Smooth Approximation to Surface Meshes of Arbitrary Topology with Locally Blended Radial Basis Functions

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1 587 Smooth Approxmaton to Surface eshes of Arbtrary Topology wth Locally Blended Radal Bass Functons ngyong Pang 1,, Weyn a 1, Zhgeng Pan and Fuyan Zhang 1 Cty Unversty of Hong Kong, mewma@ctyu.edu.hk Nanng Unversty, panon@graphcs.nu.edu.cn Zheang Unversty, zgpan@cad.zu.edu.cn ABSTRACT In ths paper, we present a new approach for smooth approxmaton of surface meshes wth arbtrary topology and geometry. The approach s based on the well-known radal bass functons (RBFs) for local shape approxmaton combned wth a blendng operator and a famly of normalzed weght functons for global surface constructon. Our method frst defnes a local approxmaton usng locally supported RBF for every vertex of the nput mesh. A sngle global smooth surface s then constructed by blendng the local approxmatons usng weght functons assocated wth mesh vertces. A proecton procedure s employed for vsualzng the global surface usng the local parameterzaton defned by barycentrc coordnates for each facet of the nput mesh. The approach provdes a robust and effcent soluton for smooth surface constructon from varous D mesh models. Keywords: surface reconstructon, RBF approxmaton, blendng, doman decomposton. 1. INTRODUCTION Polygonal meshes are wdely used n the CG (computer graphcs) and CAGD (computer aded geometrc desgn) communtes for the representaton of D shapes due to ther smplcty of data structure and flexblty n representng varous shapes wth arbtrary topology and geometry. Vertces of a mesh are n most cases sampled from a smooth surface of certan obect and the mesh provdes a dscrete approxmaton of the orgnal surface. In many applcatons, t s most mportant to have a smooth surface representaton of the underlyng obect rather than ust wth a surface mesh and how to construct a smooth surface from a gven mesh s thus a very mportant feld of study n CG and CAGD. In general, there are two classes of technques for the constructon of a contnuous surface or functon from a gven D mesh,.e. parametrc or mplct. The later uses mplct surfaces to approxmate meshes by buldng a famly of mplct real-valued scalar functons and the constructed surfaces are defned as ts zero level-sets. In lterature, one may fnd varous related work n ths area. urak[1] ftted an mplct surface from a gven pont-set usng a lnear combnaton of Gaussan blobs. Hoppe et al[] used a dstance functon to represent approxmaton of a pont-cloud. Lm et al[] dstrbuted a set of spheres n the scattered pont set by employng a Delaunay partton and a process of non-lnear optmzaton, and then used blended-unon of the spheres to construct an mplct surface from the pont set. The latest trend of mplct surface constructon s related to RBF based technques assocated wth movng least squares and level-set method nterpolatng or fttng large scattered pont-sets. Carr et al[4] constructed cranum surface from CT data by RBF nterpolatng method. Yngve et al[5] focused ther work on how to smplfy the data-set and makes t manageable n calculaton of surface constructon. Wendland[6] combned compactly supported RBF nterpolaton wth partton of unty for constructng surfaces from large scale pont-sets. orse et al[7] gave a process n usng the Wendland's method to construct a surface from a larger data set. In recent studes of surface reconstructon of unorganzed pont-set, researchers focused ther attenton on a so-called doman decomposton method[8]. The method frst dvdes the data-set nto several tractable segments and then handles them respectvely. Ohtake et al[9] used the partton of unty approach, a specal doman decomposton, and presents an algorthm to construct surface models from large sets of ponts. Tobor et al[1] presented another approach smlar to Ohake's, but they used RBFs as bass functons n local approxmatng. At the same tme, they used dfferent blendng weght functons for constructng a global surface. In [11], Ohtake et al further proposed a herarchcal approach to D scattered data nterpolaton and fttng wth compactly supported RBF.

