Multiscale Implicit Functions for Unified Data Representation

Transcription

1 KSII TRANSACTIONS ON INTERNET AND INFORMATION SYSTEMS VOL. 5, NO. 1, December Copyrght 011 KSII Mltscale Implct Fnctons for Unfe Data Representaton Seongmn Yn an Sanghn Par Department of Mltmea, Dongg Unversty Seol Repblc of Korea [e-mal: *Corresponng athor: Sanghn Par Receve Jly 14, 011; revse September 9, 011; revse November, 011; accepte December 7, 011; pblshe December 1, 011 Abstract A varety of reconstrcton methos has been evelope to convert a set of scattere ponts generate from real moels nto explct forms, sch as polygonal meshes, parametrc or mplct srfaces. In ths paper, we present a metho to constrct mlt-scale mplct srfaces from scattere ponts sng mltscale ernels base on ernel an mlt-resolton analyss theores. Or approach ffers from other methos n that mlt-scale reconstrcton can be one wthot atonal manplaton on npt ata, calclate fnctons spport level of etal representaton, an t can be natrally expane for n-mensonal ata. The metho also wors well wth pont-sets that are nosy or not nformly strbte. We show featres an performances of the propose metho va expermental reslts for varos ata sets. Keywors: Graphcs, mltmea, compter algorthms Ths research s spporte by Mnstry of Cltre, Sports an Torsm (MCST) an Korea Creatve Content Angency (KOCCA) n the Cltre Technology (CT) Research & Development Program 011, an Basc Scence Research Program throgh the Natonal Research Fonaton of Korea (NRF) fne by the Mnstry of Ecaton, Scence an Technology ( ). DOI: 10.87/ts

2 KSII TRANSACTIONS ON INTERNET AND INFORMATION SYSTEMS VOL. 5, NO. 1, December Introcton The am of mplct srface reconstrcton s to obtan an mplct fncton from nconstrcte scattere ata sng nterpolaton or approxmaton. A srface can then be expresse as the zero level-set of the fncton. Most software for srface reconstrcton s base on raal bass fnctons (RBFs) or movng least sqares (MLSs). An mplct srface s the zero level-set of a contnos scalar-vale fncton f that may be an approxmaton or nterpolaton of a ata set. We nee to fn a fncton f that satsfes f ( x ) = 0, = 1,, N to reconstrct the mplct srface from a scattere pont set X { x,, x }. In ths case, f s sally forme from a combnaton of bass fnctons, = 1 N arrange so that f has postve vales nse an negatve vales otse the obect. Technqes base on RBFs are wely se [1][][][4][5][6] e to the accracy of approxmaton, flexblty n the choce of bass fnctons, an convenence n evalatng the approxmant. A lnear system nees to be solve to fn the weghts of the RBFs. Carr et al. [1] sggeste the so-calle fast mltpole metho that s base on globally spporte RBFs bt t s har to mplement. Morse et al. [] se a compactly spporte RBF [7] to obtan a sparse nterpolaton matrx. A MLS approxmaton s a fncton that s a solton of a locally weghte least-sqares problem [7][8]. Ths powerfl metho has been combne wth a proecton operator to moel pont-set srfaces [9][10][11]. However, MLS approxmaton generates a smooth fncton: to preserve sharp featres, Fleshman et al. [1] apple MLS n a pecewse manner, whle Lpman et al. [1] se an aaptve splne space n ther MLS approxmaton that ncorporates a novel featre etecton metho for scattere ata. In ths paper, we present a ata representaton technqe base on mltscale ernels that have compactly spporte, non-symmetrc propertes [14][15]. Snce the ernels spport mlt-level representaton, gven n-mensonal ata can be converte to mplct fnctons as nfe forms wth the level of etal. Ths approach offers algorthms for varos graphcs applcatons, an spports mlt-resolton representaton that are especally sefl for real-tme splay, aaptve refnement, an progressve transmsson. Users can access fncton vales at arbtrary samples from the nfe ata strctres, an get renere mages of the ata by tratonal graphcs technqes. Also, removng nsgnfcant coeffcents by thresholng allows to compress the mplct srface. The scheme wors wth pont-sets whch are nosy or non-nformly strbte. We explan the mathematcal bacgron an theores n Secton. Secton escrbes or reconstrcton algorthm, consstng of pre-processng an evalaton stages. Secton 4 presents expermental reslts on volme ata as D an tme-varyng volmetrc ata as 4D. Fnally, Secton 5 has conclng remars an otlnes ftre wor.. Mltscale Kernels.1 Mathematcal Prelmnares Kernels are valable tools for geometrc moelng, machne learnng an nmercal analyss, an ther propertes have been wely ste n the mathematcal lteratre [7][16]. A ernel s a symmetrc fncton K : R, where can be an arbtrary nonempty sbset of R.

