Surface reconstruction from sparse data by a multiscale volumetric approach

Transcription

1 Proceedngs of the 5th WSEAS Int. Conf. on Sgnal Processng, Computatonal Geometry & Artfcal Vson, Malta, September 5-7, 2005 (pp35-40) Surface reconstructon from sparse data by a multscale volumetrc approach ATOIO CHIMIETI*, PAOLA DALMASSO, ROBERTO ERIO, GIUSEPPE PETTITI*, MASSIMILIAO SPERTIO* *VA Department CR-IEIIT CV Department IRIM Strada delle Cacce 9, Torno ITALY Abstract: - Ths paper descrbes a method for surface reconstructon from sparse three-dmensonal (3D) data that performs the reconstructon by buldng a sequence of surfaces approxmatng the data at ncreasng level of detals (LOD). The method s smple, fast and sutable for a progressve 3D data/model representaton, archvng, transmsson. The surface reconstructon s obtaned by a volumetrc method that dffers from other volumetrc methods because t does not requre mplctly or explctly nformaton on surface normals. Ths aspect s mportant n the case of nosy data sets, such as those comng from mage based methods, because normals are often estmated unrelably from 3D data. The method s based on a herarchcal parttonng of the volume data set. The workng volume s splt and classfed at dfferent scales of spatal resoluton nto surface, nternal and external voxels and ths herarchy s descrbed by an octree structure n a multscale framework. The octree structure s used to buld a multresoluton descrpton of the surface by means of compact support Radal Bass Functons (RBFs). A herarchy of surface approxmatons at dfferent LOD s bult by representng the voxels at the same octree level as RBFs havng the same spatal support. At each scale, nformaton related to the reconstructon error drves the reconstructon process at the followng fner scale. Prelmnary results on real data are presented and dscussed. Key-Words: - Surface reconstructon, sparse data, volumetrc methods, radal bass functons, multscale, octrees Introducton The reconstructon of surfaces from ts sparse three-dmensonal (3D) samples s a challengng problem both n computer vson and computer graphcs. There s a wde varety of approaches to surface reconstructon from sparse 3D data (see [4] for a survey), among whch regon growng [9], tensor votng [], dstance felds [0], algebrac and computatonal geometry methods [6], volumetrc regularzaton and mplct surface methods [3,5]. Implct surface methods [3] are an actve research area whch ncludes among others Movng Least Squares (MLS) [,2] and Radal Bass Functons (RBFs) [5]. RBF based approaches address the queston of the choce between local or global RBFs. Local RBFs lead to fast and smpler computaton but have the drawback of beng senstve to the densty of scattered data. Global RBFs on the other hand are able to deal better wth non-unform densty data but are computatonally heaver or even mpractcal for large data sets. Some authors [8,2,8] have tred to combne the advantage of both types of RBFs by usng locally supported bass functons n a herarchcal framework. A coarse-to-fne approxmaton/nterpolaton of the data s bult by sets of self-smlar local bass functons havng ther support scaled accordng to the scale of the detals consdered. The estmated reconstructon error at a coarse scale drves the reconstructon process at the followng fner scale. Ths approach s very promsng respect to the sze of the data set that can be managed, the reconstructon tme and the control on the level of detals (LOD), but t requres a herarchcal parttonng of the data set. Data set parttonng s commonly realzed by the herarchcal subdvson of the data set volume n voxels of dfferent szes accordng to a fxed subdvson scheme or an adaptve scheme, based on ether the approxmaton error, or the evaluaton cost [,8]. The herarchcal partton of the data set s commonly coded n an octree structure [4] or n a K-D tree [2]. An mportant aspect common to several of these approaches, especally those comng from the computer graphcs area s that they relay drectly or ndrectly on nformaton about the surface normals to produce a meanngful output. Generally

