PYRAMIDAL SIMPLICIAL COMPLEXES

h4icllela Bertolotto, Leila De Floriani, Paola Marzano

ABSTRACT

We propose a new mo{le] for Iepreseuting a liypersurfacedescribing a scalar field in ally dimension at ,Iifferellt levelsof detail. The model is based on a smluence of domain de-compositions into simplicial complexes and integrates themodeling characteristics of simplicial complexes wit.11 tileversatility of a multilevel descriptiml. we I)leseIlt a (hastructure an(i algorithms for efficiently encoding a]k[l nmlli}>-ulating such model The proposed (Iata structlwe is slwcifkfor applications in which a “~loll-to~~ol{~gical” lelJlesrlltatiollis required. Such applications include vollunetric data visu-alization and geometric [Iuery processing.

1 INTRODUCTION

Several important, applications ill geography, g@~kW’, lli~J-medical engineering, elect. rolllagtletics, fllli{lo(lynanli rs, me-chanical engineering, chemistry, etc. lwml recmlstl Iwtillg,analyzing, aud visuaiiziug hypersurfaces describetl by scahilfields. Examples are topographic data, msxlical inmges, (Iat,afrom finite element analysis of electrcsmaglletir systenis, air-crafts, cars, and mechanical parts, simldaticms of molecldalstructures.

A hypersurface can be represented l>y a digital IIlo(lelcharacterized by a cfomaiu partition and hy a family of funr-tions defined on such partition. Digital models are oftenintended as approximations of hypersurfmces describil 1~ nat -uraf objects or phenomena, that are eitliel sinndated or nlea-sured through a sampling process. A better precision in tlwapproximation is generally paiti in ternls of a fiuer partitionof the domain, yielding higher storage costs imd comljllta-tion times On the other IIall(l, not, all t..a<lis withiu a giveuapplication necessarily Ie(plirr the salne accmrac,y, al)(l rveua single task may need (Iiffereld [Iegrees (If resol~lt iml in tlif-ferent portions of the (lomail],

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Multilesolution models offer the posaibilit y of repre-senting and analyziug a hypersurface at different degrees ofresolution: thus, for a given application, a coarse representa-tion will he used over certain areas, while a high resolutionwill he e]npioyed ill specific areas of interest. SuCh mod-els SIIOU1{I uot illtrodllce a serious overhead with respect toIlylwrsurface nlo[lels at tile maximum ]wecision, and shoulds{l~jport efficie]it accessi I]g algoritblns.

The problem of represmlting scalar fiekis at, differentlevels of resollltioli lms been extensively studied only in thetwo-(lilllellsiollal ca..e for IIlllltiresolutioll terrain descriptionill GIS applications. Multiresolut,ion nlodels have been cle-srlibefl in t.lle litelat\ue either through a strictly hierarchi-cal structuw (see, for instance) [DeF92, Sam92, Sca92] ) orthrough a more general stratified structure [De F89].

In tlw case of t.ltlee-(lilllellsiolld data, the major appli-cation is visualization of vo]umet.sic datasets in the contextof vohune ren{kwing (see, for instance, [cJar90, Lor87, Wi191,Wil!l?h] ). Very few prcq}osals for mult irresolution xepresen-tatiolls exist ill this case: we can mention the octree-basedUIO(ICI(see [Wi192a]) am] the simplicial model proposed in[Bm!14a], whir% hm hem (kfined for arl~itrary dimensions.

Tlw niaill co]triblltion of this pal,er is in the defini-tioli of a l)yramiflal mo{lel for represelking hypersurfaces<Iefilled by multi variate scalar fields. Such model is basedcm a se{plelice of simplicial complexes IIaving their verticesat data points. Each complex represents an approximationof the hypersurface at a given level of Imecisiou and is obtailmd by applying a wfinemeut critel ion to the complexprecee[liug it in t.lw sequeuce. Different complexes are co]l-ceptnally conuectetl hy iuterfereuce relations, which describeilltersect,ions alnollg their simplices. A special case is whenilhersectiou rwlllces to iill iuclusion rel;it.ion, i.e., there is arent.aillmellt I rlat iml alllong simplices {~fdifferent levels. tnttlis c~w, t.lw IIltj(lel is callcvl hierarcliical, because it canIw represented t luough [i strictly hieral chical structure (see[S; U1L90,DeF!)2, S. WI192,Sca92]), Thus, a hierarchical modelis jlwt, a pyralui(lal Iuo{lel with special lmoperties and corre-

SPOII(IS to the situation in which the refinement crit,erioll is.al]l)lied locall,y to a I)oltioll of the domain.

The niodel we consider supports an explicit multires-olution description of a scalar field based on a decreasingSe(llwnce of tolerance vahws; it can be I)llilt from both regu-litd~ an,l irregnl:wly (Iistrihuted data, al,d it allows focusing

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on local areas of interest,. When a global xefinemeut c.riterianis used, such model presents the advantage of mailltainiliggeometric properties globally, at any level of precisiuu. Forexample, if we consider a pyramidal model based cm Del.au-uay simplicial complexes (such as that, (Iescrilwi ill [DeF89]),the empty-sphere property characterizing such cmnplcxes isglobafly satisfied at each level, while this is not true in thehierarchical case, in which geometric cent.aiumelit forces toguarantee such property only locally (see [DeF!)2]).

The contribution of this paper is in proposing an effi-cient data stmcture for encoding pyramidal n~odels a.. wellas algorithms for their construction, analysis aml manilmla-tion.

The remainder of the paper is organized w follows. Sec-tion 2 reviews the definition of simplicial complex and of (lig-ital hypersurface model; Section 3 introduces the coucept ofpyramidal model. Sectiou 4 out,liues fundamental comlm-tatioual issues in the framework of lllilltiresol~ltir>li mmlels.Section 5 describes an efficient representation for pyrami{lalmodels; in Section 6 and 7 algorithms for coustructi]lg amlmanipulating a pyramidaf model are proposed. Fiually, inSection 8, some concluding remarks are preseut.ecl.

2 PRELIMINARY DEFINITIONS

2.1 Euclidean Simplicial Complexes

In this Section, we introduce the definition of Elwli(leansimplicial complexes that, are the most, inlpurt.ant ckmw ofcomplexes used in designing digital models for hypersurfaceapproximation.

