATA MINING APPLICATION IN A STUDENT DATABASE
JOURNAL OF AERONAUTICS AND SPACE TECHNOLOGIES JULY 005 VOLUME NUMBER (5357) A DATA MINING APPLICATION IN A STUDENT DATABASE Şenol Zafer ERDOĞAN Maltepe Ünversty Faculty of Engneerng BüyükbakkalköyIstanbul
More informationVirtual machine resource allocation algorithm in cloud environment
COMPUTE MOELLIN & NEW TECHNOLOIES 2014 1(11) 27924 Le Zheng Vrtual achne resource allocaton algorth n cloud envronent 1, 2 Le Zheng 1 School of Inforaton Engneerng, Shandong Youth Unversty of Poltcal
More information) of the Cell class is created containing information about events associated with the cell. Events are added to the Cell instance
Calbraton Method Instances of the Cell class (one nstance for each FMS cell) contan ADC raw data and methods assocated wth each partcular FMS cell. The calbraton method ncludes event selecton (Class Cell
More informationCONSTRUCTION OF A COLLABORATIVE VALUE CHAIN IN CLOUD COMPUTING ENVIRONMENT
CONSTRUCTION OF A COLLAORATIVE VALUE CHAIN IN CLOUD COMPUTING ENVIRONMENT Png Wang, School of Econoy and Manageent, Jangsu Unversty of Scence and Technology, Zhenjang Jangsu Chna, sdwangp1975@163.co Zhyng
More informationLoop Parallelization
  Loop Parallelzaton C52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I,J]+B[I,J] ED FOR ED FOR analyze
More informationDEFINING %COMPLETE IN MICROSOFT PROJECT
CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMISP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,
More informationForecasting the Direction and Strength of Stock Market Movement
Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract  Stock market s one of the most complcated systems
More information1 Approximation Algorithms
CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons
More informationTechnical Report, SFB 475: Komplexitätsreduktion in Multivariaten Datenstrukturen, Universität Dortmund, No. 1998,04
econstor www.econstor.eu Der OpenAccessPublkatonsserver der ZBW LebnzInforatonszentru Wrtschaft The Open Access Publcaton Server of the ZBW Lebnz Inforaton Centre for Econocs Becka, Mchael Workng Paper
More informationNMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 7. Root Dynamcs 7.2 Intro to Root Dynamcs We now look at the forces requred to cause moton of the root.e. dynamcs!!
More informationThe eigenvalue derivatives of linear damped systems
Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by YeongJeu Sun Department of Electrcal Engneerng IShou Unversty Kaohsung, Tawan 840, R.O.C emal: yjsun@su.edu.tw
More informationTHE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek
HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo
More informationNaglaa Raga Said Assistant Professor of Operations. Egypt.
Volue, Issue, Deceer ISSN: 77 8X Internatonal Journal of Adanced Research n Coputer Scence and Software Engneerng Research Paper Aalale onlne at: www.jarcsse.co Optal Control Theory Approach to Sole Constraned
More informationThe covariance is the two variable analog to the variance. The formula for the covariance between two variables is
Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.
More informationEconomic Interpretation of Regression. Theory and Applications
Economc Interpretaton of Regresson Theor and Applcatons Classcal and Baesan Econometrc Methods Applcaton of mathematcal statstcs to economc data for emprcal support Economc theor postulates a qualtatve
More informationA R T I C L E S DYNAMIC VEHICLE DISPATCHING: OPTIMAL HEAVY TRAFFIC PERFORMANCE AND PRACTICAL INSIGHTS
A R T I C L E S DYAMIC VEHICLE DISPATCHIG: OPTIMAL HEAVY TRAFFIC PERFORMACE AD PRACTICAL ISIGHTS OAH GAS OPIM Departent, The Wharton School, Unversty of Pennsylvana, Phladelpha, Pennsylvana 191046366
More informationCalculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a twostage stratfed cluster desgn. 1 The frst stage conssted of a sample
More informationNear Optimal Online Algorithms and Fast Approximation Algorithms for Resource Allocation Problems
Near Optal Onlne Algorths and Fast Approxaton Algorths for Resource Allocaton Probles Nkhl R Devanur Kaal Jan Balasubraanan Svan Chrstopher A Wlkens Abstract We present algorths for a class of resource
More informationExtending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σalgebra: a set
More informationAnalysis of Clock Synchronization Approaches for Residential Ethernet
Analyss of Clock Synchronzaton Approaches for Resdental Ethernet Geoffrey M. Garner (Consultant) Kees den Hollander SAIT, Sasung Electroncs ggarner@cocast.net, denhollander.c.@sasung.co Abstract Resdental
More informationSCALAR A physical quantity that is completely characterized by a real number (or by its numerical value) is called a scalar. In other words, a scalar
SCALAR A phscal quantt that s completel charactered b a real number (or b ts numercal value) s called a scalar. In other words, a scalar possesses onl a magntude. Mass, denst, volume, temperature, tme,
More informationL10: Linear discriminants analysis
L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss
More informationModule 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
More informationInstitute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic
Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange
More informationThe OC Curve of Attribute Acceptance Plans
The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4
More informationThe Analysis of Outliers in Statistical Data
THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate
More informationHomework: 49, 56, 67, 60, 64, 74 (p. 234237)
Hoework: 49, 56, 67, 60, 64, 74 (p. 3437) 49. bullet o ass 0g strkes a ballstc pendulu o ass kg. The center o ass o the pendulu rses a ertcal dstance o c. ssung that the bullet reans ebedded n the pendulu,
More informationGRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 NORM
GRAVITY DATA VALIDATION AND OUTLIER DETECTION USING L 1 NORM BARRIOT JeanPerre, SARRAILH Mchel BGI/CNES 18.av.E.Beln 31401 TOULOUSE Cedex 4 (France) Emal: jeanperre.barrot@cnes.fr 1/Introducton The
More informationA hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm
Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):18841889 Research Artcle ISSN : 09757384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel
More informationWeb Servicebased Business Process Automation Using Matching Algorithms
Web Servcebased Busness Process Autoaton Usng Matchng Algorths Yanggon K and Juhnyoung Lee 2 Coputer and Inforaton Scences, Towson Uversty, Towson, MD 2252, USA, yk@towson.edu 2 IBM T. J. Watson Research
More informationCan Auto Liability Insurance Purchases Signal Risk Attitude?
Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? ChuShu L Department of Internatonal Busness, Asa Unversty, Tawan ShengChang
More informationBERNSTEIN POLYNOMIALS
OnLne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
More informationInequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.
Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.
More information1. Measuring association using correlation and regression
How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a
More informationSUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW.
SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. Lucía Isabel García Cebrán Departamento de Economía y Dreccón de Empresas Unversdad de Zaragoza Gran Vía, 2 50.005 Zaragoza (Span) Phone: 976761000
More informationA NOTE ON THE PREDICTION AND TESTING OF SYSTEM RELIABILITY UNDER SHOCK MODELS C. Bouza, Departamento de Matemática Aplicada, Universidad de La Habana
REVISTA INVESTIGACION OPERACIONAL Vol., No. 3, 000 A NOTE ON THE PREDICTION AND TESTING OF SYSTEM RELIABILITY UNDER SHOCK MODELS C. Bouza, Departaento de Mateátca Aplcada, Unversdad de La Habana ABSTRACT
More informationThe Distribution of Eigenvalues of Covariance Matrices of Residuals in Analysis of Variance
JOURNAL OF RESEARCH of the Natonal Bureau of Standards  B. Mathem atca l Scence s Vol. 74B, No.3, JulySeptember 1970 The Dstrbuton of Egenvalues of Covarance Matrces of Resduals n Analyss of Varance
More informationHow Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence
1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh
More informationLuby s Alg. for Maximal Independent Sets using Pairwise Independence
Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent
More informationThe Analysis of Covariance. ERSH 8310 Keppel and Wickens Chapter 15
The Analyss of Covarance ERSH 830 Keppel and Wckens Chapter 5 Today s Class Intal Consderatons Covarance and Lnear Regresson The Lnear Regresson Equaton TheAnalyss of Covarance Assumptons Underlyng the
More informationLinear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
More informationTailoring Fuzzy CMeans Clustering Algorithm for Big Data Using Random Sampling and Particle Swarm Optimization
Internatonal Journal of Database Theory and Alcaton,.191202 htt://dx.do.org/10.14257/jdta.2015.8.3.16 Talorng Fuzzy Means lusterng Algorth for Bg Data Usng Rando Salng and Partcle Swar Otzaton Yang Xanfeng
More informationVision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION
Vson Mouse Saurabh Sarkar a* a Unversty of Cncnnat, Cncnnat, USA ABSTRACT The report dscusses a vson based approach towards trackng of eyes and fngers. The report descrbes the process of locatng the possble
More informationMANY machine learning and pattern recognition applications
1 Trace Rato Problem Revsted Yangqng Ja, Fepng Ne, and Changshu Zhang Abstract Dmensonalty reducton s an mportant ssue n many machne learnng and pattern recognton applcatons, and the trace rato problem
More information5. Simultaneous eigenstates: Consider two operators that commute: Â η = a η (13.29)
5. Smultaneous egenstates: Consder two operators that commute: [ Â, ˆB ] = 0 (13.28) Let Â satsfy the followng egenvalue equaton: Multplyng both sdes by ˆB Â η = a η (13.29) ˆB [ Â η ] = ˆB [a η ] = a
More informationFORCED CONVECTION HEAT TRANSFER IN A DOUBLE PIPE HEAT EXCHANGER
FORCED CONVECION HEA RANSFER IN A DOUBLE PIPE HEA EXCHANGER Dr. J. Mchael Doster Department of Nuclear Engneerng Box 7909 North Carolna State Unversty Ralegh, NC 276957909 Introducton he convectve heat
More information1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 /  Communcaton Networks II (Görg) SS20  www.comnets.unbremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
More information