2 Proceedngs of the 5th WSEAS Int. Conf. on Sgnal Processng, Computatonal Geometry & Artfcal Vson, Malta, September 5-7, 2005 (pp35-40) ths nformaton s assumed avalable a pror or s estmated from the data. On the other sde, some computer vson approaches to surface reconstructon from mages such as Generalzed Voxel Colorng (GVC) [7] and space carvng [7] do not use nformaton on surface normal nor try to nfer t. They produce estmates of 3D surfaces as a partton of the surface volume n empty and surface voxels, whose centres can be assumed as surface ponts. Dependng on the technque used to measure or to estmate 3D samples of the surface (e.g. laser scanner, stereo, etc.) nformaton obtaned on surface normals can be more or less relable. Ths paper presents a volumetrc approach to surface reconstructon whch does not use ths nformaton and hence copes easly wth several sources of 3D data. The method tres to blend some of the deas concernng multscale parttonng of 3D data, local constructon of volumetrc functons, successve approxmatons to the surface drven by LOD, and GVC technques. It requres a smaller amount of nformaton wth respect to more sophstcated methods as [2], stll allowng an acceptable qualty of the surface reconstructon. The method classfes at dfferent spatal scales the volume occuped by data nto nner, outer and surface voxels. The constrants on the surface reconstructon are then defned n term of volume nstead of values of the surface normals and of the volumetrc functon at nner and outer ponts as n[5]. The algorthm bulds a descrpton of the volume around the data at dfferent spatal scales, by classfyng the volume occuped by data nto voxels of an octree herarchcal structure. The octree structure s then used to buld a multresoluton volumetrc descrpton of the surface by means of RBFs of compact support. A sequence of volumetrc functons of dfferent smoothness and LOD s generated from the octree structure. Informaton related to the partal reconstructon errors at one scale s used to drve the reconstructon process at the followng fner scale. The estmated surfaces at dfferent LOD are obtaned as the zero level set of the correspondng volumetrc functons. Prelmnary results produced by the algorthm on real data are presented and dscussed. 2. Buldng the octree structure The volume parttonng and the constructon of the octree structure starts at the coarsest scale where the workng volume s splt nto a cube made of 3x3x3 voxels of the same sze. These voxels consst of a central surface voxel where are contaned all the 3D data, surrounded by 26 empty voxels classfed as outer voxels. Other ntal confguratons are possble dependng on ntal hypothess on the surface and ts volume occupancy. The ntal surface voxel, root of the octree, s then splt nto 8 voxels of the same sze halvng the voxel sze along every dmenson [6,4] (see Fg. ). The son voxels of the root voxel are then classfed nto surface or empty voxels, accordng to the occupancy of the voxels by the 3D data. Empty voxels are the classfed nto nner and outer voxels dependng on ther adjacency to voxels of the same type. At the frst step no nner voxels are generated; these voxels are usually generated n the followng steps when are created empty voxels not adjacent to outer ones. Only the surface voxels are then splt agan and classfed proceedng toward smaller scales of spatal resoluton. Empty voxels (nner and outer) are not splt anymore, but can be reclassfed at the followng steps of the octree buldng from nner to outer voxels. The volume subdvson stops at a specfed scale or when every surface voxel contans a 3D pont only. Concernng the decomposton algorthm complexty and memory requrements, t should be noted that the maxmum number of voxels produced s lower bounded by log 8 ( ) ; n practcal cases ths bound s too conservatve. Further, the octree structure s effcently managed by ponter structures. 2 Herarchcal representaton of 3D data The algorthm makes a herarchcal parttonng of the data volume contanng the surface nto voxels of dfferent szes. These voxels are classfed nto surface, nner, and outer voxels respectvely f they contan data, or they are empty and nsde or outsde the surface. Ths descrpton of the surface and ts surroundng volume s represented by an octree structure n a multscale framework where dfferent levels correspond to dfferent scales and dfferent voxel szes (see Fg. ). The tree from the root to a specfc level contans a descrpton of the surface occupancy n the workng volume up to that spatal resoluton or scale. Fg. Volume subdvson n voxels and correspondng octree structure

