Improved Real-Time Shadow Mapping for CAD Models

Transcription

1 Improved Real-Time Shado Mapping for CAD Models Vitor Barata R. B. Barroso TeGraf, Computer Siene Dept., PUC-Rio Waldemar Celes TeGraf, Computer Siene Dept., PUC-Rio Abstrat Shado mapping is a idel used rendering tehnique for real-time shado generation. Hoever, it ma produe jagged shado borders and inorret self-shadoing artifats. When applied to CAD (Computer-Aided Design models, it presents additional hallenges due to the existene of thin and omplexsilhouette objets, high depth range, and high depth omplexit. In this ork, e present a short surve of shado mapping algorithms and investigate their suitabilit to CAD model rendering. We also present to improvements to existing tehniques: a generalied parameter for LiSPSMs (Light-Spae Perspetive Shado Maps and an adaptive -partitioning sheme.. Introdution Shados are ver important visual effets on omputer-generated senes beause the reveal information about the spatial relationships among objets and enhane the realism of the visualiation. Unfortunatel, generating fast and aurate shados is still hallenging due to the global nature of the problem. Reentl, there has been a lot of disussion about shado mapping, a idel used shado rendering tehnique due to its simpliit, effiien and generalit. The tehnique is partiularl attrative to CAD model rendering beause it does not require an expensive analsis of the model s geometr (although geometr shaders are making suh analsis heaper. As an image-spae algorithm, shado mapping [] suffers from aliasing problems due to the ited resolution available for sampling the sene. This results in jagged shado borders and the inorret selfshadoing of objets. These artifats are partiularl strong and diffiult to avoid in CAD models beause of their high depth range and depth omplexit, and the presene of man thin omplex-silhouette objets suh as bars and latties. In this paper, e present a surve of shado mapping tehniques that address the problems of selfshadoing and aliasing. Eah algorithm is analed Figure. Shado-mapped CAD Model onsidering visual qualit, omputational effiien, ease of implementation and suitabilit to CAD model rendering. Additionall, e present to improvements to existing tehniques. First, e build on the orks of Wimmer et al. [] and Zhang et al. [3] to derive a LiSPSM (Light-Spae Perspetive Shado Map parameter generalied to different angles beteen the light and vie diretions. Seond, e propose an adaptive -partitioning sheme based on the ork of Llod et al. [4,5]. B ombining these improvements ith a set of previous tehniques, e ahieved real-time highqualit shado rendering, as illustrated in Figure. The remainder of the paper is organied as follos: Setion revies and analses several existing tehniques and their suitabilit to CAD model rendering. Setion 3 presents our generalied LiSPSM parameter and our adaptive -partitioning sheme. Setion 4 presents results and additional remarks. Finall, onlusions are dran in Setion 5.. Surve of shado mapping tehniques The standard shado mapping tehnique [] first renders the sene from the viepoint of the light soure and stores the depth buffer in a texture alled shado map. Then, the sene is rendered again from the vieer s position. Eah pixel is transformed bak