2 588 In ths paper, we represent an RBF-based method that s capable of constructng a globally smooth approxmatng mplct surface from a gven surface mesh. The method s bult upon the dea of doman decomposton and makes use of the connectvty of the nput mesh. It frst constructs a local RBF approxmaton for each vertex of the ntal mesh, and then blends the local approxmatons to yeld a smooth global surface. A parameterzaton-based process s employed to vsualze the fnal surface. In the rest of the paper, Sec. provdes a theoretcal background of our method. In Sec., we present the basc dea and steps of the proposed algorthm and dscuss the vsualzaton of the constructed surface. Sec. 4 gves some expermental results produced wth our method followed by conclusons of our work n Sec. 5.. RBF-BASED SURFACE RECONSTRUCTION An mplct surface s defned as the underlyng surface that satsfes the governng equaton f ( v) =, for v R, where f s a contnuous mplct functon defned n R. Implct surface f ( v) = parttons the space nto two N halves,.e. f ( v) > and f ( v) <. For a gven mesh, suppose ts vertces V = { v = (x, y,z)} = sampled from a smooth surface S. The obectve of surface constructon from s to fnd an mplct functon f such as ts zero level-set f ( v) = approxmates S n a reasonable manner wth N = f ( v ) mn. (1) The RBF approach has proven to be very useful n shape modelng. It s bult on strct mathematcal theory and the resultng surface s a generalzed thn-plate splne nterpolatng the scattered data[7]. For a data-set V acqured by a D range scannng devce, the ponts n V should be equpped wth unt normals that ndcate the surface orentaton n order to perform RBF calculaton. Otherwse, these normals can be estmated ether from the ntal scans durng the shape acquston phase or through local least-square fttng to V. For a gven mesh, they can be drectly evaluated from the connectvty of. In order to avod trval soluton wth whch the constant functon f ( v) s acheved n the entre space of R, some off-surface ponts { v } N+ 1, named offset ponts, should be added when RBF method s used. At the same tme, value of f at each offset pont v, denoted by d, should also be estmated. In ths case, the problem of approxmatng pont-set descrbed n Equ. (1) becomes a problem n fndng a smooth mplct functon f, such as N = f ( f ( ) -d) ( v ) + v mn. () = N+ 1 The smoothness of f above can be measured by mnmzng the followng energy functon [7, 1]: f f f f f f E( f ) = dxdydz. () x y z x y x z y z The problem of mnmzng energy n Equ. () can be solved by RBF method, ts analytc soluton can be represented as [1] f ( v) = w φ ( v v ) + P( v) (4) where, w are real value weghts and φ s a RBF functon φ : R [, ), P (v) s a polynomal of degree one to represent lnear and constant parts of f, whch ensures that the resultng surface s affne nvarable wth respect to the nput data-set. In Equ. (4), the unknown weghts w and coeffcents p k of P (v) can be unquely determned by the followng nterpolatng condtons [1] f ( v ) = w φ ( v v ) + P( v ), =,1,, (5) and the followng orthogonal condtons [1] w = wx = w y = wz =. (6)

3 589 Let φ =φ( v v ), P( ) = p + p1x+ py+ pz f= f v, for =,1,,, then t s easy to derve the followng lnear system from Equ. (5) and Equ. (6): Φ C W F =, (7) T C P where, Φ = ) F, and (φ, = ( f ) 1 v and ( ) 1 x y z w p 1 x1 y1 z1 C=, w1 W =, p1 P =. (8) p 1 x y z w p Snce the coeffcent matrx of Equ. (7) s symmetrc and sem-postve, the system has unque soluton and can be solved usng LU-decomposton. Dependng on types of applcaton, we can select the RBF functon as thn-plate splne φ ( d ) =d log( d), Gaussan functon[4] φ ( d ) = exp(-cd ), mult-quadrc functon φ ( d ) = d + c [1], tr- 4 harmonc φ ( d ) = d, or Wendland's compactly support RBF functon [6] φ ( d) = (1 d ) + (4d+ 1), and so on.. PRACTICAL ALGORITHS FOR SOOTH SURFACE APPROXIATION In ths secton, we dscuss how to construct RBF mplct approxmaton from a trangle mesh wth arbtrary topology. Snce an arbtrary polygonal mesh can be transformed nto a trangle mesh through local trangulaton of the facets, the algorthm dscussed here has also been appled to general polygonal meshes. The man algorthm conssts of two steps: a) constructng