Computer representations of piecewise

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1 Edior: Gabriel Taubin Inroducion o Geomeric Processing hrough Opimizaion Gabriel Taubin Brown Universiy Compuer represenaions o piecewise smooh suraces have become vial echnologies in areas ranging rom ineracive games and eaure-ilm producion o aircra design and medical diagnosis. One o he dominan surace represenaions is polygon meshes. Mos compuer graphics applicaions require simple and eicien geomery-processing algorihms o operae on he very large polygon meshes used. In general developing hese algorihms involves undamenal conceps rom pure mahemaics algorihms and daa srucures numerical mehods and soware engineering. As an inroducion o he ield his aricle shows how o ormulae several geomery-processing operaions o solve sysems o equaions in he leas-squares sense. The equaions are derived rom local geomeric relaions using elemenary conceps rom analyic geomery such as poins lines planes vecors and polygons. Simple and useul ools or ineracive polygon mesh ediing resul rom he mos basic descen sraegies o solve hese opimizaion problems. Throughou he aricle I develop he mahemaical ormulaions incremenally keeping in mind ha he objecive is o implemen simple soware or ineracive ediing applicaions ha works well in pracice. You can implemen higher-perormance versions o hese algorihms by replacing he simple solvers proposed here wih more advanced ones. Represening Polygon Meshes We can represen polygon meshes in many ways; here I use an array-based indexed ace se represenaion in which a polygon mesh P = (V X F) is composed o a inie se V o verex indices a able o 3D verex coordinaes X = {x i: I V} indexed by a verex index and a se F o polygon aces in which a ace = (... i i n ) is a sequence o nonrepeaing verex indices. The number n o verex indices in a ace can vary rom ace o ace and o course every ace mus have a minimum o n 3 verices. Two cyclical permuaions o he same sequence o verex indices are regarded as he same ace such as (0 2) and ( 2 0) bu when a sequence o verex indices resuls rom anoher one by invering he order such as (0 2) and (2 0) we regard he wo sequences as dieren aces and say ha he wo aces have opposie orienaions. This represenaion does no suppor aces wih holes which is perecly accepable or mos applicaions. For each ace = (... i i n ) V() = (... i i n ) is he se o verex indices o he ace considered as a se. In erms o daa srucures i N V is he oal number o verices o he mesh we can assume ha he se o verex indices is {0 N V } composed o consecuive inegers saring a 0. The verex coordinaes are represened as a linear array o 3 N V loaing-poin numbers and he se o aces F is represened as a linear array o inegers resuling rom concaenaing he aces. To handle polygon meshes wih dieren-sized aces (ha is a dieren number o verex indices per ace) we append a special marker a he end o each ace such as he ineger (which is never used as a verex index) o indicae he end o he ace. For example [ ] would be he represenaion or he se o aces F o a polygon mesh composed o wo riangular aces 0 = (0 2) and = (0 2 3) and V = {0 2 3}. The paricular order o he aces wihin he linear array is no imporan bu once a paricular order is chosen we will assume ha i does no change. I will reer o a ace s relaive locaion wihin he array as is ace index. I N F is he oal number o aces hen he se o ace indices is {0 N F }. Polygon Mesh Smoohing Large polygon meshes are usually generaed by 88 July/Augus 202 Published by he IEEE Compuer Sociey /2/$ IEEE

2 measuremen processes such as laser scanning or srucured lighing ha resul in measuremen errors or noise in he verex coordinaes. In some cases sysemaic errors are generaed by algorihms ha generae polygon meshes such as isosurace algorihms. In general we mus remove he noise o reveal he hidden signal bu wihou disoring i. Algorihms ha aemp o solve his problem are reerred o as smoohing or denoising algorihms. I is probably air o say ha he whole ield o digial geomery processing grew ou o early soluions o his problem. My goal here is o oer a simple and inuiive mehodology or aacking he problem in various ways. You can expand on i wih similar approaches o ormulae oher more complex problems such as large-scale deormaions or ineracive shape design. In smoohing algorihms noise removal is consrained o changes o he values o