Efficient Free-Form Surface Modeling with Uncertainty

Transcription

1 IEEE Iteratioal oferece o Robotics ad utomatio, Mieapolis, US, pril 1996, Efficiet Free-Form Surface Modelig with Ucertaity Miguel gel García Luis asañez Divisio of Robotics ad rtificial Itelligece Istitute of yberetics Polytechic Uiversity of ataloia Diagoal 647, plata arceloa, SPIN fax: , garcia@ic.upc.es bstract This paper describes the use of a previously developed surface model for represetig 3D geometric iformatio that is subject to ucertaity. The proposed model is based o a geometric data fusio techique that allows the efficiet approximatio of arbitrary triagular meshes i space with smooth surfaces. This paper shows how the shape modifiers defied i that model ca be used to represet both global ad local ucertaity associated with the iput data. The possibility of hadlig scattered 3D poits, takig ito accout their correspodig ucertaities, makes this techique a suitable tool for modelig free-form surfaces acquired through sesig, as well as for itegratig 3D data obtaied from multiple sesors with differet associated cofideces. 1 Itroductio Free-form surface modelig is a field which ca be applied to a wide variety of disciplies, icludig robotics, computer graphics, computer aided geometric desig ad geographic iformatio systems. I mobile robotics, for example, free-form surface models ca be utilized to represet rugged terrai such that robots operatig outdoors may be able to pla their actios, safeguard their motios ad estimate their positios [9]. I additio, free-form surface models are advatageous i order to defie world models suitable for represetig complex shapes preset i the robotic workspace, especially whe o assumptios are made a priori about the ature of those shapes [7]. I fact, at the lowest level of iterpretatio ay surface ca be cosidered to be freeform, disregardig that, at subsequet stages, some of those surfaces may be idetified as well-established fuctios such as plaes or quadrics. I this sese, free-form The first author has bee supported by the Govermet of Spai uder both a FPI fellowship ad the IYT project TP complemetary grat has bee give to him by the Polytechic Uiversity of ataloia surface models costitute a rather geeral geometric descriptio tool. May differet free-form surface models have bee proposed over the last decades, especially i the areas of computer graphics ad computer aided geometric desig. survey of represetative methods ca be foud i [4]. I D/M ad computer graphics, tesor-product methods, such as -Splies or NURS, have become a stadard tool for free-form surface modelig. These models geerate smooth surfaces that approximate rectagular meshes of 3D cotrol poits. However, the eed for rectagular meshes complicates the modelig of surfaces of arbitrary topology ad geus closed surfaces with holes, for istace ad the applicatio of these techiques whe scattered data obtaied through sesig are hadled. Numerous o-tesor-product methods have also bee proposed to approximate ad iterpolate irregular meshes of 3D cotrol poits [4]. However, most of them are devised with the aim of geeratig fair surfaces they are proposed i the scope of computer graphics at the expese of efficiecy. The best models are based o recursive procedures or global optimizatio techiques. This complicates their applicatio to fields, such as robotics, with tight timig costraits. I additio, these techiques igore the presece of ucertaity sice they typically work with sythetically geerated data or with 3D poits obtaied through high resolutio scaers. I previous work [3][4], a ew techique for geeratig smooth free-form surfaces through approximatio of triagular meshes of 3D cotrol poits was proposed. Triagular meshes are appropriate for hadlig scattered iformatio ad they facilitate the itegratio of ew data as they become available to the system. techique for obtaiig triagular meshes from dese rage images was preseted i [6]. Moreover, the use of parametric fuctioals allows the represetatio of ope ad closed surfaces of arbitrary topology ad geus. That surface model possesses a set of additioal properties [5], icludig efficiecy, covex hull, affie ivariace ad shape cotrol, that make it suitable for the represetatio of

