Internatonal Conference on Recent Advances n 3-D Dgtal Imagng and Modellng, Ottawa, Canada May 12 15, 1997 Mult-Resoluton Geometrc Fuson Adran Hlton and John Illngworth Centre for Vson, Speech and Sgnal Processng Unversty of Surrey, Guldford GU25XH, UK ahlton@surreyacuk http://wwweesurreyacuk/research/vssp/3dvson Abstract Geometrc fuson of multple sets of overlappng surface measurements s an mportant problem for complete 3D obect or envronment modellng Fuson based on a dscrete mplct surface representaton enables fast reconstructon for complex obect modellng However, surfaces are represented at a sngle resoluton resultng n mpractcal storage costs for accurate reconstructon of large obects Ths paper addresses accurate reconstructon of surface models ndependent of obect sze An ncremental algorthm s presented for mplct surface representaton of an arbtrary trangulated mesh n a volumetrc envelope around the surface A herarchcal volumetrc structure s ntroduced for effcent representaton by local approxmaton of the surface wthn a fxed error bound usng the maxmum voxel sze Mult-resoluton geometrc fuson s acheved by ncrementally constructng a herarchcal surface representaton wth bounded error Results are presented for valdaton of the mult-resoluton representaton accuracy and reconstructon of real obects Mult-resoluton geometrc fuson acheves a sgnfcant reducton n representaton cost for the same level of geometrc accuracy 1 Introducton Geometrc fuson of multple sets of overlappng surface measurements to form a complete 3D obect or envronment model has receved consderable nterest [3, 5, 6, 9, 11, 12, 14, 16, 17] Reconstructon of 3D models of real obects s mportant for realstc computer generated magery and accurate reverse engneerng of CAD models Two approaches have been proposed for fuson of multple overlappng surface measurements nto a sngle representaton: mesh ntegraton [12, 9, 14, 16] and volumetrc fuson [3, 5, 6, 11, 17] Mesh ntegraton technques enable fuson of multple range mages wthout loss of accuracy In general mesh ntegraton s senstve to erroneous measurements whch may cause catastrophc falure [3, 5] Recent research has addressed the senstvty to erroneous measurements by ncorporatng vsblty constrants[9] Current mesh ntegraton technques are computatonally expensve and do not allow relable reconstructon from arbtrary surface measurements such as those obtaned from a hand-held range sensor Volumetrc fuson of surface measurements provdes a general technque for model reconstructon from surface trangulatons [3, 5, 11] Measurements are combned nto a sngle mplct surface representaton whch makes no assumptons about the geometry or topology allowng complex obects to be represented Ths approach enables elable reconstructon of complex obect models wthout loss of accuracy [5] Dscrete mplct surface representaton gves an order of magntude reducton n computatonal cost [3] Dscrete representaton causes a reducton n accuracy resultng n loss of surface detal and holes n the reconstructed model at crease edges or thn surface sectons [3, 11] Prevous dscrete representatons approxmate the mplct surface at a sngle resoluton for both complex geometrc features and smooth regons Representaton costs are prohbtvely expensve for accurate reconstructon of small surface features on large obects Ths s a crtcal for relable reconstructon of detaled envronment models and accurate reverse engneerng Mult-resoluton geometrc fuson addresses accurate reconstructon usng a dscrete mplct surface representaton The prncpal obectve s accurate reconstructon ndependent of obect sze A sngle resoluton fuson algorthm s presented n secton 2 for the constructon of a dscrete mplct representaton from a set of arbtrary trangulated meshes Ths approach enables fuson of surface measurements from both conventonal range mages and a hand-held range sensor An effcent herarchcal structure for mplct surface approxmaton wth bounded error s ntroduced n secton 3 Unlke prevous herarchcal feld functons ths representaton provdes effcent surface approxmaton based on local surface geometry for a volumetrc envelope around the obect surface Mult-resoluton fuson wth bounded error s acheved by combnng the herarchcal representaton wth the sngle resoluton algorthm Valdaton of the fuson algorthm and results for real obects are gven n secton 4