Let. Vr = {uo, wI, ..., 1)(1} be a set, of d + 1 affhlely in-

dependent points in the d-dimensional Euclideau space IE’l.The subset u of I@ formed by the poiuts, which call be ex-pressed as linear convex combination of the points of Vm, iscalled a d-simpler. The points of V“ are called vertice.~ of m,while d is the order of a. Any s-simplex ~, O < s ~ d, whichis generated by a subset of s + 1 vertices of u, is called ans-face of u; if s < d, then 7 is called a proper ~ace of u. Theinterior of a given simplex a will he denoted hy ii~t(o).

A collection Z of simplices is calletl an E~~rfidef~~~d-.qimpliciaf cf1174pkx (or, simply, ;t d.c(~flt]de.t:) WlIeII t.llef{jl-lowing conditions hold:

● for each simplex u E E, all faces of a heloug to Z;

● for each pair of simplices u, r ~ X, either o n ~ = 0 ora n T is a face of both u .aml T;

● d is the maximum among the orders of simpli ces be-longing to Z (d is called the order of X).

In particular, for d=2, a d-conlplex .aml a d-simplex are atriangulation and a hion~le, respectively. For (1=3, a d-complex aud a d-simplex are a tetr(llit:~lr(~fiz(ltI(Jrl and a tftf.e-hwh-on, respectively. Figure 1 shows m] examl~le of tetlid]e-drafization.

The union of all s-simplices of X, O s s s d, re~m(let Ias point sets, is the domain c]f X, denoted by A(X). Ad-complex Z is regulur if and ouly if, for each s-simldex r,

Figure 1: An example of tetrahdralization.

with s < d, there exists a d-simplex u such that ~ is a properface of a. In the following, we will always consider regulard-complexes. Note that a regldar d-complex X is uniquelyclmracterized by the collection of its d-simplices, since allother .s-simplims, for s < d, can be obtained as linear convexccmd>ilmtiom of vertices of d-simplices. For this reason, inthe following, except, when otherwise spticified, a d-simplex,iu a d-simp~cial complex Z, will be called a simplex. Wewill onlit the dimensiou also for complexes, i.e., we will useccnuplex iustead of d-complex, if no ambiguity arises.

Among all possible simplicial complexes, the most conl-mcmly used are Delalmay complexes: a Delaunay complexX is a conlplex such that the circumsphere of any simplexa 6 ~ tlOes not. contain any vertex of Z inside. Theseconlph-xes have imImrt.aut properties, SLICfI as the relativee[lllizillglll:ility of their simIJices (that is important for nu-mrrical iuterpolatiou), and the possibility of defining a vis-ibility orfking [EdefWa] (as it is required by visualizationteclmi(lues [Wil!)l]). Moreover, several algorithms for build-ing sllcl] coml)lexes have been proposwl in the literature[Avix3, Cig93, Ede90h, .Joe!ll, Wat81].

2.2 Digital Hypersurface Models

A d-dimtmsional hypersurface is mathematically clecribed bythe gral]ll of a function ~ : Q + El., defined over a domainO ~ I@. In practical applications, flmction ~ i~apnoriuulamwu, while its value at a finite set of points P in the[Iolllaill O is obtailmxl I)y sampling the clevatiou of the hy -pelwuface,

Let fl ~ IE’l he a compact connected domain, and letF = {~1, . . . . PN} a finite set of points of Sl at which thevail w of function f has been sampled. As the number ofsaln~ )led ]>oints can be very large, the problem of buildinga model, that approximates ~ within a certain precision ontlw (Ioluail], arises.

A fillit.e approximate description of the hypersurfacecorrespomling to the graph of J over fl can be given byt,essellatillg !2 iuto simplices, such that j can be piecewiseapproximated by usiug a different, function over each suchsimidex. To this aim, ;ve define a digitd (simplicial) hyper-

su I.ffJrr r~ioflrf (fur brevity, }Iypers?mfacr rnoflel, in the follow-illg) as a pair @ = (Z, F), where Z is a simpliciaf complexslicli that A(X) z ff, and F = {?1, fz, ..., fr} (with r equ~to the lnunber of simplices of Z) is a family of functions suchtlmt each ~, is defined over a simplex m, of ,X. If ~ is con-

154

tinuous, all functions in F must have t IIe sanle vallw u1l t Ivsfaces shared by different simplices. HeIe, we cousi[k WIIYpiecewise-linear interpolating nlode&, in wlticb the set V ofvertices of Z is always a subset, of P, a]ltl all f(ll]rt imis 01 Fare liuear interpolants of f at the vertices of the sillll}licrsof z.

Several criteria can be applied fm Iueasurillg t.fw errorin approximating a hypersurface throllgll a moflel O. tJsl I-ally, however, the maximum {Iist.auce CM-l all points 0! Fbetween function f and the fuuctions ill F is usetl. III whatfollows, given a hypersurfacx model ~, baswl oli ii colIIIJex~, we will rfenot,e by E(Z) t,lle error (nleawmxl t.fllollgll solur

given criterion) in approximating the llylwrsulface t]mollgll@ within the domain A(X). If E(X) s c, we will siIy that @satisfies precision c. In a similar way, we will deuote I,y E(cr)the approximation error within a simplex m of X, aml we willsay that a satisfies precision E if E(m) < e. III wlmt fullows,if no ambiguity arises, we will use Z a]l(l d itlt.elclla~]geal>ly.

3 HYPERSURFACE MODELING THR-

OUGH PYRAMIDAL COMPLEXES

in this %ction, we fornmlly (lefilw ]Iyl mui[lal lIy[wwlufaremodels. A pyrami(l <al Iiypwwuface IIIWIVI is c\mIIwwtl of asetluence of hypersurface Ieplese]lt ations, earn of wllirli isobtained by refiniug an iuit.ial model at i]lcreasill~ly fhierlevels of detail. We first foclls ou tile structure of dumaiusubdivision by iut.reducing t,lle collce]~t of I)y]aulitlal rolll-

plex. Here, for simplicity, we mdy {lefiIIe I)yralllitlal si]npli-ciaf complexes, since the data structure and the algorithmsproposed are specific for tfw siml~licial case, altl]otlgll tbvdefinitions could be easily ext.entletl to tlke case of genericcomplexes.

A d-dimensional pyrami<lal complex is IJase{l 011a fhiitesequence of d-complexes having \rertices ,at a set. V of I)oi])tsin Ed: each complex is obtaiu~(l from t lIe C(III1lIIPXlmxwetl-illg it, in t,fle sequence by app].yiug a refillelllellt critcliml.

