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 n my mt sust o ponts n Pont St Mtn Clultn suprposton n stn msur s sy orrsponns r nown (propos) Pont St Mtn Pont St Mtn Clultn suprposton n stn msur s sy orrsponns r nown (propos) Clultn suprposton n stn msur s sy orrsponns r nown (propos) Lst-squrs optml suprposton o orrsponn ponts
Pont St Mtn Clultn suprposton n stn msur s sy orrsponns r nown (propos) RMSD(, ) = N = ( ) Outln Introuton Pont st mtn rut or sr RNSC Gomtr sn ssoton rps Itrtv losst pont Dsusson Dstn(,) = RMSD(,) + OtrTrms rut For Sr Smpl mto: Try ll possl sts o pont orrsponns Sor t lnmnt or on rut For Sr Smpl mto: Try ll possl sts o pont orrsponns Sor t lnmnt or on Prolm: O(n m ) possl sts o m orrsponns mon n ponts Prolm: O(n m ) possl sts o m orrsponns mon n ponts 8 Pont ln RMSD =. rut For Sr Smpl mto: Try ll possl sts o pont orrsponns Sor t lnmnt or on (.., RMSD) Prolm: O(n m ) possl sts o m orrsponns mon n ponts ll ponts ln RMSD = 0. Outln Introuton Pont st mtn rut or sr RNSC Gomtr sn ssoton rps Itrtv losst pont Dsusson
RNSC Rnomly smpl st o possl orrsponns Rnomly nrt smll st o pont orrsponns Comput t lnn trnsormton or orrsponns Sor ow wll otr ponts ln tr tt trnsormton Outln Introuton Pont st mtn rut or sr RNSC Gomtr sn ssoton rps Itrtv losst pont Dsusson [Smtt0] Gomtr Hsn Dsrtz trnsormtons n sorn Gomtr Hsn Dsrtz trnsormtons n sorn Rotton & Trnslton or (,) y Pont St Pont St x Pont St [Wolson97] Pont St [Wolson97] Gomtr Hsn Dsrtz trnsormtons n sorn Gomtr Hsn Dsrtz trnsormtons n sorn y Rotton & Trnslton or (,) y Pont St x Pont St x Rotton & Trnslton or (, ) Rotton & Trnslton or (, ) Pont St [Wolson97] Pont St [Wolson97]
Gomtr Hsn Dsrtz trnsormtons n sorn Gomtr Hsn Prprossn Pont St Rotton & Trnslton or (,) y x Rotton & trnslton or ll prs o ponts n ll moluls Pont St Rotton & Trnslton or (, ) Sor orrsponns [Wolson97] Pont St n Dts Stor (molul, r. rm, proprts, pont) or vry trnsorm pont n s tl Hs Tl [Wolson97] Gomtr Hsn Gomtr Hsn Qury prossn Pont St Qury Rotton & trnslton or on pr o ponts,,, [Wolson97] 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 C pont proprts Vot or (molul, 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 Outln Introuton Pont st mtn rut or sr RNSC Gomtr sn ssoton rps Itrtv losst pont Dsusson [Wolson97]
ssoton Grps ssoton Grps G G G G Rprsnt ot ponts sts s omplt rps (G n G). (s onnt ll prs o vrts wtn pont st) [Smtt0, rown8] [Smtt0, rown8] ssoton Grps G ssoton Grp G Crt vrts n t ssoton rp or ll omptl prs o vrts n t ornl rps. Ts n l to lr numr o vrts. [Smtt0, rown8] ssoton Grps Dpt G ssoton Grp G Propnsty Consrvton Cr Hyropoty Sonry strutur typ Dstlzton 6 6 Crt vrts n t ssoton rp or ll omptl prs o vrts n t ornl rps. Comptlty oul rr to ml proprts. [Smtt0, rown8] ssoton Grps ssoton Grps G ssoton Grp G 6 G ssoton Grp G 6 6 6 Crt s twn (uv) n (wx) t s twn (u) n (w) s wll s twn (v) n (x) mt. 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 [Smtt0, rown8] [Smtt0, rown8]
ssoton Grps ssoton Grps G ssoton Grp G 6 G ssoton Grp G 6 6 6 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 [Smtt0, rown8] [Smtt0, rown8] ssoton Grps G ssoton Grp G 6 6 Fnn orrsponns: T t lrst st o orrsponn nos n t sm onurton s t mxml lqu n t ssoton rp [Smtt0, rown8] ssoton Grps Computtonl omplxty: O( n ) or n ponts NP-omplt rn n oun lortms 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 ssoton Grp 6 [Smtt0, rown8] Outln Introuton Pont st mtn rut or RNSC Gomtr sn ssoton rps Itrtv losst ponts Dsusson Itrtv Closst Ponts Gvn two pont sts [sl9] 6
Itrtv Closst Ponts Gvn two pont sts Itrtv Closst Ponts Gvn two pont sts n n ntl uss or t trnsormton tt lns tm [sl9] [sl9] Itrtv Closst Ponts ssum losst ponts orrspon Itrtv Closst Ponts ssum losst ponts orrspon: [sl9] [sl9] Itrtv Closst Ponts ssum losst ponts orrspon: n Itrtv Closst Ponts Rtn outlrs Outlr [sl9] [sl9] 7
Itrtv Closst Ponts Fn t trnsormton tt optmlly lns propos orrsponns (suprposton) (, ) = + [sl9] Itrtv Closst Ponts Itrt untl onvrn. Slt sour ponts (rom on or ot moluls). 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 nn st n trtons trtons * O(n) ponts * O(lon) to n losst pont Sl ourtsy o Szymon Rusnwz Itrtv Closst Ponts Dmo ourtsy o Szymon Rusnwz Summry rut or urt, slow RNSC pproxmt Gomtr sn Fst qury, tr slow prprossn Dstn trsol mplt n s ut szs ssoton rps Expnsv or lr pont sts Dstn trsol or ssotons Itrtv losst ponts Fst, n prt Rqurs oo ntl uss 8