q b e q 2 Sngle-Resoluton Geometrc Fuson Ths secton presents an algorthm for fuson of an arbtrary set of trangulated meshes nto a sngle resoluton dscrete mplct surface representaton Incremental trangleby-trangle transformaton usng the normal volume s used to construct a volumetrc envelope around the surface An effcent sngle resoluton volumetrc surface representaton s obtaned whch s vewpont ndependent unlke prevous methods [3] s vewpont ndependent The ncremental algorthm enables geometrc fuson of surface measurements from both conventonal range mages or a hand-held range sensor 21 Volumetrc Surface Representaton An arbtrary topology closed manfold surface can be represented n mplct form as an so-surface of a spatal feld functon,, where s any pont n Eucldean space, Thus we can represent a surface by defnng the feld functon as the sgned dstance from a pont,, to the nearest pont on the surface gvng the so-surface for all ponts on and elsewhere A dscrete volumetrc representaton can be mplemented by unform spatal subdvson nto voxels cells and local planar approxmaton for voxels near the mplct surface, Let us defne a voxel grd based on a unform spatal subdvson wth voxel centres!"$#$% gven by: ')( *+,-$/10 23 /4+65 798 :!*<;,9=>/10 2?3 /4;$5@7A8B:!*DCE,GFH/10 23 /ICJ5 798LK (1) The dscrete representaton s defned n the range JMNQJR wth unform voxel resoluton n each JMNPO spatal dmenson S T S S S The number of voxel cells n each dmenson s gven by WV R XZY"[$?R X>]A^_ `Ha VPb XcY![$ X>]9^d_ `Ha VAe XcY![$ X>]A^d_ `Ha The N volume U occuped by the voxel cell correspondng to centre denoted fg h > S $" #N S $ %h S k For a gven pont we can evaluate the correspondng voxel ndex ī by: 'l(,+nmo+ 5 798 3 * :,;mp; 5@7A8 3 * :,GCNmC 5 798 3 * K (2) Volumetrc surface representaton s acheved by local planar approxmaton of the surface n the dscrete voxel structure Gven an nput trangulated mesh q composed of a set ov vertces r s tu vavpvp t!wvpvavp tlx@y>z and trangles { }ulwvpvavp ~# vavpva WxZ>z where ~# t d tlƒ$ td We assume that s a smply connected manfold trangulaton wth no selfntersectons Thus we can estmate the local surface normal for each trangle as V} ˆ ˆŠ _" VŒ ˆ ˆ"Š _ ŽŽ VŒ ˆ ˆŠ _" VŒ ˆ ˆ"Š _ ŽŽ Vertex normals can be estmated by a weghted average of the adacent trangle normals [15]: Š ' J š k $ B (3) M+ M M- (a) Volumetrc envelope t+ r+ s+ n n n t r r- s s- (b) Normal volume Fgure 1 Volumetrc surface representaton A volumetrc envelope s defned around the mesh q whch enables us to convert to an mplct volumetrc representaton An offset surface qœ for mesh q s gven by dsplacng each mesh vertex by a dstance n the vertex normal drecton such that tbž > If we let to be a Ÿt constant offset dstance MNQJR then the dstance of all ponts on the offset surface qœ s less than or equal to the offset dstance MNQJR from the orgnal mesh q The offset surface s a contnuous mesh but may not be a smple manfold due to self-ntersecton We can now defne a volumetrc envelope around q by two offset meshes q and q such that each vertex s dsplaced by a dstance MNQJR and h MNQJR n the normal drecton respectvely The space enclosed by q and s a closed volumetrc envelope such that every pont nsde ths regon s less than MNQJR from the mesh q Ths s llustrated for a cross secton through a mesh n Fgure 1(a) For each trangle # t t ƒ t n mesh q the offset meshes q and q defne a closed normal volume, f x Œ, between # t t ƒ t and # t t ƒ t The normal volume for a trangular element s llustrated n Fgure 1(b) The concept of a volumetrc envelope and normal volume or fundamental prsm were recently used by Cohen et al[2] to defne an envelope for mesh smplfcaton wth bounded approxmaton error Each sde of the normal volume s a surface ƒ constraned by three vectors the trangle edge ƒ t ƒ t and the correspondng vertex normals and ƒ If the vertex normals are equal ƒ then the surface ƒ s a plane In general f the normals are not equal we can satsfy the constrants by defnng the surface ƒ as a blnear patch [2] The volumetrc envelope for mesh q between offset meshes q and q s equvalent to the unon of the normal volumes for all trangles t-