he verex coordinaes X. Neiher he se o verex indices V nor he aces F o he polygon mesh are allowed o change. Perhaps he simples and oldes mehod o remove noise rom a polygon mesh is Laplacian smoohing. In classical signal processing noise is removed rom signals sampled over regular grids by convoluion ha is by averaging neighboring values. Laplacian smoohing is based on he same idea: each verex coordinae x i is replaced by a weighed average o isel and is irs-order neighbors. Bu o properly describe his mehod we irs need o ormalize a ew hings. The Primal Graph o a Polygon Mesh The graph or more precisely he primal graph (we will see he dual graph laer) G = (V E) o a polygon mesh P = (V X F) is composed o he se o polygon mesh verex indices V as he graph verices and he se E o mesh edges as he graph edges. A mesh edge is an unordered pair o verex indices e = (i j) = (j i) ha appear consecuive o each oher irrespecive o order in one or more aces o he polygon mesh. In ha case we say ha he ace and he edge are inciden o each oher. The se o edges inciden o a ace is E(); n is he number o edges in his se (equal o he number o verices in he ace. For example or he ace = (0 2) i is E() = {(0 ) ( 2) (2 0)}. Noe ha one or more aces migh be inciden o a common edge. The se o aces inciden o a given edge e = (i j) is F(e) which has n e number o inciden aces. A boundary edge has exacly one inciden ace a regular edge has exacly wo inciden aces and a singular edge has hree or more inciden edges. We say ha wo verices i and j are irs-order neighbors i he pair (i j) o verex indices is an edge. For each verex index he se V(i) = {j: (i j) E} is he se o irs-order neighbors o i and n i is he number o elemens in his se. In erms o daa srucures we can represen he mesh edges (i j) as a linear array o 2N E verex indices where N E is he oal number o polygon mesh edges. To make each edge s represenaion unique his array sores eiher he pair (i j) i i < j or (j i) i j < i. Alhough his array is consruced sricly as a uncion o he aces and as such does no add any new inormaion consrucing and soring i as an addiional daa srucure is beneicial: several o he ieraive algorihms discussed here can be eicienly implemened as a linear raversal o he edge array. However implemening he same algorihms by raversing he ace array usually increases compley or creaes special cases ha complicae he algorihms. To eicienly consruc he array o edges rom he array o aces we use an addiional daa srucure o represen a graph over he se o verex indices. This graph daa srucure is iniialized wih he se o verex indices and an empy se o edges. The graph daa srucure suppors wo eicien operaions: ge(i j) and inser(i j). The operaion ge(i j) reurns he edge index assigned o he edge (i j) i such an edge ess and a unique ideniier such as which is no used as an edge index i he edge (i j) does no ye belong o he se o edges. I he edge (i j) does no ye es he operaion inser(i j) appends he pair o indices o he array o edges and assigns is locaion in he array o he edge as he unique edge index. In his way he index 0 is assigned o he irs edge creaed and consecuive indices are assigned o edges creaed laer. An eicien implemenaion o his graph daa srucure can be based on a hash able. For some algorihms i is useul o have an eicien mehod o deermine he number n e o inciden aces per edge as well as o access hose aces indices. The graph daa srucure can also be exended o suppor his uncionaliy. We can represen number n e as an addiional ield in he record used o represen he edge e in he graph daa srucure or as an exernal variable-lengh ineger array. Each value is iniialized o via he inser(i j) operaion during he graph daa srucure s consrucion and incremened during ace array raversal every ime he ge(i j) operaion reurns a valid ace index. We can represen he ses o aces F(e) inciden o he edges as an array o variable-lengh arrays indexed by he edge index and consruc hem as well during he graph daa srucure s consrucion. IEEE Compuer Graphics and Applicaions 89

