Computing Persistent Homology

Computing Persistent Homology Afr Zomorodin nd Gunnr Crlsson (Disrete nd Computtionl Geometry) Astrt We show tht the persistent homology of filtered d- dimensionl simpliil omplex is simply the stndrd homology of prtiulr grded module over polynomil ring. Our nlysis estlishes the existene of simple desription of persistent homology groups over ritrry fields. It lso enles us to derive nturl lgorithm for omputing persistent homology of spes in ritrry dimension over ny field. This result generlizes nd extends the previously nown lgorithm tht ws restrited to suomplexes of S 3 nd Z 2 oeffiients. Finlly, our study implies the l of simple lssifition over non-fields. Insted, we give n lgorithm for omputing individul persistent homology groups over n ritrry prinipl idel domins in ny dimension. 1 Introdution In this pper, we study the homology of filtered d- dimensionl simpliil omplex K, llowing n ritrry prinipl idel domin D s the ground ring of oeffiients. A filtered omplex is n inresing sequene of simpliil omplexes, s shown in Figure 1. It determines n indutive system of homology groups, i.e., fmily of Aelin groups {G i } i 0 together with homomorphisms G i G i+1. If the homology is omputed with field oeffiients, we otin n indutive system of vetor spes over the field. Eh vetor spe is determined up to isomorphism y its dimension. In this pper we otin simple lssifition of n indutive system of vetor spes. Our lssifition is in terms of set of Reserh y the first uthor is prtilly supported y NSF under grnts CCR-00-86013 nd ITR-0086013. Reserh y the seond uthor is prtilly supported y NSF under grnt DMS-0101364. Reserh y oth uthors is prtilly supported y NSF under grnt DMS-0138456. Deprtment of Computer Siene, Stnford University, Stnford, Cliforni. Deprtment of Mthemtis, Stnford University, Stnford, Cliforni. 0, 1, d,, d d 2 d, d d d 3 4 5 d Figure 1. A filtered omplex with newly dded simplies highlighted. intervls. We lso derive nturl lgorithm for omputing this fmily of intervls. Using this fmily, we my identify homologil fetures tht persist within the filtrtion, the persistent homology of the filtered omplex. Furthermore, our interprettion mes it ler tht if the ground ring is not field, there exists no similrly simple lssifition of persistent homology. Rther, the strutures re very omplited, nd lthough we my ssign interesting invrints to them, no simple lssifition is, or is liely ever to e, ville. In this se we provide n lgorithm for omputing single persistent group for the filtrtion. In the rest of this setion we first motivte our study through three exmples in whih filtered omplexes rise whose persistent homology is of interest. We then disuss prior wor nd its reltionship to our wor. We onlude this setion with n outline of the pper. 1.1 Motivtion We ll filtered simpliil omplex, long with its ssoited hin nd oundry mps, persistene omplex. We formlize this onept in Setion 3. Persistene omplexes rise nturlly whenever one is ttempting to study topologil invrints of spe omputtionlly. Often, our nowledge of this spe is limited nd impreise. Consequently, we must utilize multisle pproh to pture the onnetivity of the spe, giving us persistene omplex. d

Exmple 1.1 (point loud dt) Suppose we re given finite set of points X from suspe X R n. We ll X point loud dt or PCD for short. It is resonle to elieve tht if the smpling is dense enough, we should e le to ompute the topologil invrints of X diretly from the PCD. To do so, we my either ompute the Čeh omplex, or pproximte it vi Rips omplex [15]. The ltter omplex R ɛ (X) hs X s its vertex set. We delre set of verties σ = {x 0, x 1,..., x } to spn -simplex of R ɛ (X) iff the verties re pirwise lose, tht is, d(x i, x j ) ɛ for ll pirs x i, x j σ. There is n ovious inlusion R ɛ (X) R ɛ (X) whenever ɛ < ɛ. In other words, for ny inresing sequene of non-negtive rel numers, we otin persistene omplex. Exmple 1.2 (density) Often, our smples re not from geometri ojet, ut re hevily onentrted on it. It is importnt, therefore, to ompute mesure of density of the dt round eh smple. For instne, we my ount the numer of smples ρ(x) ontined in ll of size ɛ round eh smple x. We my then define R r ɛ(x) R ɛ to e the Rips suomplex on the verties for whih ρ(x) r. Agin, for ny inresing sequene of non-negtive rel numers r, we otin persistene omplex. We must nlyze this omplex to ompute topologil invrints tthed to the geometri ojet round whih our dt is onentrted. Exmple 1.3 (Morse funtions) Given mnifold M equipped with Morse funtion f, we my filter M vi the exursion sets M r = {m M f(m) r}. We gin hoose n inresing sequene of non-negtive numers to get persistene omplex. If the Morse funtion is height funtion tthed to some emedding of M in R n, persistent homology n now give informtion out the shpe of the sumnifolds, s well the homologil invrints of the totl mnifold. 1.2 Prior Wor We ssume fmilirity with si group theory nd refer the reder to Dummit nd Foote [10] for n introdution. We me extensive use of Munres [16] in our desription of lgeri homology nd reommend it s n essile resoure to non-speilists. There is lrge ody of wor on the effiient omputtion of homology groups nd their rns [1, 8, 9, 13]. Persistent homology groups re initilly defined in [11, 18]. The uthors lso provide n lgorithm tht wored only for spes tht were suomplexes of S 3 over Z 2 oeffiients. The lgorithm genertes set of intervls for filtered omplex. Surprisingly, the uthors show tht these intervls llowed the orret omputtion of the rn of persistent homology groups. In other words, the uthors prove onstrutively tht persistent homology groups of suomplexes of S 3, if omputed over Z 2 oeffiients, hve simple desription in terms of set of intervls. To uild these intervls, the lgorithm pirs positive yle-reting simplies with negtive yle-destroying simplies. During the omputtion, the lgorithm ignores negtive simplies nd lwys loos for the youngest simplex. While the uthors prove the orretness of the results of the lgorithm, the underlying struture remins hidden. 1.3 Our Wor We re motivted primrily y the unexplined results of the previous wor. We wish to nswer the following questions: 1. Why does simple desription exist for persistent homology of suomplexes of S 3 over Z 2? 2. Does this desription lso exist over other rings of oeffiients nd ritrry-dimensionl simpliil omplexes? 