Let C be the couvex hull of V. A ~i-(lillie]~si[)ll:tl pylurIIi-dal sirnpliciul conbplw (also pymmidfd COIIIIJIC:I:)is a rollec-t,ion S={EO, . . . Z), } of d-collll>leXes slldl that. Vj = 0, . . . . /1,V, is the set of vertices of V,, A(S, ) ~ C, aI,(lVk = 1, . ...11,Vk-l c VA. Figure 2 shows an example of a t!vo-[litllellsi{,l)alpyramidal complex.

A simplex u may belong to (liffereut complexes in t.lwsequence S: u belongs t.o a giveh compkx X, if eit.ll?r a Il<asbeen created at level I w at a level J < I, aIII I it Ims Ii(d IIeelI

I is sai(l to 1~~affected by refinmnents ocrnured betwetill levels I aIIll ,1.X, is the first complex in whirll a al~[waw,the creation Irtdof u. TIN- tl~rt.shcJd tmiut of rr c~melNIII(lsto the first level greater tlmll f at which T (lisal~lwa]s.

Pairs of sin~plices belcmgillg to Couseclltive comlkxesin the sequence S may have a spatial ilkerfvreuce, i.e. , asimplex Uk of ~, may iuk?rsect a Sllllpk fl[ {If ~,+ 1. Aspecial c~~e of pyrami(lal com]dex consists of a setlllellm- S

of simplicial coulplexes in wl]icll interference relatious be-tween pairs of simplices belongiug to conserlltive coIII]kxesreduce to inclusion relations. More precisely, giveu t WC)siul-plices uk and ul , SllCb t.h;it Ok E ~, all[] C{ 6 ~,+ 1, :111(I

itlt(uk) fl irlt(al) # 0, tb~]l al ~ Uk (regal’de(l [i> P[li]lt, sets).Moreover, the union of all t Iw f71’sof Z,+ I t.llat are inclll(lwl

@

@

@

Figlwe 2: AIL exaIIIple of p,vramidal

z~

triangulation,

iu a simplex rk of X, is a simplicial complex whose dotnaincovers IY~. SIIch a pyramiflal complex, that we will calf a hi-emwhicul (.~implicid) complex, corresponds to the situationin which the refillenlelk criterion is locally applied to a sin-

gle simplex ilwtea[l of globally to the whole domain. Figure3 SIIOWSall exalII1>le of Ilierarcl]ical sim],licial complex.

(23 ‘0

Figure 3: An example of a hierarchical simplicial complex.

III pyramidal complexes, when the refinement criterionis applied to the whole domain, geometlic properties areglobally satisfied at eaclt level. For example, if we considerI)ylallii{lal IIIo(lels based ou Delaunay complexes, the empty-Sld IWe plolwrty cl)aractei.izi]}g such complexes is globaflysatisfied at, eacl] level [DeF89].

A pyralllidal COnLId~X may be re] ,rmented by means

155

of a graph, in which interference relat,iom between pairs ofsimplices are explicitly encoded aml sinlplice+ belonging toconsecutive levels are not, duplicatd. More formally, givma P.yramldal model S = {20, . , ~), , Nub gl,ii]lll-lliise(l IelY-

)..

resentat.ion is a pair ‘P = (S’, A), wlwre

S’ is a collect.jon {Z:, ., ., Z;,} of simljlicial colll-plexes such that

1. x& = Xo

2. for every j = 1, . . . . II, Z; consists of the d-simplices of 21 except for those belollgiugalso to EJ -1.

A = {(~~,~~,(Uk,C7/)) I j > i ~j,~j = S’, ak

d-simplex of X:, U1 d-simplex Of z;, itlt(c~ ) nint(ul ) # a and ~ is the tfll’eShOkl Of uk }.

in such graph, nodes encode flef)mebic aud topologicalinformation about simplices in the same complex, while arcsrepresent, interference relations between simplices belougingto different complexes in the pyranlidal coml,lex.

Note that each simplex a belonging to cousecllt.ive rxml-plexes of a pyramidaf simplicial complex S is representedonly once in S’ (i. e., at. its creation level), thus avoi{lillg tbi-pfications. For this reason, simplices in X: are a (generallyproper) subset of the simplices forming Z,, aml they maycover a disconnected portion of the domain C!.

‘P is a Multigraph haviug S’ m its set of llo,les aml Aas its set of multiarcs. Each triple (X;, Z;, (m~, m )) E A is

an arc fron~ Z; to X; labeled with a pair (I7K,a{ ). Siuce asimplex uk belonging to a conlpkx ~, call have a Imlwsmlt,yintersection with several simplices of S] (with j > i), u~ callappear, as a first eletnentj, in the label .wsociat.e{l with severalarcs joining Z: to E;. Similarly, a simplex fl~ behging toXl can have a uon-empty iutersect,ion with several siml,licesof Z;; thus, it can appear, as second elemeut, ill the label ofseveral arcs joining Z: to E;. Since {hqdications are avoi(letl,for each tripk (~~, ~~, (uk, al)) ~ A, u~ # m.

Figure 4 shows the graph “P representi]lg tbe pyrami(laltriangulation of Figure 2.

In a hierarchical complex, a simplex al bekmging to 2,may be contained into a simplex Uk belongihg t.o X,, whereJ’ > i is the threshold of u~. Thus, ui can appear, ,SSsec-ond element, in the label of only one arc joining X: to Z;.SinlPlex uk, on the COIltIaI’Y, appeaI’S, W fil’St elelllrllt, ill

the label of alf the arcs joining E; to S; correspomlilig tothose simpfices CT1of ~j such that, al C IYk. Bawxl on thisproperty, it is possible to group alf simplices in E; that, ill-kRX?Ct, Uk d!O a ShIgk IIOdt?ail(l tO COIllleCt ~~ tO SllCh IIO{le

th’OUgh ZiSkkjk aI’C kibekd Only with Uk. 111SUCh ~ way, arcS

previously connecting E: to E; aml haviug ok as first, labelelement become a single arc Iabek(l with C7k,thlls I“t!dllcillgtbe number of arcs to be enco{led. We Id oldy avui(l [111-plications (thus, simplices created at a previous level i aldnot modified from level i to level j are not. relwate[l ), 1nltsimpfices in E; are partitioned into a collection of siulpli-cial complexes. The resulting structure is, thus, a tree (seeFigure 5).