2 complex shapes i the cotext of a robotic system that must iteract with its eviromet i real-time [7]. This paper studies the use of shape modifiers provided by this model so that the geerated surfaces may reflect global ad local ucertaity estimatios associated with the sesed 3D data. Ucertaity hadlig is a very importat issue i robotics owig to the imprecise ature of sesig. Ucertaity has bee typically approached by statistical methods, by cosiderig that sesed data are subject to oise which ca be modeled as a probability desity fuctio usually Gaussia with a give mea ad a covariace matrix. ovariace matrices are physically iterpreted as ucertaity ellipsoids [11] cetered at the positios correspodig to the sesed data. I robotics, the problem of geometric ucertaity hadlig has bee tackled from two basic perspectives. O the oe had, some authors provide the sesor data ad their related ucertaities idepedetly. For example, Kweo ad Kaade [9] geerate elevatio maps ad associated ucertaity maps, the secod oes takig ito accout the ucertaity associated with sesor measuremets. Elevatio maps are computed disregardig the estimated ucertaities. I a similar lie, Hutchiso ad Kak [8] represet ucertaity i the positio of a object by a homogeeous trasformatio matrix whose etries are expressed i terms of ucertaity variables. other importat lie is costituted by methods i which sesor data are trasformed by takig ito accout their correspodig ucertaities. Typically, the aim of these techiques is the estimatio of geometric features, such as segmet ed-poits, from sesor readigs subject to ucertaity, the latter usually modeled as Gaussia probability distributios. The Kalma filter is used to geerate the best statistical estimatio of the physical parameters the ubiased estimatio of miimum variace from the give sesory observatios cosiderig their associated ucertaities. [1], [2], [1], [12] ad [13] are some works followig this approach. This paper presets a differet approach to the problem of geometric ucertaity hadlig. Our objective is the estimatio of the exteral surface of a object by takig ito accout a collectio of poits measured from the real surface of that object, ad ucertaity estimatios associated with those poits. The aim is to obtai a surface that automatically adapts to the available kowledge by itegratig both the measured positios ad their correspodig ucertaities. Sectio 2 outlies the proposed surface model. Sectio 3 aalyzes the use of the shape modifiers defied i the model i order to represet global ad local ucertaity. Fially, coclusios are give ad future work is proposed i Sectio 4. D K E D J K E Figure 1: Polygoal patches associated with cotrol triagle. 2 Free-form surface modelig L K F J This sectio summarizes a techique for approximatig triagular meshes of cotrol poits with smooth surfaces. Further details are give i [3][4]. Every vertex X of a give triagular mesh is a surface cotrol poit defied by three spatial coordiates ad a weightig factor: X { ( x X, y X, z X, W X }. Every triagle of the cotrol mesh is the domai of a parametric surface patch S ( P. The parameters of S are the three barycetric coordiates b, b, b of a evaluatio poit P cotaied i : P b + b + b where b + b + b 1 ad b, b, b 1., ad are the 3D coordiates of the vertices of. I this paper both S ( P ad S ( b, b, b will represet the value of the surface patch S for a give evaluatio poit P. The fial surface that approximates the give triagular mesh is composed of as may surface patches S as cotrol triagles. Those surface patches are defied so that they joi smoothly. Let Patch {,, a } be the set of cotrol poits composed of ad its a atural eighbors (adjacet cotrol poits. This set is defied i R 3 ad will be called a polygoal patch. Similar patches are defied for ad. Let {,, W a } be the weightig factors associated with those cotrol poits. ccordig to this defiitio, each triagle of the mesh belogs to three polygoal patches, Fig. 1. The surface computatio correspodig to a cotrol triagle requires similar operatios applied to the three patches that cotai that triagle. Due to space limitatios, some of these formulatios will oly be show for the patch cetered at. The correspodig equatios for the patches cetered at ad are obtaied by substitutig for or ad the modifyig the remaiig idices accordigly. M N I H I H G E F H G

