INTERACTIVE IMPLICIT MODELLING BASED ON C 1 CONTINUOUS RECONSTRUCTION OF REGULAR GRIDS

Transcription

1 INTERACTIVE IMPLICIT MODELLING BASED ON C 1 CONTINUOUS RECONSTRUCTION OF REGULAR GRIDS LOIC BARTHE RWTH-Aachen, Informatk VIII Ahornstrasse 55, 5074 Aachen, Germany barthe@nformatk.rwth-aachen.de BENJAMIN MORA Equpe synthèse d mages et réalté vrtuelle, IRIT-UPS, 118 route de Narbonne, 3107 Toulouse Cedex 4, France mora@rt.fr NEIL DODGSON Ranbow Group-Unversty of Cambrdge 15 JJ Thomson Avenue, Cambrdge CB3 0FD, UK nad@cl.cam.ac.uk MALCOLM SABIN Numercal Geometry ltd. 6 Abbey Lane, Lode, Cambrdge, UK malcolm@geometry.demon.co.uk Our goal s to provde a kernel to allow nteractve and accurate modellng of volume objects. As a frst step, we present a data structure for three-dmensonal felds C 1 contnuous n the modellng space. Regular grds storng the feld values dscretely are combned wth a trquadratc approxmaton flter to defne volume objects. Ths assocaton of a grd and an approxmaton/nterpolaton reconstructon allows the feld to be defned by a C 1 contnuous real functon and the surface to be drectly vsualsed from ts own equaton. We show how accurate and hgh qualty nteractve vsualsaton s obtaned durng the modellng process, and we explan why the vsualsaton s fathful to the object defnton. We also descrbe, as an example applcaton of our data structure, how advanced Boolean operators realsed wth soft or free-form transtons are performed under the nfluence of an nteractve modellng tool. Keywords: Implct modellng, nteractve modellng, nteractve volume renderng, volume reconstructon. 1. Introducton Many algorthms for the vsualsaton of dscrete three-dmensonal volume data now exst and most of them allow nteractve vsualsaton of sopotental surfaces [1,] or of potental varatons n the volume (usng transparency or colour varaton on the surface) [3,4]. Some new hardware also ntegrates three-dmensonal texture renderng and pxel or vertex shaders, whch are effcent tools to ncrease nteractvty n volume vsualsatons [5]. A sgnfcant advantage of ths representaton s that t offers an alternatve to classcal polygon

2 renderng whle avodng some topologcal problems resultng from shape polygonalsaton [6]. Whle fast vsualsaton s avalable, technques to model such data nteractvely need to be mproved and new models proposed. Our am s to propose a unfed structure to precsely and nteractvely model and vsualse volume objects. The modeller must be able to mport objects from dfferent sources and to convert them nto ts nternal representaton. Then all operatons on objects wll be appled on ths new representaton. Objects must then be able to be saved and exported n dfferent formats. The wsh for precson and nteractvty (durng both modellng and vsualsaton processes) leads us to the followng requrements: Objects are represented as volumes. Vsualsaton s nteractve. Vsualsaton s accurate. The modellng process s accurate and nteractve. Transformatons can be appled to objects. Model and nterface are adapted to the use of Boolean composton operators wth free-form and pont-by-pont controlled transtons [7,8]. The goal of ths paper s not to propose a complete soluton, but as a frst step, to present and to justfy a kernel on whch such a modeller can be based. In the followng secton, we present an overvew of dfferent volume modellng systems, and the structure they are based on. We then justfy our choce of data structure n Secton 3. Regular grds dscretely storng the feld values combned wth a trquadratc approxmaton reconstructon are used to defne volume objects. Ths assocaton of a grd and an approxmaton/nterpolaton reconstructon allows the feld to be defned by a C 1 contnuous real functon and the surface to be drectly vsualsed from ts own equaton. In Secton 4, the trquadratc B-splne s descrbed. We present the vsualsaton method and explan how nteractve and hgh precson vsualsaton s obtaned. Ths allows us to show that the vsualsaton s fathful to the object defnton. Modellng tools and processes are presented n the ffth Secton. We dscuss the use of Boolean composton wth free-form transton and we descrbe, as an example use of our data structure, an nteractve nterface to allow the user to precsely control the modelled shape. To llustrate the work presented n ths paper, we conclude wth some examples.. Related works Dfferent famles of volume modellers exst. Some, lke the BlobTree, are essentally based on functon representaton and ther composton n a CSG tree [9]. To obtan nteractvty, only bounded skeleton prmtves are used [10,11,1] and the surface s polygonalsed for renderng. Others are based on dscrete structures lke regular or adaptve grds, and agan, the surface s polygonalsed for renderng [13]. The potental feld can be defned by ether skeleton prmtves [14] or level set methods [15,16]. All these models are targeted for a specfc category of mplct surfaces. There are also hybrd systems whch ntegrate as prmtves most of the volume representatons [17]. The prmtves are manpulated usng ther own representaton. Dfferent technques are then avalable to nteractvely render the surface: polygonalsaton, ponts, ray-castng usng trlnear nterpolaton, etc. The nteractvely vsualsed surface s always an approxmaton of the modelled object and dependng on ths approxmate precson and on the sharpness of the shape, the error can grow, consequently reducng fdelty. ADFs [18] are adaptve octrees, whch generally become deeper n proxmty to the surface. Compared wth regular grds, they stll defne a dscrete representaton of the potental feld, but they avod the lmtaton of the level of detal. They allow a fathful nteractve renderng of the surface [19] but the value of the

