Geometry. Topological Persistence and Simplification. Herbert Edelsbrunner, 1 David Letscher, 2 and Afra Zomorodian 3. 1.

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1 Dicree Comp Geom 28: (22 DOI: 1.17/ Dicree & Compaional Geomery 22 Springer-Verlag Ne Yor Inc. Topological Perience and Simplificaion Herber Edelbrnner, 1 Daid Lecher, 2 and Afra Zomorodian 3 1 Deparmen of Comper Science, De Unieriy, Drham, NC 2778, USA and Raindrop Geomagic, Reearch Triangle Par, NC, USA 2 Deparmen of Mahemaic, Olahoma Sae Unieriy, Sillaer, OK 7478, USA 3 Deparmen of Comper Science, Unieriy of Illinoi a Urbana-Champaign, Urbana, IL 6181, USA Abrac. We formalize a noion of opological implificaion ihin he frameor of a filraion, hich i he hiory of a groing complex. We claify a opological change ha happen dring groh a eiher a feare or noie depending on i lifeime or perience ihin he filraion. We gie fa algorihm for comping perience and experimenal eidence for heir peed and iliy. 1. Inrodcion The need for aomaed opological implificaion ha been ariclaed in he comper graphic and geomeric modeling lierare. Thi paper propoe a olion in hich cale i ed o ae he perience of opological aribe and o prioriize implificaion ep. Afer decribing a ne noion of opological implificaion, e mmarize he conribion of hi paper and conra hem ih prior or. Topological Simplificaion. We e homology o meare he opological complexiy of a poin e in R 3. The imple non-empy e nder hi meare are he one ha conrac o a poin. Each ch e coni of one componen and ha no oher non- Reearch by he fir and hird ahor a parially ppored by ARO nder Gran DAAG Reearch by he fir ahor a alo parially ppored by NSF nder Gran CCR

2 512 H. Edelbrnner, D. Lecher, and A. Zomorodian riial homological aribe. A general e in R 3 ha β componen, β 1 nnel, and β 2 oid. We conider opological complexiy o be expreed by β, β 1, β 2, he Bei nmber of he e. A ch, e nderand opological implificaion a a proce ha decreae Bei nmber. To do hi in a geomerically meaningfl manner, e need a ay of aeing he imporance of opological aribe. Once e hae ch a nmerical aemen, e naiely remoe aribe in he order of increaing imporance. A any momen dring hi proce, e may call he remoed aribe opological noie and he remaining one opological feare. There are hree echnical difficlie ih hi approach. The fir i he idenificaion of be expreing he non-riial opological aribe ha are meared by homology grop. The econd i he mearemen of he imporance of hee be. The hird i he eliminaion of a opological aribe ih a minimm nmber of ide effec. We oercome hee difficlie in hi paper and decribe a implificaion proce a eniioned aboe. Approach and Rel. We reric or aenion o e repreened by finie implicial complexe in R 3. For pracical reaon, moreoer, e foc on pariclar bcomplexe of Delanay rianglaion called alpha complexe 4]. We receie eenial help in oercoming ome echnical difficlie by aming a filraion hich place he complex ihin an eolionary groh proce. Gien a filraion, he main conribion of hi paper are: (i he definiion of perience for Bei nmber and non-bonding cycle, (ii an efficien algorihm o compe perience, (iii a implificaion algorihm baed on perience. Prior and Relaed Wor. A menioned earlier, e e homology grop and Bei nmber hich ere deeloped and refined dring he fir half of he enieh cenry. We refer o 9] for a decripion ha i reaonably acceible o non-peciali. Specral eqence are he by-prodc of a diide-and-conqer mehod for comping homology grop and Bei nmber 7]. Thee eqence form a frameor ihin hich or rel on perien Bei nmber may be placed. The algorihm e deelop for comping perience of non-bonding cycle i baed on he incremenal Bei nmber algorihm of Delfinado and Edelbrnner 2]. Three-dimenional alpha hape and complexe may be fond in 4]. The problem of opological implificaion a alo approached by El-Sana and Varhney 5] ing alpha hape inpired idea of geomeric groh. There i a large body of parallel or on io-rface or leel e of hree-dimenional deniy fncion. We refer o 8] for he mahemaic and o 11] for a nmerical ie. A deniy fncion i a map F: R 3 R and an io-rface F 1 (α i he preimage of a conan image ale α. The eqence of io-rface obained by increaing α repreen a groh proce imilar o ha repreened by a filraion. Specifically, implice in a filer correpond o criical poin of a deniy fncion. In hi conex, opological implificaion mean redcing he nmber of criical poin. Thi proce i relaed o moohing or implifying he graph of F, hich i a hree-dimenional manifold in R 4.

3 Topological Perience and Simplificaion 513 Oline. Secion 2 reie alpha complexe and homology grop. Secion 3 inrodce perience for Bei nmber and non-bonding cycle. Secion 4 decribe an algorihm ha compe perience. Secion 5 formlae implificaion algorihm baed on perience. Secion 6 proide experimenal eidence for he peed and iliy of hee algorihm. Secion 7 conclde he paper. 2. Bacgrond Thi ecion inrodce he bacgrond e need o define and compe opological perience. We begin ih alpha complexe, conine ih homology grop for Z 2 coefficien, and end ih he incremenal algorihm for comping Bei nmber. Alpha Complexe. A pherical ball û = (, U 2 R 3 R i defined by i cener and qare radi U 2. If U 2 <, he radi i imaginary and o i he ball. The eighed diance of a poin from a ball i πû(x = x 2 U 2. Noe ha a poin x R 3 belong o he ball iff πû(x, and i belong o he bonding phere iff πû(x =. Le S be a finie e of ball. The Voronoi region of û S i he e of poin for hich û minimize he eighed diance, Vû = {x R 3 πû(x π ˆ (x, ˆ S}. The Voronoi region decompoe he nion of ball ino conex cell of he form û Vû, a illraed in Fig. 1. Any o region are eiher dijoin or hey oerlap along a hared porion of heir bondary. We ame general poiion, here a mo for (hree in R 2 Voronoi region can hae a non-empy common inerecion. In pracice, e imlae general poiion compaionally 3]. Le T S hae he propery ha i Voronoi region hae a non-empy common inerecion, and conider he conex hll of he correponding cener, σ T = con { û T }. General poiion implie ha σ T i a -implex, here = card T 1. The dal complex of S i he collecion of implice conrced in hi manner, { } K = σ T T S, û T(û Vû. Figre 1 illrae a o-dimenional example of hi conrcion. Any o implice in K are eiher dijoin or hey inerec in a common face, hich i a implex of maller dimenion. Frhermore, if σ K, hen all face of σ are implice in K. A e of implice ih hee o properie i a implicial complex 9]. A bcomplex i a be L K ha i ielf a implicial complex. Chain, Cycle, Bondarie. Le K be a implicial complex in R 3. A -chain i a be of -implice in K. We define addiion of chain ih ineger coefficien modlo 2. In oher ord, he m of o -chain c, d i he ymmeric difference of he o e, c + d = (c d (c d,