3 76 Yn et al.: Mltscale Implct Fnctons for Unfe Data Representaton Gven a scattere pont ata set fn a contnos fncton f * N X { x,, x } an a set of vales f,, f }, we want = 1 N { 1 N * f : R R efne as a lnear combnaton of ernels = K(, x ), where the coeffcents are etermne by the nterpolaton conton =1 * f ( x ) = f, =1,, N. To mae ths lnear system non-snglar, the ernel K has to be postve efnte so that for every fnte set { x 1,, x n } of parwse stnct ponts n, the matrx ( K( x, x )) 1, s postve efnte. In general, raal ernels, terme RBFs, are wely n se, becase they are stable ner translaton an rotaton of the pont-set. In ths paper, we se the mltscale ernel (MSK) propose by Opfer [17]. Let the fncton : R R be bone, compactly spporte, an refnable, so that there s a seqence { h } of real nmbers for whch = (). Let Z be a fxe nteger an Z h Z := { } be a seqence of weghts wth = > 0 an <. Then the fncton ( x, y) := = ( Z = ( x ) ( y )) s a mltscale ernel an t s postve efnte. Some restrctons are reqre to se ths ernel n practce. Conser a fnte mltscale ernel, sch that ( x, y) := ( ( x ) ( y )), (1) = Z where the refnable fncton s chosen, so that the mas { h } s fnte (e.g. the tensor proct of nvarate B-splnes). Then s compactly spporte, an for X = { x 1,, x N } wth 1 mn x x > r, () the matrx ( ( x, x )) 1 s postve efnte, where, N spp( ) { x R : x c < r}, an r s the ras of spport of fncton. The avantage of a mltscale ernel les n ts strctre. Snce s refnable, the * nterpolant f, efne as a lnear combnaton of mltscale ernels, possesses a wavelet-le mltresolton ecomposton that permts fast evalaton: where the coeffcents f * N =1 ( c = Z = ( x, ) = ( c are gven by N =1 )) =: = s, () c := ( x ),, Z. (4) * The roghest approxmaton to f s s, whle the terms s wth hgher vales of the nex * prove sccessvely more etale representatons of f. For example, conser a fncton f ( x) = cos( x)sn( x) an 15 sample ponts (see Fg. 1-(a)). We se a qaratc B-splne wth the weghts = 1/4, = 0,, n Eqaton (1). Then we have a MSK nterpolant f * 15 = ( x, ) = s (Fg. 1-()). Moreover, s 0 s the =1 0 =0

4 KSII TRANSACTIONS ON INTERNET AND INFORMATION SYSTEMS VOL. 5, NO. 1, December roghest approxmant to the test fncton, an s0 s1 s a fner approxmant than s 0. (a) f ( x) = cos( x)sn( x) (b) s (c) 0 s0 s () 1 s0 s1 s Fg. 1. Test nterpolant evalate n 1D. Snce ( x, ) s compactly spporte, ( ( x, x )) 1, N s a sparse matrx an so the coeffcents can be calclate sng an approprate nmercal metho. Moreover, for each { Z : c 0}, there exsts a nqe x X sch that ( ) 0. Usng a -mensonal tree search, we can calclate ths x n O ( N log N) operatons, an so easly obtane. For 1, c s gven by the same eqatons 1 c := h c 1 Z x c s that nerle the fast wavelet transform algorthm. After an ntalzaton process that reqres * O ( N log N) operatons, the fncton f of Eqaton () can be evalate n O (1) operatons. Snce the magnte of s less at hgher levels, contrbton of s low. When or metho s se, the choce of approprate s crtcal for mltresolton representaton. In aton, there s a lmtaton n that reconstrcton reslts for each level are not contnos, becase mltresolton representaton epens on nteger. (5).1 Overvew. Reconstrcton Algorthms Or scheme conssts of two stages: pre-processng an evalaton. The goal of the pre-processng stage s to etermne the coeffcents an c of a MSK nterpolant from the gven n-mensonal ata set. The entre npt oman s sbve nto small sbomans herarchcally by the -tree strctre to o ths. Then, we constrct an solve lnear systems base on mltscale ernel for each sboman. Accorng to propertes of mltscale ernels, approprate ata strctres an nmercal technqes shol be mplemente to search nonzero elements from the sparse matrx an to solve the lnear system effectvely snce the lnear systems compose sparse matrces. Once coeffcents of the mplct fnctons for the hghest etal level are fon, coeffcents for the other levels can be smply calclate by Eqaton (5). Well-esgne ata strctres are neee to store an extract these coeffcents, snce they are sparse n nteger grs. Once the mltscale mplct fnctons for each sboman are constrcte, fncton vales at arbtrary poston n overall oman can be evalate by partton of nty metho that calclates a reconstrcte fncton vale by weghte sm of local fnctons for neghbor