3 potental at a random pont of the workng space has to be computed as a dstance from the surface, or as a reconstructed value from the samples of the adaptve structure. These evaluatons are tme consumng and t makes ADF structures complcated to use for our nteractve modellng purpose. Grds are not only useful to store three-dmensonal data. On each voxel of the grd, the data can be a scalar, the three components of the colour (four f the alpha channel s used), or any other multdmensonal vector data. As shown by V.V. Savchenko et al. [0], ths data structure s a template to unfy objects defned by potental felds (dscrete or not). Thus a wde varety of dfferent nput models, lke output scanner 3D volumes, reconstructed feld from scattered data [1,,3,4], reconstructed dstance feld from sosurface [5] or modelled mplct objects [6] can be ntegrated and manpulated wth a common representaton. These volume objects can usually be defned by contnuous real functons as f(x,y,z) 0 [0]. Converson methods to convert one representaton to another and detals on voxel based object representaton/manpulaton are wdely dscussed n [17,7]. Dfferent operatons can then be appled on these objects from classcal lnear transformatons and Boolean composton to more advanced algebrac operatons [7], space mappng [8], sweepng by a movng sold [9] and Boolean compostons wth soft [30,9] or free-form transtons [7,8]. 3. Volume data structure There s currently a great deal of research nterest n fndng methods to defne, model or ft volume/surface data usng real functons f: R 3 R. Amongst the reasons that motvate ths nterest, we draw attenton to the wde varety of geometrc operatons that are avalable, the straghtforward evaluaton of the poston of a pont n regards to the surface and the possblty to choose and adapt the vsualsaton accuracy from the surface equaton. A volume descrbed by a potental functon f: R 3 R s defned as follows: The set of ponts p(x,y,z) for whch f(x,y,z)=0 defnes the surface and the set of ponts p for whch f(x,y,z)<0 a defnes the nsde of the volume. Ths representaton s mplct and the generated zero sosurface s commonly called an mplct surface. Prmtves f thus defned are transformed or combned by composng them wth operators φ: R n R where n s the number of prmtves [8]. The result of an operaton F=φ(f 1,..,f n ) s a new prmtve also defned as a volume. Operatons on prmtves are organsed n herarchcal structures lke CSG trees. Leaves are the prmtves and nodes are operators. At each node, the new equaton s a composton of the combned or transformed prmtves. Ths mples that the equaton complexty grows wth the sze of the tree. Wth even a small number of levels, the functon evaluaton becomes expensve n computng tme and renderng the shape n nteractve tme becomes more dffcult. Moreover, whle a fast evaluaton of combned functons f at the surface level s suffcent to nteractvely compose prmtves wth classcal Boolean composton operators [31], Boolean operators wth soft transton requre, n addton, fast evaluaton n the transton area [11,3,9] and Boolean composton wth free-form transton requres fast evaluaton n the whole modellng space because the stretch of the transton s not predctable before t has been created. A formal vsualsaton of the potental functon s not possble n most cases, the potental functon must be sampled and, therefore, the reconstructed sosurface wll always be an approxmaton to the ntal functon. To allow the computaton tme to depend only on the equaton complexty of the appled operaton or of the mported prmtve, we store the a The sgn of the functon defnng the nsde s chosen by conventon. In ths paper, we use f(x,y,z)<0 but the nverse conventon where the nsde of the volume s defned by f(x,y,z)>0 could also be chosen.

