Algebraic Point Set Surfaces


 Avis Wilkins
 2 years ago
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1 Algebrac Pont Set Surfaces Gae l Guennebaud Markus Gross ETH Zurch Fgure : Illustraton of the central features of our algebrac MLS framework From left to rght: effcent handlng of very complex pont sets, fast mean curvature evaluaton and shadng, sgnfcantly ncreased stablty n regons of hgh curvature, sharp features wth controlled sharpness Sample postons are partly hghlghted Abstract In ths paper we present a new Pont Set Surface (PSS) defnton based on movng least squares (MLS) fttng of algebrac spheres Our surface representaton can be expressed by ether a projecton procedure or n mplct form The central advantages of our approach compared to exstng planar MLS nclude sgnfcantly mproved stablty of the projecton under low samplng rates and n the presence of hgh curvature The method can approxmate or nterpolate the nput pont set and naturally handles planar pont clouds In addton, our approach provdes a relable estmate of the mean curvature of the surface at no addtonal cost and allows for the robust handlng of sharp features and boundares It processes a smple pont set as nput, but can also take sgnfcant advantage of surface normals to mprove robustness, qualty and performance We also present an novel normal estmaton procedure whch explots the propertes of the sphercal ft for both drecton estmaton and orentaton propagaton Very effcent computatonal procedures enable us to compute the algebrac sphere fttng wth up to 4 mllon ponts per second on latest generaton GPUs CR Categores: I35 [Computer Graphcs]: Computatonal Geometry and Object Modelng Curve and surface representatons Keywords: pont based graphcs, surface representaton, movng least square surfaces, sharp features Introducton A key ngredent of most methods n pont based graphcs s the underlyng meshless surface representaton whch computes a contnuous approxmaton or nterpolaton of the nput pont set The by far most mportant and successful class of such meshless representatons are pont set surfaces (PSS) [Alexa et al 23] combnng hgh flexblty wth ease of mplementaton PSS generally defne a smooth surface usng local movng leastsquares (MLS) approxmatons of the data [Levn 23] The degree of the approxmaton can easly be controlled, makng the approach naturally well suted to flter nosy nput data In addton, the semmplct nature of the representaton makes PSS an excellent compromse combnng advantages both of explct representatons, such as parametrc surfaces, and of mplct surfaces [Ohtake et al 23] Snce ts ncepton, sgnfcant progress has been made to better understand the propertes and lmtatons of MLS [Amenta and Kl 24a,24b] and to develop effcent computatonal schemes [Adamson and Alexa 24] A central lmtaton of the robustness of PSS, however, comes from the plane ft operaton that s hghly unstable n regons of hgh curvature f the samplng rate drops below a threshold Such nstabltes nclude erroneous fts or the lmted ablty to perform tght approxmatons of the data Ths behavor sets tght lmts to the mnmum admssble samplng rates for PSS [Amenta and Kl 24b; Dey et al 25] In ths paper we present a novel defnton of movng least squares surfaces called algebrac pont set surfaces (APSS) The key dea s to drectly ft a hgher order algebrac surface [Pratt 987] rather than a plane For computatonal effcency all methods n ths paper focus on algebrac sphere fttng, but the general concept could be appled to hgher order surfaces as well The man advantage of the sphere fttng s ts sgnfcantly mproved stablty n stuatons where planar MLS fals For nstance, tght data approxmaton s accomplshed, spheres perform much better n the correct handlng of sheet separaton (fgure 3) and exhbt a hgh degree of stablty both n cases of undersamplng (fgure 2) and for very large weght functons The specfc propertes of algebrac spheres make APSS superor to smple geometrc sphere fttng It allows us to elegantly handle planar areas or regons around nflecton ponts as lmts n whch the algebrac sphere naturally degenerates to a plane Furthermore, the sphercal fttng enables us to desgn nterpolatory weghtng schemes by usng weght functons wth sngulartes at zero whle overcomng the farness ssue of prevous MLS surfaces The sphere radus naturally serves as a forfree and relable estmate of the mean curvature of the surface Ths enables us, for nstance, to compute realtme accessblty shadng on large nput objects (fgure ) Central to our framework are the numercal procedures to effcently perform the sphere ft For pont sets wth normals we desgned
2 (b) (c) Fgure 2: The undersampled ear of the Stanford bunny usng normal averagng plane ft (SPSS) wth h = 8 (b) and our new APSS wth h = 7 (c) a greatly smplfed and accelerated algorthm whose core part reduces to lnear least squares For pont sets wthout normals, we employ a slghtly more expensve fttng scheme to estmate the surface normals of the nput data In partcular, ths method allows us to mprove the normal estmaton by [Hoppe et al 992] The tghter ft of the sphere requres on average less projecton teratons to acheve the same precson makng the approach even faster than the most smple plane ft MLS Our mplementaton on the latest generaton GPUs features a performance up to 45 mllons of ponts per second, suffcent to compute a varety of operatons on large pont sets n realtme Fnally, we developed a smple and powerful extenson of [Fleshman et al 25] to robustly handle sharp features, such as boundares and creases, wth a bultn sharpness control (fgure ) 2 Related Work Pont set surfaces were ntroduced to computer graphcs by [Alexa et al 23] The ntal defnton s based on the statonary set of Levn s movng least squares (MLS) projecton operator [Levn 23] The teratve projecton nvolves a nonlnear optmzaton to fnd the local reference plane, a