The concepts of hypersurface model ((lefincxl ill Sectiou2.2) ancl of pyramidaf cwuplex (above describe(l) call 1)ecombined in order to define a pyrami( Ial hypersurface mm let.

Figlnw 4: The graph ‘P representing tlw pyramiclaf triangu-liiticm of Figuru 2: nodes of the graph are shaded.

A pyrami(lal hypersurface model is simply a hypersurfacemodel b.a..ed on a pyramidal complex. Formally, let S be apyral~lidal colllplex, and Vi = O, ... h let @i = (Z,, F’i) be ahypersurface n:o,lel defined over Ei E S. Let F.s denote theset, {F,, i = O, .,, II}. The pair Ds=(S, FS) is a PyrurniclufHyptrsIIr~mx Morlt4 (PHM). In the following, we will referto a PHM Ds and to the underlying pyramidal complex Sillt.el.cllallgeabl.y. In particular, if the pyramidal complex isdescribed by the graldi-bzwecl representation P = (s’, A),the pyramidal model will be denoted by ‘P.

Usually, a decre~sing sequence of tolerance values & =[cO, . . . . c,,] correspcmdiug to progressively finer levels of de-tail is provided. A pyramidal morlel S is defined in such away that each X, approximates the hypelsurface at precisionc, {with 0 < i < h). Tlms, Ei is called the ezpunded complez(also r:cp{msion) d precision E,, A pyramidal model satisfy-ing such requirements is said to be defined on the tolerancesequence /.

The problem of representing a hypersurface at differentlevels of resolution has been extensively studied essentiallyiu the two-di meusionaf case for producing multiresolutionterrain descriptions in G IS applications. In the literature,llllllt.ilesol~lt.io~l models have been mainly described throughhierarchical structures (see, for instance, [DeF92, Sam92,Sra!YJ]). The OIIIY existing proposaf b:wed on a pyramidafst,l.llct,lire is the llelftttt~a~ pyramiJ [DeF89]. Such represen-tation is (lefinwl ou a toleratlce sequence ? = [co,..,e~] andis bcmwd ou a pyvamid(af complex S, where each complexE, c S is a Delaunay two-dimensional simpliciaf complex(i.e., a t.liallglllatioxl). Each Xi is obtained from Xi-1 by.iteratively insertiligerror until precisiont.ur~ for encoding a

the vertex that causes the maxhlmlnc, is reached. An efficient data st ruc-two-dimemsioual pyramidaf model, as

156

Figure .5: The tree-hase(l simlllifk,l Iel,wseutat i{jlI mm r-spondiug to the llierarclkical sinil>lir-iid cc}lllplex of Fi~lue3.

well as algorithms for manipulatiu,q it, lmve Iwell I)ropuse[lin [Ber94b].

4 COMPUTATIONAL ISSUES ON AMULTIRESOLUTION MODEL

There are two Lustc operatiolis a multirtisolllt iutl nlo<M slimi-ld support, independently of the specific applicat i(,ll (i) ex-tracting a hypersurface representat imi a! a given Inerisioll,.ancl (ii) anwering spatial flueries at [liHdleld l,ldcisioli levels.Such classificat.ioll (Ioes Ilot, exlmlwt a 11Imssilde [I])eliltitlllsto be performed on a ltlllltilesollltioll nm(lel Their elticieutsolution is, however, a nlillilllal re[lllirelilellt I(JI. a luldtirrs-olution model to be effective.

(i) Extracting u hyper.suIf[Icc rcpw .stut<ition at {t mrt.,t(lv~tor vuriufdr= pwcisioll It wl. Si]lce llil:ltil.esol~itioll lno<l-els are a compact way for describing a lylwrslu.farrat different precision levels, it uIIM always lw lmssildeto efficiently extract a liypersluf:ice re~weseid :it i(}ll at,any given precisiotl. Guarantwing tile cwhilmity ofthe extracted hypelsulface is usuaIIy ii basic re(ltlile-ment. Tlw requirecl precision level may Iw:

— a constant, val{w 011 thr whole {Iolniril], {w— a value varifihle ml (Iitfemut, p.alts of t.fw (Iomaill.

In the fomwr case, the specifie{l plrcisinu vahw Illaycoincide with a predefine val Iw e, ill the giveli se-quence F = [CO,.. . cr,] CMwitfl all (m~itmrv v~ilue, i.e.,a vafue e such that E), < c < co alltl c # c, Vc, C /.Pyramidal models, as <Iescrihml hy the graph-basr(lrepresentation, explicitly repres?llt tlw I]yprtxurfareat any predefine level of precisiou ill tfw given tolel-ance sequence. He~lce, tile prol}le[ll of extracting mleof such representatio]l is trivial. On the {)t.lwr haml, illa hierarchical hyperwufnce Illo(lel vllcf)lle{l t.lm}(lgll t lWabove menttiolled tlee-l”rllX’esetltiit iuu, llifferrut colll]m-neut<s describe tbe lly[wrslwface OVFI (Iisj[}illt poltirnwof the domam. Fot tlus re.aso]i, i[l SLICI case,I a IIylw] -surface lel>lesellt.atioll at a ]ue[letitlr(l Ieve] [If Iwecisi(u)

(ii)

is ]Iot exl)licit.fy provitie[l. The proldem of extracting aI)ylw] s)ilfare Iel,lese]ltatiol] from a hierarchical modelm]cmlwl t.lm}llgll a t.ree-ha~ed repl esentation has been[hqdy invest igatr{] oldy ill the two-rlimensioual case[DeF93].

A more {Iifficult. ta+k consists of obtaining hypersur-face mo{lels at an arbitrary precision level. In Sec-tion S, we propose algorithms for extracting both arepreselitat.iol i at a predelined precision c, and a rep-resentation at an arbitrary precision c from a genericri-{linle]lsiollal pyramidal model.

A related, hut more complex, problem is the extractionof a hypersluface description according to a precision

cxiteriml that, varies over the dolllain, while presetv-illg tile continuity of the resulting representation. Toour knowledge, this problem has not been faced in theIiterat tue.

A mwIIerIF!{/.$lwtial quf ries. A flulclmnental feature in as,vst,elll for llylwl sllrfare modeling and manipulationis tile llossil~ilit,y of efficiently performing geometric(plerles 011sl}at ial (Iat a. Examples of SUC1lquel~es welwr71t l[wfitzf>v4(i. e., fk,ding tile smtity within a given(Illel’y space corkailiiug a given {piery point,), or lineir,ter~ectton (i .e,, firl,lillg all entities of a given query

space irltersect illg a give]l query lilm).