4 potental n a regular voxel grd at each level of the tree (leaves and nodes). The grd furnshes the sampled representaton of the potental feld. Thus, ths method takes advantage of the fact that drect renderng of a voxel grd s possble [33], and the loss of tme generated by the constructon of an ntermedate structure (usually a trangular mesh) to vsualse the surface s avoded. ADFs or adaptve rregular grds are not used because we need to construct the grd as fast as possble wthout the problem of expensve computatons whle nterpolatng the values n the dscrete structure to evaluate the feld at a pont p of R 3. Furthermore, the use of regular grds allow us to ntegrate most of the volume prmtves and operators n our model (as explaned n Secton ). However vsualsaton requres the nverse process that reconstructs a sgnal everywhere n the volume. Ths operaton can be performed usng the well-known convoluton operator: + ( g)( t) = f ( τ ) g( t τ ) f dτ, where f s the sampled functon and g the flter. Thus the choce of an effcent flter s essental for ths applcaton. Several knds of 3D flters for volume vsualsaton have been studed n many ways n the past [34,35,36]. The most studed are: the lnear B-splne flter, cubc B-splne flter based on BC-splnes, gaussan flters and wndowed snc flters. Even though the lnear B-splne s wdely used n the volume renderng applcatons, studes have shown that t gves poor qualty results. Better results can be obtaned wth cubc B-splne or wndowed snc flters. Indeed, the orgnal functon s better approxmated and they allow smooth transtons due to a hgh degree of contnuty. But ther complexty prohbts any use wthn nteractve software. Therefore the man problem of our approach s to fnd a flter allowng both nteractve renderngs and smooth reconstructons of the sosurfaces. A useful soluton can come from a trquadratc flter (trvarate quadratc B-splne) that allows nteractve renderngs [37] wth a hgh qualty reconstructon (Fgure 1). It has been shown that quadratcs are sgnfcantly faster to compute than cubcs (due to ther smaller support and ther lower degree) whle provdng comparable qualty n the resultng nterpolaton [38]. In addng ths flter to the grd to defne objects, the orgnal mplct representaton s reduced to a smooth C 1 contnuous representaton. We thus combne the advantages of a sampled representaton to store the potental values and a real contnuous C 1 representaton to defne and evaluate n a constant tme the value of the potental and ts frst dervatve everywhere n the modellng space, whatever the shape complexty. Thus, both the grd and the trquadratc approxmaton defne our objects. Objects are no longer defned by ther orgnal data structure. Ths ensures that f the trquadratc B-splne reconstructon s used to accurately vsualse the grd and f the renderng vewport s adapted to the grd s (a) (b) (c) Fg. 1. Renderng of the Marschner & Lobb dataset (41 3 samples) wth an deal flter (a), the trquadratc B- splne flter (b) and the usual trlnear B-splne flter (c). We can notce that trlnear B-splne nterpolaton gves poor results, especally wth surface shadng. Ths s essentally due to the erroneous shadngs provded by the separaton between the sgnal reconstructon and the gradent reconstructon (dark surfaces n c).

5 sze, no unwanted nose or detal wll appear f a more accurate vsualsaton technque s used. It s obvous that the thnness of the detals n the modelled shape drectly depends on the grd sze, and the maxmum sze of the grd depends on the machne computaton performance and the range of tme consdered as nteractve. 4. Trquadratc reconstructon Ths secton descrbes our vsualsaton method, whch s based on ray castng. The quadratc flter and ts reconstructon propertes are frst descrbed. Then we show how hgh qualty vsualsaton and nteractve renderng are provded wth the use of the trquadratc B-splne to smoothly reconstruct the potental values of a regular grd. We conclude ths secton wth some optmsatons that allow us to decrease computaton tme and reach the desred nteractvty Unvarate quadratc B-splne P P - P + P -1 P +1 Fg.. Unvarate quadratc B-splne. Before ntroducng the trquadratc B-splne, we descrbe the flter n a one-dmensonal case. Three neghbourng samples (P -1, P, P +1 ) are needed (Fgure ) n order to reconstruct the one-dmensonal sgnal wthn the nterval [-0.5, +0.5]. Now the reconstructon s made n two steps. Frst, two ponts P - and P +, whch are respectvely the mddles of the two segments [P -1, P ] and [P, P +1 ], are defned: P 1 + P P P = + P +1 P + =. Then a quadratc Bézer curve s defned from the three control ponts (P -1, P, P +1 ). Thus the reconstructon equaton, wth t=0 at -0.5 and t=1 at +0.5, can be wrtten as : f P 1, P, P + 1 or n a matrx representaton : f P + P ( t ) P. t + ( P P ). P 1, P, P + 1 P t + 1 = ( t) = [ t t 1] P P P P The C 0 and C 1 contnutes are obvous. Whle B-splne functons are nternally C contnuous, the contnuty across the knots, between segments, s only guaranteed to be C 1 by constructon. Another nterestng property s that the reconstructed sgnal does not pass through the sampled ponts. Local maxma are never reached and the hgh frequences of the sgnal are smoothed. The fnal property of note s the frst dervatve. It s analytcally gven by :

