A Hybrid System for Three-Dimensional Objects Reconstruction from Point-Clouds Based on Ball Pivoting Algorithm and Radial Basis Functions

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1 A Hybrd System for Three-Dmensonal Objects Reconstructon from Pont-Clouds Based on Ball Pvotng Algorthm and Radal Bass Functons ASHRAF HUSSEIN Scentfc Computng Department, Faculty of Computer and Informaton Scences, An Shams Unversty, Abbassa, 11566, Caro, Egypt. Abstract:- In ths paper, a new hybrd system for 3D objects reconstructon s presented. Ths system s based on the well-known Ball Pvotng Algorthm (BPA) of Bernardn et al. combned wth the Radal Bass Functon (RBF) model. The BPA can reconstruct surfaces from very large data sets n a lttle executon tme, usng small amount of memory, and exhbts lnear complexty but t produces holes n case of low samplng densty besdes t s not capable to fll ms-regstraton domans. So, the BPA s coupled wth the RBF n an ntegrated system to gan the edge of the RBF as a powerful nterpolator for generatng surface vertces wthn severe holes and/or ms-regstraton domans. Performance and accuracy of the system components are nvestgated through reconstructng benchmark objects. The results obtaned compare favorably wth other emnent algorthms. Fnally, the proposed system exhbted hgh robustness n reconstructng objects from real large clouds wth non-unform samplng and ms-regstratons. Key-Words: - Object modelng, computatonal geometry, surface reconstructon, and surface and object representatons. 1. Introducton There are many applcatons that rely on buldng accurate models of real-world objects such as sculptures, damaged machne parts, archaeologcal artfacts, and terran. Technques for dgtzng objects nclude laser range fndng, mechancal touch probes, and computer vson technques such as depth from stereo. Some of these technques can yeld mllons of 3D pont locatons on the object that s beng dgtzed. Once these ponts have been collected, t s a non-trval task to buld a surface representaton that s fathful to the collected data. Some of the desrable propertes of a surface reconstructon method nclude speed, low memory overhead, fathful reproducton of sharp features, and robustness n the presence of holes and low samplng densty. Fgure (1) llustrates a typcal example for object reconstructon problem. Few technques are avalable for automatcally modelng ths type of data. Mathematcally, the problem can be expressed as follows, gven a set of N ponts P={p 1,,p N }, whch are sampled from some unknown smooth surface U, fnd a mesh M that approxmates the surface U,.e. the mesh and the surface are near everywhere [15]. The output s ntally some mesh (mostly a trangulaton) that has the same general shape of the orgnal surface. If the generated mesh connects the orgnal nput ponts, then t nterpolates the surface. Alternatvely, the mesh may connect generated ponts other than those of the nput set. In the latter case, the mesh s sad to approxmate the surface. A common classfcaton crteron of surface reconstructon technques s to categorze the technque as ether a surface nterpolaton algorthm or a surface approxmaton one [17]. (a) (b) Fg. 1: Example of a typcal object reconstructon problem (a) Raw data of a statue wth natve holes and ms-regstratons. (b) Reconstructed object usng the present system.

2 One of the frst surface nterpolaton algorthms s the one presented by Edelsbrunner and Mücke n [11], whch uses the heurstc alpha-shapes. Amenta and Bern [1] used the propertes of the medal axs and Vorono dagram to ntroduce the Crust algorthm, whch was the frst guaranteed surface reconstructon algorthm. They extracted an approxmated trangulated surface from the Delaunay trangulaton of the nput set. Amenta et al. [] descrbed the co-cone term and used t n ther Cocone algorthm. The Cocone computes the Delaunay trangulaton of the nput set and then selects canddate trangles to construct the surface. Dey and Goswam [9] modfed the Cocone nto a new algorthm, the Tght Cocone, to solve problem nstances of specfc domans that requre hole-free surfaces. Kollur et al. [16] ntroduced a noseresstant algorthm for watertght surfaces. The latter algorthm uses a varant of spectral graph parttonng to select trangles from the Delaunay trangulaton to construct the surface. All the aforementoned algorthms can be descrbed as Vorono based algorthms and they generally suffer from slow performance and dffculty n handlng large data. To overcome ths problem, Dey et al. [1] parttoned the entre sample space nto smaller clusters usng octree subdvson. Then, they appled the Cocone algorthm on each of these clusters separately, and they called ths algorthm the Super Cocone. An nterestng nterpolaton non-vorono based algorthm s the Ball Pvotng