4 514 H. Edelbrnner, D. Lecher, and A. Zomorodian Fig. 1. Union of nine di, conex decompoiion ing Voronoi region, and dal complex. hich i commaie. The e of all -chain ogeher ih addiion form a grop denoed a C. The empy e i he zero elemen of C. There i a chain grop for eery ineger, b for a complex in R 3, only he one for 3 may be non-riial. The bondary (σ of a -implex σ i he collecion of i ( 1-dimenional face, hich i a ( 1-chain. The bondary of a -chain i he m of he bondarie of i implice, (c = σ c (σ. Each bondary operaor i a homomorphim : C C 1 and he collecion of bondary operaor connec he chain grop ino a chain complex, C 3 C2 C1 C. The ernel of i he collecion of -chain ih empy bondary and he image of i he collecion of ( 1-chain ha are bondarie of -chain, er im = {c C (c = }, = {d C 1 c C : d = (c}. A -cycle i a -chain in he ernel of and a -bondary i a -chain in he image of +1. The collecion Z of -cycle and B of -bondarie ogeher ih addiion form bgrop of C. An eenial propery of he bondary operaor i ha he bondary of eery bondary i empy, 1 (c =. Thi implie ha he grop are need, B Z C, a illraed in Fig. 2. The bondary of a erex i he empy e, hich implie ha eery -chain i alo a -cycle, Z = C. Becae K i a complex in R 3, here are no non-empy 3-cycle or 3-bondarie, ha i, Z 3 = B 3 = { }. Homology Grop. The h homology grop i he h cycle grop facored by he h bondary grop, H = Z /B. I elemen are he homology clae c + B = {c + b b B }, for all c Z. The zero elemen i + B = B, and he m of o clae C 3 C2 C1 C = Z Z 2 Z1 B2 B 1 B Z 3 = B3 Fig. 2. Chain, cycle, bondary grop and heir image nder he bondary operaor.

5 Topological Perience and Simplificaion 515 i (c + B + (d + B = (c + d + B. There i no orion in complexe embedded in hree-dimenional Eclidean pace, o he homology grop are ecor pace. A be generae a ecor pace if eery elemen i he m of elemen in he be. A bai i a minimal generaing e. In general, here are no canonical bae, b all bae hae he ame ize hich i he ran of he grop. Becae aing ymmeric difference i lie adding modlo 2, he ize of a grop i 2 raied o he poer of i ran. The h Bei nmber of K i he ran of he h homology grop, β = ran H. A H = Z /B, ran H = ran Z ran B. (1 There i a Bei nmber for each ineger, b for complexe in R 3, only he one for 2 may be non-zero. In he abence of orion, he Bei nmber nder Z 2 are he ame a hoe nder Z, according o he Unieral Coefficien Theorem for Homology 9]. We may ill e Z 2 coefficien in he preence of orion, b e ge Bei nmber ha poibly differ from hoe comped ing Z coefficien. Iniiely, a non-bonding -cycle repreen a collecion of componen of K and here i one bai elemen per componen. I follo ha β i he nmber of componen of K. A non-bonding 1-cycle repreen a collecion of non-conracible cloed cre in K, or, dally, a collecion of nnel formed by K. We can rie each nnel a a m of nnel in a bai, and β 1 i he ize of he bai. A non-bonding 2-cycle repreen a collecion of non-conracible cloed rface in K, or, dally, a collecion of oid, hich are componen of R 3 K. A before, β 2 i he ize of a bai of oid, hich i eqal o he nmber of oid. Finally, here are no 3-cycle becae K i a complex in R 3. Age Filer. We bae all formla and algorihm in hi paper on an ordering of he implice, here each prefix of he ordering conain he implice of a bcomplex. We call ch an ordering a filer. The eqence of bcomplexe defined by aing cceiely larger prefixe i he correponding filraion. Figre 3 illrae hee definiion ih a filer of 18 implice. We hin of a filraion a decribing an eolion of a complex for hich he ole elemen of change i groh. For dal complexe of a collecion of ball, e generae a filer and a filraion by lierally groing he ball. The filer i a coneqence of hi groh. We no decribe he groh model for phere e e in Secion 6 o apply or rel o eighed poin daa in R 3. For eery real nmber α R, e increae he qare radi of a ball û by α, giing û(α = (, U 2 +α. We denoe he collecion of expanded ball û(α a S(α. The α-complex K (α of S i he dal complex of S(α 4]. For example, K ( =, K ( = K, and K ( = D i he dal of he Voronoi diagram, alo non a he Delanay rianglaion of S. For each implex σ D, here i a niqe birh ime α(σ defined ch ha σ K (α iff α α(σ. We order he implice ch ha α(σ < α(τ implie σ precede τ in he filer. More han one implex may be born a a ime and ch cae may arie een if S i in general poiion. For example in Fig. 1, edge i born a he ame momen a riangle. In he cae of a ie, e order loer-dimenional implice before higher-dimenional one, breaing remaining ie arbirarily. We call he reling eqence he age filer of he Delanay rianglaion.

