Goal. Motivation - Matching. Motivation - Matching. Motivation - Morphing. Motivation Common parameterization

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1 Gol Estls to mppn twn ponts on on D sur n orrsponn ponts on rnt sur Sur Rstrton Toms Funousr COS 6, Fll 006 Prun Motvton - Mtn Motvton - Mtn Dtrmn omtr smlrty o surs Dtrmn omtr smlrty o surs Ponts Fturs D(A,B) = A FturSp + Corrsponns Rstrton provs onsstnt prmtrzton Allows or s oprtons l mtn, mn, snl prossn, t. ( n Corrsponn Prs Motvton - Morpn Motvton Common prmtrzton _ B SptlConsstny Smootly trnston rom on sur to notr Wn rstr, smply us lnr omntons o vrts... = ) Prun Alln0

Motvton Attrut Trnsr Copy ttruts rom on sur

Smntton trnsr to notr Txtur trnsr Dormton wt

Txtur trnsr Dormton wt trnsr Smntton trnsr

omplt sur Must ln sns rom rnt vws Rusnwz

notr.., to x surs ptur wt rn snnrs Rstrton

2 Motvton Attrut Trnsr Copy ttruts rom on sur to notr Txtur trnsr (low) Dormton wt trnsr Smntton trnsr Motvton Attrut Trnsr Copy ttruts rom on sur to notr Txtur trnsr Dormton wt trnsr (low) Smntton trnsr Prun Alln0 Motvton Attrut Trnsr Copy ttruts rom on sur to notr Txtur trnsr Dormton wt trnsr Smntton trnsr (low) Motvton Snnn Comn multpl sns to orm omplt sur Must ln sns rom rnt vws Rusnwz Motvton Hol Flln Us sur o on mol to ll ols o notr.., to x surs ptur wt rn snnrs Rstrton Gol Fn mnml ssmlrty msur twn surs ovr lss o possl trnsormtons Prun

Rstrton Mtos Unrlyn ssus: Trnsormton typ Sur rprsntton Dssmlrty msur Alortm strty Rstrton Mtos - Prt Trnsormton Typ R: mutul stns o ponts wtn mol r onsrv urn trnsormton x = R x + t B AB A R s rotton

3 Rstrton Mtos Unrlyn ssus: Trnsormton typ Sur rprsntton Dssmlrty msur Alortm strty Rstrton Mtos - Prt Trnsormton Typ R: mutul stns o ponts wtn mol r onsrv urn trnsormton x = R x + t B AB A R s rotton mtrx n t s trnslton vtor AB Non-r: ount or sur ormtons n t trnsormton.., An trnsormton.., Tn plt spln Rstrton Mtos - Prt Sur Rprsntton Sur srpton Ponts, ms, splns, t. Sur turs Curvtur xtrm, sl ponts, rs, t. Sp srptors Hrmon sp srptors, spn ms Rstrton Mtos - Prt Dssmlrty msur Dstn Dstns twn orrsponn ponts tr lnmnt Dormton Amount o ormton mpl y lnmnt Rstrton Mtos - Prt Alortm strty Optmzton Itrtv mtos Smult nnln Votn Pos lustrn Gomtr Hsn Gnrlz Hou Trnsorm Exmpl Rstrn Humn Bos Alortm Input St o umn rn ms St o olor tur mrrs Alln t l. T Sp o Humn Boy Sps Srp 00 Alortm Gol Dvlop orrsponn rom tmplt to trt Comput n trnsorm or vrtx Mnmz rror unton

