Author mnusript, pulish in "Pttrn Rogn., 8 (00) 97--99" DOI : 0.06/j.ptog.00.0.05 Homomorphi Alignmnt o Wight Trs Bnjmin Rynl, Mihl Coupri, Vnsls Biri Univrsité Pris-Est,Lortoir Inormtiqu Gspr Mong, Equip ASI UMR 809 UPEMLV/ESIEE/CNRS hl-006879, vrsion - Mr 0 Astrt Motion ptur, urrntly tiv rsrh r, ns stimtion o th pos o th sujt. For this purpos, w mth th tr rprsnttion o th sklton o th D shp to pr-spii tr mol. Unortuntly, th tr rprsnttion n ontin vrtis tht split lims in multipl prts, whih o not llow goo mth y usul mthos. To solv this prolm, w propos nw lignmnt, tking into ount th homomorphism twn trs, rthr thn th isomorphism, s in prior works. Thn, w vlop svrl omputtionlly iint lgorithms or rhing rl-tim motion ptur. Ky wors: Grphs, homomorphism, lignmnt, mthing lgorithm. Introution.. Motivtion Motion ptur without mrkrs is highly tiv rsrh r, s shown y Moslun t l. []: twn 000 n 006, mor thn 50 pprs on this topi wr pulish. Motion ptur is us in svrl pplitions whih hv irnt ims n onstrints: D mols nimtion, or movis FX or vio gms or xmpl, rqusts n highly urt mol, ut os not n rl-tim omputtion (olin vio prossing is ptl). Rl-tim intrtion, or virtul rlity pplitions, rqusts st omputtion, t th pri o lowr ury. This ppr is pl in th ontxt o rl-tim intrtion. Moslun t l. sri in prvious work [] th irnt stps o motion ptur. Th irst stp (ll initiliztion stp) onsists o ining th initil pos o th sujt, rprsnt hr y shp (visul hull) onstrut using multi-viw systm with n lgorithm o Shp From Silhoutt []. Emil rsss: rynl@univ-mlv.r (Bnjmin Rynl), ouprim@si.r (Mihl Coupri), iri@univ-mlv.r (Vnsls Biri) Prprint sumitt to Elsvir Mrh, 00
Most lgorithms o D pos stimtion us mnully initiliz mol, or sk th sujt to mov sussivly th irnt prts o his/hr oy [], ut svrl utomti pprohs hv n vlop, using n priori mol. This priori mol n pproximt irnt hrtristis o th sujt: Kinmti strutur: A strutur ontining ix numr o joints, with spii grs o rom, n lim lngths. Shp: A gnri humnoi mol, rprsnt y simpl shp primitivs [5]. Apprn: Th txtur o sujt sur. hl-006879, vrsion - Mr 0 This kin o omplx priori mol is iiult to mth with rl t, n ns to pt to h sujt (spilly in th s o pprn). In shp mthing, ommon pproh onsists o using sur (or ontour, in s o D shps) inormtion, vi urvtur n istn to ntroi inormtion [6, 7, 8]. This kin o pproh n not us in our s or svrl rsons: it implis omplx priori mol, urvtur is vril or rtiult shp, n th visul hull n hv vry noisy sur, u to th mtho o quisition... Our pproh Two o th most prsrv hrtristis o th rl shp y its visul hull ronstrution r its topology n th istns twn th irnt prts o th shp. A usul tool or rprsnting ths hrtristis is th sklton o th shp. A lot o pprohs using th sklton o shp hv n vlop. In motion ptur rsrh r [9, 0, ], th st tim otin or ining th initil pos is roun on son [0], whih is too slow, vn or intrtiv tim intrtion. In D shp mthing rsrh r, sklton is ommon tool o rprsnttion n omprison [, ]. A wily us mtho onsist in omprison o shok grphs [], uilt rom oth th sklton n th rius istn o th shps. Howvr, vn i this mtho n ppli in th s o D shps [5], it is not intrsting in our s, us th rius istn o th visul hull n noisy. For our purpos, th sklton is still too omplx, n w o not n ll th involv inormtion. It n osrv tht only smll prt o th sklton points r intrsting: th ning points n th intrstion points. In ition to th position o ths points, w n to know how thy r link togthr, n wht is th lngth o th sklton rnh twn thm. Consiring ths osrvtions, w n rprsnt th sklton y unroot tr (ll th t tr): th vrtis rprsnt th ning or intrstion points, n th gs rprsnt th links twn th points. In ition, w giv wight to h g, orrsponing to th gosi istn twn th points involv. Thn, our priori mol is lso n unroot wight tr (ll th pttrn tr), whr vrtis rprsnt th irnt prts o th shp (h, torso,
6 7 g h i j k l m n H A T A 0 C 6 6 F F D SHAPE SKELETON 8 o DATA TREE PATTERN TREE Figur : Exmpl o t tr quisition n xpt lignmnt with pttrn (mol) tr. From th right to th lt: th originl D shp, its sklton, th t tr omput rom th sklton, n th pttrn tr. Th numrs rprsnt th wights o th gs (i.. th gosi istns in th sklton or th t tr, n th xpt ons or th pttrn tr).th gry sh lins rprsnt th xpt lignmnt twn th t tr n th mol. hl-006879, vrsion - Mr 0 roth, hns n t), n h g rprsnts th link twn this prts, ssoit to wight, rprsnting th istn twn two prts. It n sn s vry simplii kinmti strutur, without inition o grs o rom. Our pproh to in th initil pos o th sujt is thror to in th st lignmnt twn th pttrn tr n th t tr, tht is to sy, th mthing whih involvs s w moiitions s possil, to trnsorm th t tr in th pttrn tr (s Fig. or n xmpl o our omplt piplin, n xpt lignmnt). As w know whih prt o th shp orrspon to h vrtx o th pttrn tr, n whih r th D oorints o h vrtx o th t tr, th lignmnt will giv th position o h prt o th D shp... Prolms Svrl kins o nois n ormitis n ppr in th t tr: Ghosts lims: Du to th ronstrution rom silhoutts, prts o sp nnot rv, rsulting in lims o th ojt whih o not xist on th rl mol. Our mtho must urt nough to istinguish ths ghosts lims rom rl ons. Spurious rnhs: Du to th skltoniztion lgorithm n to th mount o nois o th shp sur, rnhs without importnt topologil signiition n ppr on sklton. For xmpl, in Fig., th gs {g,h}, {l,m}, {i,j}, {j,k} r spurious rnhs. Our mtho must roust nough to work on t trs with onsqunt mount o spurious rnhs. Uslss vrtx: Vrtis with xtly two nighors r not usul to sri th topology o shp, n thn uslssly split n g (n its wight)