3 Verex Evoluion Algorihms A large amily o polygon mesh-ediing algorihms ollow hree seps. Firs or each verex index i o he polygon mesh compue a verex displacemen vecor x i. Second aer all he verex displacemen vecors are compued apply he verex displacemen vecors o he verex coordinaes = + λ x i where λ is a ixed-scale parameer (eiher user deined or compued rom he polygon mesh daa). Finally replace he original verex coordinaes X wih he new verex coordinaes X. These hree seps are repeaed or a cerain number o imes speciied in advance by he user or unil a cerain sopping crierion is me. All he algorihms discussed here belong o his amily. In erms o sorage hese algorihms require an addiional linear array o 3N v loaingpoin numbers o represen he verex displacemen. The verex coordinaes are updaed using his procedure in linear ime as a uncion o he number o verices. O course he ime and sorage compley o evaluaing he verex displacemens (o deermine he scale parameer or wheher he sopping crierion is me) mus be added o he algorihm s overall compley. In general algorihms wih linear ime and sorage compley as a uncion o polygon mesh size are he only algorihms ha scale properly or pracical use wih very large polygon meshes. Laplacian Smoohing As I menioned earlier Laplacian smoohing replaces each verex coordinae x i wih a weighed average o isel and is irs-order neighbors. More precisely or each verex index i we compue a verex displacemen vecor = ( xj ni as he average over he irs-order neighbors j o verex i (o vecors x j x i). Aer we compue all hese displacemen vecors as uncions o he original verex coordinaes X we apply he verex displacemen o he verex coordinaes wih a scale parameer in he range 0 < λ < (λ = /2 is usually a good choice). To compue verex displacemen vecors we need an eicien way o inding all he irs-order neighbors o each verex index paricularly he number o elemens in he ses o irs-order neighbors. Unorunaely he daa srucures inroduced so ar do no provide such mehods. However because each edge (i j) conribues a erm o he sums deining boh displacemen vecors x i and x j we can accumulae all he displacemen vecors ogeher while linearly raversing he array o edges. During he same raversal we also accumulae he number o each verex s irs-order neighbors so ha we can normalize he verex displacemen vecors. Descen Algorihms Le us consider he sum o he squares o he edge lenghs x i x j as a uncion o a polygon mesh s verex coordinaes: Ex ( )= xj 2 where x is he able o verex coordinaes X regarded as a column vecor o dimension 3N V resuling rom concaenaing all he N V 3D verex coordinaes x i. Noe ha as a uncion o x his uncion is quadraic homogeneous and nonnegaive deinie. Consequenly i aains he global minimum 0 when all he edges have zero lengh or equivalenly when all he verex coordinaes have he same value. As a resul saring rom noisy verex coordinaes X compuing he global minimum o his uncion does no consiue a smoohing algorihm bu aking a small sep along a descen direcion oward a local minimum (which in his case is he global minimum) is sill a valid heurisic. Gradien descen is he simples ieraive algorihm o locally minimize a uncion such as his one. The negaive o he gradien o he uncion E(x) is he direcion o seepes descen. We can look a he gradien o E(x) as he concaenaion o he N V 3D derivaives o E(x) wih respec o he verex coordinaes x i: E = ( xj )= ( xj x ). i To consruc a descen algorihm we sill need o choose a posiive scale parameer λ so ha he verex-coordinaes updae rule = E λ x i acually resuls in he value o he uncion o decrease: E(x ) < E(x). Even hough he negaive o he gradien is he seepes descen direcion selecing oo large a value o λ (called overshooing) usually resuls in he uncion s value acually increasing. In his case an ieraive algorihm based on an improper rule o choose λ migh resul in oscillaory behavior or even divergence. 90 July/Augus 202

4 Aer we choose a descen direcion v o ind he opimal value or λ we can look a he D problem o minimizing E(x + λv) wih respec o λ. In our case because he uncion E(x) is quadraic his is a simple D quadraic opimizaion problem ha we can solve in closed orm. Explicily we can compue he opimal value or λ by solving he linear equaion λ vi vj vi v j xj ij E + ( ) ( ) ( ) = 0. Because a second raversal o he lis o edges is necessary o accumulae he wo coeiciens o his linear equaion or long meshes his sep is usually avoided and replaced wih user-speciied values or λ. Alhough we can already observe a close connecion beween his mehod and Laplacian smoohing he descen direcions chosen in boh cases are no idenical. Why is ha so? The Jacobi Ieraion The Jacobi ieraion is he simples ieraive mehod o solve large square diagonally dominan sysems o linear