2 into the light spae, here it is determined to be in shado if its depth is farther from the light soure than the orresponding value in the shado map. The main problems assoiated ith shado mapping are inorret self-shadoing and aliasing artifats. In this setion, e anale existing tehniques that address these problems and disuss their suitabilit for CAD model rendering... Depth bias and sample alignment When using the shado mapping tehnique, inorret self-shadoing artifats ma appear beause of the ited preision and resolution available to sample depth values throughout the sene. Ideall, the depth of a lit pixel in relation to the light soure is equal to the orresponding value stored in the shado map. Hoever, hen a pixel is transformed into light spae, it does not align exatl ith the enter of a shado map texel [6]. Therefore, the to samples do not atuall represent the same point, hih produes rong shado test results: a lit pixel ma be onsidered to be in shado. Usuall, the self-shadoing problem is ountered b adding a small bias to the depth values stored in the shado map []. Hoever, if the bias is exaggerated, shados are visibl pushed aa from their true positions. The hoie of an appropriate value an be hard, sine it depends on fators suh as the thikness of sene objets, the relative position of the amera and the light soure, and the shado map resolution [6]. A generi bias bias an be onvenientl expressed as a funtion of the depths of the first ( and seond ( surfaes nearest to the light soure [8]. The value stored in the shado map is then given b: ( map = + bias, ( Table summaries the bias funtions proposed b different authors. The Seond-Depth tehnique, (b in Table, requires the sene to inlude onl thik solid objets. When this ondition is satisfied, the bias an be implemented easil and effiientl ith polgon frontfae ulling, and the overall result is often satisfator. Unfortunatel, hoever, CAD models often inlude non-solid objets, hih invalidate this approah. Midpoint-based tehniques, ( and (d in Table, disregard bak-faing surfaes for. The upper bound in (d is required to avoid self-unshadoing artifats hen using bak-fae ulling [8]. Although these tehniques produe the best results in a general sene, the have the drabak of requiring an additional shado map generation pass to obtain the seond surfae using depth peeling. Therefore, the are more reommended Table. Tpial bias funtions Tehnique bias (, (a Constant Bias [] offset (b Seond-Depth [6] - ( Midpoint [7] ( - / (d Dual-Depth [8] min ( ( - /, max for stati senes and uniform shado maps (ith no vie-dependent arping or partitioning, hih ma be generated onl one. Hoever, sine arping is highl reommended for CAD model visualiation (Setion.3, the additional ost of midpoint tehniques ma beome prohibitive. Beause of that, e generall reommend the use of a simple onstant bias. As a different approah to avoid self-shadoing artifats, Wang & Molnar [6] introdue the tehnique of sample alignment. Sine the amera samples do not lie on the enters of the shado map texels, the alulate a virtual sample, on the plane tangent to the objet surfae, hih does lie in the right spot. Note that the alternative, reonstruting the shado map value that mathes eah amera sample, does not ork ell, sine interpolated depth values beteen to different oluders have no useful meaning. When ombined ith depth bias, amera sample alignment greatl redues self-shadoing artifats (although some issues persist, as disussed in [6]. Therefore, the tehnique is reommended hen depth bias alone annot avoid the problem... Filtering Jagged shado borders are usuall filtered using PCF (Perentage-Closer Filter [9], hih is applied to the results of multiple shado tests. A more effiient, reentl presented alternative is the use of VSMs (Variane Shado Maps [0]. With VSMs, the shado map stores not onl the depth, but also the squared depth of eah sample. These values are filtered and then used to determine the mean µ and variane σ² of all the samples inside the filter kernel. In the final rendering pass, Chebhev s inequalit is alulated for ever pixel: P ( map pixel σ + ( µ ( In Equation (, p max is the maximum probabilit that a pixel should be lit, onsidering the distribution of depths inside the filter region. This value is used diretl as the resulting light intensit for the pixel. Unfortunatel, VSMs are not ell suited for CAD model rendering. In their paper, Donnel & Lauriten pointed out that VSMs ma produe light bleeding σ pixel p max

3 artifats [0]. This effet is partiularl strong in CAD models due to the high depth range and depth omplexit, as an be seen in Figure. In region A, the silhouette of a toer an be seen beause of a high σ² value. In region B, light bleeding ours beause the filtered depth values (µ are interpolated ith a far aa objet, inverting the result of the depth test and skipping the alulation of p max aording to the VSM algorithm. Finall, in region C, the term ( pixel - µ in ( is ver small, making p max rise and the shado disappear. Beause of suh strong light bleeding artifats, VSMs are not reommended for CAD model visualiation, and PCF should be used instead. Current researhes on Summed Area VSMs [] have attempted to einate light bleeding, but suh an approah has et to be tested. Perentage-loser filtering is alread supported diretl b some video ards ith a kernel. Larger filters are also beoming less expensive ith the inreasing fragment proessing poer of urrent graphis hardare. Krahko s implementation [] of a irular filter kernel ith jittered samples seems ver appealing, sine it effiientl einates banding effets. On the other hand, the test samples and adaptive refinement proposed in that ork ma introdue artifats in shados of omplex-silhouette objets suh as latties, and therefore this optimiation should be avoided hen performane is not at ritial levels. We reommend a 3 3 to 5 5 kernel sie. One aspet of PCF is often overlooked. The kernel should be onsidered in texel spae, not in pixel spae. In other ords, the distane beteen the enters of to ells should be that of one texel. This means the filter region ill automatiall adjust its sie to blur the aliased shado border region. Hoever, shados ith no visible aliasing ill remain unblurred as hard-shados. Unfortunatel, this is neessar to guarantee proper blur here aliasing is strong. After all, PCF is an antialiasing tehnique, not properl a soft shado one. Normall, PCF s shado tests ompare onl one amera sample ith several shado map values, hih aggravates self-shadoing artifats. In order to avoid this problem, multiple amera virtual samples should be generated on the surfae s tangent plane [6]. When a regular filter kernel is used in texel spae, onl the first virtual sample must be alulated, hile the others an be obtained b simple texture oordinate additions. For a irular jittered kernel, hoever, aligning the samples one b one an be expensive and anel the jitter effet. In this ase, e reommend alulating the virtual samples at the exat oordinates given b the PCF kernel, ithout alignment to the texel enters. This is usuall enough to prevent self-shadoing artifats ith a small onstant bias..3. Warping or parameteriation Several tehniques attempt to minimie aliasing artifats b arping the shado map to an appropriate spae before sampling the sene. In this ork, e fous on perspetive parameteriation tehniques, introdued b Stamminger & Drettakis [3]. The idea is to appl a perspetive transformation to the sene in suh a a that objets loser to the vieer (not neessaril lose to the light soure appear in a larger area of the shado map. The tehniques have onl a ver lo ost on CPU to ompute the transform matrix and an drastiall redue shado border aliasing in most ases. In order to understand perspetive arping and its benefit, e must first define an aliasing error metri. Consider Figure 3, here a surfae area an be seen b both the amera and the light soure. The amera and light beams pass through one image pixel and one shado map texel, respetivel, and hit the surfae ith angles θ and θ L, overing areas and L. Note that, hen L >, a single shado map texel determines the shado state of more than one image pixel. Therefore, e an define the aliasing error m as the ratio beteen the areas seen b a shado map texel and an image pixel [5]: m = Light beam L L os os θ L ( θ ( θ L surfae (3 θ Camera beam L L Figure. Light Bleeding ith VSMs Figure 3. Aliasing error metri