bvarate polynomal ft and teraton By omttng the polynomal fttng step, Amenta and Kl [24a] showed that the same surface can be defned and computed by weghted centrods and a smooth gradent feld Ths leads to a sgnfcantly smplfed mplct surface defnton [Adamson and Alexa 24] and faster algorthms, especally n the presence of normals [Alexa and Adamson 24] In partcular, the projecton can be accomplshed on a plane passng through the weghted average of the neghborng samples wth a normal computed from the weghted average of the adjacent normals In the followng we wll refer to ths effcent varant as SPSS for Smple PSS Bossonnat and Cazals [2] and more specfcally Shen et al [24] proposed a smlar, purely mplct MLS (IMLS) surface (b) (c) Fgure 3: Sheet separaton wth conventonal PSS compared to our new sphercal ft: Standard PSS wth covarance analyss (orange) and normal averagng (green) (b) Our APSS wthout (orange) and wth (green) normal constrants The best plane and best algebrac sphere fts for the blue pont n the mddle are drawn n blue For ths example the normals were computed usng our technque from secton 5 whch can safely propagate the orentaton between the two sheets (c) Fgure 4: A 2D orented pont set s approxmated (left) and nterpolated (rght) usng varous PSS varants: SPSS (blue), IMLS (red), HPSS (magenta) and our APSS (green) representaton defned by a local weghted average of tangental mplct planes attached to each nput sample Ths mplct surface defnton was ntally desgned to reconstruct polygon soups, but the pont cloud case was recently analyzed by Kollur [25] and made adaptve to the local feature sze by Dey et al [25] Note that n ths paper IMLS refers to the smple defnton gven n [Kollur 25] If appled wthout polygon constrants and n the case of sparse samplng, we found that the surface can grow or shrnk sgnfcantly as a functon of the convexty or concavty of the pont cloud To allevate ths problem, Alexa and Adamson [26] enforced convex nterpolaton of an orented pont set usng sngular weght functons and Hermte centrod evaluatons (HPSS) Lkewse relevant to our research s pror art on normal estmaton Many pont based algorthms, ncludng some of the aforedescrbed PSS varants, requre surface normals A standard procedure proposed by [Hoppe et al 992] s to estmate ther drectons usng a local plane ft followed by a propagaton of the orentatons usng a mnmum spannng tree (MST) strategy and a transfer heurstc based on normal angles Extendng ths technque, Mtra and Nguyen [23] take nto account the local curvature and the amount of nose n order to estmate the optmal sze of the neghborhood for the plane ft step However, the nherent lmtatons of the plane ft step and the propagaton heurstc requre a very dense samplng rate n regons of hgh curvature Intrnscally, a PSS can only defne a smooth closed manfold surface Even though boundares could be handled by thresholdng an offcenter dstance [Adamson and Alexa 24], obtanng satsfactory boundary curves wth such an approach s usually dffcult or even mpossble In order to detect and reconstruct sharp creases and corners n a possbly nosy pont cloud, Fleshman et al [25] proposed a refttng algorthm that locally classfes the samples nto multple peces of surfaces accordng to dscontnutes of the dervatve of the surface Whle consttutng an mportant progress, the method requres very dense pont clouds, t s rather expensve, and t offers only lmted flexblty to the user Also, potental nstabltes n the classfcaton can create dscontnuous surface parts A second mportant approach s the PontSampled Cell Complexes [Adamson and Alexa 26b] whch allows to explctly represent sharp features by decomposng the object nto cells of dfferent dmensons Whle ths approach allows to handle a wde varety of cases (eg, nonmanfoldness), the decomposton and the balancng of the cells nfluence on the shape of the surface demands effort by the user, makng the method unsutable for some applcatons As we wll elaborate n the followng sectons, the algebrac sphere fttng overcomes many of the aforedescrbed lmtatons by the sgnfcantly ncreased robustness of the ft n the presence of low samplng rates and hgh curvature 3 Overvew of the APSS framework Gven a set of ponts P = {p R d }, we defne a smooth surface S P approxmatng P usng a movng least squares sphercal ft to the data Our approach handles both smple pont clouds and pont
3 Normal drecton estmaton (5) usng a sphere fttng wthout normal (42) p Normal orentaton propagaton (52) usng a sphere fttng wthout normal (42) p j Projecton operator (44) usng normal constrant sphere fttng (43) m x=q p c c c u(p ) u(m) u(x) (b) (c) (d) Fgure 5: Overvew of our APSS framework Evaluaton of the normal drecton at pont p (b) Propagaton of a consstent normal orentaton from p to p j (c) Frst teraton of the projecton of the pont x onto the APSS (d) Illustraton of the scalar feld defned by APSS q Implct defnton (44) clouds wth normals The presence of surface normals leads to smplfed, more effcent, and more robust fttng algorthms (see secton 43) We recommend to estmate them from the nput pont set usng the procedures of secton 5 and secton 42 as a preprocessng step Fgure 5 presents the procedural flow usng a 2D example Startng from a smple pont cloud, we frst evaluate the normal drectons by locally fttng an algebrac sphere u at each pont p (fgure 5a) Next, a consstent normal orentaton s robustly propagated from the pont p to ts neghbor p j by approxmatng the surface nbetween usng an approxmatng sphere (fgure 5b) Gven the estmated normals, a smplfed sphercal fttng technque can be appled to compute our fnal APSS S P whch can be defned as the set of statonary ponts of a projecton operator projectng a pont onto a locally ftted sphere (fgure 5c) As an alternatve our surface S P can also be defned as the zero set of a scalar feld representng the dstance between the evaluaton