algorthm, presented by Bernardn et al. [5]. Ths algorthm ncrementally reconstructs surfaces by usng a rollng ball to construct trangles one by one and append them to the surface. Typcal examples of surface approxmaton algorthms are the algorthms presented by Hoppe et al. [15] and by Curless and Levoy [8]. They used the normal orentatons to estmate the tangent planes and approxmate the local surface. Another surface approxmaton algorthm was ntroduced by Bossonat and Cazals [4] that s based on natural neghbors nterpolaton and represents the surface mplctly as a zero-set of some pseudo-dstance functon. Gop et al. [13] suggested an algorthm that uses a plane projecton technque where the neghbors of a sample pont s computed by projectng ts close ponts onto ts tangent plane. Amenta et al. [3] suggested usng the power dagram (dual of weghted Delaunay trangulaton) n ther Power Crust algorthm. Although the Power Crust algorthm s a Vorono based algorthm, but -unlke the algorthms belongng to ths famly- t s classfed as a surface approxmaton algorthm. Another famly of surface approxmaton algorthms uses the mplct surface approach, for example [6], [] and [7], whch s sutable for reconstructng pont clouds wth local problems lke holes, low samplng densty, and ms-regstraton but not sutable for processng large clouds. To adapt these algorthms for large clouds, local support or parttonng based solutons are used (for example [18], [19] and [1]). Stll, mesh nterpolaton based methods are preferable for surface reconstructon than those based on mesh approxmaton because the former gve better accuracy and have better performance from both computatonal tme and memory usage ponts of vew. On the other hand, mesh approxmaton based methods that utlze mplct surface approach have the edge of robustness n reconstructng surfaces from cloud of ponts natvely have holes, low samplng densty and msregstraton. So, ganng the advantages of the two types of reconstructon methods can be obtaned by ntegratng methods from both of them. In ths work, a hybrd system s developed by combnng two well-known methods n order to obtan a reconstructon scheme for large data sets n the presence of low samplng densty and/or ms-regstraton domans. The Ball Pvotng algorthm [5] s used as an effcent tool for surface reconstructon whle the Radal Bass Functons (RBF) [6] s used as a powerful nterpolaton tool. The paper s organzed as follows: secton provdes the theoretcal backgrounds of the technque. In secton 3, the mplementaton s presented showng the practcal choces to obtan an easy, effcent and robust reconstructon algorthm. Secton 4 presents a study for the performance and accuracy of the present system and the applcaton of reconstructng some realstc data that have natve holes and ms-regstraton domans, and n secton 5 the work s concluded.. Theoretcal backgrounds.1. Ball pvotng algorthm The approach of Bernardn, Mttleman and Rushmeer [5] mtates the ball eraser used n α- shapes. Assume that the sampled data set P s dense enough that a ρ-ball, a ball of radus ρ, can not pass through the surface wthout touchng sample ponts. The algorthm starts by placng ρ- ball n contact wth three sample ponts. Keepng contact wth two of these ntal ponts, the ball s

3 pvotng untl t touches another pont. The pvotng operaton s depcted n Fgure (). y rejected f the dot product of the trangle normal wth the surface normal (n the vertex) s negatve, see Fgure (3). l k s k s jo σ k c k ρ γ m σ,σ j Fg. : The ball pvotng operaton. The ball of radus ρ sttng on the trangle τ formed from the vertces σ, σ j, σ s pvotng around the edge m here perpendcular to the mage. The center of the ball c j rotates around the edge m and descrbes a crcular trajectory γ. The radus of the trajectory s c j m. Whle pvotng the ball hts the pont σ k. The crcle s k s a ntersecton of ρ-ball centered at σ k wth z =. The crcle s j s the ntersecton of the pvotng ball wth z =. (mage courtesy of Fausto Bernardn [5]). The ρ-ball s pvoted around each edge of the current mesh boundary. Trplets of ponts that the ball contacts form new trangles. The set of trangles formed whle the ρ-ball walks on the surface form the nterpolatng mesh. The ball pvotng algorthm (BPA) s closely related to α- shapes. In fact, every trangle τ computed by the ρ- ball walk obvously has an empty smallest open ball b τ whose radus s less than ρ. Thus, the BPA computes a subset of the -faces of the ρ-shape of P. These faces are also a subset of the -skeleton of the three-dmensonal Delaunay trangulaton of the pont set. The surface reconstructed by the BPA retans some of the qualtes of alpha-shapes: It has provable reconstructon guarantees under certan samplng assumptons and an ntutvely smple geometrc meanng. The nput data ponts are augmented