3. Why n we ignore negtive simplies during omputtion? 4. Why do we lwys loo for the youngest simplex? 5. Wht is the reltionship etween the persistene lgorithm nd the stndrd redution sheme? In this pper we resolve ll these questions y unovering nd eluidting the struture of persistent homology. Speifilly, we show tht the persistent homology of filtered d-dimensionl simpliil omplex is simply the stndrd homology of prtiulr grded module over polynomil ring. Our nlysis ples persistent homology within the lssil frmewor of lgeri topology. This plement llows us to utilize stndrd struture theorem to estlish the existene of simple desription of persistent homology groups s set of intervls, nswering the first question ove. This desription exists over ritrry fields, not just Z 2 s in the previous result, resolving the seond question. Our nlysis lso enles us to derive persistene lgorithm from the stndrd redution sheme in lger, resolving the next three questions using two min lemms. Our lgorithm generlizes nd extends the previously nown lgorithm to omplexes in ritrry dimensions over ritrry fields of oeffiients. We lso show tht if we onsider integer oeffiients or oeffiients in some non-field R, there is no similr simple 2

lssifition. This negtive result suggests the possiility of interesting yet inomplete invrints of indutive systems. For now, we give n lgorithm for lssifying single homology group over n ritrry prinipl idel domin. 1.4 Spetrl Sequenes Any filtered omplex gives rise to spetrl sequene, so it is nturl to wonder out the reltionship etween this sequene nd persistene. A full disussion on spetrl sequenes is outside the sope of this pper. However, we inlude few remrs here for the reder who is fmilir with the sujet. We my esily show tht the persistene intervls for filtrtion orrespond to nontrivil differentils in the spetrl sequene tht rises from the filtrtion. Speifilly, n intervl of length r orresponds to some differentil d r+1. Given this orrespondene, we relize tht the method of spetrl sequenes omputes persistene intervls in order of length, finding ll intervls of length r during the omputtion of the E r+1 term. In priniple, we my use this method to ompute the result of our lgorithm. However, the method does not provide n lgorithm, ut sheme tht must e tilored for eh prolem independently. The prtitioner must deide on n pproprite sis, find the zero terms in the sequene, nd dedue the nture of the differentils. Our nlysis of persistent homology, on the other hnd, provides omplete, effetive, nd implementle lgorithm for ny filtered omplex. 1.5 Outline We egin y reviewing onepts from lger nd simpliil homology in Setion 2. We lso re-introdue persistent homology over integers nd ritrry dimensions. In Setion 3 we define nd study the persistene module, struture tht represents the homology of filtered omplex. In ddition, we estlish reltionship etween our results nd prior wor. Using our nlysis, we derive n lgorithm for omputtion over fields in Setion 4. For non-fields, we desrie n lgorithm in Setion 5 tht omputes individul persistent groups. Setion 6 desries our implementtion nd some experiments. We onlude the pper in Setion 7 with disussion of urrent nd future wor. 2 Bground In this setion we review the mthemtil nd lgorithmi ground neessry for our wor. We egin y reviewing the struture of finitely generted modules over prinipl idel domins. We then disuss simpliil omplexes nd their ssoited hin omplexes. Putting these onepts together, we define simpliil homology nd outline the stndrd lgorithm for its omputtion. We onlude this setion y desriing persistent homology. 2.1 Alger Throughout this pper we ssume ring R to e ommuttive with unity. A polynomil f(t) with oeffiients in R is forml sum i=0 it i, where i R nd t is the indeterminte. For exmple, 2t + 3 nd t 7 5t 2 re oth polynomils with integer oeffiients. The set of ll polynomils f(t) over R forms ommuttive ring R[t] with unity. If R hs no divisors of zero, nd ll its idels re prinipl, it is prinipl idel domin (PID). For our purposes, PID is simply ring in whih we my ompute the gretest ommon divisor or gd of pir of elements. This is the ey opertion needed y the struture theorem tht we disuss elow. PIDs inlude the fmilir rings Z, Q, nd R. Finite fields Z p for p prime, s well s F [t], polynomils with oeffiients from field F, re lso PIDs nd hve effetive lgorithms for omputing the gd [6]. A grded ring is ring R, +, equipped with diret sum deomposition of Aelin groups R = i R i, i Z, so tht multiplition is defined y iliner pirings R n R m R n+m. Elements in single R i re lled homogeneous nd hve degree i, deg e = i for ll e R i. We my grde the polynomil ring R[t] non-negtively with the stndrd grding (t n ) = t n R[t], n 0. In this grding, 2t 6 nd 7t 3 re oth homogeneous of degree 6 nd 3, respetively, ut their sum 2t 6 + 7t 3 is not homogeneous. The produt of the two terms, 14t 9, hs degree 9 s required y the definition. A grded module M over grded ring R is module equipped with diret sum deomposition, M = i M i, i Z, so tht the tion of R on M is defined y iliner pirings R n M m M n+m. The min struture in our pper is grded module nd we inlude onrete exmples tht lrify this onept lter on. A grded ring (module) is non-negtively grded if R i = 0 (M i = 0) for ll i < 0. The stndrd struture theorem desries finitely generted modules nd grded modules over PIDs. Theorem 2.1 (struture) If D is PID, then every finitely generted D-module is isomorphi to diret sum of yli D-modules. Tht is, it deomposes uniquely into the form ( m ) D β D/d i D, (1) i=1 3