a local nterpolaton at each vertex of the nput mesh; and b) blendng local nterpolatons to a global smooth surface usng locally-supported weght functons. The man dfference between our algorthm and other RBF based surface reconstructon from pont-set data s twofolds. Our method makes use of the connectvty nformaton of a mesh and elmnates an expensve process of computng octree-based partton of space commonly used for the constructon of global mplct surface [9]. In addton, we employ a parameterzaton-based proecton procedure to vsualze the global surface..1 Constructon of Local RBF Approxmatons K K Let v be an arbtrary vertex of a gven trangle mesh, { v } = 1 be the set of vertces ncdent to v, and { F } = 1 be the set of facets ncdent to v. An umbrella structure, denoted by U v ), s defned as v } { v } { F} (see Fg. 1), and v s called the center of the umbrella. ( { Fg. 1. An umbrella structure centered at v for local approxmaton. For an umbrella U ( v ) centered at v, the normal vector at vertex v can be estmated as: K 1 Nv = [ A( F ) N( F K )] A( F ) = = 1 1 ( F where, A ( F ) and N ) are area and normal of facet F, respectvely. As a result, we can obtan unt normal of every vertex v n mesh by formula n v = Nv / Nv. (9)

4 59 In order to evaluate offset ponts necessary for local RBF approxmaton, the longest and shortest lengths of edges ncdent to a vertex, say v, can be respectvely expressed by maxlen( v ) and mnlen( v ) as: K maxlen( v) = max{ v v } = 1 (1) K mnlen( v) = mn{ v v }. = 1 The offset ponts of v can then be defned as v v aux + aux = v = v + [ n [ n v v mnlen( v mnlen( v ).1] ).1]. The functonal value correspondng to the offset ponts are defned as f ( ± ) =± mnlen( v ).1. Here, n order to ensure that the offset ponts s suffcently close to v whle mantanng proper functonalty of the auxlary ponts and robustness of local approxmaton, we poston the auxlary ponts on the surface normal n v of v and away from the surface wth a dstance of one-tenth of the shortest edge length mnlen( v ) of v. In addton, two aux symmetrc offset ponts v ± are selected at each sde of the underlyng surface at v, whch s reasonable f the underlyng feld defned by the mplct functon n D space s smooth and contnuous. The selecton of a dfferent functonal value would certanly affect the orthogonal gradent dstrbuton of the underlyng feld. However, as two symmetrc auxlary ponts are used wth respect to the underlyng surface, the selecton would not be so senstve to the fnal constructed mplct surface. As we are only nterested n the underlyng surface, but not the feld tself, the proposed auxlary ponts should be approprate. For an umbrella centered at v, we select dscrete pont-set to construct ts local approxmaton as aux aux aux { v, v1,, vk } { v±, v1±,, vk± }. Here, we notce f ( v ) =, =,, K. Accordng to Equ. (7), the coeffcents of local RBF mplct approxmaton can be worked out easly and we denote the local surface by L v : f ( v) = v. For a trangle mesh, the valence of an nner vertex s at least. In ths case, there are at least 1 ponts used to construct a local surface. For most vertces of the mesh, the average valence should approxmately be 6,.e. there are about 1 ponts used to construct the local approxmaton of the umbrella. Ths guarantees the stablty of local constructon. In ths paper, we select the tr-harmonc, φ ( d ) = d, as RBF functon n local surface approxmaton and Fg. shows an example for local approxmaton of an umbrella. v aux (11) Fg.. Local RBF approxmaton (rght, green) of an umbrella n a cube mesh (left).. Blendng for Global Surface Constructon We develop a blendng operator, whch s smlar to that of [1] for other functons, to ntegrate local approxmatons nto a global surface. The basc requrements are shape local control and smooth blendng. At the neghborhood of v, the shape of the global surface should bascally be determned by the postons of vertces n U ( v ). Furthermore, the further a vertex n the mesh away from the center of U ( v ), the weaker t affects the shape of the approxmaton of the umbrella. When the dstance from a vertex to center of U ( v ) s larger than certan threshold, the vertex won't brng any effect on the shape at all.