3 Proceedngs of the 5th WSEAS Int. Conf. on Sgnal Processng, Computatonal Geometry & Artfcal Vson, Malta, September 5-7, 2005 (pp35-40) 3 Volumetrc regularzaton by usng RBF 3. Data approxmaton and nterpolaton d The recoverng of a functon f defned on R from the set {( ) d } g = x, y R R of functon values sampled at = random locatons s ll-posed [5]. The problem can be constraned and solved assumng some a pror knowledge on f. A commonly adopted constrant conssts n assumng that the functon s smooth. A varatonal approach [5,3,5] s then adopted and the orgnal problem s recast nto that of fndng the functon f whch mnmze a functonal G whch take nto account both the smoothness and the data values: 2 [ ] = ( ( ) ) + λφ[ ] = F f f x y f () The frst term of F takes nto account the closeness to the data and the functonal φ at the second term the smoothness of the soluton f. The trade-off between the two terms s controlled by the so called regularzng parameter λ. In the case λ = 0 the soluton leads to pure nterpolaton of the data. For a wde class of possble functonals φ the soluton f ( x ) has the form [3]: k f ( x ) = w H ( x x ) + v P( x ) (2) j = j= where the frst term conssts of a sum of bass functons H havng radal symmetry whch are centred on the x values of the 3D data and are weghted by w. The second term n (2) conssts of lnear combnaton of polynomal terms P( x ). The regularzng parameter λ can vary from sample to sample, to take nto account dfferent local trade-off between reconstructon error and smoothness. The choce of a specfc smoothness functonal φ leads to radal gaussan bass functons H of the type: 2 2 r / 2σ G( r ) = e (3) 2πσ and the polynomal term n (2) reduces to a constant p 0. The unknown weghts w and p 0 n (2) can be determned from the data [3] and the choce of λ by the lnear system of equatons: ( H + I l) w + p0 = y (4) and the constrant w 0 = =, where I the dentty matrx, and ( H ) = H ( x x ), ( l) = λ, ( y) = y, ( w ) = w j j The sosurface related to the zero level set { x f ( x ) = 0} of the volumetrc functon f gves an estmate of the surface assocated to the 3D data. 3.2 Radal bass functon of compact support Accordng to the algorthm descrpton n Sec. the volume occupancy of voxels of dfferent sze s cast nto a correspondng volume occupancy by gaussan radal bass functons of compact support (See Eq. 3) centred on the voxel centres c. In the case of wde support bass functons the system of equatons (4) becomes untractable for more than few thousands data. On the other hand a compact support of the radal bass functons means that at a gven scale the nfluence on the volumetrc functon f of the subset of 3D data contaned nto a specfc voxel s localzed around the voxel volume. Hence the compact support choce has the beneft that the system of equatons (4) becomes sparse and can be solved very effcently by standard methods [5]. For the gaussan RBF used n ths work the support of the functons depends on the parameter σ ; ths parameter changes wth the scale. 3.3 Multresoluton volumetrc reconstructon by RBF Multlevel approxmaton by adaptve doman decomposton s a promsng approach to sparse data approxmaton [5, 8]. Here the data doman decomposton n subsets produces the octree structure of Sec. 2, whch s the startng pont to buld up a multresoluton volumetrc descrpton of the surface by means of RBF of compact support. The global volumetrc functon S whch approxmates up to the scale the surface assocated to the data { } j j d j= g = x, y R R s gven by the sum of a sequence of volumetrc functons f k S ( x) = f ( x) = f ( x) + f ( x) + + f ( x ) (5) k 0 k = At a gven scale the functon f approxmates the reconstructon error due to the dfference between the actual values of the S at the voxel centres c j and the prescrbed values y j. Hence the reconstructon error propagates from the scale to the scale accordng to: f ( c ) = y S ( c ) (6) j j j where c j are the j voxel centres at the scale correspondng to the level of the octree, and

4 Proceedngs of the 5th WSEAS Int. Conf. on Sgnal Processng, Computatonal Geometry & Artfcal Vson, Malta, September 5-7, 2005 (pp35-40) Ks f j s a surface voxel, yj = Ko f j s an outer voxel, K f j s an nner voxel. The constant values Ks, Ko, K are chosen as seen n Sec. 4 whle the evaluaton of the functons f s performed as shown n Subsec Implementaton detals and choce of the parameters The choce of the parameter values and of the voxel classfcaton rules are crucal for the qualty of reconstructon process. Some parameters have standard values n the lterature, and those values have been mplctly assumed here, as the geometry of the workng volume parttons (voxels) or the shrnkng value ( 2 ) along every co-ordnate at every scale step. The constants for the functon value assgnments at the voxel centres n Eq. 7 are chosen to be K = 0, K =, K = (7) s o as n [5], but other choces can be exploted. Other parameters as the regularzng parameter λ and the support parameter σ along wth the classfcaton rules deserve a dscusson. 4. Empty voxel classfcaton The general voxel classfcaton rules have been dscussed before n Subsec. 