for d i D, β Z, suh tht d i d i+1. Similrly, every grded module M over grded PID D deomposes uniquely into the form ( n ) m Σ αi D Σ γj D/d j D, (2) i=1 j=1 where d j D re homogeneous elements so tht d j d j+1, α i, γ j Z, nd Σ α denotes n α-shift upwrd in grding. In oth ses, the theorem deomposes the strutures into two prts. The free portion on the left inludes genertors tht my generte n infinite numer of elements. This portion is vetor spe nd should e fmilir to most reders. Deomposition (1) hs vetor spe of dimension β. The torsionl portion on the right inludes genertors tht my generte finite numer of elements. For exmple, if PID D is Z in the theorem, Z/3Z = Z 3 would represent genertor ple of only reting three elements. These torsionl elements re lso homogeneous. Intuitively then, the theorem desries finitely generted modules nd grded modules s strutures tht loo lie vetor spes ut lso hve some dimensions tht re finite in size. 2.2 Simpliil Complexes A simpliil omplex is set K, together with olletion S of susets of K lled simplies (singulr simplex) suh tht for ll v K, {v} S, nd if τ σ S, then τ S. We ll the sets {v} the verties of K. When it is ler from ontext wht S is, we refer to set K s omplex. We sy σ S is -simplex of dimension if σ = + 1. If τ σ, τ is fe of σ, nd σ is ofe of τ. An orienttion of -simplex σ, σ = {v 0,..., v }, is n equivlene lss of orderings of the verties of σ, where (v 0,..., v ) (v τ(0),..., v τ() ) re equivlent if the sign of τ is 1. We denote n oriented simplex y [σ]. A simplex my e relized geometrilly s the onvex hull of + 1 ffinely independent points in R d, d. A reliztion gives us the fmilir low-dimensionl -simplies: verties, edges, tringles, nd tetrhedr, for 0 3, shown in Figure 2. Within relized omplex, the simplies must meet long ommon fes. A suomplex of K is suset L K tht is lso simpliil omplex. A filtrtion of omplex K is nested susequene of omplexes = K 0 K 1... K m = K. For generlity, we let K i = K m for ll i m. We ll K filtered omplex. We show smll filtered omplex in Figure 1. vertex edge [, ] tringle [,, ] d tetrhedron [,,, d] Figure 2. Oriented -simplies in R 3, 0 3. The orienttion on the tetrhedron is shown on its fes. 2.3 Chin Complex The th hin group C of K is the free Aelin group on its set of oriented -simplies, where [σ] = [τ] if σ = τ nd σ nd τ re differently oriented.. An element C is -hin, = i n i[σ i ], σ i K with oeffiients n i Z. The oundry opertor : C C 1 is homomorphism defined linerly on hin y its tion on ny simplex σ = [v 0, v 1,..., v ], σ = i ( 1) i [v 0, v 1,..., ˆv i,..., v ], where ˆv i indites tht v i is deleted from the sequene. The oundry opertor onnets the hin groups into hin omplex C : +1 C +1 C C 1. We my lso define sugroups of C using the oundry opertor: the yle group Z = er nd the oundry group B = im +1. We show exmples of yles in Figure 3. An importnt property of the oundry opertors is tht the oundry of oundry is lwys empty, +1 = 0. This ft, long with the definitions, implies tht the defined sugroups re nested, B Z C, s in Figure 4. For generlity, we often define null oundry opertors in dimensions where C is empty. 2.4 Homology The th homology group is H = Z /B. Its elements re lsses of homologous yles. To desrie its struture, we view the Aelin groups we hve defined so fr Figure 3. The dshed 1-oundry rests on the surfe of torus. The two solid 1-yles form sis for the first homology lss of the torus. These yles re non-ounding: neither is oundry of piee of surfe. 4

C+1 Z +1 B +1 0 δ +1 C Z B δ C 1 Z 1 0 0 B 1 Figure 4. A hin omplex with its internls: hin, yle, nd oundry groups, nd their imges under the oundry opertors. s modules over the integers. This view llows lternte ground rings of oeffiients, inluding fields. If the ring is PID D, H is D-module nd Theorem (2.1) pplies: β, the rn of the free sumodule, is the Betti numer of the module, nd d i re its torsion oeffiients. When the ground ring is Z, the theorem ove desries the struture of finitely generted Aelin groups. Over field, suh s R, Q, or Z p for p prime, the torsion sumodule disppers. The module is vetor spe tht is fully desried y single integer, its rn β, whih depends on the hosen field. 2.5 Redution The stndrd method for omputing homology is the redution lgorithm. We desrie this method for integer oeffiients s it is the more fmilir ring. The method extends to modules over ritrry PIDs, however. As C is free, the oriented -simplies form the stndrd sis for it. We represent the oundry opertor : C C 1 reltive to the stndrd ses of the hin groups s n integer mtrix M with entries in { 1, 0, 1}. The mtrix M is lled the stndrd mtrix representtion of. It hs m olumns nd m 1 rows (the numer of - nd ( 1)-simplies, respetively). The null-spe of M orresponds to Z nd its rnge-spe to B 1, s mnifested in Figure 4. The redution lgorithm derives lternte ses for the hin groups, reltive to whih the mtrix for is digonl. The lgorithm utilizes the following elementry row opertions on M : 1. exhnge row i nd row j, 2. multiply row i y 1, 3. reple row i y (row i) + q(row j), where q is n integer nd j i. The lgorithm lso uses elementry olumn opertions tht re similrly defined. Eh olumn (row) opertion orresponds to hnge in the sis for C (C 1 ). For exmple, if e i nd e j re the ith nd jth sis elements for C, respetively, olumn opertion of type (3) mounts to repling e i with e i + qe j. A similr row opertion on sis elements ê i nd ê j for C 1, however, reples ê j y ê j qê i. We shll me use of this ft in Setion 4. The lgorithm systemtilly modifies the ses of C nd C 1 using elementry opertions to redue M to its (Smith) norml form: M = 1 0... 0 0 l 0 0, where l = rn M = rn M, i 1, nd i i+1 for ll 1 i < l. The lgorithm n lso ompute orresponding ses {e j } nd {ê i } for C nd C 1, respetively, lthough this is unneessry if deomposition is ll tht is needed. Computing the norml form in ll dimensions, we get full hrteriztion of H : (i) the torsion oeffiients of H 1 (d i in (1)) re preisely the digonl entries i greter thn one. (ii) {e i l +1 i m } is sis for Z. Therefore, rn Z = m l. (iii) { i ê i 1 i l } is sis for B 1. Equivlently, rn B = rn M +1 = l +1. Comining (ii) nd (iii), we hve β = rn Z rn B = m l l +1. (3) Exmple 2.1 For the omplex in Figure 1, the stndrd mtrix representtion of 1 is M 1 = d d 1 0 0 1 1 1 1 0 0 0 0 1 1 0 1 d 0 0 1 1 0, where we show the ses within the mtrix. Reduing the mtrix, we get the norml form M 1 = d z 1 z 2 d 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0, where z 1 = d d nd z 2 = form sis for Z 1 nd {d,, } is sis for B 0. 5