into two prts, mking iiult goo mthing. This kin o vrtis n ppr whn rmoving spurious rnhs or ghosts lims. For xmpl, in Fig., vrtis j,k,m r uslss tr spurious rnhs ltion. Our mtho must l to mth two gs join y this kin o vrtx, with uniqu g. Splitt vrtx: Vrtis with mor thn thr nighors in th pttrn tr n orrspon to lustr o vrtis link y wkly wight gs in th t tr, u to th skltoniztion lgorithm. For xmpl, in Fig., vrtx T o pttrn tr mths with vrtis n in t tr. Our mtho must l to mth thm. hl-006879, vrsion - Mr 0.. Our ontriution Approhs oun in th litrtur (s St. ) o not prmit to hiv roust mthing, with rspt to ll ths prturtions. Howvr, som xisting it-s istn, th lignmnt istn [], is n intrsting wy to solv th prolm o splitt vrtis, n it prsrvs th topology uring th mthing. In ition, n oprtion sri in [0], th ut oprtion, is spilly sign or onsiring only suprt o th tr. This oprtion is xtly wht w n to voi th prolms o ghost lims n spurious rnhs, whih n onsir s uslss prts o th t tr. Th prolm o uslss vrtis nnot solv y mthos oun in th litrtur. W introu in this ppr nw kin o lignmnt, whih solvs this prolm y onsiring th homomorphism twn trs inst o th isomorphism. This ppr is orgniz s ollow: In St., w giv th si initions n nottions rlt to g-wight grphs tht will us in th squl. In St., w trmin whih it-s istn o th litrtur is th most pproprit to s our mtho. In St., w introu our min ontriution, th homomorphi lignmnt. W lso introu lgorithms to omput it iintly or root trs, s wll s unroot trs. In St. 5, w show how to us th ut oprtion [0] with homomorphi lignmnt. Finlly, in St. 6, w show th rsults o irnt xprimnttions, n omprison twn our homomorphi lignmnt istn n th lssil lignmnt istn.. Bsis notions n nottions.. Unirt grphs An unirt grph is pir (V,E), whr V is init st, n E sust o {{x,y},x V,y V,x y}. An lmnt o E is ll n g, n lmnt o V is ll vrtx. I {x,y} E, thn x n y r si to jnt or nighors. Th st o ll nighors o x is not y N(x). Th numr o vrtis jnt to vrtx v is ll th gr o v, n is not y g(v). Lt G = (V,E) n unirt grph, n lt x,y in V, pth rom x to y in G is squn o vrtis v 0,...,v k suh tht x = v 0, y = v k
n {v i,v i } E, i k. Th numr k is ll th lngth o th pth. I k = 0 th pth is ll trivil pth. Th pth is los i x = y. Th pth is simpl whn no vrtx ours mor thn on in th squn o vrtis o th pth (xpt possily x = y). A non-trivil simpl los pth in whih ll gs r istint is ll yl. A grph is onnt i or ll {x,y} V, pth rom x to y xists in G. A tr is onnt grph with no yls. A simpl pth rom x to y in tr is uniqu n is not y π(x,y). A grph with no yls is ll orst, h o its onnt omponnts ing tr. hl-006879, vrsion - Mr 0.. Dirt grphs A irt grph is pir (V,A), whr V is init st, n A is sust o V V. An lmnt o A is ll n r, n lmnt o V is ll vrtx. Lt G = (V,A) irt grph, n lt x,y in V, pth rom x to y in G is squn o vrtis s 0,...,s k suh tht x = v 0, y = v k n (v i,v i ) A, i k. Th unirt grph ssoit to G is th unirt grph G = (V,E), suh tht {x,y} E i n only i (x,y) A or (y,x) A. A vrtx r V is root o G i or ll x V \ {r}, pth rom r to x in G xists. Th grph G is ntisymmtri i or ll (x,y) A suh tht x y, (y,x) / A. Th grph G is root tr (with root r) i r is root o G, G is ntisymmtri n i th unirt grph ssoit to G is tr. A grph, whr h o its onnt omponnts is tr, is ll root orst. Lt G = (V,A) root tr. I (y,x) A, w sy tht y is th prnt o x (not y pr(x)), n tht x is hil o y. Th st o ll hilrn o y is not y C(y). Th mximum lngth o pth twn th root n ny no is ll th hight o th tr. Th vrtis on th pth rom th root to vrtx x r ll th nstors o x. W not th st o th nstors o x y n(x)... Common initions Unlss othrwis init, ll th othr initions n nottions in this ppr r similr or th two kins o grphs. W giv thm or irt grphs, th vrsions or unirt grphs n otin y rpling rs y gs. Two grphs G = (V G,A G ) n G = (V G,A G ) r si to isomorphi i thr xists ijtion : V G V G, suh s or ny pir (x,y) V G V G, (x,y) A G i n only i ((x),(y)) A G. A wight grph is triplt (V,A,ω), whr V is init st, A sust o V V, n ω mpping rom A to R. In wight tr, th wight o th uniqu pth rom x to y, not y ω(x,y), is th sum o th wights o ll rs trvrs in th pth. In this ppr, w sy tht two wight grphs (V,E,ω) n (V,E,ω ) r isomorphi whnvr th grphs (V,E) n (V,E ) r isomorphi (rgrlss o th wights). Th mrging is n oprtion tht n ppli only on rs shring - gr vrtx. Th mrging o two rs (u,v) n (v,w) in wight grph G = (V,A,ω) onsists o rmoving v in V, rpling (u,v) n (v,w) y (u,w) in A, wight y ω((u,w)) = ω((u,v)) + ω((v,w)). 5