u x f x Œ Therefore, transformng the mesh q to an approxmate volumetrc representaton can be reduced to a ncremental process of transformng the normal volume for each trangle, as follows: 1 Evaluate the normal volume, 3 2 Fnd the set of voxel centres nsde the normal volume, :, 3 3 For all voxel centres nsde the normal volume : 7 : construct a local planar surface approxmaton from the nearest pont on and the correspondng normal, ( : K : To ensure that the normal volume, f x #, encloses all ponts less than MNQJR from # we can evaluate the vertex offset dstance,, as: XZY"[ VW O Š Od!_ It s assumed that for any trangle n mesh q the angle between a trangle and adacent vertex normal s to avod the degenerate case where the normal volume reduces to zero To obtan a closed volumetrc envelope for a voxel sze S t s necessary to set the offset dstance MNQJR ES Ths gves a dscrete feld functon representaton for all ponts wth offset dstance less than S of the mplct surface? or mesh q 22 Sngle-Resoluton Fuson Algorthm Fuson of multple overlappng meshes, q u wvpvavp q x, can be acheved usng the volumetrc surface representaton ntroduced n the prevous secton For each mesh q we can defne a closed offset envelope between q and q wth offset dstance MNQJR Thus for each trangle we can defne a a normal volume f x and ncrementally transform the mesh q to a volumetrc representaton In overlappng regons of surface measurements from dfferent meshes the normal volumes wll ntersect provded the maxmum measurement error $MNQJR! MNQJR If ths condton s satsfed we can combne the feld functons from dfferent meshes to obtan a sngle volumetrc representaton Fuson of multple meshes requres an overlap test to determne f surface measurements from dfferent meshes n close spatal proxmty correspond to the same or dfferent regons of the measured obect surface Defnton of a robust overlap test s crtcal for relable surface reconstructon as dscussed n prevous work [5] As n prevous work geometrc constrants are used to estmate f overlappng measurements correspond to the same surface regon based on: 1 Spatal proxmty: dstance between overlappng measurements s less than the maxmum dstance " 5$#% 2 Surface orentaton: overlappng surface normals wth the same orentaton š('*) 3 Measurement uncertanty: lkelhood of measurement overlap based of estmates measurement uncertanty e plane M M (a) Trangle boundary (b) Multple surfaces Fgure 2 Volumetrc representaton error Spatal proxmty provdes a coarse test of measurement overlap whch has been used n prevous work [3] However, ths test s unrelable for sharp edges and for surfaces n close proxmty Surface orentaton enables relable reconstructon of crease edges and thn surface sectons for contnuous mplct surface representaton [5] The use of measurement uncertanty depends on our ablty to relably estmate measurement uncertanty For range mages ths can be estmated from the relatve orentaton of the surface normal and vewpont However, for a hand-held range sensor the vewpont s contnually changng and does not provde a relable estmate of measurement uncertanty If overlappng measurements are determned to correspond to the same surface regon they may be combned accordng to a weghted average [16, 3, 5] or maxmum confdence [9] M 23 Performance of Sngle-Resoluton Gven a mesh q we want to defne bounds for the worst case approxmaton error of transformng q nto a dscrete volumetrc representaton wth voxel sze S accordng to the procedure ntroduced n 21 Two sources of dscrete volumetrc representaton error occur Frstly, voxel ntersecton wth a trangle boundary n a non-planar regon as llustrated n Fgure 2(a) Secondly, voxel ntersecton wth multple mesh regons as llustrated n Fgure 2(b) Ths may occur at crease edges and for thn surface regons In planar mesh regons such as the nteror of a trangle the dscrete voxel representaton s exact The maxmum approxmaton error, MNQJR, occurs for multple mesh ntersectons wth a sngle voxel and s equal to the maxmum voxel sze, MQwR ES Relable reconstructon of complex surfaces therefore requres a small voxel sze The computaton and memory cost of the volumetrc representaton depend on the number of voxels nsde the offset envelope For a unform spatal subdvson of + ˆ voxels the number of occuped voxels for a planar surface s,4 -+ ˆ, a sphercal surface,4 -+ ˆ and for a free-form surface may be even hgher Consequently, for a complex surface the computaton and representaton costs are proportonal to the square (or hgher power) of the voxel sze Conversely the representaton accuracy s nversely proportonal to the voxel sze Therefore, a unform reducton n voxel sze does not provde a satsfactory mechansm for effcent volumetrc surface representaton wth bounded approxmaton error e