6 516 H. Edelbrnner, D. Lecher, and A. Zomorodian Fig. 3. The filer i he eqence of ligh implice. The correponding filraion i he eqence of complexe. Incremenal Algorihm. The ordering of implice in a filer permi a imple algorihm for comping Bei nmber of all complexe in a filraion 2]. We reie he eenial ep of he algorihm here. Sppoe he eqence of σ i, for i < m, i a filer and he eqence of K i = {σ j j i}, for i < m, i he correponding filraion. Before rnning he algorihm, he Bei nmber ariable are e o he Bei nmber of he empy complex, ha i, β = β 1 = β 2 =. The algorihm i hon in Fig. 4. Hoeer, ho do e decide heher a ( + 1-implex σ i belong o a ( + 1- cycle in K i? For + 1 =, hi i riial becae eery erex belong o a -cycle. For edge e mainain he conneced componen of he complex, each repreened by i erex e. An edge belong o a 1-cycle iff i o endpoin belong o he ame componen. Triangle and erahedra are reaed imilarly, ing he ymmery proided by complemenariy, daliy, and ime-reeral 2]. Once e decide he cycle qeion for each implex, e call a ( + 1-implex σ i poiie if i belong o a ( + 1-cycle and negaie oherie. Le β = β l be he h Bei nmber of K l, and le po = po l and neg = negl be he nmber of poiie and ineger 3 BETTI-NUMBERS ( for i = o m 1 do = dim σ i 1; if σ i belong o a ( + 1-cycle in K i hen β +1 = β ele β = β 1 endif endfor; rern (β, β 1, β 2. Fig. 4. The fncion rern he Bei nmber of he la complex in he filraion.

7 Topological Perience and Simplificaion 517 negaie -implice in K l. The correcne of he incremenal algorihm implie β = po neg +1, (2 for 2. In ord, he Bei nmber β i he nmber of -implice ha creae -cycle min he nmber of ( + 1-implice ha deroy -cycle by creaing - bondarie. Obere ha (2 i j a differen ay o rie (1. All Bei nmber are non-negaie o po neg +1 for all l. We ill ee in Secion 3 ha here exi a pairing beeen poiie -implice and negaie ( + 1-implice. Thi pairing i he ey o nderanding he perience of non-bonding cycle in homology grop. 3. Perience We ih o implify a complex hrogh he remoal of i opological aribe. We decribe a meare ha ran aribe by heir lifeime in a filraion heir perience in being a feare in he face of groh. In hi ecion e inrodce he concep of perience for Bei nmber and non-bonding cycle. We define perience abracly ing cycle and bondary grop of complexe in a filraion. To mae he abrac concree, e gie an algorihm ha pair he creaion of a non-bonding cycle ih i conerion o a bondary. Algebraic Formlaion. Algebraically, i i eay o con he poplaion of nonbonding cycle hoe lifeime exceed a gien hrehold. We ill ee laer ha hi aiic i fficien for deermining he lifeime of indiidal non-bonding cycle. We define Z l, Bl o be he h cycle grop and h bondary grop, repeciely, of he lh complex K l in a filraion. To capre perien cycle in K l, e facor i h cycle grop by he h bondary grop of K l+p, p complexe laer in he filraion. Formally, he p-perien h homology grop of K l i H l,p = Z l /(Bl+p Z l, (3 hich i ell-defined becae B l+p Z l i he inerecion of o bgrop of Cl+p and h a grop ielf. The p-perien h Bei nmber β l,p of K l i he ran of H l,p. Noe ha a e increae p, negaie implice cancel poiie implice earlier in he filraion. In oher ord, increaing p by one horen he perience of all nonbonding cycle by one. We may ill hor-lied aribe, he opological noie of he complex, by increaing p fficienly. The p-perien homology grop can alo be defined ing injecie homomorphim beeen ordinary homology grop. Obere ha if o cycle are homologo in K l, hey alo exi and are homologo in K l+p. Conider he homomorphim η l,p : H l Hl+p ha map a homology cla ino one ha conain i. The image of he homomorphim i iomorphic o he p-perien homology grop of K l, im η l,p H l,p. Finally, e noe ha Robin ha independenly formlaed an alernae b eqialen definiion of perience 1].

8 518 H. Edelbrnner, D. Lecher, and A. Zomorodian Abrac Algorihm. To meare he lifeime of a non-bonding cycle, e find hen he cycle homology cla i creaed and hen i cla merge ih he bondary grop. A poiie implex creae he cla and a negaie one merge he cla ih he bondary grop. To deec hee een, e mainain a bai for H implicily hrogh implex repreenaie. Iniially, he bai for H i empy. For each poiie -implex σ i, e fir find a nonbonding -cycle c i ha conain σ i b no oher poiie -implice. We proe ha c i exi ing indcion a follo: ar ih an arbirary -cycle ha conain σ i and remoe oher poiie -implice by adding heir correponding -cycle. Thi mehod cceed becae each added cycle conain only one poiie -implex by indcie ampion. Afer finding c i, e add he homology cla of c i a a ne elemen o he bai of H. In hor, he cla c i + B i repreened by c i, and c i, in rn, i repreened by σ i. For each negaie (+1-implex σ j, e find i correponding poiie -implex σ i and remoe he homology cla of σ i from he bai. A general homology cla of K i i a m of bai clae, d + B = (c g + B = B + c g. The chain d and c g are homologo, meaning hey belong o he ame homology cla. Each c g i repreened by a poiie -implex σ g, g < j, ha i no ye paired by he algorihm. The collecion of poiie -implice Ɣ = Ɣ(d i niqely deermined by d. The yonge implex in Ɣ i he one ih he large index and e denoe hi index a y(d. The algorihm, a hon in Fig. 5, idenifie σ j a he clpri for rning he -cycle creaed by σ i ino a -bondary. We docmen hi by appending (σ i, σ j o he li L. The perience of ha -cycle i one hor of he difference beeen indice, j i 1. Time-Baed Perience. Alernaiely, e cold define perience a he difference in birh ime of he o implice, α(σ j α(σ i. Thi ie correpond o an exenion of or preio index-baed formlaion. Le K α = {σ j α(σ j α}. Then for eery li 3 PAIR-SIMPLICES ( L = L 1 = L 2 = ; for j = o m 1 do = dim σ j 1; if σ j i negaie hen (* d = +1 (σ j ; i = y(d; L = L {(σ i, σ j } endif endfor; rern (L, L 1, L 2. Fig. 5. The fncion rern hree li of paired implice in he filer.