5 78 Yn et al.: Mltscale Implct Fnctons for Unfe Data Representaton sbomans. Users can control the level of etal by choosng an approprate level for the coeffcents se. Fg. shows the pseocoe to represent the overall processng. In general, the pre-processng stage reqres an apprecable amont of comptaton (lne : 9); bt an evalaton of the fncton vale at an arbtrary n-mensonal pont can be one mch faster than can the pre-processng steps (lne 10). 1: procere MLTSCL_KRNL_SRFC( {( x, f)} ) : m Decompose the oman : for 1, m o 4: Bl p the lnear system for f (Eqaton ()) 5: Solve the lnear system (Eqaton ()) 6: Calclate the coeffcents (Eqaton (4)) F F f 7: 9: en for 10: Ray-trace or polygonse F wth LOD 11: en procere Fg.. Pseo coe for mltscale srface reconstrcton.. Calclatng Implct Fnctons We partton the gven ata nto smaller sbsets by sng partton of nty (POU) to eal wth large npt ata on compters wth lmte memory space. In the POU metho, the global m oman s ve nto small sbomans, sch that, wth slght overlaps = 1 among the sbomans. It s necessary to overlap them to blen the local reconstrcte fnctons [18]. We can then assocate a POU wth these sbomans, n the form of a famly of m non-negatve, contnos weght fnctons w, sch that spp( w ) an w ( x) = 1 =1 for all x. For each, we constrct a local approxmaton f, an the global approxmant on s forme by weghtng these local approxmants as follows: m F( x) = w ( x) f ( x). (6) =1 The weght fncton (a) -tree base herarchcal (b) a cttng plane after oman ecomposton space partton of gea D pont ata Fg.. Herarchcal oman ecomposton n the processng. w can be fon by normalzaton of smooth fncton W, as follows:

6 KSII TRANSACTIONS ON INTERNET AND INFORMATION SYSTEMS VOL. 5, NO. 1, December w ( x) = W ( x) W ( x) sch as W s contnos at the bonary of sboman. The -tree s se to ecompose the oman, becase t s applcable to n-mensonal ata (Fg. -(a)). We mae recrsve sbvson of the -tree stop when each sboman ncles less than a certan nmber of ponts. For example, gea ata wth 4,804 ponts was parttone nto sbomans when we set each sboman to have at most 1,000 ponts (Fg. -(b)). In fact, each oman has abot 775 ponts an the stanar evaton s approxmately 10. After parttonng the oman nto sboman, a lnear system A = Y s blt p for by Eqaton () where has n ponts: x, x ) ( x, x ) y l ( 1 1 l 1 ( x, x ) l n 1 1 n 1 =. (, ) l xn xn n yn Matrx A s sparse symmetrc matrx, snce the mltscale ernel s a compactly spporte an symmetrc fncton. The only elements satsfyng ( x, x ) 0 shol be calclate to mnmze the cost of composng matrx A. We nee to fn x ncle n the mltscale r ernel spport 0.5 r where r s the ras of spport of fncton. The -tree an PARDISO [19], whch was evelope to fn a solton of sparse lnear system, an base on LU ecomposton, were se for effcent searchng an solvng the lnear system, respectvely. We extene PARDISO to an enhance parallel solver on share memory mltprocessors. After fnng soltons of the lnear systems, we can calclate the coeffcents { c : Z, l 1} efne n Eqaton (5) from hgher level 1 to lowest l. In each level, the nmber of coeffcents satsfyng c 0 s Cn where C := max{ Z : ( x ) 0}, an n s the nmber of ata ponts se n reconstrcton. The xr coeffcents n the hghest level are generate by Eqaton (4) for, sch as c 0 : x c n = ( =1 x ). Snce spp( ) { x R : x c < r} an ( r) < 0.5 mn x x, x X s nqe sch as ( ) 0. The other coeffcents c of lower levels where l 1, can be generate recrsvely by Eqaton (5). These coeffcents are sparse ata efne on nteger grs, an are neee to evalate fncton vales at arbtrary postons. We se hash tables as a ata strctre to extract the coeffcents ranomly. In ths table, the ey s nex Z of nteger grs n -menson, an retrn vale s a nonzero coeffcent c.. Evalatng Fncton Vales We fn all sbomans } { nclng x by searchng bonary volme herarchy to evalate a fncton vale at an arbtrary poston x (Fg. 4-(a)). Then, the fncton vales l (7) (8)