3 ifluece K d K P If (K, P Γ (d d D E Figure 2: omputatio of topological iflueces. (left 2D polar represetatio of the polygoal patch cetered at cotrol poit i Fig. 1. (right Use of the topological distace d K to obtai the ifluece of cotrol poit K. First, each 3D polygoal patch is mapped to a 2D regular polygo which oly depeds o the umber of eighborig poits of the origial patch. This ew represetatio has bee called the polar represetatio of a polygoal patch: Polar {,, a }. The cetral vertex is set to (.,.. The remaiig poits are defied as i ( cos ( 2π ( i 1 a, si ( 2π ( i 1 a. Next, the polar represetatio of the evaluatio poit P is computed for each of the three patches (oly show for : P b + b + b, where,, are the polar represetatios of,, i Patch. The, a ifluece factor If i, P that deotes the ifluece each i exerts upo P is computed, Fig. 2. Similar formulatios are defied for the polygoal patches cetered at ad. If (1 where d i P, d 2, represets the Euclidea 2 distace betwee two poits i R 2 ad N,2 is the quadratic basis -splie fuctio [3] with kot vector [, 1/3, 2/3, 1]. The parameter Γ i (1 cotrols the width of the bell-shaped fuctio (Fig. 2, behavig as a global shape modifier. The surface patch ( P is fially computed as S (b, b, b S (P FF (P S (P + FF (P S (P + FF (P S (P (2 where FF, FF ad FF are covex combiatio factors calculated as (oly show for Patch FF (3 ad S, S, S are parametric surface subpatches associated with the patches cetered at, ad respectively, d K i, P Γ ( d N, 2 (.5 d Γ +.5 S ( P FF ( P ( FF ( P + FF ( P + FF ( P 2 FF ( P b b b P 2 Γ Figure 3: Example of ope mesh ad recostructed surface. ll the weightig factors W X are equal to 1. Global shape modifier Γ cotrol triagles S ( P i a If (4 The previous surface patches, S, joi with 1 ( P cotiuity alog edges but with cotiuity across them. I spite of the discotiuity of derivatives across surface patches, experimetal results show good results with smooth surfaces displayig imperceptible defects, Fig. 3. Those surface patches may be optioally modified to esure first order geometric cotiuity G 1 betwee them by applyig a techique preseted i [3]. Sice this techique iterpolates the boudary curves defied by S ( P, it preserves the overall shapes defied through the shape modifiers of S ( P. Hece, the followig cosideratios are also applicable. 3 Ucertaity hadlig a i If i, P W i i i, P W i The shape of the recostructed surface ca be globally ad locally altered through shape modifiers defied i the model. These parameters may be utilized to itroduce both global ad local ucertaity i the surface recostructio process. Their behavior is studied below. 3.1 Embeddig global ucertaity Parameter Γ i (1 defies the amout of ifluece of a certai cotrol poit upo its adjacet eighbors. large value of Γ leads to the surface beig pulled eve by topologically distat vertices. The result is a icrease of smoothess, which ca be iterpreted as a bad reductio of a low-pass filter. Fig. 4 shows this effect by applyig a factor Γ equal to 1.5 istead of 1.25, which is the oe utilized i Fig. 3. This parameter is a global shape modifier that ca be used to represet the average geometrical ucertaity associated with the whole set of cotrol poits. Its behavior is aalyzed below.

4 Figure 4: Global shape modificatio of the cotrol mesh i Fig. 3. ll W X 1. Global shape modifier Γ1.5. Eq. (4 determies the relatioship betwee weight ad shape variatios. I particular, let be the cetral vertex of a 3D polygoal patch Patch with +1 cotrol poits, ad α a cotrol triagle cotaied i Patch β. ccordig to (1 to (4, cotrol poit exerts its maximum ifluece upo its correspodig surface patch whe the evaluatio poit P coicides with. I that case,. The, (4 ca be rewritte as α β i If α β i, W i i i If α β i, W i (5 ccordig to the defiitio of If give i (1, see Fig. 2, two differet iflueces are cosidered i (5: (i the ifluece exerted by the cetral vertex, If If, ad (ii the iflueces of the α β, cotrol poits adjacet to, If N If, α β i, i. Therefore, (5 may be rewritte as If + If N W i i i 1 (6 osiderig the defiitio of If ( X, P, If is the maximum of the bell-shaped fuctio Γ ( d defied i (1, ad correspods to a costat idepedet of the actual value of Γ: If Γ (.75. O the other had, If N Γ ( 1, cosiderig that the polar represetatios of all polygoal patches are regular polygos whose circumscircles have uitary radius. I this secod case, though, that ifluece depeds o the value of the global shape modifier Γ. Takig this ito accout ad cosiderig that all weightig factors are equal,... W W, (6 may be rewritte as If + If N W i i 1 Γ Figure 5: Effect upo the surface shape of the variatio of the global parameter Γ. cotrol poit with 4, 8, 12 ad 16 adjacet cotrol poits is cosidered. ( Γ ( Γ + α N ( Γ i α.75 + Γ ( 1 N ( Γ i 1 (7 which satisfies ( Γ + α N ( Γ 1. Eq. (7 expresses the way Γ affects the shape of the surface approximat. Sice Γ ( x.75 for ay x ad Γ, S ca theoretically rage betwee, whe Γ ( 1, ad whe, depedig