9 Topological Perience and Simplificaion 519 real π, e define he π-perien h homology grop of K α o be H α,π = Z α /(Bα+π Z α. (4 Noe ha e are no changing he ordering of he implice, o he implex pair do no change. Therefore, e may e he abrac algorihm aboe o compe he perience pair. In general, hoeer, e hae feer pairie differen complexe, a all implice ih birh ime α ener he filraion a ha ime. In he index-baed formlaion, he implice arrie indiidally ih differen indice. Time-baed perience i efl in he conex of io-rface of deniy fncion. Index-baed perience i appropriae for alpha complexe, a mo inereing aciiy occr in a mall range of α. We do no dic ime-baed perience in hi paper any frher, alhogh all or rel ill be alid ih lile modificaion. Vializaion. Or pairing algorihm gie a e of implex pair (σ i, σ j, each repreening a -cycle for 2. We may ialize each pair on he index axi by a half-open ineral i, j hich e call a -ineral. We ho hi in Fig. 6 for he filraion in Fig. 3. The incremenal algorihm in Secion 2 aer ha β l i he nmber of -ineral ha conain index l on he axi. We exend hi ializaion o o dimenion panned by he index and perience axe. The -ineral of (σ i, σ j i exended ino a -riangle panned by (i,, ( j,, (i, j i in he index-perience plane. The -riangle i cloed along i erical and horizonal edge and open along he diagonal connecing ( j, o (i, j i, a hon in Fig. 6. I repreen he -cycle creaed by σ i, hich i deroyed by σ j progreiely earlier a e increae p. Ame for a momen ha or algorihm i correc, hich mean β l,p i he nmber of -riangle ha conain poin (l, p. Then he perien Bei nmber are non-increaing along erical line in he index-perience plane. The ame i re for line in he diagonal direcion and for all line beeen he erical and he diagonal direcion index perience Fig. 6. Vializaion of he rel of Fncion PAIR-SIMPLICES. The riangle of and of are nbonded and no dran. The ligh and dar riangle repreen -cycle and 1-cycle, repeciely..

10 52 H. Edelbrnner, D. Lecher, and A. Zomorodian Monooniciy Lemma. β l,p β l,p heneer p p and l l l + (p p. Correcne. To proe he abrac algorihm i correc, e ho ha he pair i prodce are conien ih he perien Bei nmber defined by (3. The nmber of -riangle conaining (l, p in he index-peri- -Triangle Lemma. ence plane i β l,p. Proof. We proceed by indcion oer p. For p =, he nmber of -riangle ha conain (l, i eqal o he nmber of -ineral i, j ha conain l. Thi i eqal o he nmber of lef endpoin min he nmber of righ endpoin ha are maller han or eqal o l. Eqialenly, i i he nmber of poiie -implice σ i ih i l min he nmber of negaie ( + 1-implice σ j ih j l. Hoeer, hi i j a reaemen of (2, hich eablihe he bai of he indcion. Conider (l, p ih p > and ame indciely ha he claim hold for (l, p 1. The relean implex for he ep from (l, p 1 o (l, p i σ l+p. The perien h Bei nmber can eiher ay he ame or decreae by 1. I ill decreae only if σ l+p i a negaie ( + 1-implex, or, eqialenly, if (l + p, i he pper righ corner of a -riangle. Indeed, no oher -riangle can poibly eparae (l, p 1 and (l, p. Thi proe he claim if σ l+p i a poiie ( +1-implex or a implex of dimenion differen from + 1. No ppoe ha σ l+p i a negaie ( + 1-implex and define he -cycle d = +1 (σ l+p. There are o cae, a hon in Fig. 7: Cae 1. Ame here i a -cycle c in K l homologo o d, ha i, c d + B l+p 1. Then c bond neiher in K l nor in K l+p 1, b i bond in K l+p. If follo ha β l,p = β l,p 1 1. We need o ho ha he pair (σ i, σ l+p conrced by he algorihm aifie i l, becae only in hi cae doe he -riangle of σ l+p eparae (l, p 1 from (l, p. Recall ha σ i i he yonge poiie -implex in Ɣ(d. To reach a conradicion ppoe i > l. Then c i a non-bonding -cycle alo in K i, and becae i i homologo o d, e hae σ i c. Hoeer, hi conradic c K l a σ i / K l. Cae 2. Ame here i no -cycle in K l homologo o d. Then Z l Bl+p 1 = Z l Bl+p, and hence β l,p = β l,p 1. We need o ho ha he pair (σ i, σ l+p conrced by he algorihm aifie i > l, becae only in hi cae doe he -riangle of p -1 p i < l l i > l l+p index perience p+l Fig. 7. The ligh -riangle correpond o Cae 1 and he dar one o Cae 2.

11 Topological Perience and Simplificaion 521 σ l+p no eparae (l, p 1 from (l, p. Or ampion aboe implie ha a lea one of he poiie -implice in Ɣ(d a added afer σ l. Hence i = y(d > l. 4. Compaion In hi ecion e complee he abrac algorihm for pairing by pecifying ho o implemen line (* of Fncion PAIR-SIMPLICES. We do hi by comping he index i of he yonge poiie -implex in Ɣ(d, here d = +1 (σ j. We refer o hi compaion a cycle earch for σ j. We fir decribe he daa rcre, hen explain cycle earch, proe i correcne, and analyze i rnning ime. Daa Srcre. We e a linear array T..m 1], hich ac imilar o a hah able 1, Chaper 12]. Iniially, T i empy. A pair (σ i, σ j idenified by he algorihm i ored in Ti] ogeher ih a li of poiie implice i defining he cycle creaed by σ i and deroyed by σ j. The implice in ha li are no necearily he ame a he one in Ɣ(d. All e garanee i ha d i homologo o he m of cycle repreened by he implice in he li, and ha he li conain he yonge implex in Ɣ(d, hich i σ i a aboe. The correcne proof folloing he algorihm ill ho ha hi propery i fficien for or prpoe. The daa rcre i illraed in Fig. 8. Each implex in he filer ha a lo in he hah able, b informaion i ored only in he lo of he poiie implice. Thi informaion coni of he index j of he maching negaie implex and a li of poiie implice defining a cycle. Some cycle exi beyond he end of he filer, in hich cae e e a a bie for j. Cycle Search. Sppoe he algorihm arrie a index j in he filer and ame σ j i a negaie ( +1-implex. Recall ha Ɣ(d i he e of poiie -implice ha repreen he homology cla of d = σ j in H j 1. We earch for he yonge -implex in Ɣ(d by cceiely probing lo in T nil e find he righ one. Specifically, e ar ih a e eqal o he e of poiie -implice in d, hich i necearily non-empy, and e le i = max( be he index of he yonge member of. We ill ee laer ha if Ti] i noccpied, hen i = y(d. We can herefore end he earch and ore j and in Ti]. If Ti] i occpied, i conain a collecion i repreening a permanenly ored -cycle. A hi momen, hi -cycle i already a -bondary. We add and i o ge a ne repreening a -cycle homologo o he old one and herefore alo homologo o d. The Fncion YOUNGEST in Fig. 9 perform a cycle earch for implex σ j. A colliion i he een of probing an occpied lo of T. I rigger he addiion of and i, hich mean e ae he ymmeric difference of he o collecion Fig. 8. Hah able afer rnning he algorihm on he filer of Fig. 3.