pont and a locally ftted sphere (fgure 5d) The former defnton s well suted for resamplng, whle the latter one s convenent to raytrace the surface [Adamson and Alexa 23; Wald and Sedel 25] or to perform Boolean operatons [Pauly et al 23] The key ngredents of our framework are the computatonal algorthms to robustly and effcently ft a sphere to a set of ponts n a movng least squares fashon These methods wll be presented for the cases of a smple pont cloud as well as for a pont cloud wth normals n sectons 42 and 43 Based on the procedures for algebrac sphere fts, we wll defne and dscuss our APSS defnton n secton 44 Two sgnfcant extensons nclude normal estmaton n secton 5 and the handlng of sharp features n secton 6 4 Sphere Fttng and APSS In ths secton we wll dscuss the core mathematcal defntons and computatonal procedures of our APSS framework We wll elaborate on the sphere fttng problem and show how prevous methods can be adapted to defne a MLS surface based on sphere fts As a central computatonal algorthm, we wll present a novel and very effcent sphere fttng method takng nto account surface normals 4 General Issues Weghtng scheme Throughout the paper we wll utlze the followng generc weght functon ( ) p x w (x) = φ () h (x) descrbng the weght of the pont p for the local evaluaton poston x φ s a smooth, decreasng weght functon and h (x) descrbes the local feature sze h (x) can be constant, depend on x as n [Pauly et al 23], or only on p, e, h (x) = h as n [Adamson and Alexa 26a] As a proper choce of φ we suggest the followng compactly supported polynomal { ( x φ(x) = 2 ) 4 f x < (2) otherwse Ths functon performs a smooth data approxmaton and avods square root evaluatons for the dstance computaton Interpolaton can be acheved usng functons wth a sngularty at zero, see [Shen et al 24; Adamson and Alexa 26b] Sphercal fttng The problem of fttng a sphere to a set of ponts s not new and several methods have been proposed n the past It s mportant to dstngush geometrc approaches from algebrac ones Geometrc fttng denotes algorthms mnmzng the sum of the squared Eucldean dstances to the gven ponts (eg, see [Gander et al 994]) Geometrc sphere fts have several drawbacks Frst, because of the nonlnear nature of the problem, the soluton can only be found usng an teratve, expensve approach More mportantly, snce these algorthms are ether based on a centerradus or parametrc representaton of the sphere, they become unstable when the best ft tends to a plane Ths lmts the practcal utlty of such methods for our purposes An elegant alternatve s to substtute the geometrc dstance by an algebrac dstance as n [Pratt 987] An algebrac sphere s thus defned as the sosurface of the scalar feld s u (x) = [, x T, x T x]u, where u = [u,, u d+ ] T R d+2 s the vector of scalar coeffcents descrbng the sphere For u d+ the correspondng center c and radus r are easly computed as: c = 2u d+ [u,, u d ] T, r = c T c u /u d+ (3) wth d beng the dmenson In degenerate cases, u corresponds to the coeffcents of a plane equaton wth u representng the plane s dstance from the orgn and [u,,u d ] T beng ts normal 42 Fttng Algebrac Spheres Wthout Normals Let n be the number of ponts and let W(x) and D respectvely be the n n dagonal weght matrx and the n (d + 2) desgn matrx defned as: w (x) p T p T W(x) = p, D = w n (x) p T n p T n p n (4) Then the soluton u(x) of our algebrac sphere ft at a gven pont x R d can be expressed as: u(x) = argmn W 2 2 (x)du (5) u, u In order to avod the trval soluton u(x) =, u has to be constraned by a metrc The choce of ths constrant sgnfcantly nfluences the soluton of the above mnmzaton problem Ideally we want the algebrac ft to behave as closely as possble to a geometrc ft, but at the same tme to relably handle the planar case Among all constraned methods that have been proposed, we found Pratt s constrant [Pratt 987] the most sutable for our purposes Pratt s constrant fxes the norm of the gradent at the surface of the sphere to unt length Ths s accomplshed by (u,, u d ) 2 4u u d+ = and ensures that the algebracally ftted sphere s close
4 to the least squares Eucldean best ft, e, the geometrc ft (see also fgure 6) By rewrtng the above constrant n matrx form ut Cu =, the soluton u(x) of our mnmzaton problem yelds as the egenvector of the smallest postve egenvalue of the followng generalzed egenproblem: 2 D W(x)D u(x) = λ C u(x), wth C = T 2 (6) Note that ths method s only used to estmate the mssng normals of nput pont sets An effcent algorthm for runnng the APSS n the presence of normals wll be derved n the followng secton 43 Fttng Spheres to Ponts wth Normals Whle the problem of fttng a sphere to ponts has been nvestgated extensvely, there exsts to our knowledge no method to take specfc advantage of normal constrants at the sample postons We derve a very effcent algorthm by addng the followng dervatve constrants su (p ) = n to our mnmzaton problem (5) Note that, by defnton, t holds that kn k =, whch constrans both drecton and magntude of the normal vector Ths s especally mportant to force the algebrac dstance to be close to the Eucldean dstance for ponts close to the surface of the sphere Furthermore, the normal constrants lead to the followng standard lnear system of equatons that can be solved very effcently: W 2 (x)d u = W 2 (x) b (7) Fgure 6: The Eucldean dstance (green cone) versus an algebrac dstance (orange parabolod) to the 2D yellow crcle Pratt s normalzaton makes these two surfaces tangent to each other at the gven crcle 44 Algebrac Pont Set Surfaces Implct surface defnton The prevous sectons provde all ngredents to defne an MLS surface based on sphere fts Our APSS SP, approxmatng or nterpolatng the pont set P = {p Rd }, yelds as the zero set of the mplct scalar feld f (x) representng the algebrac dstance between the evaluaton pont x and the ftted sphere u(x) Ths