2.. When empty voxels of dfferent type become adjacent, the classfcaton depends on the scale. At scales coarser than or equal to typcal scale TYP, whch depends on the mean dstance between 3D data ponts, volume carvng s supported. Inner voxels adjacent to outer ones are by reclassfed as outer, and ths classfcaton propagates along adjacent voxels of the same or coarser scale. The partally buld octree structure s analyzed bottom up to propagate the reclassfcaton. At scales fner than TYP the reclassfcaton propagaton s partally nhbted to avod the creaton of holes (outer-nner voxel adjacency) on the surface along whch nner voxels can change to outer destroyng n ths way the surface contnuty. Empty voxels are classfed outer only f they are adjacent to at least one bgger outer voxel. Empty voxels are classfed nner only f they are adjacent to at least one bgger nner voxel. In all the other cases empty voxels are classfed as a surface voxels of partcular type because they wll never be splt further. 4.2 Support and Regularzng parameters To force the support extenson of an RBF at a gven scale to be contaned nsde the frst neghborng voxels the support parameter σ s chosen as: σ a, a = (8) where a s the length of the voxel sde at the scale. The regularzng parameter λ s crucal to control the reconstructon process. By varyng locally ts value for every voxel t s possble to stress the closeness to the data, such as near edges or sharp surface varatons, or the smoothness, such as near planar zones of the reconstructed surface. A reasonable constrant on the reconstructon process could be the preservaton of surface contnuty at scales fner than TYP, f t s supposed no holes are present at that scale and at the fner ones. Hence not only space carvng should be nhbted by the classfcaton rules on empty voxels, but also smoothness should ncreased by ncreasng λ. In any case the surface should be close to the 3D data ponts nsde the surface voxels, hence a measure of the approxmaton error has to be defned to deal wth the trade-off between smoothness and goodness of ft to the data. As a measure of the local approxmaton error of the reconstructon has been chosen the Eucldean dstance e between the pont x and the estmated surface, or more formally, between the 3D data pont x and the pont y on the sosurface f = 0 ntersected by the normal nɵ = f ( x) x to the sosurface f = f ( x ) passng though x, that s : e ( ) ( ) = x y f x f x x = e (9) For a gven surface voxel V, the maxmum e max = max e of the modul of the estmates e of the { } approxmaton errors e related to the ponts x that belong to V defnes the local maxmum devaton of the surface from the data. ote that the estmates are vald only for a frst order expanson of f ; for ths reason emax has been bounded to be at most a where a s the length of the voxel sde at the scale. To trade smoothness for accuracy, at the next scale step the values of the smoothness parameter λ for all the egth descendants of the surface voxel V (f t wll be splt) s locally set accordng to : max λ surf = (0) e Inner and outer voxels are not splt hence ther local approxmaton error does not propagates along the scales. For ths reason ther smoothness parameters are constant at a partcular scale; they only ncrease whle proceedng along the scales to compensate for the correspondng reducton

5 Proceedngs of the 5th WSEAS Int. Conf. on Sgnal Processng, Computatonal Geometry & Artfcal Vson, Malta, September 5-7, 2005 (pp35-40) of the smoothness part of the functonal φ [ f ] of Eq., accordng to : λ nner = λ () outer a 5 Reconstructon results The reconstructon performance of the algorthm has been tested on two sets of 3D data generated by laser scanners. The Horse data, made of about surface ponts, s avalable at the Large Geometrc Models Archve at Georga Inst. of Tech., whle the Bunny data, made of about surface ponts, s avalable at the at The Stanford 3D Scannng Repostory. The typcal scale TYP for the two data set was estmated by the mean nter-pont dstance of the two data sets. The two data sets were normalzed n a workng volume of unts. Startng at scale 0, where the sde of the voxel s 2, a frst surface voxel wth 26 adjacent outer voxels s refned up to scale 7, where the sde of the voxel s The Horse surface reconstructed at scale 2 s shown n Fg. 2. The surface voxels (32 voxels) are shown on the left and the correspondng reconstructed sosurface on the rght. The fnal result s obtaned at scale 7 (33000 surface voxel) and s shown n Fg. 3. The Bunny surface reconstructed at scale 5 s shown n Fg. 4. The surface voxels (3400) are shown on the left and the correspondng reconstructed sosurface on the rght. More detals are vsble respect to Fg.3, even f a surface voxelzaton s evdent. The fnal result s obtaned at scale 7 (56000 surface voxels) and the surface s shown n Fg. 5. The reconstructon tmes up to scale 7 were 20 s for the Horse data and 90 s for the Bunny data on a PentumIII GHz. The mean reconstructon errors were and for the Horse data and for the Bunny data respectvely. All the surfaces shown were obtaned by standard sosurface extracton technques [3] appled to the volumetrc functons estmated by the algorthm from the test data. Fg. 3 Horse data reconstructon at scale 7 Fg. 4 Bunny data reconstructon at scale 5 Fg. 2 Horse data reconstructon at scale 2 Fg. 5 Bunny data reconstructon at scale 7