12 522 H. Edelbrnner, D. Lecher, and A. Zomorodian ineger YOUNGEST (implex σ j = {σ +1 (σ j σ poiie}; loop i = max( ; if Ti] i noccpied hen ore j and in Ti]; exi endif; = + i foreer; rern i. Fig. 9. of σ j. The fncion rern he index of he yonge bai cycle ed in he decripion of he bondary For example, he fir colliion for he filer of Fig. 3 occr for he negaie edge. Iniially, e hae = {, } and i eqal o 4, he index of. T4] i occpied and ore 4 = {, }. The m of he o -cycle i + 4 = {, }, hich i he ne e. We no hae i = 2, he index of. Thi ime, T2] i noccpied and e ore he index of and he ne e in ha lo. Correcne. We fir ho ha cycle earch hal. Conider a colliion a Ti]. The li i ored in Ti] conain σ i and poibly oher poiie -implice, all older han σ i. Afer adding and i e ge a ne li. Thi li i necearily non-empy, a oherie d old bond. Frhermore, all implice in are ricly older han σ i. Therefore, he ne i i maller han he old one, hich implie ha he earch proceed ricly from righ o lef in T. I necearily end a an noccpied lo Tg] of he hah able, for all oher poibiliie lead o conradicion. I ae ome more effor o proe ha Tg] i he correc lo, or, in oher ord, ha g = y(d, here d = +1 (σ j i he bondary of he negaie ( + 1-implex ha riggered he earch. Le e be he cycle defined by g. Since e i obained from d hrogh adding bonding cycle, e no ha e and d are homologo in K j 1. A colliion-free cycle i one here he yonge poiie implex correpond o an noccpied lo in he hah able. Cycle earch end heneer i reache a colliion-free cycle. For example, e i colliion-free becae i yonge poiie implex i σ g and Tg] i noccpied before e arrie. Colliion Lemma. Le e be a colliion-free -cycle in K j 1 homologo o d. Then he index of he yonge poiie implex in e i i = y(d. Proof. Le σ g be he yonge poiie implex in e, and le f be he m of he bai cycle, homologo o d. By definiion, f yonge poiie implex i σ i, here i = y(d. Thi implie ha here are no cycle homologo o d in K i 1 or earlier complexe, herefore g i. We ho g i by conradicion. If g > i, hen e = f + c, here c bond in K j 1. σ g / f implie σ g c, and a σ g i he yonge in e, i i alo he yonge in c. By ampion, Tg] i noccpied a e i colliion-free. In oher ord, he cycle creaed by σ g i ill a non-bonding cycle in K j 1. Hence hi cycle

13 Topological Perience and Simplificaion 523 canno be c. Alo, he cycle canno belong o c homology cla a he ime c become a bondary. I follo ha he negaie ( + 1-implex ha coner c ino a bondary pair ih a poiie -implex in c ha i yonger han σ g, a conradicion. Hence g = i. The cycle earch conine nil i find a colliion-free cycle homologo o d, and he Colliion Lemma implie ha ha cycle ha he correc yonge poiie implex. Thi proe he correcne of cycle earch, and e may no bie i = YOUNGEST(σ j for line (* in Fncion PAIR-SIMPLICES. Rnning Time. Le d = +1 (σ j and le σ i be he yonge poiie -implex in Ɣ(d. The perience of he cycle creaed by σ i and deroyed by σ j i p i = j i 1. The earch for σ i proceed from righ o lef aring a T j] and ending a Ti]. The nmber of colliion i a mo he nmber of poiie -implice ricly beeen σ i and σ j, hich i le han p i. A colliion happen a Tg] only if σ g already form a pair, hich implie i -ineral g, h i conained inide i, j. We e he neing propery o proe by indcion ha he -cycle defined by i i he m of feer han p i bondarie of ( + 1-implice. Hence, i conain feer han ( + 2p i -implice, and imilarly g conain feer han ( + 2p g < ( + 2p i -implice. A colliion reqire adding he o li and finding he yonge in he ne li. We do hi by merging, hich eep he li ored by age. A ingle colliion ae ime a mo O(p i, and he enire earch for σ i ae ime a mo O(pi 2. The oal algorihm rn in ime a mo O( pi 2, hich i a mo O(m3. The rnning ime of cycle earch can be improed o almo conan for dimenion = and = 2 ing a nion-find daa rcre repreening a yem of dijoin e and pporing find and nion operaion 1, Chaper 22]. For =, each e i he erex e of a conneced componen. Each e ha exacly one ye npaired erex, namely he olde one in he componen. We modify andard nion-find implemenaion in ch a ay ha hi erex repreen he e. Gien a erex, he find operaion rern he repreenaie of he e ha conain hi erex. Gien an edge hoe endpoin lie in differen e, he nion operaion merge he o e ino one. A he ame ime, i pair he edge ih he yonger of he o repreenaie and reain he older one a he repreenaie of he merged e. Cycle earch i replaced by o find operaion poibly folloed by a nion operaion. If e e eighed merging for nion and pah compreion for find, he amorized ime per operaion i O(A 1 (m, here A 1 (m i he noorioly loly groing inere of he Acermann fncion 1, Chaper 22]. We may e ymmery o accelerae cycle earch for 2-cycle ing he nion-find daa rcre for a yem of e of erahedra 2]. We canno achiee he ame acceleraion for 1-cycle ing hi mehod, hoeer, a here can be mliple npaired poiie edge a any ime. The addiional complicaion eem o reqire he more caio and herefore loer algorihm decribed aboe. 5. Simplificaion In hi ecion e e informaion abo he perience of cycle o implify filraion. Simplificaion here mean reordering he implice in ch a ay ha only cycle

14 524 H. Edelbrnner, D. Lecher, and A. Zomorodian index perience Fig. 1. The -riangle ha inerec he ne axi a p = 2 hae perience 2 or larger. The implex pair repreening cycle of perience le han 2 are boxed. hoe perience i aboe ome hrehold appear in he filraion. We ge inpiraion for reordering hrogh an algorihm for comping perien Bei nmber. Comping Perien Bei Nmber. By he -Triangle Lemma in Secion 3, he p- perien h Bei nmber of K l i he nmber β l,p of -riangle ha conain he poin (l, p in he index-perience plane. To compe hee nmber for a fixed p, e inerec he -riangle ih a horizonal line a p. Figre 1 illrae hi operaion by modifying Fig. 6. The algorihm for p-perien Bei nmber i imilar o Fncion BETTI-NUMBERS gien in Fig. 4. We go hrogh he filer from lef o righ and increae β p heneer e enconer he lef endpoin of a -ineral longer han p. Similarly, e decreae β p heneer p poiion ahead of here i a righ endpoin of a -ineral longer han p. Figre 11 ho he rel of he algorihm applied o or example filraion of Fig. 3 for =. Migraion. The inerecion of he -riangle and he horizonal line a p i a collecion of half-open ineral. We inerpre hee ineral a -ineral of a implified erion β perience index 9 Fig. 11. p o 7. Perien h Bei nmber of he fir en complexe in he filraion of Fig. 3 and for perience

15 Topological Perience and Simplificaion 525 of he original filraion. Or goal i o reorder he filer o ha hi inerpreaion i alid, ha i, e ih o obain a ne filraion hoe Bei nmber are he p-perien Bei nmber of he original filraion. For each pair (σ i, σ j e moe σ j o he lef, cloer or all he ay o σ i, a hon in Fig. 1. The ne poiion of σ j i max{i, j p}. If j p i, hen σ i and σ j no longer form an ineral a hey boh occpy he ame index in he ne filer. Here, e exend he noion of a filer o allo a poibly empy e of implice a each index. We compe Bei nmber by bringing all implice of a e ino he complex a once. There i a complicaion in he reordering algorihm ha occr heneer a negaie implex aemp o moe pa one of i face. To mainain he ordering a a filer, e m moe he face along ih i coface. For example, if e increae p o 4 in Fig. 1, hen ill moe o index 11 pa i face a index 12. Moing a face along ih a implex ill no change any Bei nmber if he face repreen a cycle hoe perience i le han p. A he ime e moe i, he face i already collocaed ih i maching negaie implex, and he o cancel each oher conribion. We may hen grab he pair and moe i ih he implex, moing he pair (, ih in or example. For any moing implex, hoeer, e m alo moe all he neceary face and heir maching negaie implice recriely. Conflic. There i roble if he face of a moing negaie implex repreen a cycle hoe perience i a lea p. For inance, hen enconer he edge, he riangle hich i paired ih ha no ye reached. There i a conflic beeen or o goal of mainaining he filer propery and reordering o he ne Bei nmber are he old p-perien Bei nmber. Formally, a conflic occr heneer here are pair (σ i, σ j and (σ g, σ h ih g < i < h < j, here σ i i a face of σ h, a hon in Fig. 12. There are ( 4 2 = 6 poible ype of conflic, each idenified by he pair (dim σ i, dim σ h of he dimenion of he main paricipan. The pair (, and (, in Fig. 6 conie a conflic of ype (1, 2 and ho ha conflic do occr, alhogh or experimen in Secion 6 gge ha hey are raher rare. We parially baniae hi finding by he folloing lemma. Conflic Lemma. All conflic hae ype (1, 2. Proof. Sppoe a conflic exi in pair (σ i, σ j and (σ g, σ h, here σ i i a erex. When σ h ener he filraion, i belong o he ame componen a σ g, ince σ h complee a chain hoe bondary inclde σ g. Verex σ i, one of he erice of σ h, i npaired and herefore repreen he componen of σ h and σ g. Recall ha any componen i repreened by i olde erex, hich implie ha σ i i older han all he erice of σ g. By he filer propery, σ i i older han σ g, i.e. i < g, hich conradic he ampion ha (σ i, σ j and (σ g, σ h form a conflic. Thi proe here are no conflic of ype σ g i σ σ h j σ Fig. 12. Baic conflic configraion.

16 526 H. Edelbrnner, D. Lecher, and A. Zomorodian (, 1, (, 2, (, 3. By complemenariy and daliy, here are no conflic of ype (1, 3 and (2, 3. Difficlie in reordering may alo arie indirecly becae of he recrie nare of any reordering algorihm. For example, moing a negaie riangle may reqire moing one of i edge. Thi edge hold on o i maching riangle, hich in rn grab i needed face. Some of hee face may be npaired, and o capre hi iaion e define a recrie conflic o be a poiie implex ha i moed hen i i no collocaed ih i maching negaie implex. Exending he Conflic Lemma, e can ho ha all recrie conflic are edge. We again hae a iaion a in Fig. 12, excep ha σ i i no necearily a face of σ h. Hoeer, he moing implice all belong o he ame componen a σ h : hi i re for a face by definiion, for a maching negaie implex by he reaon gien in he proof aboe, and for all moing implice by raniiiy. Baic and recrie conflic exi in pracice, b are raher rare, a hon in Secion 6. When conflic occr, e ie he mainenance of he filer propery a iniolable, and aemp o approximae or econdary goal, achieing he correc Bei nmber. We do o hrogh conflic reolion or conflic diminion, a decribed belo. Conflic Reolion. We may reole a conflic by bdiiion. Here, e achiee he correc Bei nmber for a refined complex and i correponding filraion. Sppoe pair (σ i, σ j and (σ g, σ h form a conflic. Then σ i, σ g are edge, σ j, σ h are riangle, and σ i i a face of σ h. Le σ i = bc and σ h = abc a dran in Fig. 13. We reole he conflic by arring from he midpoin x of edge bc, bdiiding all implice ha hare bc a a common face. We replace each bdiided -implex by one ( 1-implex and o -implice. For comping perience, he order of he hree ne implice i imporan. A hon in Fig. 13, he order of he edge bx, cx ihin he ne filer i he oppoie of he riangle acx, abx. The perience algorihm prodce ne pair (x, bx and (ax, acx ha hae no effec on Bei nmber. Afer acx ener, he complex i homoopy eqialen o he old complex j before abc ener. The edge b b arring x a c a c before: σ g.... bc.... abc.... σ j afer: σg.... x bx cx Fig. 13. The conflic exi beeen moing abc oard σ g and eeping bc ahead of abc. We bdiide edge bc and order he ne implice o reole he conflic..... ax acx abx.... σ j