6 wth: f = = ( 1 t )(. P P 1 ) + t. ( P + P ) ( 1 t ). G + t. G + ' P 1, P, P G = P P G = P P The dervatve functon s n fact a lnear nterpolaton of two mddle dfference gradents. In a usual volume vsualsaton, the gradent s estmated wth a (tr)lnear approxmaton of central dfferences gradents computed at the sampled ponts usng the formula: CDIF P + 1 P 1 G + G+ MDIF G ( ) = = = G ( ). The equaton above shows that the gradent estmated wth the usual central dfference gradent s less accurate than the gradent estmaton provded by the quadratc B-splne. Indeed, the central dfference at the sampled pont s the mean value of two mddle dfference gradents G - and G +. Thus the nterpolaton of the mean values (M =1/.G - + 1/.G + ) used n the central dfference estmaton leads to a loss of nformaton n the gradent reconstructon, wth regard to the use of the drect nterpolaton of mddle dfferences G. However the quadratc reconstructon s more senstve to the data nose that s fortunately usually not present n the case of three-dmensonal sgnals generated from equatons. 4.. Trvarate quadratc B-splne The extenson to three-dmensonal volumes s done usng tensor products and the prevous results are also true whle extended to ths case. 7 volume samples (3 3 3) are now requred to reconstruct the C 1 contnuous sgnal wthn a parallelepped algned on the central sample (Fgure 3). Lke the usual trlnear nterpolaton, applyng the unvarate B-splne on the three axes performs the reconstructon. Frst 9 passes on the x axs are done provdng 9 ntermedate samples that undergo 3 passes on the y axs. It leads to 3 ntermedate results that fnally are used n one pass on the z axs. Volume samples Regon of reconstructon Fg. 3. Three-dmensonal reconstructon. The fnal reconstructed potental functon h of the sgnal wthn the parallelepped at a locaton (x, y, z) can be wrtten as: j k h P P ( x, y, z) = Sjk. x y z. (1), j, k = 0 Here the 7 S jk coeffcents are computed from the 7 grd potentals (P jk ) by usng a 7 7- transformaton matrx representatve of the trquadratc B-splne. Fortunately, a lot of matrx coeffcents are null, reducng the computatons of the S jk coeffcents.