feld s llustrated n fgure 5d: h f (x) = su(x) (x) =, xt, xt x u(x) = () If needed the actual Eucldean dstance to the ftted algebrac surface can be computed easly by ts converson to an explct form Gradent Ths mplct defnton allows us to convenently compute the gradent of the scalar feld whch s needed to obtan the surface normal or to perform an orthogonal projecton We compute the gradent f (x) as follows: f (x) = [, xt, xt x] u(x) + where w (x) β w (x) W(x) = β w (x), D = pt et etd T p p T 2e p, 2eTd p T e n b = (8) T ed n Here, {ek } denote the unt bass vectors of our coordnate system The scalar β allows us to weght the normal constrants We wll dscuss the proper choce of ths parameter subsequently We solve ths equaton usng the pseudonverse method, e: u(x) = A (x)b (x) (9) where both the (d +2) (d +2) weghted covarance matrx A(x) = DT W(x)D and the vector b (x) = DT W(x)b can be computed drectly and effcently by weghted sums of the pont coeffcents The ssue of affne nvarance of ths method deserves some further dscusson Whle the method s nvarance under translatons and rotatons s trval, the mx of constrants representng algebrac dstances and dstances between unt vectors makes t slghtly senstve to scale A smple and practcal soluton s to choose a large value w (x)h (x) for β, eg β = 6 h(x)2 where h(x) = w (x) s a smooth func ton descrbng the local neghborhood sze () Ths effectvely assgns very hgh mportance to the dervatve constrants, whch, prescrbed at gven postons and wth a fxed norm, are suffcent to ft a sphercal sosurface to the data The postonal constrants stll specfy the actual sovalue Not only does ths choce leave the fttng nvarant under scale, but t also makes t more stable, much less prone to oscllatons (fgure 3b) and less senstve to outlers Ths choce does not unbalance the relatve mportance of the postons and normals of the samples because the dervatve constrants depend on both the sample postons and normals et etd 2eT x u(x) 2eTd x () du(x) The value of the gradent u(x) = du(x) dx, dx, depends on the fttng method Wth the Pratt s constrant, the dervatves can be computed as the plane ft wth covarance analyss case (see [Alexa and Adamson 24]) Wth our normal constrant, the dervatves can be drectly computed from equaton (9): du(x) dxk = da(x) d b (x) A (x) u(x) + dxk dxk (2) Curvature The mplct defnton can be used to compute hgher order dfferental surface operators, such as curvature In practce, however, the evaluaton of the shape matrx nvolves expensve computatons Our sphere ft provdes an elegant estmate of the mean curvature readly avalable by the radus of the ftted sphere (3) Ths estmate of the mean curvature s n general very accurate, except when two peces of a surface are too close to each other In ths case the samples of the second surface can lead to an overestmaton of the curvature Note that n such cases our normal constrant fttng method stll reconstructs a correct surface wthout oscllatons (fgure 3b) The sgn of ths nexpensve mean curvature estmate s determned by the sgn of un+ and can be utlzed for a varety of operatons, such as accessblty shadng shown n fgure Projecton procedure Snce the presented APSS surface defnton s based on a standard MLS, all the projecton operators for the plane case can easly be adapted to our settng by smply replacng the planar projecton by a sphercal projecton Followng the concept of almost orthogonal projectons n [Alexa and Adamson 24] we recommend the followng procedure as a practcal recpe to mplement APSS: Gven a current pont x, we teratvely defne a seres of ponts q, such that q+ s the orthogonal projecton of x onto the algebrac sphere defned by u(q ) Startng wth q = x,
5 the projecton of x onto the surface s asymptotcally q The recurson stops when the dsplacement s lower than a gven precson threshold Ths procedure s depcted n fgure 5c Includng surface normals nto the defnton comes wth addtonal sgnfcant advantages Frst, t makes the scalar feld f (x) to be consstently sgned accordng to the nsde/outsde relatve poston of x Such consstent orentaton s fundamental to perform boolean operatons or collson detectons for nstance Furthermore, pontnormals provde a frst order approxmaton of the underlyng surface and thus allow for smaller neghborhoods at the same accuracy Tghter neghborhoods do not only ncrease stablty, but also accelerate the processng To ths end, we recommend the normal estmaton procedure descrbed n the followng secton to preprocess a raw nput pont cloud (d) (e) 5 Estmaton of Surface Normals Our method for normal estmaton draws upon the same prncple as the one proposed by Hoppe et al [992] The noveltes nclude the normal drecton estmaton and new heurstcs for the propagaton of the normal orentaton We wll focus on these two ssues 5 Normal Drecton To estmate the normal drecton n of a pont p of our nput pont set, we essentally take the gradent drecton at p of our MLS surface defnton descrbed n the prevous secton, e, n = f (p ) Here, the computaton of the algebrac sphere s based on the egenvector method descrbed n secton 42 In practce, we can approxmate the accurate gradent drecton of the scalar feld f (appendx) wth the gradent of the ftted algebrac sphere u(p ), e: e T 2e T p n s u(p )(p ) = u(p ) (3) e T d 2e T d p Ths step s llustrated n fgure 5a Although the sphere normal can dffer from the actual surface normal, the above smplfcaton s reasonable, because the approxmaton s usually very close to the accurate surface normal for ponts evaluated close to an nput sample In practce, we never experenced any problems wth t Durng ths step we also store for each pont a confdence value µ that depends on the santy of the neghborhood used to estmate the normal drecton We take the normalzed resdual value, e, d+ µ = λ/ k= λ k where λ k are the d + 2 egenvalues and λ s the smallest postve one Smlar confdence estmates