Two wight grphs G = (V G,A G,ω G ) n G = (V G,A G,ω G ) r homomorphi [6] i thr xists n isomorphism twn grph otin y mrgings on G n grph otin y mrgings on G. S Fig. or som xmpls o isomorphism n homomorphism. U U U U i j i x k l k m n y m n 5 g h g h p o p j l o hl-006879, vrsion - Mr 0 g h g x h i k m n y p j l o i m D D D D D 5 k n 5 Figur : First row: U is otin rom U y mrging on x. U is otin rom U y mrging on y. U n U r isomorphi, U n U r homomorphi. Son row: D is otin rom D y mrging on x. D 5 is otin rom D y mrging on y. D n D r isomorphi, D n D 5 r homomorphi, D n D r not isomorphi (r rs hv irnt orinttions).. Eit-s istns Th prolm o ompring grphs (in prtiulr trs) ours in ivrs rs suh s omputtionl iology, img nlysis n strutur tss. Howvr, th grphs onsir in ths omins r most otn with ll vrtis. Hr, h notion will introu in th s o grphs with wight gs/rs. In this stion, w rviw irnt it-s istns propos in th litrtur, n slt th on tht is th st pt our ims. p j l o.. Eit oprtions A lot o mthos hv n vlop or tr omprison, s th mximl ommon sutr [7, 8], or th pproximt sutr homomorphism [9]. An pproh wily us to ompr trs is to srh or squn o simpl primitiv oprtions (ll it oprtions) tht trnsorms tr into th othr n tht hs miniml ost. 6
For grph G = (V,A,ω), lssil it oprtions r: Rsiz: Chng th wight o n r = (u,v) A. Dlt: Dlt n r = (u,v) A n mrg u n v into on vrtx. Insrt: Split vrtx in two vrtis, n link thm y nw r. S Fig. or illustrtions. Th ost o ths it oprtions is givn y ost untion γ(w,w ), whr w (rsptivly w ) is th totl wight o th rs involv in th oprtion or (rsptivly, tr) its pplition. As onsqun, th ost o rsizmnt n not y γ(w,w ), whr w is th ormr wight o th rsiz r n w is th nw wight, th ost o ltion n not y γ(w,0), whr w is th wight o th lt r, n th ost o n insrtion, γ(0,w), whr w is th wight o th rt r. Furthrmor, w ssum tht γ is mtri. Typilly, γ(w,w ) = w w or (w w ). hl-006879, vrsion - Mr 0 U U Rsizmnt 5 Rsizmnt 5 u u U U v v 9 9 g g Dltion Insrtion Dltion Insrtion w w D D D D Figur : Exmpls o it oprtions: on th top, on unirt grphs. On th ottom, on irt grphs. 9 g 9 g.. Choi o it-s istn Vrious it-s istns hv n in, using irnt onstrints on th it oprtions orr or on th rsulting orrsponn twn th vrtis sts, n irnt initions o oprtions. Ths it-s istns n lssii, s propos y Wng t l. [0]: it istn [], lignmnt istn [, ], isolt-sutrs istn [, 5, 6], n top-own istn [7, 8, 9]. 7
... Eit istn Lt G th grph tht rsults rom th pplition o n it oprtion s to grph G ; this is writtn G G vi s. Lt S squn s,s,...,s k o it oprtions. W sy tht S trnsorms grph G to grph G i thr is squn o grphs G 0,G,...,G k suh tht G = G 0, G = G k n G i G i vi s i or i k. Th ost o th squn S, not y γ(s), is th sum o osts o th onstitunt it oprtions. Th istn rom G to G, not y δ(g,g ), is th minimum ost o ll squns o it oprtions tking G to G. For our purpos, this kin o it-s istn nnot us, us th ssoit mthing os not prsrv topologil rltions twn trs. In ition, th lgorithm s on this istn is th on with th highst tim omplxity. hl-006879, vrsion - Mr 0... Isolt-sutrs istn Isolt-sutrs istn is n it-s istn suh tht two isjoint sutrs in th pttrn tr will lwys mth with two isjoint sutrs in th t tr.... Top-own istn Top-own istn is n it-s istn suh tht n r (p,p ) in th pttrn tr n mth n r (, ) in th t tr, only i (pr(p),p) mths (pr(),). Isolt-sutrs istn n top-own istn nnot lwys mth ll th mol tr, ut only suprts, most otn unonnt. Howvr, w will s in th nxt sustion tht it is not th s or lignmnt istn... Alignmnt istn In [], Jing t l. propos similrity msur twn vrtx-ll trs, tht w trnspos hr or g-wight grphs. Lt G = (V,A,ω ) n G = (V,A,ω ) two wight grphs. Lt G = (V,A,ω ) n G = (V,A,ω ) wight grphs otin y insrting rs wight y 0 (zro) in G n G, suh tht thr xists n isomorphism I twn (V,A ) n (V,A ). Th st o ll oupls o rs A = {(, ); A, A, = I( )} is ll n lignmnt o G n G. Th ost C A o A is th sum o th osts o ll oprtions us to lign G n G : th insrtions o rs wight y 0 (zro) (whih is r, u to non-vrition o wights), n th rsizmnt or h r A to th wight o I( ). Mor ormlly : C A = γ(ω ( ),ω ( )). (, ) A Th miniml ost o ll lignmnts rom G n G, ll th lignmnt istn, is not y α(g,g ). Th lignmnt istn is spil s o 8
th it istn, whr ny insrtion ours or ny ltion. As rsult, α(g,g ) δ(g,g ). Alignmnt istn is intrsting in our s or thr rsons: it tks into ount topologil rltions twn trs, it n omput in polynomil tim, n it nls to rmov gs, rgrlss o th rst o th grph, solving th prolm o splitt vrtis.... Illustrtion W onsir only th prolm o splitt vrtis in this xmpl. In Fig., w show th wy to otin n lignmnt with miniml ost: only on g insrtion in th pttrn tr, on th vrtx T, rprsnting th torso. Th rsulting lignmnt, rprsnt y ott lins, hs ost o. Th orrsponing vrtis mthing is th ollowing: H mths with, T with {,}, A with, A with, C with, F with n, n F with o. hl-006879, vrsion - Mr 0 A G F A G F 6 6 H T 0 C H T 0 0 T C 6 6 A 6 G F 6 n o A 6 Figur : Exmpl o lignmnt: G (rsp. G ) is otin rom G (rsp. G ) y insrtions o gs (hr G = G ). Th ott lins rprsnt on o th possil lignmnts o G n G. G 6 F n o. Homomorphi lignmnt istn I w now tk into ount th prolm o uslss -gr vrtis, w n s tht th lignmnt strtgy os not work nymor (s Fig.5). In this xmpl, th split o th lg gs o th t tr in svrl prts involvs tht th st lignmnt is otin y mthing th rms o th pttrn (gs {T,A } n {T,A }) with th prts o th lgs in th t tr whih r losst o thm, in rgr o thir wights (gs {m,o} n {j,n}). Th ost o th lignmnt, shown y th ott lins, is qul to 6 n is miniml. For th purpos o solving th uslss vrtx prolm, w propos nw lignmnt strtgy. 9