17 Topological Perience and Simplificaion 527 σ i σ j index perience Fig. 14. Reordering ih conflic. cx replace bc and he riangle abx replace abc in he filer. Coneqenly, he algorihm prodce pair (σ g, abx and (cx, σ j. A cx i no a face of abx, e hae remoed he conflic and preered he Bei nmber of a refined filraion. Conflic Diminion. Ofenime, implice hae rcral meaning in a filraion, and conflic ignal properie of he rcre he implice decribe. We may no ih o amper ih hi rcre hrogh bdiiion, a ch acion may no hae any meaning ihin or filraion. For example, in alpha complex filraion, implice are ordered according o a pariclar groh model. The ordering of he ne implice pecified by bdiiion in Fig. 13 migh no hae a correponding e of eighed ball ha old generae he filraion nder he groh model. We may aemp o redce he effec of conflic on Bei nmber iho eliminaing he conflic. Recall ha a implex pair (σ i, σ j define a -cycle hich may be ialized by a -riangle, a in Fig. 14. Wheneer σ i occr in a conflic, e allo i o be dragged o a ne locaion. Thi clearly change he Bei nmber of he reordered filraion, o hey no longer mach he p-perien Bei nmber of he original filraion. We alo allo σ j o moe faer dring reordering, heneer σ i i moed. Thi mehod creae a region of he -cycle ih he ame area a he -riangle, a hon in Fig. 14. Therefore, e allo each -cycle o hae he ame effec on Bei nmber a i old in he abence of conflic, b a differen ime. Lazy Migraion. We end hi ecion by decribing an alernae mehod for reordering. Or moiaion for formlaing perien homology in (3 a o eliminae cycle ih lo perience. A a coneqence of he formlaion, he lifeime of eery cycle i redced regardle of i perience, leading o he creaion of -riangle. A poibly more iniie goal old be o eliminae cycle ih lo perience iho changing he lifeime of cycle ih high perience. In oher ord, e replace -riangle by -qare a illraed in Fig. 15. In analogy o p-perien Bei nmber, e define γ l,p a he nmber of -qare ha conain he poin (l, p in he index-perience plane. Figre 16 illrae ho hee nmber change a e increae perience from p = o 7. Noe ha e can eaily read off perien cycle from he graph. We may alo implify complexe ing γ l,p by only collaping -ineral of lengh a mo p, leaing oher -ineral nchanged.

18 528 H. Edelbrnner, D. Lecher, and A. Zomorodian index perience Fig. 15. Alernaie ializaion of he rel of Fncion PAIR-SIMPLICES. The qare of and are nbonded and no hon. The ligh qare repreen -cycle and he dar qare repreen 1-cycle. 6. Experimen We hae implemened he algorihm decribed in hi paper and creaed a prooype baed on he alpha hape ofare of 4]. Thi ecion preen experimenal iming rel and proide eidence for he claim ha perience eparae opological noie from feare. Daa. We hae applied he ofare o a ariey of daae and ho he rel for fie repreenaie e in hi ecion. Three e repreen moleclar rcre ih eighed poin and o repreen rface of macrocopic hape ih neighed poin. In each cae e fir compe he poibly eighed Delanay rianglaion and hen he age filer of ha rianglaion. The daa poin become erice or -implice of he rianglaion. Table 6 gie he ize of he daa e, heir Delanay rianglaion, and age filer. Timing. We ime only he porion of he ofare ha i direcly relaed o comping perience. We diingih for ep: maring implice a poiie or negaie, and γ perience index 9 Fig. 16. Nmber γ for he fir en complexe in he filraion of Fig. 3.

19 Topological Perience and Simplificaion 529 Table 1. Size of daa e, heir Delanay rianglaion, and age filer. Nmber of -implice Toal G 318 2,322 3,978 1,973 8,591 Z 1,296 11,41 2,98 9,992 42,787 D 7,774 6,675 15,71 52,88 226,967 B 42, ,664 68,445 34,91 1,31,511 S 54, , , ,2 1,642,39 G i Gramicidin A, a mall proein. Z i a porion of a periodic zeolie rcre. D i a porion of DNA. B i a iny cbe of microcopic bone rcre. S i a canned and reampled Bddha ae. adding -cycle for =, 1, 2. Recall ha he compaion of perience can be acceleraed for =, 2 ing a nion-find daa rcre. A baniaed in Table 2, hi improemen bme adding - and 2-cycle in he maring proce, hrining he ime for hee o ep o eenially nohing. The rel gge a poible linear dependence of he rnning ime on he ize of he daa, hich i banially faer han he cbic dependence proed in Secion 4. Of core, e need o diingih orcae from aerage rnning ime. Afer acceleraing ih nion-find, he loe porion of he algorihm i adding 1-cycle. We poe he deailed analyi of he algorihm a an open problem, and e alo a for a differen and more efficien algorihm, if i exi. Saiic. Or cbic pper bond in Secion 4 folloed from he oberaion ha he -cycle creaed by σ i goe hrogh feer han p i colliion and he lengh of i li bil p dring hee colliion i le han ( + 2p i. We can explain he apparenly linear rnning ime docmened in Table 2 by hoing ha he aerage nmber of colliion and he aerage li lengh are boh conan. Table 3 and 4 proide eidence ha hi migh indeed be he cae. We do no ho aiic for -cycle in Table 4 a eery -cycle i repreened by a li of lengh 2. Recall ha he nmber of colliion and he lengh of li i bonded from aboe by he perience of cycle. Table 5 ho ha he aerage perience i coniderably larger han he aerage nmber of colliion and li lengh. Table 5 alo gie eidence ha conflic are indeed rare. Feare Deecion. We conclde hi ecion by ing perience for deecing feare of he daae G. Thi daae conain he ime-aeraged moleclar dynamic rcre of Gramicidin A, a pepide ha form a channel for ion and aer moemen acro lipid membrane. The primary opological feare of hi daa i a nnel ha rn hrogh he molecle, a hon in Fig. 17. We ho he graph in Fig. 18 for comparion. While here i coniderable opological noie a p =, a implificaion proce hich eliminae 1- cycle of perience le han 2688 cceed in eparaing he nnel from he remaining opological aribe deeced by mearing homology. We ho hi implificaion for hree complexe in Fig. 19.