We my use similr proedure to ompute homology over grded PIDs. A homogeneous sis is sis of homogeneous elements. We egin y representing reltive to the stndrd sis of C (whih is homogeneous) nd homogeneous sis for Z 1. Reduing to norml form, we red off the desription provided y diret sum (2) using the new sis {ê j } for Z 1 : (i) zero row i ontriutes free term with shift α i = deg ê i, (ii) row with digonl term i ontriutes torsionl term with homogeneous d j = j nd shift γ j = deg ê j. The redution lgorithm requires O(m 3 ) elementry opertions, where m is the numer of simplies in K. The opertions, however, must e performed in ext integer rithmeti. This is prolemti in prtie, s the entries of the intermedite mtries my eome extremely lrge. 2.6 Persistene We end this setion with y re-introduing persistene. Given filtered omplex, the ith omplex K i hs ssoited oundry opertors i, mtries M i, nd groups C i, Z i, B i nd H i for ll i, 0. Note tht supersripts indite the filtrtion index nd re not relted to ohomology. The p-persistent th homology group of K i is H i,p = Z i / (B i+p Z i ). (4) The definition is well-defined: oth groups in the denomintor re sugroups of C l+p, so their intersetion is lso group, sugroup of the numertor. The p- persistent th Betti numer of K i is β i,p, the rn of the free sugroup of H i,p. We my lso define persistent homology groups using the injetion η i,p : Hi H i+p, tht mps homology lss into the one tht ontins it. Then, im η i,p H i,p [11, 18]. We extend this definition over ritrry PIDs, s efore. Persistent homology groups re modules nd Theorem 2.1 desries their struture. 3 The Persistene Module In this setion we te different view of persistent homology in order to understnd its struture. Intuitively, the omputtion of persistene requires omptile ses for H i nd H i+p. It is not ler when suint desription is ville for the omptile ses. We egin this setion y omining the homology of ll the omplexes in the filtrtion into single lgeri struture. We then estlish orrespondene tht revels simple desription over fields. Most signifintly, we illustrte tht the persistent homology of filtered omplex is simply the stndrd homology of prtiulr grded module over polynomil ring. A simple pplition of the struture theorem (Theorem 2.1) gives us the needed desription. We end this setion y illustrting the reltionship of our strutures to the persistene eqution (Eqution (4).) Definition 3.1 (persistene omplex) A persistene omplex C is fmily of hin omplexes {C } i i 0 over R, together with hin mp s f i : C i C i+1, so tht we hve the following digrm: C 0 f 0 C 1 f 1 C 2 f 2. Our filtered omplex K with inlusion mps for the simplies eomes persistene omplex. Below, we show portion of persistene omplex, with the hin omplexes expnded. The filtrtion index inreses horizontlly to the right under the hin mps f i, nd the dimension dereses vertilly to the ottom under the oundry opertors. 3 3 3 C 0 f 0 2 C 1 f 1 2 C 2 f 2 2 2 2 2 C 0 f 1 1 C 2 1 1 1 1 C 0 0 f 0 C 1 1 f 0 C 1 0 f 1 C 2 0 f 2 f 2 Definition 3.2 (persistene module) A persistene module M is fmily of R-modules M i, together with homomorphisms ϕ i : M i M i+1. For exmple, the homology of persistene omplex is persistene module, where ϕ i simply mps homology lss to the one tht ontins it. Definition 3.3 (finite type) A persistene omplex {C i, f i } (persistene module {M i, ϕ i }) is of finite type if eh omponent omplex (module) is finitely generted R-module, nd if the mps f i (ϕ i, respetively) re isomorphisms for i m for some integer m. As our omplex K is finite, it genertes persistene omplex C of finite type, whose homology is persistene module M of finite type. We showed in the Introdution how suh omplexes rise in prtie. 6

3.1 Correspondene Suppose we hve persistene module M = {M i, ϕ i } i 0 over ring R. We now equip R[t] with the stndrd grding nd define grded module over R[t] y α(m) = M i, i=0 where the R-module struture is simply the sum of the strutures on the individul omponents, nd where the tion of t is given y t (m 0, m 1, m 2,...) = (0, ϕ 0 (m 0 ), ϕ 1 (m 1 ), ϕ 2 (m 2 ),...). Tht is, t simply shifts elements of the module up in the grdtion. Theorem 3.1 (orrespondene) The orrespondene α defines n equivlene of tegories etween the tegory of persistene modules of finite type over R nd the tegory of finitely generted non-negtively grded modules over R[t]. The proof is the Artin-Rees theory in ommuttive lger [12]. Intuitively, we re uilding single struture tht ontins ll the omplexes in the filtrtion. We egin y omputing diret sum of the omplexes, rriving t muh lrger spe tht is grded ording to the filtrtion ordering. We then rememer the time eh simplex enters using polynomil oeffiient. For instne, simplex enters the filtrtion in Figure 1 t time 0. To shift this simplex long the grding, we must multiply the simplex using t. Therefore, while exists t time 0, t exists t time 1, t 2 t time 2, nd so on. The ey ide is tht the filtrtion ordering is enoded in the oeffiient polynomil ring. We utilize these oeffiients in Setion 4 to derive the persistene lgorithm from the redution sheme in Setion 2.5. 3.2 Deomposition The orrespondene estlished y Theorem 3.1 suggests the non-existene of simple lssifitions of persistene modules over ground ring tht is not field, suh s Z. It is well nown in ommuttive lger tht the lssifition of modules over Z[t] is extremely omplited. While it is possile to ssign interesting invrints to Z[t]-modules, simple lssifition is not ville, nor is it ever liely to e ville. On the other hnd, the orrespondene gives us simple deomposition when the ground ring is field F. Here, the grded ring F [t] is PID nd its only grded idels re homogeneous of form (t n ), so the struture of the F [t]-module is desried y sum (2) in Theorem 2.1: ( n ) m Σ αi F [t] Σ γj F [t]/(t nj ). (5) i=1 j=1 We wish to prmetrize the isomorphism lsses of F [t]-modules y suitle ojets. Definition 3.4 (P-intervl) A P-intervl is n ordered pir (i, j) with 0 i < j Z = Z {+ }. We ssoite grded F [t]-module to set S of P- intervls vi ijetion Q. We define Q(i, j) = Σ i F [t]/(t j i ) for P-intervl (i, j). Of ourse, Q(i, + ) = Σ i F [t]. For set of P-intervls S = {(i 1, j 1 ), (i 2, j 2 )..., (i n, j n )}, we define Q(S) = n Q(i l, j l ). l=1 Our orrespondene my now e restted s follows. Corollry 3.1 The orrespondene S Q(S) defines ijetion etween the finite sets of P-intervls nd the finitely generted grded modules over the grded ring F [t]. Consequently, the isomorphism lsses of persistene modules of finite type over F re in ijetive orrespondene with the finite sets of P-intervls. 3.3 Interprettion Before proeeding ny further, we rep our wor so fr nd relte it to prior results. Rell tht our input is filtered omplex K nd we re interested in its th homology. In eh dimension the homology of omplex K i eomes vetor spe over field, desried fully y its rn β i. We need to hoose omptile ses ross the filtrtion in order to ompute persistent homology for the entire filtrtion. So, we form the persistene module orresponding to K, diret sum of these vetor spes. The struture theorem sttes tht sis exists for this module tht provides omptile ses for ll the vetor spes. In prtiulr, eh P-intervl (i, j) desries sis element for the homology vetor spes strting t time i until time j 1. This element is -yle e tht is ompleted t time i, forming new homology lss. It lso remins non-ounding until time j, t whih time it joins the oundry group B j. Therefore, the P-intervls disussed here re preisely the so-lled -intervls utilized in [11] to desrie persistent Z 2 -homology. Tht is, while omponent homology groups re torsionless, persistene ppers s torsionl nd free elements of the persistene module. 7