were used before by several authors (eg [Pauly et al 24]) n combnaton p p h Fgure 7: Illustraton of the BSP neghborhood of a pont p : each neghbor defnes a halfspace lmtng the neghborhood The blue ponts correspond to a nave knn wth k = 6 (b) Illustraton of Ψ j The angle between the gradent of the ftted sphere (green arrows) and the normal drecton (shown n red here) serves as a measure for the relablty of the normal propagaton p j (b) c p p j m (b) (c) (f) Fgure 8: Closeup vew of the tp of the ear of fgure 2 after recomputng the normals usng our method A few normals are ncorrectly orented and eventually break the APSS reconstructon (b) when usng a weght radus of h = 2 A smoothed manfold reconstructon (c) can stll be obtaned when ncreasng the weght radus sgnfcantly (h = 35) Fgures (d) and (e) show the normals after 5 and 3 teratons of our normal optmzaton procedure respectvely (f) APSS reconstructon of (e) wth h = 2 wth covarance analyss to perform adaptve resamplng We wll use ths confdence value n the next secton for our propagaton algorthm 52 Propagaton of Orentaton Our second contrbuton concerns the propagaton of normal orentaton Followng [Hoppe et al 992], we use a mnmum spannng tree (MST) strategy whch s ntalzed by settng the normal orentaton of a sngle pont n contact wth the pont set s boundng box to pont outsde The major dfference of our approach les n the propagaton method and the weght functon for the MST To propagate the orentaton from a pont p of normal n to a pont p j of normal n j we locally ft an algebrac sphere at m = p +p j 2 halfway between the two ponts Fgures 3c and 5b serve for llustraton The sphere gves us a relable approxmaton of the surface connectng the two ponts It hence allows us to relably transfer a gven normal orentaton even across sharp features or along two close surface sheets The normal n j of the pont p j s flpped f s u(m) (p ) T n s u(m) (p j ) T n j < The purpose of the weght functon of the MST s to favor edges along whch t s safe to transfer the normal orentaton usng the prevous procedure The MST assgns an edge  j f and only f the two nvolved samples p and p j are neghbors Unlke Hoppe s knearest neghborhood defnton (knn), we utlze a somewhat more elaborate BSP neghborhood defnton Ths defnton removes from a large knn of the pont p every pont that s behnd another neghbor, e, every pont p j such that there exst a thrd pont p h satsfyng (p p h ) T (p j p h ) < Fgure 7a depcts an example Ths neghborhood defnton allows us to select relably the closest neghbors n each drecton, even n cases of strongly nonunform samplng In addton t avods that the propagaton procedure jumps over relevant samples To compute the weght of an edge  j we consder the followng two terms: The frst one takes the sum of the confdence values of ts adjacent vertces: µ j = µ + µ j The ratonale of ths heurstc s
6 to push the samples wth the lowest confdence values to the bottom (e the leaves) of the tree The second term Ψ j s computed by frst fttng a sphere at the center m of the edge and by quantfyng the dfference between the gradent of the sphere and the estmated normal drecton Ψ j s defned as the dot product between the two vectors, e, Ψ j = ( s u(m) (p ) T n + s u(m) (p j ) T n j )/2 As llustrated n fgure 7b, ths confrms our ntuton that the propagaton of normal drectons s less obvous when the two vectors devate strongly We fnally weght each edge  j by a weghted sum of µ j and Ψ j, an emprcal choce beng 8µ j + Ψ j For some rare dffcult cases, we propose to use the actual normals to further optmze the normals wth low confdence Ths s accomplshed by usng our stable sphercal fttng method from secton 43 For the ft, we weght each normal constrant accordng to ts respectve confdence value Ths ensures that samples of hgher confdence have ncreased mportance for the ft To further optmze accuracy, we process the samples startng wth the ones havng the hghest average confdence of ther neghborhood Our normal estmaton procedure s depcted fgure 8 6 Sharp Features In ths secton we present a new approach to handle sharp features such as creases, corners, borders or peaks We wll explan t n the context of APSS, but the dscussed technques can easly be used wth any PSS defnton Our approach combnes the local CSG rules proposed n [Fleshman et al 25] wth a precomputed partal classfcaton, and we present extensons for the robust handlng of borders and peaks Durng the actual projecton of a pont x onto the surface, we frst group the selected neghbors by tag values Samples wthout tags are assgned to all groups Then, the APSS s executed for each group and the actual pont s projected onto each algebrac sphere Next, we detect for each par of groups whether the sharp crease they form orgnates from a boolean ntersecton or from a unon and apply the correspondng CSG rule We refer to [Fleshman et al 25] for the detals of ths last step, and focus on the novel aspects of our approach 6 Taggng a Pont Cloud In a frst preprocessng step or at run tme, we compute a partal classfcaton of the samples by assgnng a tag value to each sample n the adjacency of a sharp feature Two adjacent samples must have the same tag f they are on the same sde of the dscontnuty and a dfferent tag otherwse Ths classfcaton can be acheved n several ways from fully automatc to entrely manual processng Tags can easly be set automatcally durng a CSG operaton [Pauly et al 23; Adams and Dutré 23] Tags can also be convenently panted by usng a brush tool Fgure 9 dsplays a smple tag stroke Such nteractve pantng provdes full user control over the poston of sharp features and t can be appled very effcently Fgure 9: A sharp crease s obtaned by pantng two strokes on an ntally smooth pont cloud n, n, same tag same tag x n (x) Fgure : Left: A cubc Bézer curve connects two samples to propagate the taggng and to correctly algn the