wth approxmate surface normals computed from the range maps. The surface normals are used to dsambguate cases that occur when dealng wth mssng or nosy data. Areas of densty hgher than ρ present no problem to the algorthm, but mssng ponts create holes that cannot be flled by the pvotng ball. Any post-process hole-fllng algorthm could be employed; n partcular, BPA can be appled multple tmes wth ncreasng ball rad. To handle possble ambgutes, the normals are used as stated above. When pvotng around a boundary edge, the ball can touch an unused pont lyng close to the surface. Therefore a trangle s τ c jo n σ o x Fg. 3: The normal check (a D case). The rghtmost pont s boundary and the ball pvots around ts edge and touches a hdden pont, but the dot product of the trangle normal wth the vertex normal s negatve, so the trangle s rejected. (mage courtesy of Fausto Bernardn [5]) Ths algorthm s very effcent because t exhbts lnear tme and storage requrements. However ts dsadvantages are the selecton of the radus ρ of the ρ-ball and ncapablty to handle data that have natve problems lke low samplng densty, holes and ms-regstratons... Radal bass functons Gven the set of N dstnct ponts P= {p 1,,p N }of dmenson d: p k R d, and the set of values {h 1,,h N }, we want to fnd a functon f: R d R wth f ( p ) = h (1) In order to obtan a radal bass functon reconstructon of the pont set P, a functon f satsfyng the equaton f k ( p) = ω φ( p p ) + π ( p) =1, () has to be found. We denote here p, p j the Eucldean dstance, ω the weghts, φ : R R a bass functon, and π a polynomal of degree m π p = c π p dependng on the choce of φ. ( ) ( ) wth { π } Q α a bass n the d-dmensonal null α = 1 space contanng all real-valued polynomals n d varables and of order at most m, hence m + d Q =. d The bass functon φ has to be condtonally postve defnte [1], and some popular choces proposed n the lterature are shown below: bharmonc φ ( r ) = r wth π of degree 1 (3) 3 pseudo-cubc φ ( r ) = r wth π of degree 1 (4) 3 trharmonc φ( r ) = r wth π of degree (5)

4 The bharmonc and trharmonc splnes are known as the smoothest nterpolators n the sense that they mnmze certan energy functons and nterpolate the data [6]. So, the bharmonc splne s used n the present work. Gven a set of nodes N { p} R 3 N and a set of functon values { h } = 1 R, = 1 the bharmonc RBF f(p) satsfes the nterpolaton f p = h and mnmzes condtons ( ) f ( p) f ( p) f ( p) f f = 3 R x f + x y + ( p) f ( p) f ( p) y + x z + z + y z dx (6) s a measure of the energy n the second dervatve of f. As we have an under-determned system wth N+Q unknowns (ω and c) and N equatons, socalled natural addtonal constrants for the coeffcents ω are added, so that ω c1 = ωc =... = ωcq = (7) The equatons (1), (), and (7) determne the followng lnear system: Ax = b (8) T Φ P A = P (9) Φj = φ ( p, p j ) = N j = N P = π α p = N = α... Q x = ω ω,...,, c, c,..., ( ) [ ] T 1, ω N 1 c Q b = h1, h,..., h N,, 1443,..., (1) Q tmes The soluton vector x s composed of the weghts ω and the polynomal coeffcents c for equaton () and represents a soluton of the nterpolaton problem gven by (1). 3. Reconstructon scheme 3.1. Hybrd reconstructon To overcome the dsadvantages of the Ball Pvotng Algorthm, a hybrd system s developed by combnng BPA and RBF and works as shown T n Fgure (4). The system calls the BPA teratvely by defnng the trangulaton doman. The RBF soluton vectors (whch contan the weghts ω and the polynomal coeffcents c ) are computed once after the dentfcaton of holes and/or msregstraton domans and ther neghborhood zones from the ntal reconstructed surface obtaned by the BPA or gven as an nput to the present system. Each teraton, holes are dentfed (whch le wthn that dentfed prevously) then the RBF s employed as nterpolator to unformly generate surface ponts wthn the holes and the BPA s used to trangulate the generated ponts. The process of callng BPA and generatng sampled ponts usng RBF surface nterpolator s repeated untl reachng a perfect surface. End Yes Cloud of ponts or mesh contanng holes to be flled RBF_Sol_Vec_Comp= No Defne BPA workng doman Apply BPA Identfy holes loops Nholes> Yes RBF_Computed=1 No Compute RBF soluton vector for each hole loop RBF_Sol_Vec_Comp=1 Apply nterpolaton For each hole doman Man teraton loop Fg. 4: The hybrd system for 3D object reconstructon 3.. Hole or ms-regstraton dentfcaton The applcaton searches for every edge whch les n only one trangle and constructs a lst of these edges. Startng wth a seed vertex whch les on any edge wthn the constructed lst, the applcaton can trace a seres of edges untl t dentfes a closed loop defnng a hole. The dentfed loop of edges s removed from the lst. The process of tracng and removng edges s repeated untl dentfyng all the holes.