7 4.3. Fast Vsualsaton By now the vsualsaton of the mplct surface crossng ths parallelepped reconstructon can be reduced to the vsualsaton of a C 1 contnuous surface defned by the equaton h(x,y,z)=0, wth x, y and z n [0..1]. However t s not obvous how to fnd the projecton of ths surface on the mage plane at nteractve rates. The proposed soluton allows fast renderng whle preservng a hgh qualty of vsualsaton. It s based on the fact that every ray gong through ths parallelepped can be wrtten n a parametrc form: x = x + t. ( x x ), y = y + t. ( y y ), z = z + t ( z z ) () By substtutng the coordnates n the prevous equaton, the reconstructon along the ray can be wrtten as a sxth degree polynomal: ( t) 6 h = C. t 0 t 1. (3) = 0 Ths polynomal should be solved n order to fnd the ntersecton wth the sosurface. However there s no algorthm that allows the accurate computaton of such a soluton n a very short tme. The Sturm theorem could be used n order to effcently approxmate the roots, but ts mplementaton s currently too expensve to be used n an nteractve vsualsaton tool. Thus a less elegant but more effcent soluton whch samples many values (fxed to 16 here) along the ray s used. Once an ntersecton has been detected (.e. h(t -1 ) < 0 < h(t ) ), the fnal value of t s lnearly nterpolated from the last two samples. Ths allows us to obtan an accurate approxmaton of the ntersecton between the ray and the sosurface. Fnally, the shadng process requres the surface normal whch s obtaned by computng from the ntersecton pont usng the followng equaton: h d j S jk x y z k dx, j, k = 0 P P( x, y, z)= j d S jk x y z k dy, j, k = 0 d j S jk x y z k dz, j, k = 0 The accurately raytraced surface s the zero sosurface reconstructed from the grd by the trquadratc B-splne. Ths solves one of our requrements: The vsualsed shape s a fathful representaton of the modelled one Optmsatons Because many computatons are requred by the quadratc reconstructon, the vsualsaton process must be hghly optmsed. Our optmsed algorthm allows nteractve renderng tmes for the vsualsaton of 56 3 voxels n a 51 vewport usng a 1.0 GHz AMD Athlon processor. Furthermore ths algorthm allows nteractve settngs of the sosurface threshold. A complete descrpton s gven n [37], hence we wll just descrbe the man optmsatons. The fast computaton of coeffcents C of the polynomal equaton (Eq. (3)) represents the man requrement to perform the nteractve renderng. They depend on both the S jk coeffcents (Eq. (1)) and the ray parameters (Eq. ()). A naïve raycastng algorthm would compute the S jk values for every voxel crossed by the ray. A more sutable approach s to use a sorted representaton of the object where voxels are projected n a front-to-back order. Then the S jk values are computed only once for all the rays gong through the voxel. Ther 1 0.

8 computaton requres 316 addtons and 80 multplcatons per voxel. The evaluaton of the C coeffcents from the ray parameters and the S jk values requres an addton of 74 multplcatons and 108 addtons per ray. But most of these multplcatons depend only on the ray poston, and by usng a large set of precomputed rays (nstead of the real rays), the number of multplcatons s reduced to 108 per ray. A Mn-Max octree s also used n order to compute only the voxels of nterest. The octree s run n a front-to-back order and the nodes that do not enclose the mn-max values are skpped. The leaf nodes contan the mn and max values of the 7 samples needed for the reconstructon. These samples also represent an absolute boundary of the reconstructed sgnal wthn the voxel. Trees are often used n computer graphcs. They ncrease raytracers performances whle fndng the ntersecton between the ray and the sampled surface n O(Log(n)) steps [39]. Our mplementaton of an octree provdes the same mprovement. The use of the Mn-Max octree as an addtonal structure may seem to be a contradcton wthn our approach, but t allows our renderng technque to be nteractve. As we can see n Table 1, renderng tme s not anymore prncpally dependent on the volume sze, but on the zoom on the surface and on the mage sze. Notably, the vsualzaton speed manly depends on the number of rays ntersectng the surface. A compromse allowng both fast constructon of the Mn-Max octree (n 1.6 seconds) and nteractve renderng can be proposed, as shown wth the grey cells n Table 1 (grds of 18 3 voxels and a renderng wndow of 51 pxels). For our modellng purpose, we use ths confguraton. Less nteractve but more accurate renderng s possble n a larger mage to temporally check small detals on the surface f necessary. Table 1. Mn-Max octree computaton tmes (n sec) for dfferent szes of the grd (n voxels) and correspondng renderng tmes (n sec/frame) for dfferent zooms on the surface and dfferent szes of the renderng wndow (n pxels). The grey cells show a compromse allowng both fast computaton of the Mn-Max octree and nteractve renderng. These tmes are obtaned usng a 1.0 GHz AMD Athlon processor wth 51 Mbytes of DDR memory. Grd s sze n number of voxels Mn-Max octree n sec Zoom, wndow s sze n pxels and renderng n seconds per frame Potental functon manpulaton The data structure and ts vsualsaton method are now well defned. Note that nteractvty durng the modellng process depends on the sze of the grd and the computaton complexty of the operators whereas nteractve vsualsaton essentally depends on the sze of the renderng wndow, whatever the grd. The computer used to test the modeller has a 1.8 GHz AMD Athlon XP processor and 1.0 Gbytes of SDR memory at 133 MHz. As suggested n Secton 4.4, the standard grd s