7 80 Yn et al.: Mltscale Implct Fnctons for Unfe Data Representaton f (x) can be calclate from each local mplct fncton efne n { } (Fg. 4-(b)). The fnal vale s the weghte sm of local fncton vales: F( x) = w ( x) f ( x) (Fg. 4-(c)). We se ecay fnctons n the partton of nty propose by Tobor et al [18] as weght fnctons. These evalaton steps are one only for the nces, sch that ( x ) 0, to enhance rn-tme processng spee. The nces satsfyng the above conton were se for the hash tables eys. In aton, the fncton f = s s a complete nterpolant. We can = l control LOD by mofyng the pper bon, as follows: ~ f = s, l <. = l (a) oman search from the herarchcal strctre (b) calclatng local reconstrcte fncton vales (c) weghte smmaton of fncton vales Fg. 4. Evalaton step for fnng a fncton vale. Fg. 5 shows the reslts of a smple test vsalzaton n whch a mlt-resolton representaton of gtal elevaton moel (DEM) ata wth resolton was proce by an MSK nterpolaton, base on the tensor proct of qaratc B-splnes wth (1.8) =, l =, = 0. Mltresolton reconstrcton was one wth for levels, an accorng to Eclean stance between camera an evalaton pont, the egree of etal can be selecte atomatcally. Srface normals se n renerng were calclate from the graent of the mplct fnctons. Ths program ran n real-tme. 4.1 Unorganze Pont-sets 4. Expermental Reslts In ths secton, we explan how to create mplct srfaces from scattere ata sets, sch as gea (8,68 ponts), bnny (5,947 ponts), an sqrrel (9,995 ponts). If only gven ponts are se for srface reconstrcton, the lnear system n Eqaton (8) has st a trval solton. To avo ths problem, the ponts, whch have an approprate offset to normal recton from orgnal ponts an assgne fncton vales -1 an 1, are se as atonal npt ata. If the orgnal nmber of ponts s n, then the nmber of processe ponts s n. We sbve D space to mae the leaf noe n the -tree to have less than 1,000 ponts, each corresponng oman was overlappe wth aacent omans at abot 16% ratos. A local mplct fncton s blene wth neghbor fnctons n C contnty by partton of nty. Table 1 shows the parameters se n the experments. Fncton s the tensor proct of cbc B-splnes. The parameter shol be etermne for reconstrcte fnctons to spport level of etal representaton expermentally. The ata epenent parameters, an l, shol be careflly

8 KSII TRANSACTIONS ON INTERNET AND INFORMATION SYSTEMS VOL. 5, NO. 1, December chosen to mae the lnear system a postve efnte an sparse matrx. level polygonal renerng textre mappng 1 4 Fg. 5. Mltscale reconstrcton of DEM ata. Table 1. Parameters se for test D scattere pont-sets reconstrcton experments an pre-processng tmes for orer of B-splne n secons. pre-processng parameter orer of B-splne ( s) l = 1 gea 4 1 s = sqrrel 4 1 s = The choce of of Eqaton (1) s a crtcal sse n the applcaton of MSKs. If we se too small, then the conton expresse by Eqaton () s not satsfe an the nterpolaton matrx s snglar. However, f s too large, then there s naeqate spport for MSK an the nterpolant wll be of low qalty. We ntroce a scale mltscale ernel to overcome ths problem, as follows: ( x/ s, y/ s) = = l ( Z ( x/ s ) ( y/ s )). Scale parameters s are etermne to garantee the nonsnglarty of the nterpolaton matrx, as follows:

9 8 Yn et al.: Mltscale Implct Fnctons for Unfe Data Representaton where spp( ) { x R : x c < r}. 1 s = mn x x / r, (9) After a sample volme ata wth nform gr strctre was create by evalatng fncton vales f ( p) for all voxel ponts p = ( x, y, z), an sosrfacng algorthm, sch as Marchng Cbe, was se to compte an sosrface from the volme. Fg. 6 shows mages of srfaces reconstrcte from the gea an bnny ata sets at fferent levels of etal ( level =,, 4, an 6). Or metho ffers from the other technqes, sch as [4][18], n that any manplaton operatons, sch as selecton, an eleton, are not necessary on npt ata to create mlt-level mplct fnctons. However, few artfacts appear on renere srfaces of the lowest level at the bonary of sbomans. The hgher the level, the more smlar are the renere mages, becase the weght approaches zero as the level ncreases. Ths, an algorthm that atomatcally fns optmal shol be evelope. In general, the pre-processng tme s proportonal to the nmber of ponts an the orer of the B-splne (see Table 1). Althogh we mplemente the pre-processng algorthm wthot sophstcate optmzaton technqes sch as explotng GPU or parallelzaton, t taes abot 1 mnte (gea, sqrrel) or less than 6 mntes (bnny) for cbc B-splne. Fg. 6. Srface reconstrcton from scattere D pont ata at for fferent levels of etal (level =,, 4, an 6).