5 3.3. RBF computatons After the dentfcaton of the closed loops of holes and/or ms-regstraton domans, the RBF computatons are performed for each of these loops. For each loop, the neghborhood zone depth should be defned prory (the zone depth s a user selected value as a rato of the hole loop largest dagonal). The vertces wthn ths zone are remarked by a smple tracng process through edges untl reachng the predefned zone depth as shown n Fgure (5). The remarked vertces are employed n the RBF computatons to get the soluton vector (whch contans the weghts and the polynomal coeffcents) usng equaton (8). The z-components of these vertces wll be consdered as the functon values n the rght hand sde of the equaton. Ths process s appled once to get the soluton vector for each hole or ms-regstraton doman. The nterpolaton process s appled to fll each hole or ms-regstraton doman by generatng surface vertces usng dfferent x and y coordnates values, wth nearly unform dstrbuton, then the correspondng z-coordnates are obtaned by the same equaton usng the computed soluton vector. Identfed hole boundary loop Identfed hole boundary loop b Largest dagonal Largest dagonal Fg. 5: Tracng of neghborhood zone vertces Selectng x and y coordnates values s the key parameter n generatng surface vertces wth nearly unform dstrbuton wthn each hole or ms-regstraton doman. If the dstrbuton of the generated surface vertces s not suffcently unform, more than one teraton wll be needed untl reachng perfect surface as mentoned n Fgure (4). In such case, the nterpolaton process wll be repeated to generate surface vertces wthn the dentfed holes usng the same soluton vector computed at the frst teraton. c a d g e f 4. Applcatons and results All results presented n ths secton were performed on Pentum IV.4 GHz PC computng platform wth 51 MB RAM runnng wndows XP. Source codes are developed usng Mcrosoft vsual C/C++ verson 6.. Also, Intel Math Kernel Lbrary was employed to mprove the performance of matrx operatons BPA performance verfcaton The performance of the BPA s verfed aganst other emnent algorthms, namely Crust, Power Crust and Tght Cocone, n reconstructng bench mark objects of [14] wth dfferent resolutons. These algorthms are developed on the same platform and usng the same programmng tools to get proper conclusons. Fgures (6.a) to (6.c) show the varaton of computatonal tme versus model sze for the three selected objects for comparson (Bunny, Dragon and Happy Buddha) whle Fgures (6.d) to (6.f) show the correspondng varaton of memory usage. All the algorthms exhbted lnear performance ndependent of the object shape but BPA bested others from both computatonal tme and memory usage ponts of vew. Fgure (7) shows the reconstructed meshes usng the dfferent algorthms for the 1889 vertces Bunny model. The Crust output (Fgure 7.a) s good but t s not a manfold. Besdes beng watertght, the Power Crust output (Fgure 7.b) may be consdered the best, except that the mesh s very dense and not trangular. The Tght Cocone output (Fgure 7.c) s also watertght, but shows a problem at the left ear end (a trangle s not connected to the rest of the ear). The Ball Pvotng output (Fgure 7.d) shows also problems at both of ears whch can be consdered as holes n the obtaned mesh. 4.. Hybrd system applcaton and accuracy assessment The system s tested va the reconstructon of the Londog object used by Ohtake et al. [18], Fgure (8.a), after makng an artfcal hole n ts head to assess the accuracy of the present system n fllng holes and ms-regstraton domans. The orgnal model contans 493 unformly sampled ponts. The mesh s reconstructed ntally by applyng the BPA usng a sngle ball (ts radus s estmated as the average samplng dstance) n 8. secs. Then, the artfcal hole s successfully dentfed by ts