z 3 Mult-Resoluton Geometrc Fuson In ths secton we ntroduce a new mult-resoluton dscrete volumetrc surface representaton The obectves of the mult-resoluton approach are: Dscrete mplct representaton of a mesh q wth bounded approxmaton error MNQJR for a closed volumetrc envelope q MQwR Accurate representaton of surface features wth a mnmum voxel sze S Effcent representaton of smooth surface regons wth voxels sze )S Fuson of multple overlappng meshes, q udwvpvpva q x Constructon wthout buldng an ntermedate sngle resoluton representaton at S Representaton cost ndependent of the sze of the nput trangulaton The smplest algorthm for constructng the multresoluton representaton wth bounded approxmaton error s to frst buld a sngle unform representaton at the hghest resoluton S A mult-resoluton representaton wth bounded error,, can then be constructed by ncrementally replacng the hghest resoluton voxels wth correspondng lower resoluton voxels whle lmtng the surface approxmaton error However, sngle resoluton representaton s assumed to be prohbtvely expensve n both computatonal and storage costs A mult-resoluton representaton whch s ndependent of the sze of the nput trangulaton s necessary to ensure that the cost does not contnue to ncrease as new trangles that overlap surface regons already represented are added If explct references to the nput trangles are stored n the volumetrc representaton then the representaton cost s proportonal to the number of nput trangles Cost proportonal to the sze of the nput trangulaton s prohbtvely expensve for large sets of nput trangulatons wth overlap In ths secton we ntroduce a procedure for sequental addton of new trangulated meshes nto a herarchcal volumetrc representaton whle mantanng an upper bound on the surface approxmaton error Ths uses the normal volume approach ntroduced n secton 2 to transform each trangle nto a volumetrc representaton For each voxel nsde the normal volume at the hghest resoluton we determne the largest voxel sze n the mult-resoluton representaton that gves a local surface approxmaton wthn the error bound, 31 Mult-Resoluton Volumetrc Representaton Octrees provde a herarchcal volumetrc data structure for effcent representaton of feld functons for pont clouds Fgure 3 Mult-resoluton representaton n [13] In ths secton we extend the classcal octree representaton to enable effcent herarchcal surface representaton Octree splnes [7] provde a technque for effcent mplct representaton of an obect surface,, as a sgned dstance feld functon, A dscrete feld functon representaton s stored for all ponts nsde a boundng box around the surface, e for the entre space occuped by the obect The feld functon s constructed from a set of ponts unformly sampled on the surface, The herarchcal octree structure stores a dscrete approxmaton of the feld functon wth hgh spatal resoluton near the surface and decreasng resoluton wth ncreasng dstance from the surface The cost of ths representaton wth unform re-samplng of the surface s therefore proportonal to the surface area dvded by the voxel resoluton as dscussed n secton 23 Ths does not satsfy our obectve of effcent mult-resoluton volumetrc mplct surface representaton The new mult-resoluton volumetrc surface representaton presented n ths secton s based on an herarchcal spatal decomposton of a volumetrc envelope around the surface Ths s llustrated n Fgure 3 whch shows a cross secton through the mplct surface, wth detaled features represented at hgh voxel resoluton and smooth surface regons represented at low resoluton The surface approxmaton error s for all ponts nsde the volumetrc envelope of sze MNQJR An octree volumetrc representaton can be defned wth levels wvpvavp" such that the voxel sze at level s gven by S S where S s the voxel sze at the hghest resoluton, For octree subdvson the voxel centres are gven by: 'l( *,-/ 0 2?3 /+65 798B:W*,9= / 0 2?3 /;$5 798 :W*,GF?/ 23 0 /ncj5 7A8K (4) S The volume enclosed by each voxel s f The octree subdvson gves a smple 8:1 correspondence between voxels at level and level for a volumef whch s gven by:
u u f u ' (5) k ī ī As for the sngle-resoluton volumetrc surface representaton we defne a closed volumetrc offset envelope for the mesh q Let the offset dstance be MNQJR MNQJR at level To ensure that at level the feld functon s defned for all ponts nsde the offset envelope at level we must set MNQJR S MNQJR The normal volume f x ~#$ for a trangle # at level s the volume between # and # where trangle vertex t t MNQJR > Thus we can transform a mesh, q, to a sngle resoluton volumetrc representaton at level by sequentally transformng each trangular element The unon of the normal volumes wll form a closed volumetrc envelope around the mesh q Note the maxmum surface approxmaton error for a sngle-resoluton representaton at level s gven by MNQJR ES The problem now remans how to buld an effcent multresoluton representaton wth bounded approxmaton error for all ponts nsde an offset dstance MNQJR We ntroduce a sequental procedure for addng each trangle # n a mesh q nto the mult-resoluton volumetrc representaton Each voxel n the representaton contans a local planar approxmaton,, of the surface nsde that voxel For each trangle # we frst defne the set of voxels at level wth centres nsde the normal volume: vpvpva u vavpv x ~# Next we determne the correspondng voxel for u at level such that u The maxmum approxmaton error for trangle # at level s then computed A smple functon TEST can be defned to determne the maxmum error, for approxmaton of trangle }# by the local planar surface approxmaton of voxel 1 Evaluate the nearest pont on voxel plane, ( : K to each tr- angle vertex, :J: š 3 gvng errors, : : š 3 2 Evaluate the maxmum approxmaton error as: ' + : : š If ths procedure s repeated recursvely for correspondng voxels at levels LwvPvAvP + we can determne the maxmum level n the exstng representaton for whch Ths provdes a mechansm for transformng a trangle # nto a mult-resoluton dscrete volumetrc approxmaton wth bounded error Repeatng ths process sequentally for each trangle n mesh q wll result n a dscrete multresoluton volumetrc representaton The volumetrc representaton s defned for an offset envelope, MNQJR, around the mesh q wth bounded approxmaton error 32 Mult-Resoluton Fuson Algorthm In ths secton we ntroduce a procedure for fuson of multple overlappng trangulated meshes nto a sngle multresoluton representaton Fuson of multple overlappng meshes combnes the sngle-resoluton fuson algorthm ntroduced n secton 22 wth the mult-resoluton representaton presented n the prevous secton We apply a bottomup procedure for addng new voxels nto a herachcal representaton whch ensures that the maxmum error remans bounded The bottom-up procedure starts at the lowest level n the voxel herarchy and tests the approxmaton error for vavpvp" Ths bottom-up procedure s cautous as the frst prorty s ensure that the maxmum approxmaton error remans bounded An alternatve s to use a greedy top-down algorthm to approxmate new trangles at the hghest possble level n the mult-resoluton representaton and to refne the representaton as new trangles are ntroduced A top-down approach s desrable as t gves a consderable reducton n the computatonal cost of buldng the mult-resoluton representaton However, t has been found that the greedy top-down approach does not allow bounded approxmaton error Therefore, a more expensve bottom-up algorthm has been developed For a set of meshes, q @ wvpvpva +, we transform each trangle # nto a dscrete volumetrc approxmaton usng the normal volume approach as follows: 1 Evalutate the normal volume at level zero:!, 3 2 Fnd the set of voxel centers nsde the normal volume: : ī : ", 3 3 For each voxel nsde ī ntroduce a local planar approxmaton of trangle at the hghest level, #, for whch $% and ī usng functon ADD k Steps 1 and 2 above are the same as for constructng a sngle resoluton representaton A recursve nserton procedure, ADD, s ntroduced at step 3 to approxmate # at the hghest level n the octree whch gves a representaton wth bounded error The followng procedure s used to test the addton of new voxels nto the representaton at level to approxmate # or recursvely to test the addton at the next level, Note the recursve procedure s a bottom-up search to fnd the hghest level n the exstng representaton that gves bounded local approxmaton error However, the addton of new voxels s cautous If all sub-voxels at level n the exstng representaton are empty for a correspondng voxel at level then the new trangle s represented at level Functon ADD for recursve mult-resoluton trangle approxmaton s mplemented as follows: 1 Fnd the correpsondng voxel at level # / 0 where 2 If s occuped by a planar approxmaton ( TEST the maxmum approxmaton error, for : : K,