A rnldt i]esolut.iou ]uo(lel should allow finding an effi-cielkt so]llt.ion to SIIC1lproblems at any level of detail.

ReseaIclI efforts Ilave been devoted to the defitionaml sollltiorl of simtial queries fol the hierarclricaf caseilk t.wo-(llnmuswlls. Irl particular, space-based hiexar-clkical Ili{xlels, s! ICl[ as qlladtcee-lwxsed models, natu-rally offel a spat iid imlex, that allow performing geo-lnetric qwat iolw with high efficiency through stan-(Iard lpm[ltree algorithms [Samftt)]. Querying tecll-uiipws Ilavr- heel) proposed in [DeF94] for hierarchi-cal st rrlct Itles Imw-(1 on triangulations, that achievegoo[l lw;ictical Iwtmvioul. For generic two-dimensional~}yralllitlal lIIc,[lels, a first attenlIJt to the solution ofgrolnrt Iic ,Illelies Ilas hewn proposed in [Ber94h]. Thefiehl is Imillvrst igate{l fol both the hierarchical and thegetlel ill case ill IIiglwr dimensions.

Data structru es for el)coding a mldt.iresolution modelsIIoI1l(I he tlesigtlrtl accol{li]ig to t.lle olwr.ations to be pel.-

fornw(l ml t.lw rno(lstl, siuce the topological and interferenceccmtmlt may Iw (Iiffel ellt accordirlg to different cornputa-tiolial re{lllilelllellts. The ha+ic operations, however, mustIW possible ill t lie cm]t.ext of ally nrldt irresolution model, and,tlielef{)le, they Illlwt l~e slll~l~olt.e(l hy a]iy (Iata structure.

O]w of the nlain applications we arw interesteci in is vol-ume rell[lerillg. To this ailll, we are developing a prototypes,yst,eln, briefly {lrsclihwl ill tlm last. %ctiou. In this context,a data st.rlmtrue not. encoding any topological relation (sucha+ a(ljacr-llcies illvolvir]g ellt,ities in tlw model) is sufficient(see Section 5). O], the contrary, whrmever tbe possibilityof rmvigat illg tlw struct.ul e t Imougb afljacencies is relevant,topological irlfot;llat iml slIoIIld he encoded by means of amore CUIIIIkX (Iata Mcllctrwe.

157

5 A DATA STRUCTURE FOR PYRAMI-DAL MODELS

In this Section, we describe a new (Iata structure fur en-coding a pyxamidaf model S = {XIJ, , Z),} based oli asequence ~ of precision, which permits the extraction of ex-pansions as well as the solutiou of the point, Iocaticm pro})-Iem, at tiny precision, with a low collll)lltatio]l;tl cost. SuclIdata structure, called SeqIJence of .List.~ oj Sinqdices (SLS),is int,emled for visualization p~upsrs; tl i[w, it eucmlrs aminimum amount. of topological information.

The SLS implements the graph-lmsed relwestmt.aticnlP = (S’, A) of S illustrated ill Sect.iou 3. Each coml,iexZ; E S’ is represented just. as a list. L, of simpfices: themodel is thus encoded through a sequence L of lists of si m-pfices, one for each precision level. For each simplex a, anexplicit link to any simplex iutersectiuk u aml Ilelol}gil}g tothe complex corresponding to the threshohl vallle of a ismaintained. Each labeled arc (S{, E;, (m~, m ) ) iu “P is RII-coded in the SLS as a liuk betweeM a~ aII(l ml.

For every simplex u, the following ild’ornmtioll are sto-red: its d + 1 vertices V(u), its cleatioh level c(fr), its ellwl.

J!?(a), and its threshold value s(C), whirll is mwle,l to rx-tract the expanded complex at a givtw lmx.ision vallw (s~eSection 6). The sequence of fists is kel)t sortetl ill LIecreasillgorder with respectr to the precision level (im-recasilig iutltixlevel), and the simphces belonging to each list. are sortcxl indecreasing order with respect, to the t.hreshohl vallw. Tlwvertices of the simplices composing L are stcmxl in a ~l(ll)idlist V.

An SLS also encodes int,erferemce links between lmirs ofintersecting simplices belonging to consecutive levels, col.re-sponding to arcs in the Multigraph. These liuks allow a“vertical” traversal of the mo{lel. For each siml~lex ~, westore a list inter(u) of references, oue for each siml&x ill-tersecting u and belonging to L,(”). Imlee(l, simplices inter-fering with a are those intersecting a WI(I Iwlollgillg to tlwfist correspomiing to its threshokl level s(a).

The spatial complexity of the Se(liwm:e of lists of silll-pfices can be evaluated as follows. Let r) denote the munlwrof points inserted in the model. Let tis., an[l t), ileuote thenumber of simplices created at. level i aml the m unl N-r of ver-tices of the expanded complex at level i (with O <, i < h),respectively. Note that, nh = r), The munher of sllill~licesin the expanded complex at. level i is, iu t] Ie wnrst. Crlw-,

1%’ ) [R.aj91].O(ni

The list, V of vertices involves (d + 2)rJ itenls c,f infor-mation, since, for each vertex, its d + 1 coor(liuates togdwr

with the reference to the next vertex [ire st.oretl. For eachsimplex created at a generic level i (aml tfaw, stored at Ievrli), we ?naintain d + 5 information correspomlillg to l’(~)(d+ 1 ~tems), c(a), E(a), and s(o), pb,s a refere,~ce to tlwnext sinmlex in the list,. Thus, iust the swmence of lists of

simplices- requires (d + .5) ~~=o”tts,. Let. N, = ~~={) )1.s,;such quantity cfeuot.es the totaf Munher of simplices store{ 1in the data structure. N. can be expressed in t,ernls of thenumber Nh of simplices of the complex XI, at the maximumlevel of precision. Indeed, N, < N), + llt~,,,,,~, where n,,,,,,, isthe maximum number of simplices gelmrntetl at each level.The result is b~ed on the fact, that, each comIdex is gener-ated by the insertion of at, least oIie point, iu the ron Lldex

at tilt- lweviolw kwel, Note that both )t~nr and Nh are, in

*J), although in practicaf cases, rim.=tfw worst. case, O(?)l ~mig,llt, be cousitlerably smaller.