A F G H T 0 C 6 6 A F C 0 0 C 6 F F 6 0 0 C G T 0 T A H T 0 T A 6 G 7 j h n m 8 o 6 G 7 j 0 h n m 8 o Figur 5: Exmpl o lignmnt, u to uslss -gr vrtis. hl-006879, vrsion - Mr 0.. Mrging krnl Consiring tht mrging on vrtx v on th grph G = (V,A,ω) os not t th gr o ny vrtx in V \ {v} (y inition o mrging oprtion) n thror th possiility o mrging this vrtx, th numr o possil mrgings rss y on tr h mrging. In onsqun, th mximl siz o squn o mrging oprtions, trnsorming G into nothr grph G = (V,A,ω ) is qul to th initil numr o possil mrgings in G. It n rmrk tht ny squn o mrging oprtions o mximl siz yils th sm rsult. Th grph rsulting o suh squn on G is ll th mrging krnl o G, n is not y MK(G). Th ollowing proposition is strightorwr: Proposition. Two grphs G = (V,A,ω ) n G = (V,G,ω ) r homomorphi i MK(G ) n MK(G ) r isomorphi... Homomorphi lignmnt istn Lt G = (V,A,ω ) n G = (V,A,ω ) two wight grphs. Lt G = (V,A,ω ) n G = (V,A,ω ) wight grphs otin y lting rs in G n G, suh tht thr xists n homomorphism twn G n G (not nssrily uniqu). Lt G = (V,A,ω ) n G = (V,A,ω ) th mrging krnl o G n G, rsptivly. From proposition, thr xists n isomorphism I twn G n G. Th st o ll oupls o rs H = {(, ); A, A, = I()} is ll n homomorphi lignmnt o G with G (s igur 6). Th grph G is ll th lt grph o H. Th grph G is ll th right grph o H. Th ost C H o H is th sum o th osts o ll oprtions us to homomorphilly lign G n G : th ltion o rs in G n G, to otin G n G rsptivly, n th rsizmnt or h r A to th wight o H( ). Mor ormlly : 0
C H = (, ) H γ(ω (),ω ( )) + A \A γ(ω ( ),0) + A\A γ(0,ω ( )). This miniml ost o ll homomorphi lignmnts twn G n G, ll th homomorphi lignmnt istn, is not y η(g,g ). hl-006879, vrsion - Mr 0... Illustrtion Lt us onsir gin th pttrn n t trs us in Fig. 5. In Fig.6 is shown th wy to otin homomorphi lignmnt with miniml ost: only on g ltion in th t tr (g {,}), is nssry to otin homomorphism. Th mrging krnl o G is qul to G, n th mrging krnl o G is otin y mrging on vrtis j, h n m. Th rsulting homomorphi lignmnt, rprsnt y ott lins, hs ost o. Th orrsponing vrtis mthing is th ollowing: H with, T with {,}, A with, A with, C with, F with n, n F with o. 6 A G F H T 0 C A G 7 j h 6 6 F n m 8 o A G F H T 0 C 6 6, A 6 G 7 j h F n m 8 o A G F H T 0 C 6 6 A, Figur 6: Exmpl o homomorphi lignmnt: G (rsp. G ) is otin rom G (rsp. G ) y ltions o gs. G = MK(G ) n G = MK(G ). Th ott lins rprsnt on o th possil homomorphi lignmnts o G n G. 6 G 6 F n o.. Algorithm or root trs In orr to omput homomorphi lignmnt istn with goo omplxity, w will us ottom-up pproh. This pproh, wily us in th litrtur on it-s istns, is s on rormultion o th istn twn trs in trms o oth th istn twn thir sutrs n th mthing o thir roots. Thn, using this rormultion, th omputtion strts y th istns twn th simplst sutrs (i. th ls), n gos up to th root. Th istns twn sutrs r kpt in mmory, voiing runny o omputtion. Howvr, thr r two min irns to tk into ount in th s o homomorphi lignmnt istn: th t tht inormtion is on rs, rthr
hl-006879, vrsion - Mr 0 thn o vrtis, n th t tht n r n mth with st o rs mrg togthr (u to mrging krnl pplition). Th irst prolm involvs to tk into onsirtion svrl r wights in th rormultion (on or h r twn th root n its hilrn), inst o th uniqu vlu link to th root, s in th litrtur. Th son prolm involvs to tk into ount not only th sutrs o th tr, ut lso thos whih n gnrt y som mrging. Our pproh to solv ths two prolms is to us spil kin o tr, th root o whih hs only on hil. Thn, w mk rormultion o th homomorphi lignmnt istn twn this kin o trs, in untion o sutrs with th sm proprty. By this wy, thr is only on nw r to tk into onsirtion t h stp, th on twn th root n its hil. Morovr, w onsir tht this r n th rsult o mrging. In omintion with th rursiv ntur o th ottom-up pproh, it will l to onsir sutrs with ll possil mrgings. Finlly, w lso n rormultion o th homomorphi lignmnt istn twn lssi trs in untion o th on twn th sutrs sri ov.... Dinitions n nottions First, w n to in mor ormlly th prtiulr kin o tr sri ov. Lt T = (V,A,ω) wight tr root in r T. W not y T(v),v V, th sutr o T root in v (s Fig.7). Lt v n nstor o v, w not y T ut (v,v ) th sugrph o T otin rom T(v ) y rmoving ll omplt sutrs whih o not ontin vrtis o T(v). W not y T(v,v ) th tr otin rom T ut (v,v ) y mrging on h vrtx n n(v) \ {v,v}. W sy tht T(v,v ) is th sutr o T root in v prun in v, n w ll this kin o trs prun tr (s Fig.7 or n xmpl). W not y F(T, v) th prun orst, th onnt omponnts o whih r th trs T(p,v), or ll p C(v) (s Fig.7 or n xmpl). By us o nottion w lso not y F(T,v) th st o ll onnt omponnts o this orst.... Rormultions In orr to us ottom-up pproh, w hv to xprss th homomorphi lignmnt istn: twn trs in untion o th istns twn thir prun orsts (Prop. ). twn prun trs in untion o th istns twn thir prun sutrs (Prop. ). twn prun orsts in untion o th istns twn thir prun sutrs (Prop. 5).