o Γ. 1 + i Γ ( 1.75 i, I the first case, which correspods to Γ 1, the cotrol mesh is iterpolated although the resultig surface loses its smooth appearace. The secod case correspods to Γ ad leads to a fial degeerated surface. I practice, appropriate values for Γ rage betwee 1.5 ad 2.. Fig. 5. plots the fuctio ( Γ for Γ ragig betwee 1. ad 2. ad differet umber of adjacet cotrol poits. ( Γ reflects the shape of the approximatig surface sice it determies the relative ifluece of cotrol poit i the surface subpatch ( P with respect to its eighborig cotrol poits. Whe ( Γ 1, the surface shape is etirely determied by. I that case, the fial surface iterpolates. O the other had, ( Γ implies that the approximatig surface does ot deped o, oly o the weighted average of its eighborig cotrol poits. The graph i Fig. 5 shows that, for large values of Γ, the model favors the geeratio of a cosesus surface that teds to adapt to the shape defied by groups of cotrol poits rather tha idividuals. Hece, Γ may be used to represet the average ucertaity associated with the positios of all the cotrol poits of the mesh. Whe the ucertaity is low, the approximatig surface will reflect Γ ( Γ ( 1

5 Figure 6: Local shape modificatio of the cotrol mesh show i Fig. 3. ll W X 1 except for the cetral peak which is 25. Global shape modifier Γ1.25. the positios of the cotrol poits, which are cosidered to be accurate. Whe the ucertaity is high, the surface will reflect the overall shape defied by sets of poits. I this case, isolated poits that stray from the domiat tred will be filtered. Eq. (7 establishes the relatioship betwee surface shape ad Γ. lthough Γ may be directly utilized as a ucertaity variable, i some cases it may be desirable to defie other types of relatioships betwee shape variatio ad ucertaity. This issue will be discussed i Sectio Embeddig local ucertaity The weightig factors W i i (4 behave as local shape modifiers. The approximatig surface teds to pass closer to those cotrol poits with larger weights. For istace, Fig. 6 shows the approximatio of the same cotrol mesh used i Fig. 3, but givig the cetral peak a weightig factor equal to 25, four times lower that the other cotrol poits. The surface shape is oly altered i the area defied by the cotrol triagles surroudig the modified vertex. These local shape modifiers act by icreasig or decreasig the attractio forces associated with idividual vertices. Therefore, they ca be used to represet local ucertaity associated with the mesh cotrol poits. Their behavior is aalyzed below. s i the precedet sectio, let be the cetral vertex of a 3D polygoal patch Patch ad α β a cotrol triagle cotaied i Patch. Followig the same reasoig above, the value of the approximatig surface that correspods to cotrol poit is α β. The effect of eighborig cotrol poits ca be homogeized by doig W i W N for i i (4 to (6. Thus, (6 ca be rewritte as Figure 7: Effect of weight variatio upo the surface shape. is the weightig factor associated with. W N is the weightig factor associated with the eighbors of. + α N i α N / W N i 1 (8 which satisfies + α N 1 ad lim 1. α expresses the way W affects the shape of the approximat ear. I fact, will rage betwee ( 1 i 1, i whe, ad whe 1, as a fuctio of. Fig. 7 plots the fuctio ( / W N assumig that If Γ (.75, If N Γ ( 1.45, W N 1 ad several values of. These values of If ad If N correspod to a fuctio Γ ( d with Γ1.25, Fig. 2(right. Whe is equal to oe, the surface iterpolates cotrol poit. This would oly occur i the limit, whe. O the other had, a value of equal to zero meas that does ot exert ay ifluece upo the resultig surface. I that case, the surface would etirely deped o the cotrol poits adjacet to. ccordig to this formulatio, the variatio of a weighig factor produces a o-liear variatio of the shape associated with its correspodig cotrol poit. s show i Fig. 7, whe the weightig factor of a cotrol poit is large (low ucertaity with respect to its eighbors, the surface shape will highlight the positio of this poit. lteratively, whe that weightig factor ratio is low (high ucertaity, the surface shape will ted to igore that poit, approachig the 3D positio obtaied by averagig its eighbors. If W If + If N W N If N W N If + If N W N