equaions Ax = b where he ih equaion is solved independenly or he ih variable keeping he oher variables ixed and resuling in a new value or he ih variable. Aer we use his mehod o deermine he new values or all he variables he old variables are replaced by or displaced in he direcion o he new variables and he process repeas or a ixed number o ieraions or unil convergence. Under cerain condiions he mehod converges o he soluion o he sysem o equaions. In he conex o opimizing a quadraic uncion E(x) he sysem o equaions o be solved corresponds o making he uncion s gradien equal o zero. Even when he perormance uncion is no a quadraic uncion o he variables we can use his mehod o consruc a properly scaled descen algorihm. This generalized Jacobi ieraion is equivalen o minimizing he uncion E(x) wih respec o he ih variable independenly. I we wrie he new value or he variable x i as he old value plus a displacemen x i + x i in he case o he sum o square edge lenghs uncion deermining he displacemen x i reduces o solving he equaion E ( + ) x x x i x i x N V = ( + xj)= ni ( xj 0 = i which resuls in he Laplacian-smoohing displacemen vecor. We can deermine he opimal value or he parameer λ by minimizing E(x + λ x) wih respec o λ as described earlier alhough λ = /2 usually works well in pracice. How o Fix Laplacian Smoohing Laplacian smoohing is a simple easy-o-implemen algorihm. I produces smoohing bu when oo many ieraions are applied he shape o he polygon mesh undergoes signiican and undesirable deormaions. As I menioned his is because he uncion E(x) being minimized has a global minimum (acually ininiely many bu unique modulo a 3D ranslaion) ha does no correspond o he resul o removing noise rom he original verex coordinaes. Any converging descen algorihm will approach ha minimum which is why we observe signiican deormaions in pracice. In our case in Laplacian smoohing all he verex coordinaes o he polygon mesh o converge o heir cenroid: N N V. V i= In he lieraure his problem is reerred o as shrinkage. Many algorihms based on dieren mahemaical ormulaions ranging rom signal processing o parial dierenial equaions have been proposed over he pas 5 or more years o deal wih and solve he shrinkage problem bu I will no survey hese algorihms here. For he sake o simpliciy I ake he viewpoin ha he shrinkage problem is a direc resul o he wrong perormance uncion being minimized. Consequenly I address he shrinkage problem by modiying he perormance uncion being minimized. However aer consrucing each new perormance uncion I ollow he same simple seps described earlier o minimizing he uncion wih respec o each variable independenly o obain a properly scaled descen vecor and hen updaing he variables as in Laplacian smoohing by displacing he verex coordinaes in he direcion o his descen vecor. Finally I repea he process or a predeermined number o seps or unil convergence based on an error-olerance sop es. Verex Posiion Consrains The mos obvious way o preven shrinkage is o updae all he verex coordinaes. More ormally we pariion he se o verex indices V ino wo disjoin ses: a se V C o consrained verex indices and a se V U o unconsrained verex indices. IEEE Compuer Graphics and Applicaions 9

5 We also pariion he vecor o verex coordinaes x ino a vecor o consrained verex coordinaes x C and a vecor o unconsrained verex coordinaes x U. We keep he same sum o squares o edge lenghs uncion E(x) = E(x U x V) bu we regard i as a uncion o only he unconsrained verex coordinaes x U wih he consrained verex coordinaes x V regarded as consans. As such his uncion is sill quadraic and nonnegaive deinie bu i is no longer homogeneous. Generally his uncion sill has a unique minimum (modulo a ranslaion in his case) ha has a closed-orm expression and he minimum does no correspond o placing all he verices a a single poin in space. I we apply he same approach I described earlier o compue a descen direcion by minimizing E(x U) wih respec o each unconsrained variable independenly we end up wih he same descen vecors as in Laplacian smoohing and he same descen algorihm bu here only he unconsrained verex coordinaes. So his algorihm s compuaional cos is roughly he same as ha or Laplacian smoohing. Unorunaely we sill see shrinkage. In general i is no clear which verices should be consrained and which should be ree o move bu wihin an ineracive modeling sysem ha allows or ineracive