respectve normals Rght: Defnton of a peak sample p of axs a and angle b Its normal n (x) depends on the actual evaluaton pont x n j, Automatc Taggng by Local Classfcaton Another way to automatcally tag the pont cloud would be to use the local classfcaton scheme of Fleshman et al [25] Whle beng very effcent, ths method, however, cannot guarantee a consstent global classfcaton Therefore, we developed the followng groupng method that guarantees global consstency The local classfcaton s appled to each sample n a breadth frst order of the Eucldean mnmum spannng tree, each sample countng the number of tmes t s assgned a gven tag value If only one group s found (smooth part) then the algorthm contnues wth the next sample Otherwse, the tag value of each group s determned from the tag value wth the maxmum count of all samples In addton, each local group must have dfferent tag values If requred, new tags are created Next, the group tags are propagated to ther samples The fnal tag value of a sample s the one of maxmum count Taggng from Sharp Ponts A common way to represent sharp features n pont sampled geometry s to use sharp ponts, e, ponts wth multple normals [Pauly et al 23; Wcke et al 24; Guennebaud et al 25] Usually, two normals are assgned to a pont on a crease and three normals to a corner pont We handle such explct representatons wthn our framework by the followng two step taggng procedure: The frst step focuses on taggng the normals of the sharp samples only For smplcty, we wll frst only consder the case of edge ponts, e, ponts havng two normals We start wth any sharp sample, assgn a dfferent tag value to each of ts normals, and propagate the tag values along the crease lnes nto each drecton Our propagaton method s manly based on the constructon of an nterpolatng cubc Bézer curve between sharp samples Ths enables us to safely and naturally handle hgh curvature of the crease lnes Such a curve can be constructed easly by takng the cross product of the two normals of the samples as the curve s end tangent vectors Fgure shows the dea We scale the tangent vectors such that ther length s equal to the thrd of the Eucldean dstance between the two samples and orent them such that the length of the resultng curve s mnmal The best possble successor of a sharp pont p along a crease lne s thus the sharp pont p j that leads to the shortest curve connectng t to p In practce, we approxmate the length of the Bézer curve by takng the length of ts control polygon We also propagate the tags of the pont s normals through tangent orentaton If the normals of the successor are already tagged, the propagaton for ths branch stops, and the correspondng tag values are marked Otherwse, the procedure contnues by searchng the best successor of p j n the outgong drecton untl all sharp samples are tagged Handlng corner ponts s accomplshed consderng them as three dstnct edge samples, and when a corner s reached, a new tag value s assgned to the thrd normal n j, b a p
7 The sharpness control of a crease feature by a user parameter can easly be added to our concept To ths end we nsert all samples nto all groups but assgn a lower weght to a sample when nserted nto a group wth a tag dfferent from ts own one We defne a global or local parameter α [, ] whch modulates the weght of these samples For α = we obtan a fully sharp feature, for α = the feature s completely smoothed snce all groups are weghted dentcally Values nbetween permt a contnuous transton from smooth to sharp (fgure 2) (b) Fgure : Illustraton of boundares handled usng clppng samples and an nterpolatory weght functon (b) Illustraton of peaks handled wthout any treatment (left part) and wth our specal peak samples (rght part) The second step nvolves a propagaton of the tag values to the samples close to a sharp edge, e, where dstance s defned by the actual weghtng scheme Taggng only the samples close to a sharp crease specfcally enables us to deal wth nonclosed creases as n fgure 9 If p s a smooth pont and p j ts closest sharp pont havng a nonzero ntersecton of nfluence rad, the tag value of p yelds as the tag value of the normal of p j defnng the plane closest to p A result of ths procedure s depcted n fgure rght 62 Extensons Object boundares can be treated usng clppng samples A set of clppng ponts defnes a surface that s only used to clp another surface In order to smplfy both the representaton and the user control, we place such clppng samples at the postons of boundary samples Smlar to the above noton of sharp ponts t s suffcent to add a clppng normal to each sample at the desred boundary A clppng normal s set orthogonal to the desred boundary curve and should be tangent to the surface Fgure a llustrates the concept Obvously, clppng samples may also be tagged to defne boundary curves wth dscontnutes Fnally, the evaluaton of the surface or the projecton onto the surface only requres a mnor addtonal modfcaton Clppng samples are always treated n separate groups and the reconstructed clppng surface s only used to clp the actual surface by a local ntersecton operaton Automatc detecton of boundares n a pont cloud can easly be done by analyzng the neghborhood of each sample, such as n [Guennebaud et al 25] Handlng peak dscontnutes also deserves some specal attenton We propose to explctly represent a peak by a pont p equpped wth an axs a and an angle b, e, by a cone, as llustrated n fgure Durng the local fttng step, we take as the normal n (x) of such a peak sample the normal of the cone n the drecton of the evaluaton pont x n (x) thus yelds as the unt vector a rotated by angle π2 b and axs a (x p ) We also guarantee that the surface wll be C at the peak by attachng an nterpolatng weght functon to peak samples The results n fgure b demonstrate that our method produces hgh qualty peaks 7 Results 7 Implementaton and Performance