The spatiaf overhead due to the encoding of interfer-ence links is equal to ~~=-01 (~~~; ns, (. j, ), and this quan-

tity is bouded from above by O(hn&).

6 A CONSTRUCTION ALGORITHM

In this Section, we proposgan algorithm for constructing,an SLS st.artiug fronl a set P of representative points and atolerance sequence ? = [e., . . . Sh]. The description is para-metric witl: respect to the criterion for point selection andto the technique used for the refinerneut of a complex afterpoiut ilwert.iou.

The algorithm worfw ou increasing finer precision lev-els: at a grllelic level i, the expamled mnnplex at level i -1is rrfine~l t,luoll~ll tile iterative application of a refinement

o]}eration ulltil .mI exl)aude(l complex ~, satisfying c, is o~

t.ailmd. All IIewly cleat,ell simpfices generated at level i arest.o)etl ill L,; fnr e,acll of t,flenl t,he creation level is set t,o i,

w hilw t.fw tfm-slmhl vallw is set to h + 1. In this way, sini-lllices crcate(l at, level I and not. affect,etl by the refinementlnocess at ally further level will have a threshold value equaf

to II+ 1, correspon~ling to tile expanded complex at the finest

precision CJ,. Tile tlmeshold value of simplices deleted at, alevel j will be e[lual to j: these simplices do not belong toal].y expall~ led coltlplex at any further level.

The biuic steps of the construction algorithm can besulnlllarize(l as follows:

1.

~

3,

4.

5.

Au iuitial simplicial complex ZO ~overing the domain(i.e., tlie couvex hldl of the set P of data points) isCl)llll>llte(l that, satisfies precision co,

For each simplex a E ZO, we set c(fl) = O, s(u) = h+l,a]d we ilwert. u illt.o list LO.

For each i from 1 to h,

(a) the expamled complex Z, -1 is refined into a conl-plex Y, sat,isfyillg precision c,;

(1)) fur each a c X, \ Z,_,, we set c(u) = i, s(a) =

II + 1, all(l we insert, a into list, L,.

(c) for each r e X,_ I \ X,, we set. s(u) = i.

For each i = O,..., h, we order silaplices in fist, L, by{Iecreasiug threshokl value.

For each i from 1 to h – 1, for each simplex u E C,such that S(r) # h + 1 (i.e., for ench u ti~at has been(Ieletetl), we a..sign to i?lter(u) all simplices in L~(aliutersectillg u.

Nc,tim that, the or(lering of the lists (step 4) that com-pose the structure can he done only ouce the structure hasl,een built,: indeed, the threshold vafue of alf simpfices iskuown only at that time. Since the algorithm is paramet-ric with respect to the refinement criterion, we evaluate itstime complexity in terms of the input Imrameters (number?I of illsert~fl Iwillts, cart Iiualit,y N of the set, of sampled

158

poi[lt, s) iul[l of tile tillle rollll)lexity 7}, EF {~1 tilt l)ltjf~tllllf’

inipleltle]kti]]g SIdI critt.li{nl. We (’FIII ~IssliuIt- t Ilat Tf/EF is

a flulcticm of N A]N1of t Iw IJIIIIIIWI ol !WLt icrs ul tlw c{JlIi-]dex it apldies Let. ,,, lw t lie I,III,,IW, {f ,wlt ices of t Ii,exlmn{lecl co]nl)lex X,. Note that, at, a grlie] ic level f (withO < ~ ~ 11), tile Illllllbel 1/,., of Iiewly ct.eatefl ,sillll~lices,aml, collsexpwntiy, the muulwv of (Ielete<l sillllJicrs is, ill tlw

QJworst case, o(r), ~ 2 ). Let.,,,,,,,, be tllr 111.axillltlltl [IIiIiIl>eIof simplices generatrtl iit each level. The t ilne cOUI1~lrxit yfol coltstructing tile sr(lllel]ce of lists uf sillll)lices is ~ivel] I,y

h h-l

o(~(~REF(TI,, ~v)+ !1,<, +11.$, log !1.., ) + ~(rl..,n s,+,)) =,=0 ,=()

t,——o((~TR&F(fl,, N’)) + Ilfl,,, r,, k% rJr,,(,, + hi:,,,,,).

,=0where tllr logaritllnlic tet]ll is {llIr to tile oltlr]it]g []1 lists,

all(l tlle last. terul is due to settill~ illted’ewlwv Iiuks. Notrtilat O(log ,l,,,,,=)=O(log !, ). hfo[ef,vr,. ,asslllllillg tlmt tlwinterfelenm test call he pelft)l me<l ill c(jIlstaILt tilne, ill tfwworst case, a siln])lrx m c]eate(l at level I lIIiIy illtelsect evrly

siIllplex at level s(17), a]ltl ,s(fl) lIIay I)e e(llIal t,u ~(c)+l (i. e.,

every simplex I)elongs UII1,V t<) its cleati{~ll level). TIIIIS, tl]ecost fot set tillg all iI1trl few:lce litlics is l,[)IIII(Iv(I fl(}lll al>ove

IV o(Mrl(u)

7 EXTRACTION OF EXPANDED COM-

PLEXES

[n this Section, we first plesrl)t all alg{,] it IIUI f“(,l ext ractiugthe expan[id complex at a given in.~cisi{ni c,, wlwre c, isone of the vafues in swlueuce ~. TiLtiu, we 1)1t-st-llt all alg O-lithm fol extracting tile exlmll(le(i coltll)lrx ;it iill~ ad)itl alyl]wcisiou; such algolitll]ll exploits il]t.elfew]ire links.