g h i j h i j h i h i j T T() T(, ) F(T, ) Figur 7: From th lt to th right: tr T, sutr o T root in, sutr o T root in n prun in, n prun orst o T with origin. hl-006879, vrsion - Mr 0 In ition, w hv to rormult ths istns in th spil ss whr t lst on o th trs is mpty (Prop. ). In th squl w onsir two wight trs P = (V P,A P,ω P ) n D = (V D,A D,ω D ), root rsptivly in r P n r D. Proposition. W hv : η(p,d) = η(f(p,r P ), F(D,r D )). Proo. Sin th root o tr nnot limint y ny oprtion (mrging, ltion) involv in th inition o η, it my sn tht ny homomorphi lignmnt H o P with D hs orrsponing lignmnt H o F(P,r P ) with F(D,r D ) o sm ost, n th onvrs lso hols. Proposition. Lt i V P \ {r P },j V D \ {r D },i n(i),j n(j), Proo. Strightorwr. η(, ) = 0 η(p(i,i ), ) = η(f(p,i), ) + γ(ω P (i,i),0) η(f(p,i ), ) = η(p(i,i ), ) i C(i ) η(,d(j,j )) = η(, F(D,j)) + γ(0,ω D (j,j)) η(, F(D,j )) = η(,d(j,j )). j C(j ) Proposition. Lt i V P \ {p},j V D \ {},i n(i),j n(j). η(p(i,i ),D(j,j )) = η(p(i,i ), ) + η(,d(j,j )), γ(ω min P (i,i),ω D (j,j)) + η(f(p,i), F(D,j)), min j C(j){η(P(i,i ),D(j,j )) + η(, F(D,j) \ D(j,j))}, min i C(i){η(P(i,i ),D(j,j )) + η(f(p,i) \ P(i,i), )}.
Proo. Lt H n homomorphi lignmnt o P(i,i ) with D(j,j ), n lt H L = (V L,A L,ω L ) n H R = (V R,A R,ω R ) th lt n right grphs o H, rsptivly. Thr r svn possil ss:. H is n mpty st (it is hpr to rmov oth P(i,i ) n D(j,j ) thn to lign thm).. {(i,i),(j,j)} H.. A R, ing otin y mrging (j,j) with othr rs. In this s, thr is on n only on hil j o j, suh (j,j ) is mrg with (j,j) in, n thn ll D(j,j),j C(j) \ {j } r lt.. A L, ing otin y mrging (i,i) with othr rs. In this s, thr is on n only on hil i o i, suh (i,i ) is mrg with (i,i) in, n thn ll P(i,i),i C(i) \ {i } r lt. hl-006879, vrsion - Mr 0 Css,,, justiy, rsptivly, th lins,,, o th xprssion o η(p(i,i ),D(j,j )) in th proposition. Th thr lst ss nnot l to ttr homomorphi lignmnt: 5. Th ltion o (i,i) n (j,j) nnot prrr to th rsizmnt (possil s ), us γ(ω P (i,i),0) + γ(0,ω D (j,j)) γ(ω P (i,i),ω D (j,j)). 6. I (i,i) ws lt, n not (j,j), thn only on P(i,i),i C(i) is lign with D(j,j ), th othr ing rmov. It is lss xpnsiv to mrg (i,i) with (i,i ) (possil s ), us γ(ω P (i,i),0) + γ(ω P (i,i ),ω D (j,j)) γ(ω P (i,i ),ω D (j,j)). 7. Th ltion o (j,j) is mor xpnsiv thn th mrging o (j,j) (possil s ), or th sm rsons s ov. Proposition 5. A F(P,i),B F(D,j), η(a,b) = min D(j,j) B {η(a,b \ {D(j,j)}) + η(,d(j,j))}, min P(i,i) A {η(a \ {P(i,i)},B) + η(p(i,i), )}, min P(i,i) A,D(j,j) B {η(a \ {P(i,i)},B \ {D(j,j)}) +η(p(i min,i),d(j,j))}, min P(i,i) A,B B {η(a \ {P(i,i)},B \ B ), +η(f(p,i ),B ) + γ(ω P (i,i ),0)}, min A A,D(j,j) B {η(a \ A,B \ {D(j,j)})+ η(a, F(D,j )j) + γ(0,ω D (j,j ))}. Proo. Lt H n homomorphi lignmnt o A F(P,i) with B F(D,j), n lt P(i,i) A n D(j,j) B. Thr r iv possil ss:. D(j,j) is not lign with lmnt o A,
. P(i,i) is not lign with lmnt o B,. P(i,i) is lign with D(j,j),. isjoint suprts o P(i,i) r lign with lmnts o B, 5. isjoint suprts o D(j,j) r lign with lmnts o A. Css,,,, 5 justiy, rsptivly, th lins,,,, 5 o th xprssion o η(a,b) in th proposition. hl-006879, vrsion - Mr 0... Algorithm Using th rormultions in ov, w n now sign our lgorithm ollowing th ottom-up pproh. Algorithm : Homomorphi Alignmnt Distn or Root Trs Dt: pttrn root tr P, t root tr D Rsult: η(p,d) = η(f(p,r P ), F(D,r D )); // Prop. gin orh p V P, in suix orr o orh A F(P,p) o Comput η(a, ); // Prop. orh p n(p) \ {p} o Comput η(p(p,p ), ); // Prop. orh V D, in suix orr o orh B F(D, ) o Comput η(,b); // Prop. orh n() \ {} o Comput η(,d(, )); // Prop. orh p V P, in suix orr o orh V D, in suix orr o orh A F(P,p) o orh B F(D, ) o Comput η(a,b); // Prop.5 orh p n(p) \ {p} o orh n() \ {} o Comput η(p(p,p ),D(, )); // Prop. n... Complxity W not y N th mximum gr o vrtx in th pttrn tr or in th t tr, H th mximl hight o oth trs n S th mximl siz (numr o vrtis) o oth trs. W rll tht st o siz n hs n susts. Th tim omplxity o omputing h rormultion us in th lgorithm is: 5