9 composed of 18 3 float values of storage eght Mbytes each and the vsualsaton s done n a 51 vewport (larger vewports are avalable for less nteractve but fner renderng). Tmes gven n ths secton refer to ths confguraton. In order to model complex objects, prmtves have to be combned or transformed. In ths secton we show how operators are appled and we dscuss the lmtatons caused by the bounded representaton of the potental feld n a grd. We also recall how free-form accurate blendng s performed and controlled n order to ntroduce requrements for a sutable nterface. Ths allows us to show how our data structure can be used n an nteractve nterface to precsely model volume objects. The applcaton s an example gven to llustrate our model s capabltes Operators Operators φ comp defned b as a functon of prmtves f are drectly appled on the grds representng the prmtves. The potental feld resultng from the operaton s stored n a new grd and the modelled object s defned by ts approxmaton by the trquadratc flter. Ths approach allows us to modfy or compose objects n a tme ndependent of the shape complexty. However, grds are large data structures: Ther storage requres a lot of memory. To defne fne objects n a large enough modellng space, we use 18 3 float values grds (smaller or larger grds can be used dependng on the processor speed). It s nconcevable to store all the grds representng each node or leaf of the tree: Even f a lot of memory s avalable, objects defned by many operatons are lable to overflow the maxmum avalable storage space. For ths reason, t s unsutable to modfy the object by actng on prmtves from whch t s defned. These prmtves beng nodes or leaves stuated at lower level of the tree. Actually, f a node or a leaf of the current object s modfed, all the computaton wll have to be done from the leaves to the current node. Ths complete evaluaton of the tree can be a very long process. For ths reason, t s preferable to drectly apply operators on the current object to be able to modfy t n an nteractve tme (f operators allowng the desred modfcaton are avalable). Ths s mportant to take nto account when modellng. Indeed, to lmt backtracks n the tree and long computatons, the user has to carefully and accurately model hs object operaton by operaton. Space mappng operator φ map transforms the space n whch the potental feld s defned. Whle deformatons lke taperng, twstng or bendng can be appled on volume objects when ther equaton s known [40,9], t becomes delcate when the volume s defned n a bounded space wthout the knowledge of the potental values outsde the boundares and when the operator apples tself on the whole modellng space. Even for basc operators lke translaton or rotaton, how can we compute all the values of the grd representng a volume after ts translaton where at least metrc, and maybe curvature, of the potental feld have to be conserved? We do not gve a soluton to ths problem, whch s llustrated n two dmensons n fgure 4. Specfc research wll have to be done to explore ths queston and to propose sutable solutons. An ntroducton to ths problem can be found n [41] whch catalogues a varety of possble solutons. Approaches lke level set methods [4] may also be worth nvestgatng. Thus, only bounded space mappng operators [43] havng ther bounded box ncluded n the modellng space are able to be drectly used on our volume objects. A soluton to avod ths b Offsettng, Boolean composton wth or wthout soft transtons and metamorphoses are elements of ths operator set.

10 problem s to use only bounded prmtves [9,44,9,14] sampled n an nfnte grd [14]. The values outsde the boundares are a known constant scalar and unknown areas are avoded. Moreover, even f the grd s nfnte, the vsualsaton could reman nteractve (renderng speed depends mostly on the vewport sze). Volumes defned by bounded potental felds can be used as nput objects, but mechansms to restrct all prmtves to ths representaton are out of the scope of ths paper. Object Boundares of the modellng space Translaton vector Unknown area Fg. 4. It s an open queston how to compute the potental feld values n the unknown area whle conservng regular varaton of the potental feld. Because each prmtve s stored n ts own grd, the use of n-ary operators leads us to store at least the n grds, whch s a memory greedy operaton. To mnmse the number of stored grds, we only use unary and bnary operators. As shown n [9], ths does not consequently restrct the varety of avalable operatons. Moreover, because t s defned as the sum of potental functons havng specfc feld varatons, operators lke the classcal n- ary blendng operator used for blobs can be decomposed and appled as a successon of bnary blendng operators [14]. Boolean composton operators wth soft transton are consdered as powerful and useful [10,45,0,9] for mplct modellng, therefore, one of our man preoccupatons s to propose an accurate and nteractve tool to realse them. As shown n [7,8], pont-by-pont control of the blend s possble and the classcal smooth and regular curved or nflated transton can be extended to a free-form transton. Our free-form composton operator [8] s defned by a two-dmensonal potental functon g: R R as follows: ( f, f ) = g( f f ) comp φ 1 1,. After composton, the resultng shape s defned by g(f 1,f )=0 and the appled operator depends on the form of the curve g(x,y)=0. Ths curve s controlled by control ponts and the classcal Boolean composton operators wth soft transtons (unon, ntersecton and dfference) are generalzed and extended to a unque free-form operator that combnes, sculpts or creates mplct prmtves (Fgure 5). Functons g used to create the curve preserve the metrc of the combned prmtves outsde the part of the feld affected by the blend and keep regular varatons nsde, whch makes ths approach very relevant for composton of volume objects [46]. Furthermore, control ponts can be drectly selected from a Eucldean space. In an adapted nterface, ths allows the user to accurately and easly defne and control the form of the transton from ts modellng space. Such an nterface s presented n the followng secton and a complete descrpton of ths model s gven n [7,8].