6 boundary loop as shown n Fgure (8.b). Dfferent values for the hole neghborhood zone depth are consdered. The values consdered are 5%, 7% and 1% of the hole loop largest dagonal (HLD). The RBF soluton vector s computed for the three cases n 9, 16 and 7 secs usng the drect method [6]. Fnally, the teratve process between the BPA and RBF s appled untl achevng a perfect surface. Only one teraton s performed n 6 secs to get surface wth fnal qualty as shown n the same fgure. The accuracy of the present system n fllng holes and ms-regstraton domans s assessed by comparng the reconstructed surface for the artfcal hole and the orgnal surface (for each of the consdered values of the hole-neghborhood zone depth). The dfference between the two surfaces at each pont wth the same x and y coordnates s consdered as the dfference n the magntude of the z-coordnate. The contours representng the dstrbuton of the dfference between surfaces are shown n Fgures (8.c) to (8.e) for the three consdered values. It s found that ncreasng the hole-neghborhood zone depth decreasng the accuracy of fllng and ncreasng the smoothness Reconstructon of realstc data The robustness of the present system n reconstructon of 3D objects from large cloud of ponts that have natve holes and/or msregstraton s confrmed usng a realstc cloud of ponts for a statue of an ancent Egyptan man. The model contans 6,461 ponts wth many msregstraton domans n ts head, face, shoulders, hands and the base as shown n Fgure (9.a). The mesh s reconstructed ntally by applyng the BPA usng sngle ball n 91 secs. Then, holes and msregstraton domans are dentfed. The value of the neghborhood zone depth for the holes and msregstraton domans s taken as 5% of HLD of the correspondng holes or ms-regstraton domans. The RBF soluton vectors for all the dentfed holes are computed n 194 secs. Then, three teratons are performed n 89 secs to get surface wth fnal qualty. Fgures (9.b), (9.c) and (9.d) compare the reconstructed object usng the present system aganst the results of the reconstructon usng the Tght Cocone algorthm and the multlevel partton of unty mplcts of [18]. It can be shown that the Tght Cocone, Fgure (9.b), exhbted flat fllng for the ms-regstraton domans whle the reconstructed object usng the mult-level partton of unty mplcts of [18] suffers from spkes n the ms-regstraton domans besdes losses n the object detals as shown n Fgure (9.c). On the other hand, the present system, Fgure (9.d), exhbted curved fllng for the msregstraton domans wth contnuty n the normals dstrbuton at ther boundares as shown n Fgure (9. e). Hence, the present system s more sutable for reconstructon of complcated surfaces from large clouds sufferng from holes and/or msregstraton domans. 5. Conclusons A hybrd system for surface reconstructon based on the Ball Pvotng Algorthm (BPA) and Radal Bass Functons (RBF) was developed and presented. The present system employs the Ball Pvotng algorthm as a surface reconstructon tool has the capablty of processng very large data sets n a lttle executon tme and usng small amount of memory. To overcome the weakness of the BPA n the presence of holes, ms-regstratons and low sample densty, the RBF s employed by the system to unformly nterpolate ponts representng the mssng surface parts. The RBF soluton vector s computed once after dentfyng the holes and/or ms-regstraton domans from the ntal reconstructed surface obtaned by the BPA. Then, the process of the nterpolaton and callng the BPA to trangulate the dentfed holes and/or msregstraton domans are appled teratvely untl reachng perfect surface. The accuracy and robustness were verfed n reconstructng surfaces from realstc data wth non-unform samplng and ms-regstratons. References: [1] Amenta N. and Bern M., Surface reconstructon by Vorono flterng, In Proc. of ACM SIGGRAPH 98, 1998, pp [] Amenta N., Cho S., Dey T. and Leekha N., A smple algorthm for homeomorphc surface reconstructon, In Proc. of 16 th ACM Sym. on Comp. Geom.,, pp [3] Amenta N., Cho S. and Kollur R., The power crust, unon of balls, and the medal axs transform, In proc. of the 6 th ACM Sym. on sold modelng and applcatons, 1, pp [4] Bossonat J. and Cazals F., Smooth surface reconstructon va natural neghbor nterpolaton of dstance functons, In Proc. of the 16 th annual sym. on Comp. Geom.,, pp. 3-3.