4 Optmzton Vrls Alortm vw s optmzton prolm Gvn n ntl tmplt sur wt vrts v Corrsponn n trnsormton mtrs T Currnt stt s T v or ll (s rm) Fn vlus o T to mnmz otv unton Attmpts to n oo t (lu) o tmplt (yn) to trt (r) Otv Funton Otv Funton s tr wt trms Dt rror Smootnss rror Mrr rror Wll us rnt wts n ps o pross Multstp / Mult-rsoluton ttn pross Otv Funton Mrr Error Msurs stn twn pr-ll mrrs Corrsponns st up orn Otv Funton Dt Error Dt rror trm rqurs urrnt mt to los to trt Uss stn rom trnsorm vrtx to t trt sur Wt y onn msur (rom snnn) Hol rons v wt = 0 Sums totl rror Dstn unton Uss trnsorm tmplt vrtx Ts mnmum stn to omptl vrts n trt Otv Funton Smootnss Msurs smootnss o ormton ppl to tmplt E_s msurs n n T_I twn nt vrts Enours smlrly-sp turs to mpp to otr Alortm Prour Mnmz rror unton usn L-BFGS-B lortm Qus-Nwton mto wt lmt mmory us M our psss ovr t ( low rs, rs) Ft mrrs (low rs, α = 0, β =, γ = 0) Rt usn t trm (low rs, α =, β =, γ = 0) Rpt n rsoluton ( rs, α =, β =, γ = 0) Rt usn promnntly t trm ( rs, α = 0, β =, γ = ) Uss Fronus norm (vtor L norm)

5 Exmpl Alnn Brns Alortm Input St o umn rn surs Prll rrn rn ms (low rsoluton) Trnsormton typs In rns, w s omott ormton (lol unorm strt) wn lnn turs Alortm Gol Corrsponn rom tmplt to trt Inty prtulr turs n t rn (yr, sul) Mnmz rror unton Otv Funton Exmpl Alnn Pont Sts Mnmzs rror twn vrtx n tur pont G G Euln stn msur Sur Norml Mt Ftur Mt Consr r trnsormtons Assoton Grp Assoton Grp G G G Assoton Grp G Rprsnt ot ponts sts s omplt rps (G n G). (s onnt ll prs o vrts wtn pont st) Crt vrts n t ssoton rp or ll omptl prs o vrts n t ornl rps. Ts n l to lr numr o vrts.

6 Assoton Grp Assoton Grp G Assoton Grp G 6 G Assoton Grp G Crt vrts n t ssoton rp or ll omptl prs o vrts n t ornl rps. Comptlty oul rr to ml proprts. Crt s twn (uv) n (wx) t s twn (u) n (w) s wll s twn (v) n (x) mt. Assoton Grp Assoton Grp G Assoton Grp G 6 G Assoton Grp G Crt s twn (uv) n (wx) t s twn (u) n (w) s wll s twn (v) n (x) mt. For ts xmpl, lnt s t only onsrton Crt s twn (uv) n (wx) t s twn (u) n (w) s wll s twn (v) n (x) mt. For ts xmpl, lnt s t only onsrton Assoton Grp Assoton Grp G Assoton Grp G 6 G Assoton Grp G Crt s twn (uv) n (wx) t s twn (u) n (w) s wll s twn (v) n (x) mt. For ts xmpl, lnt s t only onsrton T t lrst st o orrsponn nos n t sm onurton s t mxml lqu n t ssoton rp (.., t lrst sust o t ssoton rp or w ll nos r onnt to on notr). 6

7 Assoton Grp Itrtv Closst Ponts (ICP) Computtonl omplxty: O( n ) or n ponts NP-omplt Fn t Mxml Clqu{ rturn Clqus(mpty, ll nos) } Clqus(X, Y){ (no no n Y-X s onnt to ll o X){ rturn X; }ls{ y = no n Y onnt to ll o X; rturn Lrst(Clqus(X unon y, Y}, Clqus{X, Y-y}); } } 6 Assoton Grp 6 [Smtt0, Brown8] Assum losst ponts orrspon Avo nn on-to-on orrsponns R oy trnsormtons Gry optmzton prour Strt wt rou uss or lnmnt Itrtvly rn trnsorm [Bsl9] Itrtv Closst Pont Gvn two pont sts Itrtv Closst Pont Gvn two pont sts n n ntl uss or t trnsormton tt lns tm A B [Bsl9] [Bsl9] Itrtv Closst Pont Assum losst ponts orrspon Itrtv Closst Pont Assum losst ponts orrspon: A B A B [Bsl9] [Bsl9] 7