Our interprettion lso llows us to s when e + B l is sis element for the persistent groups H l,p. Rell Eqution (4). As e B l for ll l < j, we now tht e B l+p for l + p < j. Along with l i nd p 0, the three inequlities define tringulr region in the index-persistene plne, s drwn in Figure 5. The region gives us the vlues for whih the -yle e is sis element for H l,p. In other words, we hve just shown diret proof of the -tringle Lemm in [11], whih we restte here in different form. Lemm 3.1 Let T e the set of tringles defined y P- intervls for the -dimensionl persistene module. The rn β l,p of Hl,p is the numer of tringles in T ontining the point (l, p). Consequently, omputing persistent homology over field is equivlent to finding the orresponding set of P- intervls. (i, 0) (j, 0) index (l) p > 0 (i, j i) (i, 0) (j, 0) (i, j i) persistene (p) l > i l+p < j Figure 5. The inequlities p 0, l i, nd l+p < j define tringulr region in the index-persistene plne. This region defines when the yle is sis element for the homology vetor spe. 4 Algorithm for Fields In this setion we devise n lgorithm for omputing persistent homology over field. Given the theoretil development of the lst setion, our pproh is rther simple: we simplify the stndrd redution lgorithm using the properties of the persistene module. Our rguments give n lgorithm for omputing the P- intervls for filtered omplex diretly over the field F, without the need for onstruting the persistene module. This lgorithm is generlized version of the piring lgorithm shown in [11]. 4.1 Derivtion We use the smll filtrtion in Figure 1 s running exmple nd ompute over Z 2, lthough ny field will do. The persistene module orresponds to Z 2 [t]-module y the orrespondene estlished in Theorem 2.1. Tle 1 reviews the degrees of the simplies of our filtrtion s homogeneous elements of this module. d d d d 0 0 1 1 1 1 2 2 3 4 5 Tle 1. Degree of simplies of filtrtion in Figure 1 Throughout this setion we use {e j } nd {ê i } to represent homogeneous ses for C nd C 1, respetively. Reltive to homogeneous ses, ny representtion M of hs the following si property: deg ê i + deg M (i, j) = deg e j, (6) where M (i, j) denotes the element t lotion (i, j). We get d d d 0 0 t t 0 M 1 = 0 1 t 0 t 2 t t 0 0 0, (7) t 0 0 t 2 t 3 for 1 in our exmple. The reder my verify Eqution (6) using this exmple for intuition, e.g. M 1 (4, 4) = t 2 s deg d deg = 2 0 = 2, ording to Tle 1. Clerly, the stndrd ses for hin groups re homogeneous. We need to represent : C C 1 reltive to the stndrd sis for C nd homogeneous sis for Z 1. We then redue the mtrix nd red off the desription of H ording to our disussion in Setion 2.5. We ompute these representtions indutively in dimension. The se se is trivil. As 0 0, Z 0 = C 0 nd the stndrd sis my e used for representing 1. Now, ssume we hve mtrix representtion M of reltive to the stndrd sis {e j } for C nd homogeneous sis {ê i } for Z 1. For indution, we need to ompute homogeneous sis for Z nd represent +1 reltive to C +1 nd the omputed sis. We egin y sorting sis ê i in reverse degree order, s lredy done in the mtrix in Eqution (7). We next trnsform M into the olumnehelon form M, lower stirse form shown in Figure 6 [17]. The steps hve vrile height, ll lndings hve width equl to one, nd non-zero elements my only our eneth the stirse. A oxed vlue in the figure is pivot nd row (olumn) with pivot is lled 8

0 0 0 0. 0 0 0 Figure 6. The olumn-ehelon form. An indites non-zero vlue nd pivots re oxed. pivot row (olumn). From liner lger, we now tht rn M = rn B 1 is the numer of pivots in n ehelon form. The sis elements orresponding to non-pivot olumns form the desired sis for Z. In our exmple, we hve M 1 = d z 1 z 2 d t 0 0 0 0 t 1 0 0 0 0 t t 0 0 0 0 t 0 0, (8) where z 1 = d d t t, nd z 2 = t 2 t 2 form homogeneous sis for Z 1. The proedure tht rrives t the ehelon form is Gussin elimintion on the olumns, utilizing elementry olumn opertions of types (1, 3) only. Strting with the left-most olumn, we eliminte non-zero entries ourring in pivot rows in order of inresing row. To eliminte n entry, we use n elementry olumn opertion of type (3) tht mintins the homogeneity of the sis nd mtrix elements. We ontinue until we either rrive t zero olumn, or we find new pivot. If needed, we then perform olumn exhnge (type (1)) to reorder the olumns ppropritely. Lemm 4.1 (Ehelon Form) The pivots in olumnehelon form re the sme s the digonl elements in norml form. Moreover, the degree of the sis elements on pivot rows is the sme in oth forms. Proof: Beuse of our sort, the degree of row sis elements ê i is monotonilly deresing from the top row down. Within eh fixed olumn j, deg e j is onstnt. By Eqution (6), deg M (i, j) = deg ê i. Therefore, the degree of the elements in eh olumn is monotonilly inresing with row. We my eliminte nonzero elements elow pivots using row opertions tht do not hnge the pivot elements or the degrees of the row sis elements. We then ple the mtrix in digonl form with row nd olumn swps. The lemm sttes tht if we re only interested in the degree of the sis elements, we my red them off from the ehelon form diretly. Tht is, we my use the following orollry of the stndrd struture theorem to otin the desription. Corollry 4.1 Let M e the olumn-ehelon form for reltive to ses {e j } nd {ê i } for C nd Z 1, respetively. If row i hs pivot M (i, j) = t n, it ontriutes Σ deg êi F [t]/t n to the desription of H 1. Otherwise, it ontriutes Σ deg êi F [t]. Equivlently, we get (deg ê i, deg ê i + n) nd (deg ê i, ), respetively, s P- intervls for H 1. In our exmple, M1 (1, 1) = t in