10 KSII TRANSACTIONS ON INTERNET AND INFORMATION SYSTEMS VOL. 5, NO. 1, December Table shows tme performances for evalaton at sample ponts n approprate resolton grs to generate a volme to be sosrface. The expermental envronment s a general prpose Wnows estop wth Intel Core 7.80 GHz, an 4.0 GB memory. Table. Evalaton tmes from mplct fncton n secons (Intel Core 7 processor,.8 GHz: man memory, 4 GB). level of etal gea ( ) sqrrel ( ) bnny ( ) (a) MSK (b) FastRBF Fg. 7. Comparson of renere mages of srfaces generate by MSK an FastRBF. We analyze or metho's tmng performance compare to the FastRBF toolbox that mplemente RBF by the fast mltpole metho [1]. Two tmng performances were consere on han ata wth 15,000 scattere ponts: (1) reconstrcton of mplct fnctons, an () evalaton of fncton vales. All ponts n the gven ata were se n srface reconstrcton wthot the center recton technqe for srface reconstrcton by FastRBF. 1 The mltscale ernel create by the tensor proct of qaratc B-splne s se for or 4 metho. The pre-processng tmes of two methos are.8 (MSK) an 11.5 (FastRBF) secons, so FastRBF s twce as fast as MSK. We harnesse Intel Threang Blng Blocs [0] for mlt-core programmng to enhance CPU processng spees (t taes 49.4 secons to pre-process the same ata wthot mlt-core programmng). In aton, f the state-of-the-art technqes sch as sparse lnear solvers [1], real-tme hashng [], an parallel - trees [] mplemente on hgh-performance GPUs are employe n or pre-processng stage, the tmng performances wol be remarably mprove. Conversely, the evalaton tmes of fncton vales n or metho are better than for FastRBF (see Table ). Theoretcally, both methos can be evalate n O(1) operatons after a prelmnary comptaton reqrng O ( N log N) [1][17]. Fg. 7 shows the renere mages of srfaces evalate by the methos. It s ffclt to notce the fferences between them. The maxmm reconstrcton errors max y f x ) between orgnal npt an 4 4 evalaton ponts are (MSK) an (FastRBF). MSK nterpolaton performs poorly on a non-nform ata set f ts ras of spport s chosen (

11 84 Yn et al.: Mltscale Implct Fnctons for Unfe Data Representaton globally. A non-nform ata set can be hanle by MSKs wth a fferent ras n each cell of a partton: the scale parameter s can then be etermne from Eqaton (9) sng ponts x from each cell. Fg. 8-(a) shows the mofe non-nform ata set of gea. The sample ata set was create by removng ranomly selecte ponts, a lnear fncton proportonal to the postve recton of an axs was se to control the nmber of ponts to be elmnate. Many small parttons ten to concentrate n the regons where the ensty of ponts s relatvely hgher. Conversely, the parts where pont strbton s sparser are ve by some bgger parttons. Approprate ernels wth fferent ra of spport for each partton can be ynamcally selecte. Fg. 8-(b) to Fg. 8-() show srfaces reconstrcte n fferent levels. Even thogh the gven ata set s non-nform, the renere mages are sffcently goo. Table. Comparson of evalaton tmes between MSK (level 8) an FastRBF n secons on han scattere pont ata (S: 9 646, M: , L: ). resolton of samples S M L FastRBF MSK (a) non-nformly strbte npt ponts (b) level (c) level () level 6 Fg. 8. Mltscale srface reconstrcton from non-nform scattere ponts. (a) ncomplete mesh wth holes (b) hole-fllng reslt by or metho (4,566 vertces) Fg. 9. Reconstrcton of polygonal mesh wth partally mssng parts. Or metho can be apple to reconstrct an ncomplete polygonal mesh. Fg. 9-(a) an 9-(b) show an npt test mesh of 4,566 vertces wth partally mssng polygons, an the resltant mages after applyng or srface reconstrcton metho, respectvely. In ths (1.6) experment, the parameters are as follows: =, l =, = 16, an s a tensor proct of cbc B-splne. Larger spports of reconstrcton fnctons can nterpolate we