6 I order to use those weightig factors as a meas to represet local ucertaity, it may be desirable to defie relatioships amog ucertaity ad shape variatio differet to the o-liear fuctio determied by (8. This issue is described below. 3.3 ustomized ucertaity hadlig Eq. (7 ad Eq. (8 express the relatioships betwee the shape of the surface approximat ear the cotrol poits of a give triagular mesh ad the global ad local shape modifiers defied i the model that ca be used to represet global ad local ucertaity respectively. I some cases, though, it may be preferable to express this relatioship as a user-defied fuctio differet from the oe defied by the surface model. If the relatioship betwee surface shape ad a certai parameter x is give by a o-liear fuctio α f ( x, such as (7 or (8, it is possible to obtai a liear shape variatio with respect to a secod parameter y by defiig 1 1 x as x f ( y, sice i that case, α f ( f ( y y. I this way, the shape variatio associated with a cotrol poit is a liear fuctio of y. I tur, if y is a fuctio of a third parameter u (for ucertaity, y g ( u, the the shape variatio will be fially determied by that userdefied fuctio g, α g ( u. Thus, it is possible to relate ucertaities to shape variatios i a customized way. osiderig the represetatio of global ucertaity i Sectio 3.1, the previous priciple may be applied as follows. For 1 < Γ 3, Γ ( 1 from (1 takes the form Γ ( 1 9. The, i (7 is α ( Γ Γ 2Γ 2 ( Γ ( Γ 2 6Γ + 3 (9 If ( Γ i (9 is substituted for a parameter y, Γ ca be obtaied by solvig the quadratic equatio ( u ( Γ 2 6uΓ + 3u. If Γ is defied i this way, ( y i (7 is a liear fuctio of y. Similarly, if i (8 is substituted for a parameter y, the local weight of a cotrol poit ca be obtaied as (1 With defied i this way, ( y i (8 is a liear fuctio of y. I both cases, y ca be defied as a arbitrary fuctio, g, of a ucertaity variable u. 4 oclusio y If N W N ( 1 y If This paper presets the applicatio of a previously developed free-form surface model to the represetatio of geometric iformatio that is subject to ucertaity. That model allows the approximatio of triagular meshes of 3D cotrol poits with smooth surfaces. Global ad local ucertaity associated with the cotrol poits ca be directly embedded ito the recostructio process through shape modifiers defied i the model. The behavior of those parameters has bee studied, as well as the way to cotrol the fial surface shape through user-defied ucertaity fuctios. pedig problem of iterest is the itegratio of measuremets that are subject to ucertaity with data cosidered to be reliable. I order to do that, the proposed surface model must be complemeted with iterpolatio capabilities. 5 Refereces [1] N. yache ad O. D. Faugeras, Maitaiig represetatios of the eviromet of a mobile robot. IEEE Tras. o Robotics ad utomatio, vol. 5, o. 6, pp , December [2] J. L. rowley, World modelig ad positio estimatio for a mobile robot usig ultrasoic ragig, IEEE It. of. o Robotics ad utomatio, pp , [3] M.. García, Smooth approximatio of irregular triagular meshes with G1 parametric surface patches. It. of. o omputatioal Graphics ad Visualizatio Techiques, lvor, Portugal, pp , December [4] M.. García, Efficiet surface recostructio for scattered poits through geometric data fusio. IEEE It. of. o Multisesor Fusio ad Itegratio for Itelliget Systems, Las Vegas, US, pp , [5] M.. García, Recostructio of visual surfaces from sparse data usig parametric triagular approximats. IEEE It. of. o Image Processig, usti, US, pp , [6] M.. García, Fast approximatio of rage images by triagular meshes geerated through adaptive radomized samplig. IEEE It. of. o Robotics ad utomatio, Nagoya, Japa, 6 pages, [7] M.. García, hierarchical world model represetatio supportig heterogeeous multisesory itegratio. ccepted for publicatio at It. of. o dvaced Robotics, Sat Feliu de Guíxols, Spai, 11 pages, September 2-22, [8] S.. Hutchiso ad.. Kak, Extedig the classical I paradigm to robotic assembly plaig. IEEE It. of. o Robotics ad utomatio, pp , 199. [9] S. Kweo ad T. Kaade, High resolutio terrai map from multiple sesor data. IEEE Tras. o Patter alysis ad Machie Itelligece, vol. 14, o. 2, pp , February [1] J. J. Leoard, H. F. Durrat-Whyte ad I. J. ox, Dyamic map buildig for a autoomous mobile robot. The Iteratioal Joural of Robotics Research, vol. 11, o. 4, pp , ugust [11] Y. Nakamura ad Y. Xu, Geometrical fusio method for multisesor robotic systems. IEEE It. of. o Robotics ad utomatio, pp , [12] F. Nashashibi ad M. Devy, 3D icremetal modelig ad robot localizatio i a structured eviromet usig a laser rage fider. IEEE It. of. o Robotics ad utomatio, pp. 2-27, 1993, [13] Z. Zhag ad O. Faugeras, 3D world model builder with a mobile robot. The Iteratioal Joural of Robotics Research, vol. 11, o. 4, pp , ugust 1992.