Eqution (8). As deg d = 1, the element ontriutes Σ 1 Z 2 [t]/(t) or P- intervl (1,2) to the desription of H 0. j i M M +1 m 1 x m m xm+1 j i = 0 Figure 7. As +1 = 0, M M +1 = 0 nd this is unhnged y elementry opertions. When M is redued to ehelon form M y olumn opertions, the orresponding row opertions zero out rows in M +1 tht orrespond to pivot olumns in M. We now wish to represent +1 in terms of the sis we omputed for Z. We egin with the stndrd mtrix representtion M +1 of +1. As +1 = 0, M M +1 = 0, s shown in Figure 7. Furthermore, this reltionship is unhnged y elementry opertions. Sine the domin of is the odomin of +1, the elementry olumn opertions we used to trnsform M into ehelon form M give orresponding row opertions on M +1. These row opertions zero out rows in M +1 tht orrespond to non-zero pivot olumns in M, nd give representtion of +1 reltive to the sis we just omputed for Z. This is preisely wht we re fter. We n get it, however, with hrdly ny wor. Lemm 4.2 (Bsis Chnge) To represent +1 reltive to the stndrd sis for C +1 nd the sis omputed for Z, simply delete rows in M +1 tht orrespond to pivot olumns in M. Proof: We only used elementry olumn opertions of types (1,3) in our vrint of Gussin elimintion. Only the ltter hnges vlues in the mtrix. Suppose we reple olumn i y (olumn i) + q(olumn j) in order to 9

eliminte n element in pivot row j, s shown in Figure 7. This opertion mounts to repling olumn sis element e i y e i +qe j in M. To effet the sme replement in the row sis for +1, we need to reple row j with (row j) q(row i). However, row j is eventully zeroed-out, s shown in Figure 7, nd row i is never hnged y ny suh opertion. Therefore, we hve no need for row opertions. We simply eliminte rows orresponding to pivot olumns one dimension lower to get the desired representtion for +1 in terms of the sis for Z. This ompletes the indution. In our exmple, the stndrd mtrix representtion for 2 is d t t 2 M 2 = d 0 t 3 d 0 t 3. t 3 0 t 3 0 To get representtion in terms of C 2 nd the sis (z 1, z 2 ) for Z 1 we omputed erlier, we simply eliminte the ottom three rows. These rows re ssoited with pivots in M 1, ording to Eqution (8). We get d ˇM 2 = z 2 t t 2, z 1 0 t 3 where we hve lso repled d nd with the orresponding sis elements z 1 = d d nd z 2 =. 4.2 Algorithm Our disussion gives us n lgorithm for omputing P- intervls of n F [t]-module over field F. It turns out, however, tht we n simulte the lgorithm over the field itself, without the need for omputing the F [t]- module. Rther, we use two signifint oservtions from the derivtion of the lgorithm. First, Lemm 4.1 gurntees tht if we eliminte pivots in the order of deresing degree, we my red off the entire desription from the ehelon form nd do not need to redue to norml form. Seond, Lemm 4.2 tells us tht y simply noting the pivot olumns in eh dimension nd eliminting the orresponding rows in the next dimension, we get the required sis hnge. Therefore, we only need olumn opertions throughout our proedure nd there is no need for mtrix representtion. We represent the oundry opertors s set of oundry hins orresponding to the olumns d d d d 0 1 2 3 4 5 6 7 8 9 10 4 5 6 10 9 d Figure 8. Dt struture fter running the lgorithm on the filtrtion in Figure 1. Mred simplies re in old itli. of the mtrix. Within this representtion, olumn exhnges (type (1)) hve no mening, nd the only opertion we need is of type (3). Our dt struture is n rry T with slot for eh simplex in the filtrtion, s shown in Figure 8 for our exmple. Eh simplex gets slot in the tle. For indexing, we need full ordering of the simplies, so we omplete the prtil order defined y the degree of simplex y sorting simplies ording to dimension, reing ll remining ties ritrrily (we did this impliitly in the mtrix representtion.) We lso need the ility to mr simplies to indite non-pivot olumns. Rther thn omputing homology in eh dimension independently, we ompute homology in ll dimensions inrementlly nd onurrently. The lgorithm, s shown in Figure 9, stores the list of P-intervls for H in L. COMPUTEINTERVALS (K) { for = 0 to dim(k) L = ; for j = 0 to m 1 { d = REMOVEPIVOTROWS (σ j ); if (d = ) Mr σ j ; else { i = mxindex d; = dim σ i ; Store j nd d in T [i]; L = L {(deg σ i, deg σ j )} } } for j = 0 to m 1 { if σ j is mred nd T [j] is empty { = dim σ j ; L = L {(deg σ j, )} } } } Figure 9. Algorithm COMPUTEINTERVALS proesses omplex of m simplies. It stores the sets of P-intervls in dimension in L. When simplex σ j is dded, we he vi proedure REMOVEPIVOTROWS to see whether its oundry hin d orresponds to zero or pivot olumn. If the hin is empty, it orresponds to zero olumn nd d 10

hin REMOVEPIVOTROWS (σ) { = dim σ; d = σ; Remove unmred terms in d; while (d ) { i = mxindex d; if T [i] is empty, re; Let q e the oeffiient of σ i in T [i]; d = d q 1 T [i]; } return d; } Figure 10. Algorithm REMOVEPIVOTROWS first elimintes rows not mred (not orresponding to the sis for Z 1 ), nd then elimintes terms in pivot rows. we mr σ j : its olumn is sis element for Z, nd the orresponding row should not e eliminted in the next dimension. Otherwise, the hin orresponds to pivot olumn nd the term with the mximum index i = mxindex d is the pivot, ording the proedure desried for the F [t]-module. We store index j nd hin d representing the olumn in T [i]. Applying Corollry 4.1, we get P-intervl (deg σ i, deg σ j ). We ontinue until we exhust the filtrtion. We then perform nother pss through the filtrtion in serh of infinite P-intervls: mred simplies whose slot is empty. We give the funtion REMOVEPIVOTROWS in Figure 10. Initilly, the funtion omputes the oundry hin d for the simplex. It then pplies Lemm 4.2, eliminting ll terms involving unmred simplies to get representtion in terms of the sis for Z 1. The rest of the proedure is Gussin elimintion in the order of deresing degree, s ditted y our disussion for the F [t]-module. The term with the mximum index i = mx d is potentil pivot. If T [i] is non-empty, pivot lredy exists in tht row, nd we use the inverse of its oeffiient to eliminte the row from our hin. Otherwise, we hve found pivot nd our hin is pivot olumn. For our exmple filtrtion in Figure 8, the mred 0-simplies {,,, d} nd 1-simplies {d, } generte P-intervls L 0 = {(0, ), (0, 1), (1, 1), (1, 2)} nd L 1 = {(2, 5), (3, 4)}, respetively. 