12 KSII TRANSACTIONS ON INTERNET AND INFORMATION SYSTEMS VOL. 5, NO. 1, December areas an fnally the holes are flle, snce the lower lmt l s set a relatvely small vale. However, that cases the lnear system to be ense, as well as ncreasng comptatonal costs. We ae nose to the bnny ata set wth pertrbaton ntentonally. Fg. 10 shows the test ata an reconstrcte srfaces. The renere srface s clearly affecte by nose, bt the slhoette, featres are escrbe aeqately, an the reconstrcte srfaces exhbt the characterstcs of nosy ata. The e-nosng effect n reconstrcte srfaces at level 4 (Fg. 10-(b)) s better than that n srfaces at level 6 (Fg. 10-(c)). An approprate mem-level reconstrcton of mplct srfaces create from nosy ata set can be mch more sefl, especally f the moel appears n a small porton of the entre splaye mage. (a) test ata wth nose (b) lower level showng e-nosng effect on (level 4) (c) hgher level wth notceable nose (level 6) Fg. 10. Srface reconstrcton from scattere ponts wth nose. 4. D Volmetrc Data Sets Volme vsalzaton extracts an reners meanngfl nformaton from volmetrc ata consstng of fncton vales f at voxel grs v (,, ). In general, the volme renerng process reqres access of fncton vales f ( p) at arbtrary sample ponts p ; they can be calclate by varos nterpolaton technqes. In ths secton, we emonstrate that or metho can effcently fn nterpolate fncton vales at any ponts, an mch better renere mages can be generate. We renere bcyball volme ata efne on a nform voxel gr sng Marchng Cbe algorthm. Fg. 11-(a) an (b) show sosrfaces for an sovale of 0.5, an the vertces of the srfaces were generate from trlnear nterpolaton on an 18 grs respectvely. We apple or srface reconstrcton metho to the volme ata wth voxel coornates (,, ) an fncton vales f. Mltscale ernel, whch conssts of v 1 cbc B-splne tensor proct, was se for ths experment. After fnng fncton vales from reconstrcte mplct fnctons at 18 sample ponts on nform grs, we can rener sosrfaces from the new volme (See Fg. 11-(c)-(f)). We can fn from the mages that nterpolaton base on mltscale ernel s mch better than commonly se trlnear methos, an the renerng reslts show lttle alasng. The fference n qalty s e to the trlnear nterpolaton conserng only eght voxel vales srronng a sample pont, whle n contrast, mltscale base mplct fnctons are create by all voxel vales ncle n the ernel spport se. Frthermore, the level of etal representaton can be sefl n varos tratonal volme renerng technqes, sch as ray-castng an splattng. In aton, f or metho s apple to D volme ata efne on a non-nform rreglar gr, whch reqres an atonal process or complex hanlng, the ser can effectvely vsalze the ata by ranomly accessng the fncton vales at any pont. The renere mages n Fg. 1 were create for engne volme ata of resolton ner smlar contons se to vsalze bcyball n Fg. 11. The pre-processng tmes for bcyball an engne volme ata are 41 an 179 secons respectvely, Table 4 evalates performance for the access fncton

13 86 Yn et al.: Mltscale Implct Fnctons for Unfe Data Representaton vales. (a) orgnal ( ) (b) tr-lnear nterp. ( 18 ) (c) level 4 ( 18 ) () level ( 18 ) (e) level ( 18 ) (f) level 1 ( 18 ) Fg. 11. Mltscale reconstrcton of a volme ata bcyball ( ). (a) orgnal ( 64 ) (b) tr-lnear nterp. ( ) (c) level 4 ( ) () level ( ) (e) level ( ) (f) level 1 ( ) Fg. 1. Mltscale reconstrcton of a volme ata engne ( 64 ).

14 KSII TRANSACTIONS ON INTERNET AND INFORMATION SYSTEMS VOL. 5, NO. 1, December Table 4. Evalaton tmes n secons an maxmm errors as a measre of nterpolaton qalty from or mltscale fncton of the test volme ata. level 1 4 Evalaton tmes (n secons) Maxmm errors ( max y f x ) ) ( 4. Tme-varyng Data Sets bcyball ( 18 ) engne ( 18 ) bcyball ( ) engne ( 64 ) It s natral that or technqe can be apple to srface representaton of n-mensonal ata, snce mltscale ernels spport n-mensonal ata nterpolaton an approxmaton. In ths secton, we show that or metho can be sefl for 4D ata, sch as tme-varyng D volmetrc ata set. We constrcte three volme ata sets, whch have voxels wth fncton vales f ( x, y, z) from qaratc fnctons n Table 5 at nform gr ponts ( x, y, z) of 64 where 1. x, y, z 1., for ths test. We ae atonal tmng parameter t to the three volmes, an assgne tmestep vales 0.0, 1.0,.0 to t respectvely, to mae the ata tme-varyng. Fg. 1 shows sosrfacng reslts wth f ( x, y, z, t) = 0 at the three tmesteps. After creatng 4D mplct fnctons from tme-varyng volme sng mltscale ernels from tensor 1 proct of lnear B-splne, a fncton vale f ( x, y, z, t) at an arbtrary pont ( x, y, z) an tme t can be accesse n the same manner. We can mae a morphng anmaton by evalatng the fncton vales at fxe sample ponts for t = t t (0.0 t.0). Fg. 14 shows that the obect can be nterpolate smoothly from 4D fncton f ( x, y, z, t) = 0 n tmng axs t. In ths case, the colors of the obect were nterpolate accorng to t. It too abot 400 secons to pre-process the three volmes ( 64 ), Table 6 shows the tmng performance to evalate fncton vales at 64 sample ponts. Table 5. Fnctons for test tme-varyng volme ata. tme t fncton f ( x, y, z) x /1.5 y /1.0 z /0.5 x y z x y z (1.5cos(( z a/)/ a)) a, a = 1.0 (a) ellpso ( t = 0.0 ) (b) sphere ( t = 1.0 ) (c) oval ( t =.0 ) Fg. 1. Test tme-varyng volme ata.