(a) If $ % (b) Else If ' % 3 Else If then FUSE then INSERT s empty: wth at level # (a) Fnd the set of occuped sub-voxels at level # nsde : : : r (b) If there are no occuped sub-voxels INSERT at level # (c) Else If the number of occuped sub-voxels ) : FUSE the sub-voxels : : to form a new canddate voxel: r TEST approxmaton error of : A If $ % ADD and REMOVE sub- voxels B Else f ' % INSERT at level # The above algorthm uses the followng sub-procedures INSERT, REMOVE, FUSE, TEST and ADD INSERT and RE- MOVE modfy the current representaton by nsertng or deletng voxels from the structure repspectvely FUSE uses the fuson algorthm defned for sngle-resoluton volumetrc data to combne data for overlappng surface measurements TEST evaluates the maxmum approxmaton error for trangle W# as defned n the prevous secton ADD s a recursve call to the same procecedure for a voxel at the next hghest level Ths enables a correspondng voxel at level to approxmate multple sub-voxels at level provded the bound on the approxmaton error s satsfed Approxmaton of multple sub-voxels at a hgher level gves a reducton n representaton cost Note the bottom-up herarchcal approxmaton allows a new voxel at level whch wll always gve a surface approxmaton error less than Lower level voxels are only replaced at a hgher level f the approxmaton error s bounded The procedure ntroduced above enables fuson of multple meshes nto an effcent mult-resoluton volumetrc representaton For multple overlappng meshes to be ntegrated correctly the maxmum measurement error n the nput mesh must be less than the offset dstance, MNQJR, at level Ths condton ensures that the normal volumes for overlappng surface measurements ntersect and can be fused nto a sngle representaton An equvalent assumpton s made for the sngle-resoluton volumetrc fuson Sngle resoluton mplct surface polygonsaton [8, 4] has been used throughout ths work to reconstruct an explct trangulated model for vsualsaton Future work wll address mult-resoluton adaptve polygonsaton [1] to obtan an effcent trangulaton 4 Results In ths secton we frst verfy the valdty of the multresoluton representaton and fuson algorthm ntroduced n secton 3 Results are then presented for the use of multresoluton geometrc fuson for reconstructon of 3D surface models of real obects Surface measuremetns are obtaned usng both a conventonal range mage sensor and a hand-held range sensor A comparson s gven of the representaton cost for sngle and mult resoluton approaches 41 Valdaton of Mult-Resoluton Representaton To verfy the accuracy of the mult-resoluton representaton we can construct a volumetrc representaton for a sngle mesh q The dstance between the orgnal mesh and the mplct surface, o, gves a measure of the representaton error The error at every vertex n the orgnal mesh s evaluated and used to compute the root mean square (rms) and maxmum representaton error Note the representaton accuracy s expected to be lowest at the mesh vertces Volumetrc representaton errors for a sngle nput mesh are gven n Table 1 for sngle-resoluton and Table 2 for multresoluton resoluton Throughout ths paper the error tolerance for mult-resoluton representaton s equal to the maxmum error at the lowest resoluton LS The followng observaton can be made from these results The representaon error for mult-resoluton representaton s bounded There s an order of magntude reducton n representaton cost between sngle and mult resoluton for the same maxmum error Although there s a correspondng ncrease n the rms error for mult-resoluton For sngle-resoluton the representaton cost s approxmately proportonal to the square of the resoluton For mult-resoluton representaton cost reducton for dfferent numbers of levels depends on the level of surface detal, e for accurate representaton of the edge voxels at the hghest resoluton are requred Fgure 4 shows the reconstructed model of the edge and sphere for sngle and mult resoluton sphere The edge approxmaton accuracy s approxmately equal for sngle and mult resoluton However, the surface s smoother for sngle resoluton as expected from the lower rms error 42 Valdaton of Mult-Resoluton Fuson The representaton accuracy for fuson of multple overlappng meshes s verfed by comparng the error for fuson at sngle and mult resoluton Results are presented n Table 3 for two smple obects captured usng a hand-held range sensor A factor 3 5 reducton n representaton cost s acheved for mult-resoluton fuson wth three or fve levels The maxmum representaton error remans bounded The rms representaton error s approxmately double for fve levels compared to three levels