5 591 Let us consder a bounded doman Ω n R and a set of local non-negatve compactly supported weght functons { w ( v ) =,, N} wth Ω supp( w ). These weght functons can always be normalzed such that the sum of all weghts equals to 1 at all postons of Ω,.e., the partton of unty property commonly requred for actng as bass functons[1]. Let { ϕ } denote the normalzed weght functons of { w ( v)} defned on sub-domans of Ω, we have w ( v) ϕ ( v) =, =,1,, N. (1) N = w ( v) 1 oreover, let us consder the case that the supported center of w s located at vertex v n the ntal mesh. The normalzed weght functons { ϕ } wll be used as the blendng functon for defnng a global approxmaton from local approxmatons assocated to ndvdual vertces of the ntal control mesh. The global approxmaton on Ω can be defned as follows [9]: f ( v) = ϕ ( v) f ( v). (1) v In the paper, for approxmaton purposes, we use locally supported weght functons on sphere doman n blendng weghts, whch are defned by quadratc B-splne B (t) : v v w ( v ) = B. (14) maxlen( v) If an nterpolaton s requred, we can use the nverse-dstance sngular weghts: where w ( a) R as (maxlen( v ) - ) v v + ( v ) = (15) maxlen( v) v v + a, fa> =, otherwse (16) Fg.. Blendng of local RBF approxmatons: A surface patch correspondng to the shaded facet on global approxmaton s defned as a blendng surface of a set of local RBF approxmatons As mentoned above, for each vertex v of mesh, ts support functon s defned n the form of Equ.(14) wth the center and radus of the supported doman beng v and maxlen( v ), respectvely. In ths case, the unon of all supported domans of vertces of s exactly Ω that covers the entre mesh. As an example, we dscuss the representaton of Equ. (1) n the vcnty of an arbtrary facet Δ vv1v of mesh. Let us frst consder the representaton at arbtrary vertex v nearby Δ vv1v. We note that, only those local approxmatons, whose support doman covers the vertex, can affect the value of f (v). In other words, f and only f the dstance from the center of an local approxmaton L v to v s less than the control radus of L k v,.e., k maxlen( v k ), the approxmaton can brng an effect on f (v). So, Equ. (1) becomes f ( v) = ϕ ( v) f ( ) (17) v v < maxlen( v ) v v

6 59 where, the weght functon ϕ can be calculated by Equ. (14) and Equ. (1). On the other hand, on a global surface generated by blendng local approxmatons, each facet of has a correspondng patch on the global surface and the complete global approxmaton s ust a unon of all ndvdual patches correspondng to the parametrc doman of ndvdual facets of. Based on Equ. (17), we can fnd all local approxmatons for defnng each patch. In Fg., trangle Δ vv1v s a facet of and ts correspondng surface patch on global approxmaton s determned by the local approxmatons of L v, L v, L 1 v, L v, L v, and so on. These local approxmatons can be found by the 4 connectvty of or an optmzed k d -tree research strategy.. Global Surface Vsualzaton Global approxmaton of the nput mesh s an mplct surface represented by a sngle and complex mplct functon. At present, there are two types of algorthms to render mplct surfaces. Ray-tracng based method s only applcable to off-lne renderng because t nvolved heavy computaton, whereas, polygonzaton method mplements rapd vsualzaton of mplct surfaces by convertng them nto polygonal meshes. Among varous methods reported so far, the optmzed archng Cubes (C) based algorthms can provde rapd and effcent polygonzaton and archng Trangles based method can gve hgh qualty of trangulaton. Velho [14] presented a smple and effcent algorthm through a physcal process of smulatng partcle movement. Jn et al [15] combnes subdvson and physcal smulaton process to render the so-called subdvson nterpolatng mplct surfaces. Based on Velho's and Jn's work, we propose a pecewse renderng method of the global approxmaton. For a trangular facet Δ vv1v of mesh, we frst partton the planar doman Δ vv1v nto n sub-trangles as llustrated n Fg.4 by unformly subdvdng the edges nto n segments. For each vertex of the sub-trangles, the correspondng vertex on the underlyng mplct surface are then computed usng a mappng or proecton procedure. A refned surface mesh of the