η(p(i,i ), ) n η(,d(j,j )) O(). η(a, ) n η(, B) O(N). η(p(i,i ),D(j,j )) O(N). η(a,b) O(N N ). Comining ths omplxitis with th irnt loops o th lgorithm, w otin tht th totl omputtion is in O(S (N N +H N)) tim omplxity. I th mximl gr is oun, th totl omputtion is in O(S H ) tim omplxity. hl-006879, vrsion - Mr 0.. Algorithm or unroot trs First, lt us giv n xprssion o th homomorphi lignmnt istn twn unroot trs, in untion o th istns twn ll thir possil root vrsions. Lt G = (V,E,ω) wight tr, lt r V, w not y G r, th irt wight tr root in r, suh tht G is th unirt grph ssoit to G r (s Fig.8). G G G g Figur 8: A tr G n th root trs G n G. Proposition 6. Lt P = (V P,E P,ω P ) n D = (V D,E D,ω D ) two wight trs. W hv: η(p,d) = min i VP,j V D {η(p i,d j )}. g g Proo. Lt G = (V,E,ω) grph, n r V vrtx o G. Noti tht: mrging ourring on vrtx v V \ {r} in G n our in G r, ltion, insrtion, rsizmnt, n ivision ourring in G n our in G r. On th othr hn, or h optiml homomorphi lignmnt H o P in D, it is sy to s tht thr xists p V P n V D, suh tht p n r not t y mrging. For xmpl, i D = (V D,E D,ω D ) is sugrph suh s η(p,d) = η(p,d ), p n n hosn s -gr vrtis o V P n 6
V D, rsptivly. As rsult, η(p,d) = η(p p,d ). Sin th homomorphi lignmnt is mor onstrin in th s o root trs thn in th s o unroot trs, w n ssur thn η(p,d) η(p,d ), V P, V D. To sum up, knowing tht η(p,d) η(p,d ), V P, V D, n tht thr xists p V P n V D, suh tht η(p,d) = η(p p,d ), w onlu tht η(p,d) = min i VP,j V D η(p i,d j ).... Niv lgorithm Lt P = (V P,E P,ω P ) n D = (V D,E D,ω D ) two wight trs. From proposition 5, w propos irst, niv lgorithm to omput η(p, D), onsisting o omputing th homomorphi lignmnt istn or ll oupls o wight root trs w n otin rom P n D, n kping th minimum rh. hl-006879, vrsion - Mr 0 Complxity. Lt P = (V P,E P,ω P ) n D = (V D,E D,ω D ) two wight trs. As th numr o root trs w n otin rom n unirt tr is qul to th numr o vrtis o this grph, th numr o oupls o wight root trs w n otin rom P n D is qul to V P V D. Th totl omputtion is thn in O( V P V D ( P D ( D P + P D ) + h P h D ( P + D ))) tim omplxity, whr h P (rsptivly, h D ) is th mximl hight o root tr otin rom P (rsptivly, D). I th mximl gr is oun, th totl omputtion is in O( V P V D h P h D ) tim omplxity.... Optimiz lgorithm It is sy to s tht th ov lgorithm omputs th lignmnt o suprts o P n D mor thn on tim. Using ynmi progrmming n n pt orr o nvigtion in th tr, w n voi uslss omputtion. Lt P = (V P,E P,ω P ) n D = (V D,E D,ω D ) two unirt wight trs. W not y F(P,,),, V P th st o root trs P r, r V P, suh tht is n nstor o in P r. W not y n(p,,),, V P th st o vrtis x V P suh tht x is n nstor o in t lst on root tr in F(P,,). W not y C(P,,),, V P th st o o vrtis x V P suh tht x is hil o in t lst on root tr in F(P,,). It is sy to s tht or omputing η(p p (i,i ),D (j,j )) n η(f(p p,i), F(D,j)), w n to know η(p p (i,i),d (j,j)) or ll i C(P,i,p), j C(D,j,). W n strt y omputing η(p p (i,i ),D (j,j )) n η(f(p p,i), F(D,j)), or ll i (rsptivly, j) ing l o P p (rsptivly, D ), whih hv no hil, y inition, n ontinu itrtivly with ll vrtis whih hv ll thir hilrn lry omput. An pt orr o nvigtion or tr T = (V,E,ω) n otin y th ollowing lgorithm. 7
hl-006879, vrsion - Mr 0 Algorithm : Orr o nvigtion omputtion lgorithm (omput- Orr) Dt: unroot tr T Rsult: list o oupls o vrtis L gin list o oupls o vrtis L LIFO quu o vrtis Q orh v V o i g(v) = thn push v in Q n whil Q o pop v o Q orh w N(v) o i (v,w) / L thn i x N(v) \ {w},(x,v) L thn push w in Q (v,w) t th n o L Complxity. Lt T = (V,E,ω) n unroot tr. Eh vrtx o gr (on) will put in th quu, thn, or h g, h o its vrtis will put twi in th quu. As V = E +, th omplxity o omputorr is in O( E ). Finl lgorithm. W n omput Homomorphi Alignmnt or unroot trs with improv omplxity, using this orr o nvigtion. 8
Algorithm : Homomorphi Alignmnt Distn or Unroot Trs Dt: pttrn root tr P, t root tr D Rsult: η(p, D) gin list o oupls o vrtis L P omputorr(p) list o oupls o vrtis L D omputorr(d) orh (p,p ) L P o orh A F(P p,p) o Comput η(a, ); // Prop. orh p n(p,p,p ) \ {p} o Comput η(p p (p,p ), ); // Prop. hl-006879, vrsion - Mr 0 orh (, ) L D o orh B F(D,) o Comput η(,b); // Prop. orh n(d,, ) \ {} o Comput η(,d (, )); // Prop. orh (p,p ) L P o orh (, ) L D o orh A F(P p,p) o orh B F(D,) o Comput η(a,b); // Prop.5 orh p n(p,p,p ) \ {p} o orh n(d,, ) \ {} o Comput η(p p (p,p ),D (, )); // Prop. Comput η(p,d); // Prop. n Prop.6 n Complxity. Lt P = (V P,E P,ω P ) n D = (V D,E D,ω D ) two wight trs. Th initiliztion o L P n L D is in O( V P P + V D T ) tim omplxity. Osrv tht, or p V P, n(p,p,p ) \ {p} = V P. As rsult, p N(p) (p,p ) L P n(p,p,p ) \ {p} = ( V P ) V P. Comining th tim omplxitis o omputing th rormultions (givn in th S...) with th loops o th lgorithm, n tking into ount th ov 9