15 88 Yn et al.: Mltscale Implct Fnctons for Unfe Data Representaton t = 0.0 t = 0. t = 0.4 t = 0.6 t = 0.8 t = 1.0 t = 1. t = 1.4 t =1.6 t = 1.8 t =.0 Fg. 14. Reconstrcton of tme-varyng volme ata ( t = 0.). Table 6. Tmng performance for reconstrcton of tme-varyng volme ata ( 64 ) n secons. 4.4 Thresholng level 1 4 tme We can se a threshol, an n partclar a soft threshol, n MSK nterpolaton, as we can wth wavelets. We can scar coeffcents below a certan threshol, snce many coeffcents c are very small an nsgnfcant. We efne a postve threshol t ( ) as t ( ) =, where s the average of c, s the stanar evaton of c, an s a ser-controllable parameter. For each, we choose a threshol t ( ) an se c ~, efne as c t( ), f c t( ) ~ c = c t( ), f c t( ) 0, otherwse, nstea of c n Eqaton (). Table 7 shows the percentage of coeffcents remanng after ths thresholng process for the sqrrel ata set. Fg. 15 shows srfaces reconstrcte wth thresholng coeffcents of abot 8% ( = 1/ 8) at a fxe level of etal (level=11). Even thogh many coeffcents have been remove, we can fn that the renere mage qaltes of reconstrcton srfaces are very smlar. If the effectve encong scheme that spports fast ranom access to the remanng coeffcents can be mplemente, then t wol be very sefl for varos real-tme compter graphcs applcatons.

16 KSII TRANSACTIONS ON INTERNET AND INFORMATION SYSTEMS VOL. 5, NO. 1, December Table 7. Percentage of coeffcents remanng after thresholng the sqrrel ata set ( s the thresholng parameter). level / / / / (a) no thresholng (117 MB) (b) after thresholng (5 MB) where = 1/ 8 Fg. 15. Removng coeffcents by thresholng at a fxe level of etal (level=11). 5. Conclsons an Ftre Wor We presente a technqe to represent mlt-mensonal ata of mplct fnctons n a nfe form. They allow sers to control the level of etal n renerng varos types of graphcs ata, snce the generate mplct fnctons are base on mltscale ernels that prove a robst representaton wth mltple levels of etal. We showe that or metho col effectvely be apple to tme-varyng volmetrc ata, as well as D volme. In aton, the propose nfe ata representaton can sonly reconstrct srfaces of non-nformly strbte, nosy or scattere pont-sets, an ncomplete meshes wth holes. Even thogh pre-processng of or metho s slower than that of other srface reconstrcton technqes, sch as RBFs, bt evalaton of fncton vales at arbtrary sample ponts from the calclate mplct representaton s faster. By employng effectve technqes sch as mlt-core programmng to enhance the tmng performances, the spees for pre-processng an evalaton ncrease by abot two or threefol. We are crrently mplementng algorthms to rece the comptatonal costs of ths metho, an schemes to encoe the coeffcents that reman after thresholng effectvely, to aress varos compter graphcs applcatons. We beleve that ths approach can be apple to fnamental algorthms for mesh eformaton, srface compresson, ege etecton, an scentfc vsalzaton of commonly se n-mensonal ata sets. References [1] J. Carr, R. Beatson, J. Cherre, T. Mtchell, W. Frght, B. McCallm, an T. Evans, Reconstrcton an Representaton of D Obects wth Raal Bass Fnctons, n Proc. of ACM SIGGRAPH 001, pp , 001. Artcle (CrossRef Ln)