constructed mplct surface s fnally produced usng the newly computed vertces on the underlyng mplct surface wth the same topologcal connecton as that of the sub-trangles for fnal vsualzaton. In ths paper, the vertces of sub-trangles are named subdvson-ponts and all the sub-trangles form a subdvson net. In [14] and [15], the proecton or mappng of the subdvson-ponts onto the underlyng surface were drven by smulatng partcles' free movement along gradent descendng drecton of the surface functon. In the paper, however, we use the normals attached to each vertex of the nput mesh to fnd the proected ponts of the subdvson-ponts onto the global approxmaton surface. Ths approach avods an expensve procedure for computng the gradents of mplct surfaces. Fg. 4. Patch-wse renderng of mplct surface: 1) a facet of the nput mesh after doman partton wth corner surface normals and estmated surface normal at an nteror pont (left); ) the parametrc doman wth correspondng surface patch on the underlyng surface for vsualzaton (rght). For any subdvson pont v of facet Δ vv1v, t s easy to evaluate ts barycenter coordnates, ( c, c1, c), about ponts v, v 1 and v, the vertces of the facet. We defne the normal assocated to v as cnv + c 1nv + c 1 nv nv =. (18) cn + cn + cn v 1 v1 v Followng Sec..1, we know that n v ponts outwards of the surface. If a pont v s not on the surface, we can fnd a pont, at whch the sgn of the mplct functon s opposte to that of v, along ether the postve or negatve drecton

7 59 of n v. Once two ponts at dfferent sdes of the surface are found, an ntersect pont, named n, of the lne segment p connectng the two ponts and the mplct surface can be obtaned usng a bsecton search method. The ntersect n wll be taken as the proected pont of v on the surface for vsualzaton. Obvously, the resoluton of polygonzaton of the global approxmaton can be adusted by ncreasng or decreasng the number of subdvsons on edges of the facet. As the normals of a mesh are predefned, the normal vector at each subdvson-pont s also unquely defned. The poston of the ntersect pont on the surface can then be represented by a sngle pure scalar, whch s the dstance from v to the ntersecton pont. The dstance s postve when the drecton from v to ntersecton pont s concdent to the drecton of the normal of v, otherwse negatve. One should notce that, two neghborng facets, Δ vv1v and Δ vv1v, have common global functon along ther shared edge v v 1, whch s the blended sum of the same local approxmatons weghted by the same weght functons, so that global surface s contnuous along the shared edge. In addton, because the normals of subdvson ponts on a common edge have dentcal drecton n Δ vv1v and Δ vv1v respectvely (subdvson ponts on v v 1 have the same barycenter coordnates n Δ vv1v and Δ vv1v ), ther proected ponts should also be the same. Hence, there should be no crack left on the global surface. It should also be noted that doman subdvson dscussed n ths secton and the estmaton of local surface normal usng equaton Equ. (18) would only affect the vsualzaton of the underlyng surface. The underlyng mplct surface, however, remans the same. The underlyng mplct surface s unquely defned by the local RBF approxmatons and the blendng functons used for constructon the global approxmaton. 4. IPLEENTATION, RESULTS AND DISCUSSIONS We have mplemented the algorthm presented n the prevous secton usng C++ on an Intel Celeron PC wth 1.4GHz CPU and 51 memory under the WnXP operatng system. The algorthm conssts of two parts as followng: 1) Constructon of local approxmatons: For every vertex v of the nput mesh, compute ts n v, maxlen( v ) and mnlen( v ); Defne two offset ponts for v usng n v and mnlen( v ); Construct the local approxmatng surface for v from vertces n U ( v ) and ther offset ponts. ) Vsualzaton of the global approxmaton by renderng the pecewse surface patch for each facet, Δ vv1v, of the nput mesh: Dvde Δ vv1v nto n sub-trangles; Evaluate: a) all the subdvson ponts, ther barycenters and normals about the vertces v, v 1 and v ; and b) proected ponts of the subdvson ponts on the global surface; Replace the vertces of subdvson net on the facet wth the correspondng proected ponts and