0 0 U U nput sze res mem rms max * (Mb) %* %* 2 Edge 9660 ) ) 033 21 1 30 ) ) 067 05 1 16 ) 13 01 13 46 2 Sphere 1490 ) ) 001 68 02 07 ) ) 002 17 02 8 ) 004 05 03 4 Table 1 Sngle-resoluton error nput lev mem rms max * (Mb) %* %* nput Sze res Memory Usage(Mb) * ' 0 ' ' bunny 40K ) ) 04 21 7 6 solder 175K ) ) 04 36 13 12 fan 50K ) ) 084 41 11 9 dwarf 420K ) ) 12 83 17 8 Table 4 Fuson for complex obects + Edge 9660 033 06 12 70 04 12 90 03 37 136 Sphere 1490 001 05 1 3 005 3 18 004 40 63 L Table 2 Mult-resoluton error Fgure 5 3D models for complex obects 43 3D Model Reconstructon Fgure 4 Reconstructon for sngle(top) mult(bottom > ) representaton nput lev res mem rms max * (Mb) %* %* Ths secton presents results for the mult-resoluton fuson of surface measurements for complex real obects Table 4 gves the representaton cost for four obects for fuson wth dfferent numbers of levels A factor three reducton n representaton cost s obtaned for mult-resoluton fuson Fgure 5 shows the reconstructed 3D models for mult-resoluton fuson wth There s no vsble reducton n the qualty of the reconstructed model wth the mult-resoluton fuson approach The bunny and solder models were constructed from multple range mages from Cyberware[16] and NRC[10] scanners The fan and dwarf models were constructed from hand-held range sensor data collected usng the 3D Scanners, ModelMaker system Cube 14K 0 032 22 7 7 93 4 13 115 Sphere 32K 0 02 48 10 25 87 9 56 137 Table 3 Mult-resoluton fuson + L
5 Conclusons A sngle-resoluton geometrc fuson algorthm has been presented whch allows volumetrc mplct surface representaton of a set of trangulated meshes An ncremental trangle-by-trangle procedure s ntroduced for convertng an arbtrary mesh to a volumetrc representaton The normal volume s used for effcent transformaton of each trangle to a local volumetrc representaton The unon of the normal volumes form a volumetrc offset envelope around the surface Ths approach has the advantage over prevous vewpont based technques that the volume s ndependent of the surface orentaton allowng effcent representaton Ths algorthm enables fuson of surface measurements from both conventonal range mages and a hand-held range sensor Measurements from a hand-held range sensor are a seres of unstructured strpes wth overfoldng A herarchcal octree structure has been developed for effcent volumetrc surface representaton wth bounded approxmaton error The surface s locally represented by the largest voxel sze wth a planar patch that fts the surface measurements wthn an error tolerance Ths enables effcent approxmaton for a volumetrc envelope around the surface Prevous herachcal mplct surface representatons [1, 7] approxmated the feld functon throughout the entre regon of space occuped by the obect Results demonstrate that the herarchcal approxmaton acheves an order of magntude reducton n representaton cost wth accurate representon of local surface geomtry for complex features Mult-resoluton geometrc fuson s achved by combnng the herachcal volumetrc surface representaton wth the ncremental surface to volume transformaton A cautons bottom-up algorthm s ntroduced to obtan effcent surface representaton whlst mantanng a tolerance on the surface approxmaton error Ths algorthm gves computatonal cost approxmately equal to the sngle resoluton reconstructon at the hghest resoluton Results demonstrate that ths approach acheves a sgnfcant reducton n representaton cost wthout a sgnfcant reducton n accuracy Mult-resoluton reconstructon of real obect models usng surface measurements from both range mages and a hand-held range sensor gves a factor three reducton n representaton cost Further research s requred to dentfy more effcent algorthms for constructng the mult-resoluton representaton and achevng further reductons n representaton costs 6 Acknowledgements Ths reasearch was supported by EPSRC Grant GR/K04569 Fnte Element Snakes for Depth Data Fuson The authors wsh to thank 3D Scanners Ltd,UK for testng and feedback of fuson algorthms In partcular Peter Champ who dentfed lmtatons of sngle resoluton fuson References [1] J Bloomenthal Polygonzaton of mplct surfaces Computer Aded Geometrc Desgn, 5:341 355, 1988 [2] J Cohen, A Varshney, D Manocha, G Turk, H Weber, P Agarwal, F Brooks, and W Wrght Smplfcaton envelopes In SIGGRAPH, pages 119 128, 1996 [3] B Curless and M Levoy A volumetrc method for buldng complex models from range mages In Computer Graphcs Proceedngs, SIGGRAPH, 1996 [4] A 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