6 Proceedngs of the 5th WSEAS Int. Conf. on Sgnal Processng, Computatonal Geometry & Artfcal Vson, Malta, September 5-7, 2005 (pp35-40) 6 Conclusons Ths paper descrbes a multscale volumetrc approach to surface reconstructon from non-unform data whch s based on a herarchcal parttonng of the volume data set nto an octree structure. The surface s reconstructed from the 3D data as a sequence of surfaces approxmatng dfferent level of detals n the space of spatal scales. Informaton related to reconstructon error propagates along the scales to drve the reconstructon process. A test on the performance of the method has been made on commonly used data sets avalable on the web. The results show the algorthm s effectve to produce from the orgnal 3D data a sequence of reconstructons that converge to a fnal good-qualty surface. An mportant feature of the algorthm s ts ablty to ncrementally refne the reconstructon. When the object represented by 3D data should be archved or transmtted at lower LOD than the fnest one, the algorthm output a sequence of nformaton able to upgrade an ntal rough representaton up to the desred LOD. Ths s partcular attractve for applcatons to progressve transmsson of 3D data [4]. The multscale approach, the octree data structure and the choce of local support RBFs allow both effcent and fast surface reconstructon. Further nvestgatons and mprovements are necessary to understand n deep how the nter-relatons between decomposton rules, local error and smoothng parameters affect the fnal qualty of the reconstructon. References: [] M. Alexa, J. Behr, D. Cohen-Or, S. Fleshman, D.Levn, C. T. Slva, Pont set surfaces, IEEE Vsualzaton 200 [2]. Amenta, Y. Kl, Defnng pont-set surfaces, ACM SIGGRAPH 2004 Proceedngs [3] J. Bloomenthal,(Edtor), Introducton to mplct surfaces, Morgan Kaufmann, 997 [4] R. M. Bolle, B. C. Vemur, On three-dmensonal surface reconstructon methods, IEEE Trans. on Patt. Anal. Mach. Int. Vol. 3, o., pp. -3, January 99 [5] M. D. Buhmann, Radal Bass Functons : Theory and Implementatons, P. G. Carlet (Edtor), A. Iserles (Edtor), R. V. Kohn (Edtor), M. H. Wrght (Edtor) [6] H. H. Chen, T. Huangh, A survey of constructon and manpulaton of octrees, Comp. Vs., Graph., Im. Proc., n.43, pp , 988 [7] W.B. Cullbetson, T. Malzbender, G. Slabaugh, Generalzed Voxel Colorng, ICCV Workshop on Vson Algorthms: Theory and Practce, pp.00-5, Sprnger, 999 [8] A. Iske, J. Levesley, Multlevel scattered data approxmaton by adaptatve doman decomposton, to appear n umercal Algorthms, Technsche Unverstät München, January 29th, 2004 [9] H. Hoppe T. DeRose, T. Duchamp, Surface reconstructon from unorganzed ponts, SIGGRAPH 992 Proceedngs, pp. 7-78, July 992 [0] B. Curless, M. Levoy, A volumetrc method for buldng complex models from range mages, SIGGRAPH 996 Proceedngs, pp , August 996 [] C.K. Tang, G. Medon, and M.S. Lee, Tensor Votng, n Perceptual Organzaton for Artfcal Vson Systems, K. Boyer and S. Sarkar Eds., Kluwer Academc Publshers, Boston, [2] Y. Ohtake, A. Belyaev, H.P. Sedel, A mult-scale approach to 3D scattered data nterpolaton wth compactly supported bass functons, Shape Modellng Internatonal 2003 Proceedngs [3] F. Gros, M. Jones, T. Poggo, Prors, stablzers and Bass Functons: from regularzaton to radal,tensor and addtve splnes, A.I. Memo n. 430, C.B.C.L. n. 75, Massachussets Insttute of Technology A.I. Lab, June 993 [4] H. Samet, A. Kochut, Octree approxmaton and compresson methods, 3DPVT'02 Proceedngs, Padua, Italy, pp , June [5] H. Q. Dnh, G. Turk, G. Slabaugh Reconstructng Surfaces By Volumetrc Regularzaton Usng Radal Bass Functons,IEEE Trans. on Patt. Anal. Mach. Int. Vol. 24, o. 0, pp , October 2002 [6] F. Bernardn, J. Mttleman, H. Rushmeer, C. Slva, G. Taubn, The Ball-Pvotng Algorthm for Surface Reconstructon, IEEE Transactons on Vsualzaton and Computer Graphcs, Vol. 5, o.4, Oct/Dec 999 [7] A. Yezz, G. Slabaugh, R. Cpolla, A surface evoluton approach of probablstc space carvng, 3DPVT'02 Proceedngs, Padua, Italy, pp , June [8] P. Dalmasso, R.erno, Herachcal 3D surface reconstructon based on radal bass functons, 3DPVT 04 Proceedngs, Thessalonky, Greece, Sept