8 Itrtv Closst Pont Assum losst ponts orrspon: A B n B A Itrtv Closst Pont Rtn outlrs A B Outlr [Bsl9] [Bsl9] Itrtv Closst Pont Fn t trnsormton tt optmlly lns propos orrsponns (suprposton) ( A, B) = A A A B + B B B A [Bsl9] Itrtv Closst Pont Itrt untl onvrn. Slt sour ponts (rom on or ot surs). Mt to ponts n t otr molul. Wt t orrsponns. Rt outlr pont prs. Comput n rror mtr or t urrnt trnsorm 6. Mnmz t rror mtr w.r.t. trnsormton Computtonl omplxty O( * nlon) or n ponts pr st n trtons trtons * O(n) ponts * O(lon) to n losst pont Sl ourtsy o Szymon Rusnwz ICP Alnn Surs (Sns) Strt wt mnul ntl lnmnt ICP - Alnn Surs (Sns) Improv lnmnt usn ICP Sl ourtsy o Szymon Rusnwz Sl ourtsy o Szymon Rusnwz 8

9 ICP - Alnn Surs (Sns) Assum losst ponts orrspon, omput t st trnsorm ICP - Alnn Surs (Sns) n trt to n lnmnt Convrs to som lol mnmum Corrt strtn poston los nou Sl ourtsy o Szymon Rusnwz Sl ourtsy o Szymon Rusnwz Pos Clustrn Gnrl mto Enumrt possl trnsormtons Vot or st on Mtos Pos lustrn Gomtr Hsn Gnrlz Hou Trnsorm Pos Clustrn Dsrtz trnsormtons n sorn Pont St A Pont St B [Wolson97] Pos Clustrn Dsrtz trnsormtons n sorn Pos Clustrn Dsrtz trnsormtons n sorn Rotton & Trnslton or (,) y y Pont St A x Pont St A x Rotton & Trnslton or (, ) Pont St B [Wolson97] Pont St B [Wolson97] 9

10 Pos Clustrn Dsrtz trnsormtons n sorn Pos Clustrn Dsrtz trnsormtons n sorn Rotton & Trnslton or (,) y Rotton & Trnslton or (,) y Pont St A x Pont St A x Rotton & Trnslton or (, ) Rotton & Trnslton or (, ) Sor orrsponns Pont St B [Wolson97] Pont St B [Wolson97] Gomtr Hsn Prprossn Gomtr Hsn Qury prossn Rotton & trnslton or ll prs o ponts n ll moluls Rotton & trnslton or on pr o ponts, Pont St n Dts Pont St Qury,, Stor (ot, r. rm, proprts, pont) or vry trnsorm pont n s tl Hs Tl [Wolson97] [Wolson97] Gomtr Hsn Prprossn For trpl o ponts Comput rrn rm For pont Trnsorm pont nto rrn rm Hs (molul, r. rm, proprts, pont) Qury prossn Coos ny trpl o ponts Comput rrn rm For pont Trnsorm pont nto rrn rm For ntry n s n or trnsorm pont Vot or (ot, r. rm) Gomtr Hsn Prprossn omplxty O(n ) or n ponts pr nn st O(n ) possl trpls * O(n) trnsormtons pr trpl Qury omplxty O(m) * nsz or m ponts n qury nn st trpl * O(m) trnsormtons pr trpl * nsz s prossn pr trnsormton [Wolson97] 0

11 Summry Drnt mtos or rnt Trnsormton typs Sur rprsnttons Dssmlrty msurs

Registration of Point Sets with Rigid Transformations

Registration of Point Sets with Rigid Transformations Pont St Mtn Rstrton o Pont Sts wt R Trnsormtons Gol s to omput smlrty msur or pr o ttrut pont sts Toms Funousr Prnton Unvrsty CS6, Fll 00 Dstn(,) =? Pont St Mtn Clln s to n orrsponn ponts Sust o ponts