17 90 Yn et al.: Mltscale Implct Fnctons for Unfe Data Representaton [] H. Dnh, G. Tr, an G. Slabagh, Reconstrctng Srfaces sng Ansotropc Bass Fnctons, n Proc. of Internatonal Conference on Compter Vson 001, pp , 001. Artcle (CrossRef Ln) [] S. Morse, T. Yoo, D. Chen, P. Rhengans, an K. Sbramanan, Interpolatng Implct Srfaces from Scattere Srface Data sng Compactly Spporte Raal Bass Fnctons, n Proc. of Internatonal Conference on Shape Moelng an Applcatons 001, pp , 001. Artcle (CrossRef Ln) [4] Y. Ohtae, A. Belyaev, an H.-P. Seel, A Mlt-scale Approach to D Scattere Data nterpolaton wth compactly spporte bass fnctons, n Proc. of Shape Moelng Internatonal 00, pp , 00. Artcle (CrossRef Ln) [5] I. Tobor, P. Reter, an C. Schlc, Effcent Reconstrcton of Large Scattere Geometrc Datasets sng the Partton of Unty an Raal Bass Fnctons, n Proc. of WSCG 004, pp , 004. Artcle (CrossRef Ln) [6] X. W, M. T. Wang, an Q. Xa, Implct Fttng an Smoothng sng Raal Bass Fnctons wth Partton of Unty, n Proc. of the 9th Internatonal Conference on Compter Ae Desgn an Compter Graphcs, pp , 005.Artcle (CrossRef Ln) [7] H. Wenlan, Scatte Data Approxmaton, Cambrge Unversty Press, 005. Artcle (CrossRef Ln) [8] D. Levn, The Approxmaton Power of Movng Least-sqares, Mathematcs of Comptaton, vol. 67, no. 4, pp , Artcle (CrossRef Ln) [9] M. Alexa, J. Behr, D. Cohen-Or, S. Fleshman, D. Levn, an C. Slva, Comptng an Renerng Pont set Srfaces, IEEE Transactons on Vsalzaton an Compter Graphcs, vol. 9. No. 1, pp. -15, 00. Artcle (CrossRef Ln) [10] S. Fleshman, D. Cohen-Or, M. Alexa, an C. Slva, Progressve Pont Set Srfaces, ACM Transactons on Graphcs, vol., no. 4, pp , 00. Artcle (CrossRef Ln) [11] D. Levn, Geometrc Moelng for Scentfc Vsalzaton, Spnger-Verlag, pp [1] S. Fleshman, D. Cohen-Or, an C. Slva, Robst Movng Least-sqares Fttng wth Sharp Featres, ACM Transactons on Graphcs, vol. 4, no., pp , 005. Artcle (CrossRef Ln) [1] Y. Lpman, D. Cohen-Or, an D. Leven, Data-epenent MLS for Fathfl Srface Approxmaton, n Proc. of the ffth Erographcs symposm on Geometry processng, pp , 007. Artcle (CrossRef Ln) [14] J. Manson, G. Petrova, an S. Schaefer, Streamng Srface Reconstrcton sng Wavelets, Compter Graphcs Form, vol. 7, no. 5, pp , 008. Artcle (CrossRef Ln) [15] C. Waler, B. Schölopf, an O. Chapelle, Implct Srface Moellng wth a Globally Reglarse Bass of Ccompact Spport, Compter Graphcs Form, vol. 5, no., pp , 006. Artcle (CrossRef Ln) [16] R. Schabac an H. Wenlan, Kernel Technqes: From Machne Learnng to Meshless Methos, Acta Nmerca, vol. 15, pp , 006. Artcle (CrossRef Ln) [17] R. Opfer, Mltscale Kernels, Avances n Comptatonal Mathematcs, vol. 5, pp , 006. Artcle (CrossRef Ln) [18] I. Torbor, P. Reter, an C. Schlc, Mlt-scale Reconstrcton of Implct Srfaces wth Attrbtes from Large Unorganze Pont Sets, n Proc. of Shape Moelng Internatonal, pp. 19-0, 004. Artcle (CrossRef Ln) [19] O. Schen an K. Gartner, On Fast Factorzaton Pvotng Methos for Sparse Symmetrc Inefnte Systems, Elec. Trans. Nmer. Anal, vol., pp , 006. [0] W. Km an M. Voss, Mltcore Destop Programmng wth Intel Threang Blng Blocs, IEEE Software, vol. 8, no. 1, pp. -1, 011. Artcle (CrossRef Ln) [1] L. Batos, G. Camon, an B. Levy, Concrrent Nmber Crncher: a GPU Implementaton of a General Sparse Lnear Solver, Internatonal Jornal of Parallel, Emergent an Dstrbte Systems, vol. 4, no., pp. 05-, 009. Artcle (CrossRef Ln) [] D. Alcanta, A. Sharf, F. Abbasnea, S. Sengpta, M. Mtzenmacher, J. Owens, an N. Amenta, Real-tme Parallel Hashng on the GPU, ACM Transcatons on Graphcs, vol. 8, no. 5, 009.

18 KSII TRANSACTIONS ON INTERNET AND INFORMATION SYSTEMS VOL. 5, NO. 1, December Artcle (CrossRef Ln) [] N. Naasato, Implementaton of a Parallel Tree Metho on a GPU, Jornal of Comptatonal Scence, 011. Artcle (CrossRef Ln) Seongmn Yn receve hs B.E. an M.E. egree n Mltmea Engneerng from Dongg Unversty, Seol, Korea, n 009 an 011, respectvely. He s crrently a research assocate at Defense Informaton System Management Grop, Korea Insttte for Defense Analyses. Hs research focses on geometrc moelng, real-tme renerng, an hgh performance comptng. Sanghn Par receve a B.S. n Mathematcs from Sogang Unversty, Seol, Korea n 199. He also earne hs M.E. an Ph. D. n Compter Scence from Sogang Unversty n 1995 an 000, respectvely. He s crrently an assocate professor at Department of Mltmea, Graate School of Dgtal Image an Contents, Dongg Unversty, Seol, Korea. Hs research nterests are compter graphcs, scentfc vsalzaton, an hgh performance comptng. Before onng Dongg Unversty, he was a research staff member at Comptatonal Vsalzaton Center, Insttte for Comptatonal Engneerng an Scences, Unversty of Texas at Astn