The retrieval of the simpiires of tlw exl,all{ii+i c{)IIII~iexZ, (i.e., the expanded cmnpiex at i,recisimi s, ) call iw {irmrby oimerviug tile follmviug fact. Let rr 1w a siltlljirx c1ratr~lat Ievei k, if -i = s(u) is its tloeslmi[i !’allle (Cl?ill Is J >

k), a belougs to ali exlmu(iwl c~)lni~iexes fl[,lll levrl k tolevei J – 1. For tflis reaw,]i, sillll)lices f[wlliitlg tlw exlmll{lr(iCOIUplCX Z, have a ~leiltiull leVrl less t fl.111(1I V(]llill t {1# :Ul(i iitinwhoiti value strictly greater t.imlk i Tlwse sillll,iices callbe retrieve,i by simpiy &-almil,g Se,llwl,ce L I’1(1111t IIe fiI st.

to the i’” list, anti i>y Ietlwl,iug {,IIIY t II[,sv sillll,lices wit])a threshoki value strictly greater thall ~. Siuce wit Itlli eac J1list L, simplices are solt e(i ill {Iecwasillg or{irr wit ii Iesl, ectto the threshoki vallle, we scat) each list froln t I)e I]egillllillgto tile first simpiex with a tillesli(Ai vaille less tllall 01”e(lllni

The worst-raw tinw colnplexit,y (,f ti,is aig,nitl,,l) is

,=0

where each r,si (O ~ ~ < i) lms IWVI)rel)lacr[i wi!i] tlw

maximum nunI x-r of simpiices iu S,.

lll{~(iei I)ililts a rt)llll~lfx (caile{i F:lpand+d complez at preci-.sif~t~c ), wlii{.11 is ll(jt tlf-ce>sat ii,y sillllJici al. Il](iee(l, extlalwillts call lay UII t Ilr Iwilil,l:il y of sollle simplex as we wilislwcify iater Sllcll c{ JIII]Jle.x is rIJImxxse[l of .simp iices of tileI,yliulli(lal ]Ilo{lei satisl”yillg a g,ive[l plrcision c, such t,llat,c,+l < C < c1 (Witil E,+i, c, E ~). ill this case, we Ileeti tollavigate tile ~>yrallli[lal st I ucture.

To colnlmte t im expall[ied conlplex Xc at precision e,we cousi(iel the graph wlmse nodes are either simplices intlw expall[le(i complex X, at precision c, or simpiices in L,+ I

(i e, ~reate~i at ir~ei ~+ 1) :n(i wilose arcs are the interfer.euce h]:ks cullllectlllg SIICII sllu I)iices. Let ~ be sucil graph.Tlw algolitl,l,, is imw(i or) tile fact timt tile simplices of Gwflicil {10 llot heimlg to Z, are:

a) tlkose sillll)iires of S, ]Lut satisfyill* e;

l>) si]ll])lices of X, satisfyil]jz, E i)ut, illt, erferixlg witil somesil[l]~lices cwatrti at level I + 1 wilicll l[lllst he incimle(lill Xc Iwcalwe of tlw refiueoleut of at least one siul-l)lrx of tyl,e a): SIICI1silll~,iices bel(,llg to tile connectedrollll]{)llel]t of at ieast {IIIe silnplex of type a) in ~,

Tluw, ill or(let to comlnlte X, , tile foii[~wing steps are per-follllr(i:

(1)

(?)

I.(JIIIIIIItIJall s]lll]]li[rs ml, , firn ill X, uot satisfying c;

r[jltll)llte tile collilet.tetl colllpollellt~ 01, , CPOf ~ CO1l-taiilillg crl , .0,,,. h’otr tilat 1, < ln, since either each

~J (wit]] 1 < .1 < III ) detelIllilles a contpOnellt oc at

]rast t\vo silik]dices ml, . ..0.,, i)eiollg to the same colll-l,(lIietlt

~c is compose(i of a]i tlw simplices in c1, . . . . CP belongingto L,+, I,IIw ali the sin~l Jices in X, satisfying c and not be-iougillg to ally of c], . . . . rP. In the example of Figure 6,L,+ i is compose[i of tlial,gies .1, 1(, L, Af, N, O, P, Q, R, S, T.[f we itssll],le tl:at t.lw oIIiy tliallgles in X, (Figure 6(a)) that(10 Imt satisfy c ale {I all(i i, X{ is composed of triangles.1, [(, L, Af, Ar, O, P, S, T (Iwlcn]gillg to the connected conl-I,t)lwllts (II’ (1 .alIfl i, we Figiwe 6(c)) ami fI, c, ~, g, h. TheSiilllV Ieslllt Wolll(l I)e {)l)tailir(i if eitiwr {1or e [iid not satisfy1)1rcisi(,ll c, ilbt e;i[i of {/. ,.4s ,alw.a(iy obsvl.vecf, the expan(ie(iconildex at all arl)itl m v IIIecisio]t may not, be a simplicialCOIIIldeX. Ill this exalnlde, Z, (see Figltle 6(ti)) is not, a tri-:ulgulatioll Iwrtmwe y is ]mt .1 tl iallgle all,y more, ilaviug fourveltires (Ill its lN)IIII{iWy.

Tile worst-case t in,e coml~iexity of the proposed algo-ritfllll call Iw evaillate[i I),y cousi~ierillg that the construc-t.i{m {)1 ~ Ciill I w (lolIe ill t iole I>roport ional to tile ciinleu-

l*Jsi(nl of ~, tfiat is {)(r~, ,1~,+, ). Tl,r cost of step (1) isiilwar ill tlw IIIIIIIIWI of iI1tr] fe] ruce i)llks het.ween level iall{l level i + 1. illl(l is I,igl,el tfmll tile cost of step (2). [ntil~ w{)lst C:WV, eacl} siltll>lex ill E, illtrrsects all si[llplicesill L,+ 1. TIIIW, tiw Inaxillllilll Iol]uhel (If int.erfererlce iinks

L+Jis 0(71, )/,.,+, ), ti,at is O((rI,r\,+,)~~j). Note tilat., inSIWIIcase, t Iwt e wm Ii(i he jlwt one colule[-ted component andtfllw X, wOILI<Ic{jiltci,le wit 11~,+, . Tlw total time conlplex-

ity is tlwl, O((rf, ~t,+,)[~J).

Tlw algorithm for extractil!~ au alqm,xi)lmt i,nl of tlwil,vpersurface at all arhitrm y plerisioll c fl [)111t lw 1)~’lalrliflal

159

@@’

(J) (h)

(c)

@

K z,MN

JLh

Po

11c I

L!

s

(cl)

Figure 6: Expansions at, precision c, (a) imd c,+ I (1)) of a pyramidal triangulation aml comesponding graph G (c). Li+l iscomposed of triangles .J, A’, L, M, IV, O, P, Q, R, S, T; ((l) expansiou at an ilkermefliate precision c.