11 Fg. 5. Dfferent free-form operators appled on two orthogonal cylnders. The top row shows classcal unon, ntersecton and the two dfference operators wth soft transtons. The followng row llustrates more advanced possbltes offered by our free-form operator. In the operator pcture, the vertcal axs represents x=0 (whch corresponds to the surface f 1=0 when g s composed wth f 1 and f ) and the horzontal axs represents y=0 (whch corresponds to the surface f =0 when g s composed wth f 1 and f ). 5.. Applcaton n a modellng nterface Durng the object renderng process, both frame-buffer and Z-buffer are computed from the grd and the trquadratc reconstructon. Once these buffers are ntalsed, the standard threedmensonal object vsualsaton lbrary OpenGL s used to buld nterface components n the scene, mnmally affectng the nteractvty of the vsualsaton. Indeed, f specfc graphcs hardware s used, the processor s mostly releved of nterface component manpulaton and renderng tasks (other three-dmensonal object vsualsaton lbrares accelerated by graphcs hardware could be used wth the same effcency). To apply bnary operators on prmtves, at least three grds are necessary: One for each prmtve and one for the resultng object. In our nterface, we use a supplementary grd to temporarly store a volume. Thus, a modelled object can be stored whle another s created. The temporary object can be used later as a prmtve to be combned wth the current one. At ths tme, our renderng method only allows the vsualsaton of a sngle grd. A new prmtve s postoned wth regards to the current object, n vsualsng them n the same grd usng the classcal unon operator (realsed wth the mn functon [31]). It takes 0.10 seconds to compute the grd. The classcal ntersecton and dfference operators are also avalable and obvously, the rendered object resultng from one of these compostons can be selected as a new prmtve. The sharp transton normally created between the combned objects s, n grds reconstructed by a trquadratc B-splne, a small oscllatng transton the sze of whch depends on the grd resoluton (Fgure 6). We pont out that the same type of oscllatons can be found around rdges or at the level of very thn surfaces (Fgure 7).

12 (a) (b) (c) (d) (e) Fg. 6. Sharp transton between two cylnders obtaned usng Rcc s Mn-Max unon Boolean operator [31]. Grd Zoom out Zoom n 18 3 (a) (b) 64 3 (c) (d) Fgures (a),(b),(c) and (d) show the oscllatons along the sharp transton reconstructed from the grd and Fgure (e) shows the varatons of the potental feld n a D secton. Fg. 7. On the left, an object modelled wth our nterface. The zoom on the area delmted by the whte rectangle s shown on the rght. We can clearly see the oscllatons on the boundary of thn surfaces. (a) (b) (c) Fg. 8. Illustraton of oscllatons generated by the three-drectons quadratc Box-splne reconstructon. Fgures (a), (b) and (c) show respectvely the reconstructon of thn detals, sharp transtons and rdges. Fgure (d) shows the Box-splne drectons. As we can see, no oscllaton appears along these drectons. (d)