osrvtion, w otin tht th totl omputtion tim o this lgorithm is in O(S (N N + S N)) omplxity, with N n S in s in S... I th mximl gr is oun, th totl omputtion is in O(S ) tim omplxity. 5. Cut oprtion hl-006879, vrsion - Mr 0 For th purpos o rmoving spurious rnhs without ny ost, w propos to intgrt th ut oprtion in our lignmnt. In [0], Wng t l. propos nw oprtion llowing to onsir only prt o tr. Lt G = (V,A,ω) wight tr. Cutting G t n r A, mns rmoving, thus iviing G into two sutrs G n G. Th ut oprtion onsists o utting G t n r A, thn onsiring only on o th two sutrs. Lt K sust o A. W us Cut(G,K,v) to not th sutr o G ontining v n rsulting rom utting G t ll rs in K. In th s o root tr, w onsir tht th root r G o G nnot rmov y th ut oprtion, n thn w us th nottion Cut(G,K) = Cut(G,K,r G ). In th s o root orst, w onsir tht th root o h root tr omposing th root orst nnot rmov y th ut oprtion, n thn w us th sm nottion thn ov: Cut(G,K). S Fig.9 or som xmpls o ut oprtion. g h i j g h i j g h i G Cut(G, {{, }}, ) Cut(G, {{, }, {h, j}}, ) k o t k o t l p q u l q u m n r s n s F Cut(F, {(l,m),(o, p)}) Figur 9: Exmpls o ut oprtions. Top row: on n unroot tr. Bottom row: on root orst. 0
Our min prolm n stt s ollows: Givn wight tr P = (V P,A P,ω P ) (th pttrn tr) n wight tr G D = (V D,A D,ω D ) (th t tr), in η ut (P,D) = min K AD,v V D {η(p,cut(d,k,v)} n th ssoit homomorphi lignmnt. In th s o root trs n root orsts, η ut (P,D) = min K AD {η(p,cut(d,k))}. 5.. Intgrtion o ut oprtion in our lgorithm It n sn tht th ut oprtion n intgrt in th homomorphi lignmnt y rpling th ltion o omplt sutrs y thir ut. In orr to otin th lgorithms or th omputtion o η ut, w just n to rpl: η y η ut in th rormultions n lgorithms o th Sus.. n. Prop. y th ollowing: hl-006879, vrsion - Mr 0 Proposition 7. Lt i V P \ {p},j V D \ {},i n(i),j n(j), η ut (, ) = 0 η ut (P(i,i ), ) = η ut (F(P,i), ) + γ(ω P (i,i),0) η ut (F(P,i ), ) = η ut (P(i,i ), ) η ut (,D(j,j )) = 0 η ut (, F(D,j )) = 0. i C(i ) It is intrsting to rmrk tht th omplxity is not moii. 6. Exprimnttion 6.. Usg o homomorphi lignmnt Thr r svrl wys to us th homomorphi lignmnt: I w hv no priori knowlg oth on pttrn tr P n on t tr D, w n to us th homomorphi lignmnt on th two unroot trs. In this s, th omplxity is in O( V P V D ). I w wnt to sur tht spii vrtx v in th t tr is lign with spii vrtx w in th t tr (or xmpl i w r sur thn th vrtx o th h in th t is th on with th highst z-oorint), w n us th homomorphi lignmnt twn P v n D w. In this s, th omplxity is in O( V P V D h P h D ), whr h P (rsptivly, h D ) rprsnts th hight o P (rsptivly, D). I w wnt to sur tht spii vrtx v in th mol tr is lign (i.. thr is no mrging on v), w n us th homomorphi lignmnt twn P v n D, y sussivly omputing th homomorphi lignmnt twn P v n D w, or ll w V D, n using optimiztions s in... In this s, th omplxity is in O( V P h P V D ).
I w wnt to sur tht spii vrtx v in th t tr is lign (i.. thr is no mrging or ut on v), w n us th homomorphi lignmnt twn P n D v, using th sm mtho s ov. In this s, th omplxity is in O( V P V D h D ). In our pplition, onsisting to in th initil pos o sujt, w n t lst ssum tht th torso o th sujt will lign. Thn, w n us th mthing with O( V P h P V D ) or th initiliztion. For th trking, i prt o th sujt is stti, w n us th lst lignmnt o this prt or otin mthing in O( V P V D h P h D ). hl-006879, vrsion - Mr 0 6.. Rsults Our mol tr ontins svn vrtis, rprsnting h, torso, roth, th two hns n th two t. Exprimntlly, th t tr otin rom th sklton o th visul hull hs gr oun y, n its numr o vrtis is twn svn n twnty, with Gussin proility rprtition ntr on tn. All th rsults hv n otin on omputr with prossor Intl(R) Cor(TM) Qu Q800 t. GHz n Go o RAM, with Linux, Uuntu 9.0. Th lgorithms hv n implmnt in C++. 6... Sp Protool. To in th vrg omputtion tim o involv lgorithms, w hv rnomly gnrt pttrn trs, n or h on, t trs, yiling 0 pirs o trs. Eh pttrn tr hs svn vrtis, on o whih hs gr qul to. Eh t tr hs t lst on -gr vrtx. Th rsults or oth lignmnt n homomorphi lignmnt lgorithms, or th irnt kins o trs, r shown in igur 0. Disussion. For our purpos o initilizing th pos o sujt, w n s tht in th vrg s ( V = ), th homomorphi lignmnt n omput vry quikly (rquny 00), vn in th s o unroot trs. Howvr, w prr to us th homomorphi lignmnt with root pttrn tr, us o its ttr sp, without loss o prision. Th min osrvtion w n o, in rgr o th rsults, is th wk irn o prormns twn lignmnt n homomorphi lignmnt in prti, vn i thr is signiint irn o omplxity twn lignmnt n homomorphi lignmnt. It is u to smll siz o th trs involv hr, n to th t tht, vn i th gr o th trs is oun, n y this wy, n onsir s onstnt, th omputtion o th lignmnt (or homomorphi lignmnt) o suorsts tk lrg prt o th omputtion tim. This omputtion prt ing similr or oth lgorithms, th rst (whr th omplxity irs) is lss signiint in rgr o th sp.