In addton to a software mplementaton, we accelerated the projecton operator descrbed n secton 44 on a GPU Our mplementaton reles on standard GPGPU technques and computes all the steps of the projecton, ncludng neghbor queres, the fttng step and the orthogonal projecton, n a sngle shader We use grds of lnked lsts as spatal acceleraton structures Our mplementaton performs about 45 mllon projecton teratons per second on a NVda GeForce 88GTS for APSS, compared to 6 mllon for SPSS As detaled n table, our APSS projecton operator converges about two tmes faster than SPSS makng APSS overall about 5 tmes faster for the same precson As we can see, a sngle teraton s usually suffcent to obtan a reasonable precson, whle SPSS acheves a smlar accuracy only after 2 to 3 teratons # ter APSS SPSS 2e4 94e e5 462e4 3 9e5 67e e5 753e e5 395e e5 232e5 Table : Comparson of the convergence of APSS and SPSS The values ndcate the relatve average precson obtaned after the gven number of teratons for a typcal set of models The results of MLS surfaces depend sgnfcantly on the weght functons Although we tred to fnd best settngs for each method, other weght functons could lead to other, potentally better results In partcular, we performed all our comparsons on unform pont clouds usng the weght functon descrbed n secton 4 wth a constant weght radus h (x) = h r where h s an ntutve scale factor and r s the average pont spacngmoreover, our IMLS mplementaton corresponds to the smple defnton gven n [Kollur 25]: f (x) = w (x)(x p )T n = w (x) (4) In order to convenently handle arbtrary pont clouds we suggest to use h (x) = h r where r s the local pont spacng that s computed as the dstance to the farthest neghbor of the pont p usng our BSP neghborhood defnton An open source lbrary of our APSS framework s avalable at: 72 Analyss of Qualty and Stablty Approxmaton and nterpolaton qualty Fgure 4 compares the ablty of our sphercal MLS to perform tght approxmaton and convex nterpolaton For nterpolaton we used φ (x) = log(x)4 for both APSS and IMLS and φ (x) = /x2 for the two others The values of h used for ths fgure are summarzed n the followng table (nterpol/approx): Fgure 2: Illustraton of the sharpness control, from left to rght: α =, α = 5, α = 5, α = SPSS 335 / na HPSS na / na IMLS 95 / 95 APSS 95 / 325
8 (b) (c) (d) Fgure 3: Intal dense chameleon model from whch 4k samples have been unformly pcked (depcted wth red dots) Ths low sampled model s reconstructed usng APSS (h = 9) (b), SPSS (h = 8) (c) and IMLS (h = 26) (d) Stablty for Low Samplng Rates One of the central advantages of our sphercal MLS s ts ablty to robustly handle very low samplng denstes as llustrated n fgure 3 The fgure compares the approxmaton qualty of a unformly sampled 4K chameleon model for APSS, SPSS and IMLS We can clearly see the superor qualty of APSS We computed the texture colors from the orgnal dense model In order to quanttatvely evaluate the robustness under low samplng and the approxmaton qualty, we teratvely created a sequence of subsampled pont sets P, P, by frst decomposng the current pont set P nto dsjunctve pont pars usng a mnmum weght perfect matchng (see [Waschbu sch et al 24] for a smlar scheme) A subsequent projecton of the par centers onto the actual PSS S yelds a reduced pont set P+ The graph n fgure 4 shows the relatve average dstance of the ntal pont set P onto varous PSS S as a functon of the number of samples Note that the graph s gven on a logarthmc scale The results demonstrate that the APSS s about three tmes more precse than the planar MLS varants Furthermore, the planar versons break already at much hgher samplng rates (crosses n the dagram), whle APSS allows for stable ft at very low samplng rate The nstablty of Levn s operator s manly due to the polynomal projecton whch requres a relatvely large and consstent neghborhood The dstrbuton of the error s shown for an example n fgure 5 Stablty for Changes of Weghts Fgure 6 compares the stablty of APSS versus SPSS as a functon of the sze of the weght rad We utlzed a hgher order genus model wth very fne structures and a low samplng densty As can be seen SPSS exhbts undesred smoothng already at small rad and breaks quckly at larger szes In ths example, APSS always delvers the expected result, and the surface stays reasonably close to orgnal data even for large rad 73 Unwanted Extra Zeros Our sphercal fttng algorthm can exhbt an ncreased senstvty wth respect to unwanted statonary ponts away from the surface We refer to [Adamson and Alexa 26a] for a dscusson of the planar ft Wth spheres, such extra zeros can specally occur close to the boundary of the surface defnton doman D when for nstance a small set of ponts yelds a small and stable sphere that les entrely nsde D D s the unon of the balls centered at the sample postons p and havng the radus of the weght functon (h r ) We handle such spurous extra zeros by lmtng the extent of D, e by both reducng the ball rad and removng regons n whch fewer than a gven number (eg, 4) of samples have nonzero weghts We found that ths soluton s workng well n practce, but admt that ndepth theoretcal analyss s needed as part of future work 8 Concluson and Future Work We have demonstrated that the planar ft utlzed n conventonal PSS mplementatons can successfully be replaced by sphercal fts Our sphere ft MLS has numerous sgnfcant advantages and overcomes some of the ntrnsc lmtatons of planar MLS In partcular, we obtan an ncreased stablty for low samplng rates, a tghter ft, a better approxmaton qualty, robustness, and forfree mean curvature estmaton Fast numercal procedures and effcent mplementatons allow for the processng of very complex, hghqualty models and make APSS as fast as smple MLS Extensons of the method handle dscontnutes of varous knds An apparent natural extenson of the method would be to employ ellpsods, especally for hghly ansotropc objects, such as cylnders Algebrac ellpsodal fts, however, do not naturally degenerate to planes In contrast, our method handles such objects correctly, as can be seen on the saddle confguraton of fgure rght Future work comprses applcatons of the representaton n nteractve modelng We beleve that the method s wellsuted for deformatons of pont sampled geometry where the surface normals could be estmated drectly from the gradent operator Another mportant ssue s the theoretcal analyss of both unwanted extra zeros and the samplng requrements for our settng Fnally, we want to explore ts extenson to nonmanfold geometry as well as realtme raytracng of dynamc APSS Average dsplacement E E2 E3 APSS SPSS IMLS Levn's E4 E5 E6 53k 76k 38k 9k 95k 47k 4k 2k # pts Fgure 4: Logarthmcally scaled graph of the average projecton error for the Chnese Dragon model (b) % 6% (c) Fgure 5: Chnese Dragon model (5 K) approxmated wth APSS and colorcoded magntude of the dsplacement between the nput samples and a low resoluton model (9 K) usng APSS (b) and SPSS (c)
9 B OISSONNAT, JD, AND C AZALS, F 2 Smooth shape reconstructon va natural neghbor nterpolaton of dstance functons In Proceedngs of the 6th Annual Symposum on Computatonal Geometry, ACM Press, D EY, T K, AND S UN, J 25 An adaptve MLS surface for reconstructon wth guarantees In Proceedngs of the Eurographcs Symposum on Geometry Processng 25, D EY, T K, G OSWAMI, S, AND S UN, J 25 Extremal surface based projectons converge and reconstruct wth sotopy manuscrpt F LEISHMAN, S, C OHEN O R, D, AND S ILVA, C T 25 Robust movng leastsquares fttng wth sharp features ACM Transactons on Graphcs (SIGGRAPH 25 Proceedngs) 24, 3, Fgure 6: A sparsely sampled pont model (9 K) handled wth APSS (top row) and wth SPSS (botton row) usng dfferent szes of the weght rad: 6, 4, 8 G ANDER, W, G OLUB, G H, AND S TREBEL, R 994 Leastsquares fttng of crcles and ellpses BIT Numercal Mathematcs 34, 4, Acknowledgments G UENNEBAUD, G, BARTHE, L, AND PAULIN, M 25 Interpolatory refnement for realtme processng of pontbased geometry Computer Graphcs Forum (Proceedngs of Eurographcs 25) 24, 3, Ths work was supported n part by the ERCIM Alan Bensoussan Fellowshp Programme We would lke to thank all revewers for ther nsghtful comments and dscussons of the method, as well as Maro Botsch and Ronny Pekert for proof readng the paper The models of fgures 5 and 6 are provded courtesy of INRIA and SensAble by the Shape Repostory The tree of fgure s provded courtesy of Vncent Forest References A DAMS, B, AND D UTR E, P 23 Interactve boolean operatons on surfelbounded solds ACM Transactons on Graphcs (SIGGRAPH 23 Proceedngs) 22, 3, H OPPE, H, D E ROSE, T, D UCHAMP, T, M C D ONALD, J, AND S TUETZLE, W 992 Surface reconstructon from unorganzed ponts In Proc of ACM SIGGRAPH 92, ACM Press, 7 78 K AZHDAN, M, B OLITHO, M, AND H OPPE, H 26 Posson surface reconstructon In Proceedngs of the Eurographcs Symposum on Geometry Processng 26, KOLLURI, R 25 Provably good movng least squares In ACMSIAM Symposum on Dscrete Algorthms, 8 8 L EVIN, D 23 Meshndependent surface nterpolaton Geometrc Modelng for Scentfc Vsualzaton, 8 87 A DAMSON, A, AND A LEXA, M 23 Approxmatng and ntersectng surfaces from ponts In Proceedngs of the Eurographcs Symposum on Geometry Processng 23, M ITRA, N J, N GUYEN, A, AND G UIBAS, L 24 Estmatng surface normals n nosy pont cloud data Internatonal Journal of Computatonal Geometry and Applcatons 4, 4 5, A DAMSON, A, AND A LEXA, M 24 Approxmatng bounded, nonorentable surfaces from ponts In Proceedngs of Shape Modelng Internatonal 24, IEEE Computer Socety O HTAKE, Y, B ELYAEV, A, A LEXA, M, T URK, G, AND S EI DEL, HP 23 Multlevel partton of unty mplcts ACM Transactons on Graphcs (SIGGRAPH 23 Proceedngs) 22, 3, A DAMSON, A, AND A LEXA, M 26 Ansotropc pont set surfaces In Afrgraph 6: Proceedngs of the 4th nternatonal conference on Computer graphcs, vrtual realty, vsualsaton and nteracton n Afrca, ACM Press, 7 3 PAULY, M, K EISER, R, KOBBELT, L P, AND G ROSS, M 23 Shape modelng wth pontsampled geometry ACM Transactons on Graphcs (SIGGRAPH 23 Proceedngs) 22, 3 A DAMSON, A, AND A LEXA, M 26 Pontsampled cell complexes ACM Transactons on Graphcs (SIGGRAPH 23 Proceedngs) 25, 3, A LEXA, M, AND A DAMSON, A 24 On normals and projecton operators for surfaces defned by pont sets In Proceedngs of the Eurographcs Symposum on PontBased Graphcs, A LEXA, M, AND A DAMSON, A 26 Interpolatory pont set surfaces  convexty and hermte data Submtted paper A LEXA, M, B EHR, J, C OHEN O R, D, F LEISHMAN, S, L EVIN, D, AND S ILVA, C T 23 Computng and renderng pont set surfaces IEEE Transactons on Computer Graphcs and Vsualzaton 9,, 3 5 A MENTA, N, AND K IL, Y 24 Defnng pontset surfaces ACM Transactons on Graphcs (SIGGRAPH 24 Proceedngs) 23, 3, A MENTA, N, AND K IL, Y 24 The doman of a pont set surface In Proceedngs of the Eurographcs Symposum on PontBased Graphcs 24, PAULY, M, M ITRA, N J, AND G UIBAS, L 24 Uncertanty and varablty n pont cloud surface data In Proceedngs of the Eurographcs Symposum on PontBased Graphcs, P RATT, V 987 Drect leastsquares fttng of algebrac surfaces In Proc of ACM SIGGRAPH 87, ACM Press, S HEN, C, O B RIEN, J F, AND S HEWCHUK, J R 24 Interpolatng and approxmatng mplct surfaces from polygon soup ACM Transactons on Graphcs (SIGGRAPH 24), WALD, I, AND S EIDEL, HP 25 Interactve ray tracng of pont based models In Proceedngs of the Eurographcs Symposum on Pont Based Graphcs 25 WASCHB U SCH, M, G ROSS, M, E BERHARD, F, L AMBORAY, E, AND W U RMLIN, S 24 Progressve compresson of pontsampled models In Proceedngs of the Eurographcs Symposum on PontBased Graphcs 24, 95 2 W ICKE, M, T ESCHNER, M, AND G ROSS, M 24 CSG tree renderng of pontsampled objects In Proceedngs of Pacfc Graphcs 24, 6 68
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