4.3 Disussion From our derivtion, it is ler tht the lgorithm hs the sme running time s Gussin elimintion over fields. Tht is, it tes O(m 3 ) in the worst se, where m is the numer of simplies in the filtrtion. The lgorithm is very simple, however, nd represents the mtries effiiently. In our preliminry experiments, we hve seen liner time ehvior for the lgorithm. 5 Algorithm for PIDs The orrespondene we estlished in Setion 3 elimintes ny hope for simple lssifition of persistent groups over rings tht re not fields. Nevertheless, we my still e interested in their omputtion. In this setion, we give n lgorithm to ompute the persistent homology groups H i,p of filtered omplex K for fixed i nd p. The lgorithm we provide omputes persistent homology over ny PID D of oeffiients y utilizing redution lgorithm over tht ring. To ompute the persistent group, we need to otin desription of the numertor nd denomintor of the quotient group in Eqution (4). We lredy now how to hrterize the numertor. We simply redue the stndrd mtrix representtion M i of i using the redution lgorithm. The denomintor, B i,p = B i+p Z i, plys the role of the oundry group in Eqution (4). Therefore, insted of reduing mtrix M+1 i, we need to redue n lternte mtrix M i,p tht desries this +1 oundry group. We otin this mtrix s follows: (1) We redue mtrix M i to its norml form nd otin sis {z j } for Z i, using ft (ii) in Setion 2.5. We my merge this omputtion with tht of the numertor. (2) We redue mtrix M i+p +1 to its norml form nd otin sis { l } for B i+p using ft (iii) in Setion 2.5. (3) Let N = [{ l } {z j }] = [B Z], tht is, the olumns of mtrix N onsist of the sis elements from the ses we just omputed, nd B nd Z re the respetive sumtries defined y the ses. We next redue N to norml form to find sis {u q } for its null-spe. As efore, we otin this sis using ft (ii). Eh u q = [α q ζ q ], where α q, ζ q re vetors of oeffiients of { l }, {z j }, respetively. Note tht Nu q = Bα q + Zζ q = 0 y definition. In other words, element Bα q = Zζ q is elongs to the spn of oth ses. Therefore, oth {Bα q } nd {Zζ q } re ses for B i,p mtrix M i,p +1 from either. = Bi+p Z i. We form We now redue M i,p +1 to norml form nd red off the torsion oeffiients nd the rn of B i,p. It is ler from the proedure tht we re omputing the persistent groups orretly, giving us the following. Theorem 5.1 For oeffiients in ny PID, persistent homology groups re omputle in the order of time nd spe of omputing homology groups. 11

6 Experiments In this setion, we disuss experiments using n implementtion of the persistene lgorithm for ritrry fields. Our im is to further eluidte the ontriutions of this pper. We loo t two senrios where the previous lgorithm would not e pplile, ut where our lgorithm sueeds in providing informtion out topologil spe. χ = ( 1) s. We use the Morse funtion to ompute the exursion set filtrtion for eh dtset. Tle 3 gives informtion on the resulting filtrtions. 6.1 Implementtion We hve implemented our field lgorithm for Z p for p prime, nd Q oeffiients. Our implementtion is in C nd utilizes GNU MP, multipreision lirry, for ext omputtion [14]. We hve seprte implementtion for oeffiients in Z 2 s the omputtion is gretly simplified in this field. The oeffiients re either 0, or 1, so there is no need for orienting simplies or mintining oeffiients. A -hin is simply list of simplies, those with oeffiient 1. Eh simplex is its own inverse, reduing the group opertion to the symmetri differene, where the sum of two -hins, d is + d = ( d) ( d). We use 2.2 GHz Pentium 4 Dell PC with 1 GB RAM running Red Ht Linux 7.3 for omputing the timings. 6.2 Dt Our lgorithm requires persistene omplex s input. In the introdution, we disussed how persistene omplexes rise nturlly in prtie. In Exmple 1.3, we disussed generting persistene omplexes using exursion sets of Morse funtions over mnifolds. We hve implemented generl frmewor for omputing omplexes of this type. We must emphsize, however, tht our persistene softwre proesses persistene omplexes of ny origin. Our frmewor tes tuple (K, f) s input nd produes persistene omplex C(K, f) s output. K is d-dimensionl simpliil omplex tht tringultes n underlying mnifold, nd f : vert K R is disrete funtion over the verties of K tht we extend linerly over the remining simplies of K. The funtion f ts s the Morse funtion over the mnifold, ut need not e Morse for our purposes. Frequently, our omplex is ugmented with mp ϕ : K R d tht immerses or emeds the mnifold in Euliden spe. Our lgorithm does not require ϕ for omputtion, ut ϕ is often provided s disrete mp over the verties of K nd is extended linerly s efore. For exmple, Figure 11 displys tringulted Klein ottle, immersed in R 3. For eh dtset, Tle 2 gives the numer s of -simplies, s well s the Euler hrteristi Figure 11. A wire-frme visuliztion of dtset K, n immersed tringulted Klein ottle with 4000 tringles. K len filt (s) pers (s) K 12,000 1,020 0.03 < 0.01 E 529,225 3,013 3.17 5.00 J 3,029,383 256 24.13 50.23 Tle 3. Filtrtions. The numer of simplies in the filtrtion K = i s i, the length of the filtrtion (numer of distint vlues of funtion f), time to ompute the filtrtion, nd time to ompute persistene over Z 2 oeffiients. 6.3 Field Coeffiients A ontriution of this pper is the generliztion of the persistene lgorithm to ritrry fields. This ontriution is importnt when the mnifold under study ontins torsion. To me this ler, we ompute the homology of the Klein ottle using the persistene lgorithm. Here, we re interested only in the Betti numers of the finl omplex in the filtrtion for illustrtive purposes. The non-orientility of the Klein ottle is visile in Figure 11. The hnge in tringle orienttion t the prmetriztion oundry leds to rendering rtift where two sets of tringles re front-fing. In homology, the non-orientility of the Klein ottle mnifests itself s torsionl 1-yle where 2 is oundry (indeed, it ounds the surfe itself.) The homology groups over Z re: H 0 (K) = Z, H 1 (K) = Z Z 2, H 2 (K) = {0}. 12