8 ANSWERING SPATIAL QUERIES

h this Section, we briefiy discuss the sullltioli of spatialcluenes within a pyramidal mwlel and, ill particular, we seehow the SLS stmcture allows the sohltion of the Imild. k)ca-tion problem at, a giveu precisicm value. Givtm a ipwly l,oithp c lEd+l, a multiresolution pyramidal model ~~defille(l olla tolerance sequence F = [cO, .. . c,,], and a plecisiou V,ilU~c such that, e,+ ~ < c ~ c,, the point locat.iou at. precisiouc consists of determining the sinlpiex u of the Inwlel suchthat:

(1) a contains p;

(2) E(a) <c;

(3) a is the simplex with the largest erro,, anlmlg all si*u-plices satisfying conflit.ions (1) alltl (2).

The purpose is to determine a simldex which is “gootl e]lcni-gh”, contains the given poiut ad satisfies the giveu preci-sion, by avoiding unnecessary refinenwnt.s. Note that c Ina.yeither be a predefine value in the tolerance seip KWCeS onwhich the model is defined or au arbitrary precisiou value.

The idea of the algorithm for solving the poiut luca-tion problem consists of first locating the giveu Ix}iut p iuthe coarsest complex XO. This task call lw acconllJlislle[l hyscanning list. Lo until a silnplrx m containing p is rtiarhe[l.The process is iterated on the simplices illtelferillg with cuntil a simplex satisfying plecisioll e is [Ietelmlille[[. Figluw7 shows an example of point, location IIsillg tlw SLS il~itastructure encoding the pylan Ii(hl tl’l{lllgl l]ilt.i{)ll of Figlne ‘2.

The time complexity of the algorithm (Iepencls on the num-ber d visitwl silntdices, that, iu the worst case, can be equal

to~;;; tl.s,[*J

. =O(in, +l ), if the inciusion test can be per-

fcmntsl in constal)t, tilne.

Note that, ill order to efficiently locate p in the rootcolllplex XO, we Cold(l use a “navigation” process that, start-ing from a ramlom simplex in EO, moves towards p througha~ljilcellt, simplices, thus avoiding a complete visit of Zo.Ouce p has been located in a simplex u that does not satisfyI,recision E, the llavigat,iou is iteratively performed in the listiutrr(c) of simplices intersecting u. This approach permitsto (lisregard some simplices imstead of completely scanningthe lists at. each level. Of course, the application of thisprocess is possihlc only when adjacency relations betweensilnl~hctis witlli~i the stanw level are encoded. However, tileworst-case time comldexi ty of the afgorithm would not beaffecte( 1, alt,lmllgh experillielltal results have shown a goodlmact, ical Jwlf’orlllallce,

The netxl for topological relations is relevant if we wantto solve more complex queries such as line intersection. Instlch ciwe, a[ ljaceucies het.ween simplices at the same leveluot @ speed up the location of an endpoint of the givensegmeut 1, hut also help finding simplices intersected by 1at. each level (see Figure 8). Without any topological infor-mation, we woldtl sc.w each list of simplices, while, by ex-I>lnit.illg a{ljareucies, cully simpiices actually intersected by1 wt,ldtl Iw consi,lemxl. The same hokls for the more gen-eral IWO1kn~ of il lt.ersecting au arbkrary k-simplex with asiltll)lirial colllplex,

160

9 CONCLUDING REMARKS

P

We lmve Iwesellt Ml a formal definition of pyramidal modelfor l[l~lltilesoltltioll Ilypersurface clescril~tion in any dinlen-sicm. We have proposed an efficient representation (the SLSstructure) for ellccxliug a simplicial-trased pyramidaf model.We have also described algorithms for efficiently manipulat-illg a pyramitlal mmlel IwMed on the SLS structure.

A prototype system for terrain m(,rieling and process-ing, }Mwed cm a l]yramiclal Delalmay triangulation, has beendeveloped, M desclil w(1 ill [Be194b]. We are currently work-ing w] the iltll)leltlellt:it.ioll of a prototype svstem based on

F’

Figure 7: An example of lwillt Iocatiou Iwillg t lie SLS [Lit iistructure encoding tile pyralnitla] triallglllat iull of Figllie 2.Visitled triangles are highligllteue(l.

Figure 8: Navigaticm throllgh a(ljacellt triiu)gles.

D~lalulay co]l~plexes t.llat allow; repr~~ent~tion, visualiza-t ion, aml aualysis of a t lllee-(iiltlellsiotlal scalar field at, dif-felellt levels of Iesollltioll. Tile kernel of t.lle system includestile data strllcture ald the construction algorithm as well asaccessing algoritfuns for manipulating t.lle model, i.e., the al-goritlllu for ext rart illg a represerrtat ion at a given precisionlevel, all,l t Ile algorit 11111for ext Iactillg isosurfaces. We wzultto ext.ell{l the syst vlll to also illcltlde algorithms for directvolllnle rf-ljtlelitlg Imswl 011 rnn pyrami[lal structure. lnter-artive almlysis of voll]:lle (Iata is supl]{mted by our mode]allll algorit.l,l,,s, si]we itlel,tificatiol, of lelevant zones (e.g.,relative ext I elila for filiite elentent aualysis, special tissuesItw Inr,lir:i] ,Iat ii), all[l ff,clwillg o]) a]eas of interest are maderasy iu](l rtficield till oilgll o111[Illlltires[)lut.ioll approach,

hlt,l rove] , we ale rlllrel]tly ilivestigat.illg possible inl-Iwrn,elllellts ill tfle Iil[le complexity of tile algorithm pro-I)ose(l fol cc,llst.r Ilct illg a I)yraulidal si)liplicial complex bynlaillt. ainirlgt Iw Ilist ory of t lW updates of the complex.

Fllt.rile [levelol,llleIlts of t Ile work ,Iescribed in this pa-l]el illclll(le tile (Iesigll akI[l analysis of data st,rllctules er~cod-

illg the Itlillilllid lIUIIIIIel of topological information (such as

a{ljace]]cy relations I wt Wswll simplices) that permit to effi-

rirntl,v solve colnplf-x spatial operations at rfifferent levels oflesc)llltioll (sllrll its Iilw interswtiorl, etc.).

ACKNOWLEDGEMENT

Tllis~J~olkl lasl>ee]ls llI}I>olt e(li>.vt .lleSt.lategicPr eject “Kno-wledge thlollgll lnlags?s: All al,l,licati[,n to Cultural Her-it:ige” c)f the Italiau Natirmal Research (!ouucil throughcon-trar! N. 94. f)4221. ST74.

REFERENCES

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