render the net. Fg. 5 to Fg. 7 llustrate some examples produced usng the algorthms presented n ths paper. Fg. 5 shows fve smple examples. Illustratons of top row are meshes to be approxmated and llustratons of the mddle and bottom rows are flat shadng wth wre-frame and Phong shadng renderngs of ther global surfaces, respectvely. The ntal control mesh of the frst example (the frst column on left hand sde) s a cube, where the valance of the center vertex of each facet s 4 and ts neghborng facets are coplanar. Wth the second example (the second column), the vertex at the ntersecton pont of three planes has complex local geometry. In the thrd example (mddle column), the ntal control mesh s an open mesh consstng of a convex peak and a concave valley. The vertex between the peak and the valley vertces also has complex local structure. The surface meshes n two columns on rght hand sde take on complex topologes. The results show that all the above shapes can be handled wth the presented algorthm. p

8 594 Fg. 5. Illustratons of the constructed surfaces from fve dfferent nput meshes (top row) wth partcular vertces or topology and ther approxmatng surfaces (mddle and bottom rows). For a gven mesh, because the hghest valance of ts vertces s a constant, the computng tme needed for the constructon of a local RBF approxmaton of one umbrella has ts upper lmt, whch s ndependent of the number of vertces of the nput mesh. Hence, the computng tme for the constructon of local approxmatons has lnear complexty, O (N ), wth respect to number N of vertces of the nput mesh. In the step of vsualzaton, the tme used for renderng a sngle patch correspondng to a partcular facet s O ( n ), where n s the number of subdvsons for each edge, whch s agan ndependent to the number N of the vertces of the nput mesh. As a result, the total tme complexty for the vsualzaton of the global approxmaton s O ( N) O( n ). When n s decded, whch s usually not large, the tme needed n the vsualzaton s agan lnear to the number N of vertces of the nput mesh. Fg. 6 and Fg. 7 present two examples, respectvely, constructed usng the proposed method. The ntal meshes of former are created by C based samplng procedure from exstng smooth surfaces and the control meshes of later are bult by mesh smplfer from the dense cat and horse models, whch are commonly used n the CG and CAGD communtes. These examples demonstrate the capablty of the developed algorthms. Tab. 1 summarzes some parameters of the ntal control meshes shown n the fgures and t also provdes the computng tme used to construct the local RBF approxmatons wth respect to meshes of dfferent resoluton scales. Tab. shows tme needed on the second phase for the vsualzaton of the global approxmaton of varous meshes, where N -segs ndcates the number of subdvsons for each edge for vsualzaton. From the tables, one can fnd that t s very fast for constructng the local RBF approxmatons for the vertces of the nput meshes and the man computng tme of our algorthm s spent on vsualzaton of the resultng surface. The tme costs are proportonal to the resoluton scale of the fnal meshes n the scale of O ( n ), at the same tme, t s lnearly proportonal to the number N of vertces of the ntal nput mesh.

9 595 (a) Jack-1 (b) Jack- (c) Jack- (c) Ddymous-1 (c) Ddymous- (c) Ddymous- Fg. 6. Surface constructon of two practcal examples generated usng varous resoluton control meshes. The ntal meshes (top row) are generated usng C method from smooth surfaces. (a) Cat-1 (b) Cat- (c) Cat- (d) Horse-1 (e) Horse- (f) Horse- Fg. 7. Surface constructon of two practcal examples commonly used n the CAGD and graphcs communtes, the control meshes (top row) are obtaned from ntal dense meshes through mesh smplfcaton. esh Jack-1 Jack- Jack- Ddymous-1 Ddymous- Ddymous- V/T 89/ /6 /66 6/416 97/ /998 Tme esh Cat-1 Cat- Cat- Horse-1 Horse- Horse- V/T 51/9 145/7 65/ /88 44/ /97 Tme Tab. 1. Parameters of the meshes shown n Fg. 6 and Fg. 7 and tme consumed n constructng ther local RBF approxmatons (V/T: vertces/trangles; Tme: ms). esh Ddymous-1 Ddymous- Ddymous- Horse-1 Horse- Horse- -segs segs segs segs Tab.. Computng tme n vsualzaton phase of the algorthm (Tme unt: ms).