numer s of -simplies 0 1 2 3 4 K 2,000 6,000 4,000 0 0 0 E 3,095 52,285 177,067 212,327 84,451 1 J 17,862 297,372 1,010,203 1,217,319 486,627 1 χ Tle 2. Dtsets. K is the Klein ottle, shown in Figure 11. E is potentil round eletrostti hrges. J is supersoni jet flow. F β 0 β 1 β 2 time (s) Z 2 1 2 1 0.01 Z 3 1 1 0 0.23 Z 5 1 1 0 0.23 Z 3203 1 1 0 0.23 Q 1 1 0 0.50 Tle 4. Field oeffiients. The Betti numers of K omputed over field F nd time for the persistene lgorithm. We use seprte implementtion for Z 2 oeffiients. Note tht β 1 = rn H 1 = 1. We now use the height funtion s our Morse funtion, f = z, to generte the filtrtion in Tle 3. We then ompute the homology of dtset K with field oeffiients using our lgorithm, s shown in Tle 4. Over Z 2, we get β 1 = 2 s homology is unle to reognize the torsionl oundry 2 with oeffiients 0 nd 1. Insted, it oserves n dditionl lss of homology 1-yles. By the Euler-Poinré reltion, χ = i β i, so we lso get lss of 2-yles to ompenste for the inrese in β 1 [16]. Therefore, Z 2 -homology misidentifies the Klein ottle s the torus. Over ny other field, however, homology turns the torsionl yle into oundry, s the inverse of 2 exists. In other words, while we nnot oserve torsion in omputing homology over fields, we n dedue its existene y ompring our results over different oeffiient sets. Similrly, we n ompre sets of P-intervls from different omputtions to disover torsion in persistene omplex. Note tht our lgorithm s performne for this dtset is out the sme over ritrry finite fields, s the oeffiients do not get lrge. The omputtion over Q tes out twie s muh time nd spe, sine eh rtionl is represented s two integers in GNU MP. 6.4 Higher Dimensions A seond ontriution of this pper is the extension of the persistene lgorithm from suomplexes of S 3 to omplexes in ritrry dimensions. We hve lredy utilized this pility in omputing the homology of the Klein ottle. We now exmine the performne of this lgorithm in higher dimensions. For prtil motivtion, we use lrge-sle time-vrying volume dt s input. Advnes in dt quisition systems nd omputing tehnologies hve resulted in the genertion of mssive sets of mesured or simulted dt. The dtsets usully ontin the time evolution of physil vriles, suh s temperture, pressure, or flow veloity t smple points in spe. The gol is to identify nd lolize signifint phenomen within the dt. We propose using persistene s the signifine mesure. The underlying spe for our dtsets is the fourdimensionl spe-time mnifold. For eh dtset, we tringulte the onvex hull of the smples to get tringultion. Eh omplex listed in Tle 2 is homeomorphi to four-dimensionl ll nd hs χ = 1. Dtset E ontins the potentil round eletrostti hrges t eh vertex. Dtset J reords the supersoni flow veloity of jet engine. We use these vlues s Morse funtions to generte the filtrtions. We then ompute persistene over Z 2 oeffiients to get the Betti numers. We give filtrtion sizes nd timings in Tle 3. Figure 12 displys β 2 for dtset J. We oserve lrge numer of two-dimensionl yles (voids), s the o-dimension is 2. Persistene llows us to deompose this grph into the set of P-intervls. Although there re 730,692 P- intervls in dimension 2, most re empty s the topologil ttriute is reted nd destroyed t the sme funtion level. We drw the 502 non-empty P-intervls in Figure 13. We note tht the P-intervls represent ompt nd generl shpe desriptor for ritrry spes. β 2 f 90 80 70 60 50 40 30 20 10 0 0 50 100 150 200 250 f Figure 12. Grph of β f 2 for dtset J, where f is the flow veloity. 13

Figure 13. The 502 non-empty P-intervls for dtset J in dimension 2. The mlgmtion of these intervls gives the grph in Figure 12. For the lrge dt sets, we do not ompute persistene over lternte fields s the omputtion requires in exess of 2 gigytes of memory. In the se of finite fields Z p, we my restrit the prime p so tht the omputtion fits within n integer. This is resonle restrition, s on most modern mhines with 32-it integers, it implies p < 2 16 1. Given this restrition, ny oeffiient will e less thn p nd representle s 4-yte integer. The GNU MP ext integer formt, on the other hnd, requires t lest 16 ytes for representing ny integer. 7 Conlusion We elieve the most importnt ontriution of this pper is reinterprettion of persistent homology within the lssil frmewor of lgeri topology. Our interprettion llows us to: 1. estlish orrespondene tht fully desries the struture of persistent homology over ny field, not only over Z 2, s in the previous result, 2. nd relte the previous lgorithm to the lssi redution lgorithm, therey extending it to ritrry fields nd ritrry dimensionl omplexes, not just suomplexes of S 3 s in the previous result. We provide implementtions of our lgorithm for fields, nd show tht they perform quite well for lrge dtsets. Finlly, we give n lgorithm for omputing persistent homology group with fixed prmeters over ritrry PIDs. Our softwre for n-dimensionl omplexes enles us to nlyze ritrry-dimensionl point loud dt nd their derived spes. One urrent projet uses this implementtion for feture reognition using novel lgeri method [2]. Another projet nlyzes the topologil strutures in high-dimensionl dt set derived from nturl imges [7]. Yet nother pplies persistene to derived spes to rrive t ompt shpe desriptors for geometri ojets [3, 5]. Future theoretil wor inlude exmining invrints for persistent homology over non-fields nd defining multivrite persistene, where there is more thn one persistene dimension. An exmple would e tring Morse funtion s well s density of smpling on mnifold. Finlly, we hve reently reimplemented the lgorithm using the generi prdigm. This implementtion will soon e prt of the CGAL lirry [4]. Anowledgments The first uthor thns Herert Edelsrunner nd John Hrer for disussions on Z-homology, nd Leo Guis for providing support nd enourgement. Both uthors thn Anne Collins for her thorough review of the mnusript nd Ajith Msrenhs for providing simpliil omplexes for dtsets E nd J. The originl dt re prt of the Advned Visuliztion Tehnology Center s dt repository nd ppers ourtesy of Andre Mlgoli nd Milen Miono of the Lortory for Astrophysis nd Spe Reserh (LASR) t the University of Chigo. Figure 11 ws rendered in Stnford Grphis L s Snlyze. Referenes [1] BASU, S. On ounding the Betti numers nd omputing the Euler hrteristi of semi-lgeri sets. Disrete Comput. Geom. 22 (1999), 1 18. [2] CARLSSON, E., CARLSSON, G., AND DE SILVA, V. An lgeri topologil method for feture identifition, 2003. Mnusript. [3] CARLSSON, G., ZOMORODIAN, A., COLLINS, A., AND GUIBAS, L. Persistene rodes for shpes. In Pro. Symp. Geom. Proess. (2004), pp. 127 138. [4] CGAL. Computtionl geometry lgorithms lirry. http://www.gl.org. [5] COLLINS, A., ZOMORODIAN, A., CARLSSON, G., AND GUIBAS, L. A rode shpe desriptor for urve 14