13 It would be useful f these undesred oscllatons could be avoded. The unform trquadratc B-splne s also a trvarate (by tensor product) quadratc Box-splne [49] and ths phenomenon s a known problem of Box-splne reconstructon. However, around rdges or thn detals, t has smooth reconstructon propertes f the feature s along one of the Boxsplne drectons (Fgure 8). In our case, grd axes are Box-splne drectons. Ths oscllaton phenomenon s smlar to the lateral artfact n subdvson surfaces [47] and a soluton could be to use a local reconstructon based on Box-splnes havng more dfferent drectons. More research and experments are needed n ths area. It s also possble to produce a sharp transton wth a C 1 dscontnuty on the surface whle mantanng a C 1 contnuous potental feld everywhere else n the potental feld [30,8]. At the grd level, t generates smoother samples but as shown n Fgure 9, there s no mprovement of the oscllaton effects (wth respect to Fgure 6). Fg. 9. Sharp transton between two cylnders obtaned usng the unon Boolean operator proposed by Barthe et al. n [8]. Ths operator provdes a potental feld that s C 1 contnuous when both functons f 1 and f are not equal to zero. On the rght, the 3D vsualsaton from a grd composed of 18 3 voxels and on the left, a D secton of the potental feld. As descrbed n secton 5.1, when free-form transtons are used n Boolean operators, the form of the blend can be controlled by ponts selected n the modellng space. Dependng on the operator complexty, the computaton tme of the grd vares followng Table. Table. Tme of computaton of a grd whle usng a Boolean operator wth a free-form soft transton dependng on the number of control ponts used to defned ts form. Number of control ponts Computaton of a plane secton of 56 pxels, crossng the grd Computaton of the grd sec 0.14 sec sec 0.15 sec sec 0.16 sec sec 0.17 sec sec 0.0 sec 0.40 sec 0.3 sec To allow the maxmum nteractvty whle selectng the control ponts, we vsualse the varaton of the potental feld n a plane secton crossng the grd. The plane s frst postoned n the modellng space, wth respect to the combned prmtves (Fgure 10a). Then the secton s rendered n a vewport of 51 pxels (Fgure 10b) n approxmately 0.08 seconds. The trquadratc reconstructon s also used here to fathfully represent the feld values of the volume. Dfferent colours are used to dentfy the dfferent relevant areas of the

14 volumes,.e. nsde/nsde respectvely object1/object, nsde/outsde, outsde/nsde and outsde/outsde. The transton s selected wth the mouse n ths plane secton (Fgures 10c and 10e) and the result s nteractvely vsualsed wth a transparency effect. When desred, the resultng surface can be vsualsed n the three-dmensonal modellng space (Fgures 10d and 10f). The delay to vsualse the frst 3D pcture of the resultng object s equal to the sum of computaton tmes of both grd and Mn-Max octree. In general, ths tme vares between 1.5 and.5 seconds (on the computer used for our modeller, the computaton of the Mn- Max structure s reduced to 1.0 second). Once the grd and the Mn-Max octree are bult, the object s nteractvely vsualsed wth several frames per second. Two-dmensonal plane secton representng the potental feld values can be used to ncrease the nteractvty (Table ) and control the potental feld varatons whle applyng any composton operators wth soft transtons. For bounded operators, only a small regon of the secton need to be vsualsed and real tme control of the form of the transton can be expected. (a) (b) (c) (d) (e) (f) Fg. 10. The plane secton s postoned n the modellng space wth respect to the two combned cylnders (a), and the secton of the potental feld s vsualsed n a two-dmensonal vewport (b), n whch the transton s nteractvely selected and the result s vsualsed usng a transparency effect (c). The correspondng object s accurately vsualsed n the three-dmensonal modellng space (d). More complcated operatons can be acheved when dfferent forms of the curve g are selected n the two-dmensonal vewport (e) and our renderng method provdes an accurate and fathful three-dmensonal vsualsaton (f). 6. Concluson and future work The combnaton of a dscrete representaton of potental felds wth an adequate reconstructon generates a contnuous approxmaton of the feld n a bounded area. A regular grd allows a fast storage of the potental values and the trquadratc approxmaton B-splne generates a C 1 contnuous functon. Our renderng algorthm provdes a fast, accurate and smooth vsualsaton of the feld sosurfaces reconstructed wth the trquadratc B-splne. Thus, the defnton of volume objects usng the assocaton of these two components allows us to obtan a fathful vsualsaton and provde nteractve modellng tools.

15 These propertes of our volume objects have been successfully llustrated wth the applcaton of advanced Boolean composton operators wth free-form transtons n an nteractve modellng software. For these reasons, the proposed modellng kernel fulfls most of our requrements. Further studes can now to be done to mprove and complete ths approach. In partcular, research needs to be undertaken n the followng areas: How to apply space mappng operators. Potental felds are stored wth float values (coded wth 3 bts). As suggested by Adzhev and al. [17], the values can be mapped n short nteger and coded wth 16 bts. Our renderng algorthm already supports such a grd and the 16 bts left could then be used to store colour components. Dfferent reconstructons could be appled on the grd values to try to avod alasng n the approxmaton of sharp edges. A local update of the data structure and of the rendered mage would ncrease the nteractvty whle applyng local operators.

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