10 CONCLUSION In ths paper, we propose an algorthm for constructng an approxmatng surface from a trangle mesh wth arbtrary topology and geometry usng compactly supported radal bass functons. Our algorthm frst constructs a local RBF approxmaton for each umbrella structure centered at an ndvdual vertex of the nput mesh. The local approxmatons are then blended to form a global approxmaton of the underlyng geometry usng locally defned weght functons. The vsualzaton makes use of the correspondence, a knd of parameterzaton, between facets of a mesh and ther correspondng patches on the resultng surface. Results show that our algorthm can cope wth open or closed mesh wth arbtrary topology n the same way. Both the constructon process and the vsualzaton procedure are pretty fast. Compared wth exsted algorthms for constructng approxmatng surface from meshes, our algorthm has lnear tme complexty to the number of vertces of the nput meshes. 6. ACKNOWLEDGENT The work presented n ths paper s supported by the Research Grants Councl of Hong Kong SAR through research grant #CtyU 111/E. 7. REFERENCES [1] urak, S., Volumetrc shape descrpton of range data usng blobby model, Computer Graphcs, Vol. 5, No. 4, 1991, pp 7 5. [] Hoppe, H., DeRose, T., and Cuchamp, T., Surface reconstructon from unorganzed ponts, n Proc. of AC SIGGRAPH 9, 199, pp [] Lm, C. T., Turkyyah, G.., Ganter,. A. and Stort, D. W., Implct reconstructon of solds from cloud pont sets, n AC Symposum on Sold odelng and Applcatons. AC Press, New York, USA, [4] Carr, J. C., Beatson, R. K., Cherre, J. B., tchell, T. J., et al, Reconstructon and representaton of d obects wth radal bass functons, n AC SIGGRAPH Conference on Computer Graphcs, Los Angeles, CA, August 1, pp [5] Yngve, G. and Turk, G., Creatng smooth mplct surfaces from polygonal meshes, Graphcs, Vsualzaton, and Useablty Center. Georga Insttute of Technology, Tech. Rep. GIT-GVU-99-4, [6] Wendland, H., Fast evaluaton of radal bass functon: ethods based on partton of unty, n Approxmaton Theory X: Wavelets, Splnes, and Applcatons, Chu C. K., Schumaker L. L. and Stockler J., Eds. Nashvlle TN.: Vanderblt Unversty Press,, pp [7] orse, B. S., Yoo, T. S., Rhengans, P., et al, Interpolatng mplct surface from scattered surface data usng compactly supported radal bass functons, n Proc. of Shape odelng Internatonal, 1, pp [8] Babuska, I., The partton of unty method, Internatonal Journal of Numercal ethods n Engnererng, No. 4, 1997, pp [9] Ohtake, Y., Belyaev, A., Alexa,., Turk G. and Sedel H.-P., ult-level partton of unty mplcts, AC Transactons on Graphcs, Vol., No.,, pp [1] Tobor, I., Reuter, P. and Schlck, C., Effcent reconstructon of large scattered geometrc datasets usng the partton of unty and radal bass functons, Laboratore Bordelas de Recherche en Informatque, Unverste Bordeaux-1, Tech. Rep. RR-11-,. [11] Ohtake, Y., Belyaev, A. and Sedel, H.-P., d scattered data nterpolaton and approxmaton wth multlevel compactly supported rbfs, Graphcal odels, Vol. 67, No., 5, pp [1] Turk, O. G., odellng wth mplct surfaces that nterpolate, AC Transactons on Graphcs, Vol. 1, No. 4,, pp [1] Chaturved, A. K. and Pegl, L. A., Procedural method for terran surface nterpolaton. Computer & Graphcs, Vol., No. 4, 1996, pp [14] Velho, L., Smple and effcent polygonzaton of mplct surfaces, Journal of Graphcs Tools, Vol. 1, No., 1996, pp 5 4. [15] Jn, X., Sun, H. and Peng, Q., Subdvson nterpolatng mplct surfaces, Computer & Graphcs, No. 7,, pp