rquny (Hz) 600 500 00 00 00 00 0 Frqunis or Vp = 7, on root trs 0 5 0 5 0 rquny (Hz) 500 00 00 00 00 0 Frqunis or Vp = 7, on root pttrn tr, unroot t tr 0 5 0 5 0 hl-006879, vrsion - Mr 0 rquny (Hz) 00 00 00 00 0 Frqunis or Vp = 7, on unroot trs 0 5 0 5 0 V Alignmnt Homomorphi Alignmnt Figur 0: Frqunis o lignmnt n homomorphi lignmnt or vril sizs o t tr, or th irnt rooting ss. 6... Aury Protool. Eh xprimnt onsists o th ollowing:. W rnomly gnrt pttrn tr P.. W gnrt t tr D rom P in two stps: wight vrition: w rnomly ltr th g wights y givn mount o vrition. struturl nois: w rnomly nw vrtis y thr wys, orrsponing to th irnt typs o nois: splitting n xisting vrtx, n linking th two prts y 0-wight g (splitt vrtis), ing nw -gr vrtx y th split o n g (uslss -gr vrtis), n ing nw -gr vrtx, link y n rnomly wight g (spurious rnhs n ghost lims).. W omput th lignmnt n th homomorphi lignmnt twn P n D.. Th prision is givn y th prntg o goo mthings, tht is th prntg o pttrn tr vrtis tht mth with thir quivlnt in th
t tr (whih is known, oring to th ov protool to gnrt th t tr). Th rsults, otin y vrging prisions ovr 000 xprimnts, r shown on Fig.. % o goo mth 00 80 60 0 0 0 0 50 00 50 00 50 00 00 80 60 0 0 0 0 50 00 50 00 50 00 00 80 60 0 0 0 0 50 00 50 00 50 00 00 00 00 hl-006879, vrsion - Mr 0 % o goo mth 80 60 0 0 0 0 50 00 50 00 50 00 % o struturl nois 80 60 0 0 0 0 50 00 50 00 50 00 % o struturl nois Alignmnt 80 60 0 0 0 0 50 00 50 00 50 00 % o struturl nois Homomorphi Alignmnt Figur : Prision with irnt prnts o wight vrition, or irnt sizs o pttrn tr. First row: pttrn tr with 0 vrtis. Son row: pttrn tr with 0 vrtis. Columns, rom lt to right: 0%, 0% n 50% o wight vrition. Disussion. Th irst importnt point whih n osrv is tht homomorphi lignmnt giv lwys prt mthings whn only struturl nois ours. In th othr ss, th homomorphi lignmnt glolly givs mor urt mthing thn th lignmnt, spilly in rgr o th struturl nois. Howvr, in s o high wight vrition n low struturl nois, th rsults o lignmnt r ttr thn thos o homomorphi lignmnt. It is u to spil s o sutr n wight vrition, whih ts th ury o homomorphi lignmnt ut not th lignmnt (s Fig. or n xmpl). In th s o our pplition, w mpirilly osrv wight vrition twn 0 n prnt, n struturl nois twn 0 n 00 prnt. For th mjority o ths ss, homomorphi lignmnt giv snsily ttr rsults. 6... Pos Initiliztion W hv hk th lignmnt on svrl typs o visul hulls, with irnt rsolutions (6 n 8 ), with or without spurious rnhs n ghosts lims. Som o ths rsults r shown in Fig.. Th ghost lim n mth y th lgorithm only i its position on th sklton is th sm s nothr lim, n i it hs pproximtly th sm lngth. In th othr ss, it is sussully rmov.
P P 0 0 Alignmnt Homomorphi Alignmnt 8 8 9 D D Figur : Spil s whr lignmnt is ttr thn homomorphi lignmnt. First olumn: P is th pttrn tr, D is otin rom P y wight vrition o 0% on {, }, n y ing nw -gr vrtx, link to y n g wight y. Son olumn: optiml lignmnt. P = P n D = Cut(D, {{, }}, ). Aury is qul to 00%. Lst olumn: homomorphi lignmnt. P n D r lry homomorphi. P = P n D = MK(D). Aury is qul to 50%. P 0 D hl-006879, vrsion - Mr 0 Th spurious rnhs o not istur th goo lignmnt o th mol on th t. Th only s o lignmnt hs n otin on vry low qulity visul hull, whr th lngth o lgs ws shortr thn th lngth o rms. This s n solv y rooting th mol tr on th h, n th t tr on th vrtx with highst z-oorint. T H C A A F F A T C H A H A T A A A C F F F F F F Figur : Exmpls o rsults otin on D shps with irnt rsolutions n noiss: grn voxls rprsnt th points mthing with pttrn tr. C T H 7. Conlusion In this ppr, w hv introu nw typ o lignmnt twn wight trs, th homomorphi lignmnt, tking into ount th topology n voiing th nois inu y spurious rnhs, splitt n uslss -gr vrtis. W hv lso vlop svrl roust lgorithms to omput it with goo omplxity, whih nl its pplition in rl tim or motion ptur purpos. In utur works, w will tk into ount mor usul inormtion on th mol, suh s sptil oorints o t vrtis, n inlu thm in our lgorithm, or ttr roustnss. Finlly, using this lignmnt, w will propos nw st mtho o pos initiliztion or motion ptur pplitions. 5
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