7 [5] Bernardn F., Mttleman J., Rushmeer H., Slva C. and Taubn G., The ball-pvotng algorthm for surface reconstructon, IEEE Trans. n Vs. and Comp. Graph. 5, 4, 1999, pp [6] Carr J., Beatson R., Cherre J., Mtchell T., Frght W., McCallum B., and T. Evans., Reconstructon and representaton of 3D objects wth radal bass functons, In Proc. of ACM SIGGRAPH 1, 1, pp [7] Carr J., Beatson R., McCallum B., Frght W., McLennan T., and Mtchell T., Smooth surface reconstructon from nosy range data, In Proc. of Graphte 3, 3, pp [8] Curless B. and Levoy M., A volumetrc method for buldng complex models from range mages, In Proc. of SIGGRAPH 96, 1996, pp [9] Dey T. and Goswam S., Tght Cocone: A water-tght surface reconstructor, In Proc. of 8 th ACM Sym. on sold modelng and applcatons, 3, pp [1] Dey T., Gesen J. and Hudson J., Delaunay based shape reconstructon from large data, In Proc. Of IEEE Sym. n parallel and large data vsualzaton and graphcs, 1, pp [11] Edelsbrunner H. and Mücke D., Threedmensonal alpha shapes, ACM TOG 13, 1, 1994, pp [1] Franke R., Scattered data nterpolaton: tests of some methods, Mathematcs of Computaton 38, 157, 198, pp [13] Gop M., Krshnan S. and Slva C., Surface reconstructon based on lower dmensonal localzed Delaunay trangulaton, In Proc. of Eurographcs 19, 3,. tme n secs memory n MB crust Crust power Power Crust tght Tght Cocone pvot Ball Pvot x 1 crust Crust power Power Crust tght Tght Cocone pvot Ball Pvot nput sze tme n secs crust Crust power Power Crust tght Tght Cocone pvot Ball Pvot [14] ep [15] Hoppe H., Derose T., Duchamp T., McDonald J. and Stuetzle W., Surface reconstructon from unorganzed ponts, In Proc. of SIGGRAPH 9, 199, pp [16] Kollur R., Shewchuk J., O'Bren J., Spectral surface reconstructon from nosy pont clouds., In Proc. of Eurographcs Geometry Processng Sym., SGP4, 4, pp [17] Peng L. and Shamsuddn S., 3D object reconstructon and representaton usng neural networks, In Proc. of the nd nternatonal conference on computer graphcs and nteractve technques n Australasa and Southe East Asa, 4, pp [18] Ohtake Y., Belyaev A., Alexa M., Turk G., and Sedel H., Mult-level partton of unty mplcts, In Proc. of ACM SIGGRAPH 3, 3, pp [19] Ohtake Y., Belyaev A., and Sedel H., Multscale approach to 3D scattered data nterpolaton wth compactly supported bass functons, In Proc. of Shape Modelng Internatonal, 3, pp [] Turk G. and O bren J., Modellng wth mplct surfaces that nterpolate, ACM TOG 1, 4,, 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, WSCG 4 1,3, x 1 nput sze 9 8 Crust crust 7 Power power Crust 6 Tght tghtcocone 5 Ball pvot Pvot x 1 nput sze (a) (b) (c) x 1 nput sze memory n MB crust Crust power Power Crust tght Tght Cocone Ball Pvot pvot x 1 nput sze tme n secs memory n MB Crust crust power Power Crust tght Tght Cocone pvot Ball Pvot nput sze x 1 (d) (e) (f) Fg. 6: Comparson between dfferent algorthms from computatonal tme and memory usage ponts of vew. (a, d) Bunny Model. (b, e) Dragon Model. (c, f) Happy Buddha Model.

8 (a) (b) (c) (d) Fg. 7: Reconstructed Stanford Bunny from 1889 vertces usng dfferent algorthms. (a) Crust. (b) Power Crust. (c) Tght Cocone. (d) BPA (a) Identfed hole after BPA reconstructon (b) Fnal qualty reconstructon (c) 5% HLD (d) 7% HLD (e) 1% HLD Fg. 8: Accuracy assessment of the present system for three consdered values of the hole-neghborhood zone depth. (a) Londog after BPA reconstructon and hole dentfcaton. (b) fnal qualty reconstructon usng neghborhood zone depth of 5% HLD (c,d,e) contours of the offset dstance between the generated surface and the orgnal surface for the three consdered values of the hole-neghborhood zone depth.

9 (a) (b) (c) (d) (e) Fg. 9: Reconstructon of real scanned data wth natve holes and ms-regstraton domans of an ancent Egyptan man statue. (a) the scanned cloud of 6,461 ponts wth ms-regstraton domans ndcated by the red arrows. (b) reconstructed surface usng the Tght Cocone algorthm. (c) reconstructed surface usng mult-level partton of unty mplcts of [18] (d) reconstructed surface usng the present system.(e ) surface-normal dstrbuton over the reconstructed object usng the present system.