verex selecion his eecively smoohes ou seleced porions o a polygon mesh which is useul in pracice. Raher han keeping he consrained verices a heir original posiions we can assign hem new arge posiions in which case he consrained verices can be updaed irs and hen kep ixed during he algorihm ieraions. Unorunaely i he consrained verex displacemens are large compared wih he average edge lengh his algorihm migh produce noiceable shape ariacs during he ieraions. An alernaive is o swich o a so-consrains sraegy in which all he variables are ree o move again and he consrains are saisied in he leassquares sense by adding one or more erms o he uncion being minimized. For example in our case we consider his uncion Ex ( )= xj + µ x iv C where µ is a posiive consan he second sum is over he consrained verices and x 0 i is a arge consrained-verex posiion provided as inpu daa o he algorihm. By minimizing each variable independenly we obain he same expression as in Laplacian smoohing or he displacemens x i corresponding o he unconsrained verices and i = 0 ( )+ + xj µ ( ni µ or he displacemen x i corresponding o he consrained verices. Face-Cenroid Consrains To produce accepable resuls we mus consrain many o he verices. Raher han imposing consrains on verex posiions we impose similar consrains on some or all o he ace cenroids. The inuiion here is ha he ace cenroids are weighed averages o he ace verex coordinaes because o he smoohing process so he problem is how o ranser ha smooh-shape inormaion back rom he ace cenroids o he verex coordinaes. Coninuing wih a so consrains approach we can consider he ollowing perormance uncion which looks very similar o he one used o impose so verex consrains: Ex ( )= xj + µ x x F C where F C is he subse o consrained aces (i could be all he aces). For each we express he cenroid x as he average o he ace verex coordinaes: x = n x i + + x ( i n ) so ha we can regard he overall uncion as a uncion o only he verex coordinaes and x 0 as he arge 3D poin value or he ace cenroid. For example x 0 could be he ace cenroid s iniial value beore we apply any smoohing. Even hough we would sar he algorihm wih he erm o he perormance uncion corresponding o he acecenroid consrains idenical o zero i could become nonzero aer one or more ieraions while he overall uncion decreases. By applying he generalized Jacobi sraegy o minimizing wih respec o each variable independenly we obain he ollowing expression or each displacemen x i: = n + µ i FC () i n ( xj + n x 0 µ x F i C () ( ) where F C(i) is he subse o consrained aces conaining verex index i. These displacemens 92 July/Augus 202

6 and normalizaion acors can be accumulaed as in previous algorihms by iniializing o zero raversing he array o edges raversing he array o consrained aces and hen normalizing. Once we compue he displacemens we can updae he verex coordinaes as in Laplacian smoohing. Face-Normal Consrains None o he consrains discussed so ar allow or direc conrol o local surace orienaion. A smoohing algorihm ha can selecively conrol his orienaion is a useul ool in an ineracive polygon-mesh-ediing sysem and ye anoher possible way o preven he shrinkage problem o Laplacian smoohing. To conrol surace orienaion we mus inroduce surace-normal vecors ino he perormance uncion o be minimized. As we have done or he ace cenroids one possibiliy is o derive an expression or a ace-normal vecor as a uncion o he ace verex coordinaes and hen add an error erm o he perormance uncion or all or some o he aces. Because doing so resuls in nonlinear equaions o solve or he displacemen vecors we propose a simpler alernaive. We consider he ollowing perormance uncion ( ) Ex ( )= xj µ u ( xj ( ij ) E F N ( ij ) E ( ) where he irs erm is he sum o square edge lenghs as in all he previous perormance uncions and he second erm is a sum over a subse F N o aces in which we wan o impose he ace-normal consrain. For each such ace-edge pair we impose as a so consrain ha he acenormal vecor u be orhogonal o he ace boundary vecor x j x i. The user provides ace normal vecors u as addiional inpu o he algorihm. However our polygon mesh represenaion does no orce aces o be planar. For his o happen he so consrain mus be saisied or all he ace boundary vecors o each ace. Noe ha wih he consrained ace-normal vecors regarded as consans his perormance uncion is also quadraic and homogeneous in he verex coordinaes. In