20 53 H. Edelbrnner, D. Lecher, and A. Zomorodian Table 2. Rnning ime in econd. Add -cycle Toal Mar 1 2 Wiho UF Wih UF G Z D B S All iming ere done on a Micron PC ih a 266 MHz Penim II proceor and 128 MB random acce memory, rnning he Solari 8 operaing yem. Table 3. Maximm and aerage nmber of colliion for he fie daa e. -Cycle 1-Cycle 2-Cycle Max Ag Max Ag Max Ag G Z D B , S , , Table 4. Maximm and aerage lengh of cycle li, oer all li (ag, and all final ored li (agf. 1-Cycle 2-Cycle Max Ag Agf Max Ag Agf G Z D B 1, S 4, , Table 5. Aerage perience of cycle and nmber of conflic in implifying he filraion. Aerage perience Nmber of conflic -Cycle 1-Cycle 2-Cycle Baic Recrie G Z 1, D 7, , B 59, , S 95, ,

21 Topological Perience and Simplificaion 531 Fig. 17. Side and op ie of he moleclar rface for Gramicidin A, preening he channel for ion ranpor. β 1 l,p p l γ 1 l,p p l Fig. 18. Graph of β l,p 1 and γ l,p 1 of Gramicidin A ampled ono an 8 by 8 grid.

22 532 H. Edelbrnner, D. Lecher, and A. Zomorodian Fig. 19. Side and op ie of complexe K 715, K 1431, K 2682 of Gramicidin A are hon in he lef hree colmn. The correponding 2688-perien complexe are hon on he righ. 7. Fre Direcion We inrodce he noion of opological perience for a filraion in R 3 in hi paper and gie algorihm for aeing perience and implifying he filraion. We plan o apply hee idea o he analyi of hree ype of daae: (i Moleclar daa are narally repreened by he age filer of heir Delanay rianglaion. While differeniaing opological noie from feare i a generally efl faciliy, e beliee i i mo ignifican for analyzing he rcre of molecle. (ii Simplifying hape hile preering feare i a he core of he rface reconrcion problem died in comper graphic, a ell a compaional geomery. We beliee perience can help in aomaic reconrcion. (iii We plan o apply perience o deec hierarchical or oher complex ype of clering in ery large daae, ch a hoe expeced from he crren effor of mearing he locaion of galaxie in he niere. There i a forh and le direc applicaion o rface reconrcion hrogh iorface exracion. Io-rface are idely ed in medical imaging here olme deniy daa i common. A mooh deniy fncion ha finiely many criical poin of for ype, correponding o he for differen dimenion of implice in a hree-dimenional complex. We can e he perience algorihm o meare he ignificance of eery criical poin. A more difficl a i ing hi informaion for aomaic denoiing he deniy fncion and i io-rface. Acnoledgmen We han Jeff Ericon and John Harer for helpfl dicion dring early age of hi paper. We alo han Daniel Hon for he zeolie daae Z, Thoma LaBean for he DNA daae D, and he Sanford Graphic Lab for he Bddha daae S. To generae he bone daae B, e ampled an io-rface generaed by Dominiqe Aali. The olme daa

23 Topological Perience and Simplificaion 533 a proided by Françoie Peyrin from CNRS CREATIS in Lyon and a ied from Synchroron Radiaion Microomography from he ID19 beamline a ESRF in Grenoble. We generaed Fig. 17 ing he Proein Explorer 6]. Reference 1. T. H. Cormen, C. E. Leieron, and R. L. Rie. Inrodcion o Algorihm. The MIT Pre, Cambridge, Maache, C. J. A. Delfinado and H. Edelbrnner. An incremenal algorihm for Bei nmber of implicial complexe on he 3-phere. Comp. Aided Geom. Deign, 12: , H. Edelbrnner and E. P. Müce. Simlaion of impliciy: a echniqe o cope ih degenerae cae in geomeric algorihm. ACM Tran. Graphic, 9:66 14, H. Edelbrnner and E. P. Müce. Three-dimenional alpha hape. ACM Tran. Graphic, 13:43 72, J. El-Sana and A. Varhney. Topology implificaion for polygonal iral enironmen. IEEE Tran. Vial. Comp. Graphic, 4: , E. Marz. Proein explorer 1.8b. hp://.ma.ed/microbio/chime/explorer. 7. J. McCleary. Uer Gide o Specral Seqence. Pblih or Perih, Wilmingon, Delaare, J. Milnor. More Theory. Princeon Unieriy Pre, Princeon, Ne Jerey, J. R. Mnre. Elemen of Algebraic Topology. Addion-Weley, Redood Ciy, California, V. Robin. Comping opology a mliple reolion: fondaion and applicaion o fracal and dynamic. Ph.D. hei, Deparmen of Applied Mahemaic, Unieriy of Colorado, Jne J. A. Sehian. Leel Se Mehod. Cambridge Unieriy Pre, Cambridge, Receied March 19, 21, and in reied form Ocober 4, 21. Online pblicaion Ocober 29, 22.