his case he displacemen vecors saisy he ollowing linear equaions: ni i + µ nn FN i jv i = () ( ) ( xj + µ nn ( xj FN () i jv ( i ) where I is he 3 3 ideniy marix and V( i) = V() V(i) is he se o verices ha belong o ace and are irs-order neighbors o verex i (here are exacly wo such verices when he ace is known o conain verex i). The 3 3 marix on he le side muliplies x i which is symmeric and posiive deinie and can be accumulaed during he mesh raversal along wih he oher sums. We can easily inver i using Cholesky decomposiion. Smoohing Face-Normal Vecors Assume ha we have a ace-normal vecor u or every ace o he polygon mesh. We concaenae hese ace-normal vecors o orm a vecor u o dimension 3N F which we consider no a userprovided consan bu a new variable. O course he variables u and x are no independen. Bu raher han imposing heir relaions as hard consrains we regard hem as independen variables and represen heir relaions as so consrains as in he case o ace-normal consrains. To remove noise rom he ace-normal vecor we consider he perormance uncion Eu ( )= u ug ( g ) E* 2 which is he sum over he dual-mesh edges o he square dierences o ace-normal vecors. The se o dual-mesh edges E* consiss o pairs ( g) o aces sharing a regular edge (one ha has exacly wo inciden aces). Formally a mesh s dual graph has he mesh s aces as dual-graph verices and he dual-mesh edges as dual-graph edges. I is imporan o noe he similariy beween his perormance uncion and he sum o square edge lenghs. I we iniialize he ace-normal vecors rom he verex coordinaes we can use his perormance uncion o irs remove noise rom he ace-normal vecors and hen use he smoohed ace normal as ace-normal consrains in a second smoohing process applied o he verex coordinaes. This second process o smoohing he verex coordinaes wih ace-normal consrains is an inegraion o smoohed ace normals. Applying our sraegy o his perormance uncion we can compue a displacemen vecor u or he ace-normal vecors. Because aer we apply hese ace-normal displacemens he ace-normal vecors are no longer uni lengh we normalize he updaed ace-normal vecors o uni lengh and hen perorm he ace-normal inegraion sep. An alernaive is o consider he ollowing perormance uncion IEEE Compuer Graphics and Applicaions 93

7 (a) (b) (c) Figure. Example. A sample applicaion implemens all conceps described in his aricle including (a) smoohing (b) applying an algorihm o he mesh in Figure a wihou consrains and (c) using a selecion panel o speciy muliple subses o verices and aces. and apply our minimizaion sraegy o he variables x and u ogeher. In his way we can deermine displacemen vecors x i and u as uncions o x and u and hen updae boh variables a once ollowed by ace-normal uni lengh normalizaion. I leave he deails o hese derivaions o you. Noe ha he wo approaches discussed here allow or hard consrains o be applied o a subse o ace-normal vecors. As in he case o hard verex-coordinae consrains he consrained values are jus no updaed. Also we can impose so ace-normal consrains by adding ye anoher erm o he las perormance uncion (he sum over a subse o aces o square errors beween acenormal vecors and arge ace-normal vecors). We have considered a number o ways o add consrains o Laplacian smoohing. All hese sraegies can be combined ino a single general polygon-mesh-smoohing ha allows or hard and so verex posiion consrains on disjoin subses verices and so ace-cenroid consrains on a subse o aces and hard and so ace-normal consrains on a disjoined subse o aces. Overall he perormance uncion has six erms and he conribuion o each can be conrolled hrough six corresponding weighs. An implemenaion o wha I described in his aricle wrien in Java appears in Figure. You can download he soware a hp://mesh.brown. edu/opimizaion along wih a more deailed descripion o is operaion. By selecively seing he weighs o some o he six erms o he perormance uncion o zero and by consraining all he verex posiions or all he ace-normal vecors no o change you can implemen all he algorihms discussed in his aricle including Laplacian smoohing. Gabriel Taubin is a proessor in Brown Universiy s School o Engineering and he edior in chie o his magazine. Conac him a aubin@brown.edu. Exu ( )= 2 xj + ( u( xj ) 2 ( ij ) E F ( ij ) E ( ) + γ u ug 2 ( g ) E* Seleced CS aricles and columns are also available or ree a hp://compuingnow.compuer.org. 94 July/Augus 202