4 light map 0 t near amera φ φ 0 n f 0 n f Figure 4. Perspetive arp P fit to a frustum V Equation (3 is often used to lassif aliasing into perspetive aliasing ( L / and projetion aliasing (os(θ L /os(θ [3]. The seond tpe is potentiall unbounded and depends on the orientation of eah surfae of the sene in relation to the light and vie diretions. Sine this effet is also less notieable and not avoided b perspetive arping, it is usuall ignored in the analsis of suh algorithms. No, onsider Figure 4, here V is the vie frustum. The light and vie diretions are perpendiular in this ase. Perspetive arping tehniques first fit a frustum P around the vie volume (and objets that ast shados into it. Before updating the shado map, the appl the perspetive transformation assoiated ith P to the sene. In the figure, V has a field-of-vie of φ, near plane n and far plane f. Similarl, P has a field-of-vie of φ, near plane n and far plane f. A generi point on the entral axis of V, hih is at a distane from the amera, is mapped to a distane from point, the enter of projetion of P. The oordinates (x,, orrespond to the light spae defined in the LiSPSM paper [], hose origin gives the position of. The shado map oordinates (s,t are assumed to be aligned ith (x,, respetivel. The strength of the perspetive arp is ontrolled b the parameter n. When n = n, e have the PSM (Perspetive Shado Map tehnique [3]. When n, the arp loses strength and beomes orthogonal, leading to the standard shado mapping. The LiSPSM (Light-Spae Perspetive Shado Map parameter n LiSPSM = n + f n is the one that minimies M, the maximum error in the diretion: M = max [ n, f ] (4 It an be shon [,5] that, hile PSM minimies the maximum error in the x diretion, it makes the error in the diretion gro linearl ith, hih produes strong aliasing on surfaes not lose to the near plane. On the other hand, LiSPSM minimies the maximum error in the diretion hile also keeping a d P L ( L C V far φ amera 0 n /3 n f /3 n f f n n (a perpendiular light (b parallel light Figure 5. Vie frustum, as seen b the light soure, ith -partitioning small maximum error in the x diretion. Furthermore, LiSPSM guarantees that the error in the diretion is equal on both the amera s near and far planes, hih is benefiial for -partitioning (Setion.4. For those reasons, e reommend the use of LiSPSM for CAD model rendering and other appliations..4. Partitioning Partitioning tehniques subdivide the sene aording to some riteria and generate a different shado map for eah partition. The shado map resolution alloated to eah partition ma be the same or be deided b an aliasing error metri. In this setion, e fous on fae and -partitioning as desribed b Llod et al. [4,5]. Compared ith other tehniques [4,5], these are simpler to implement, require feer rendering passes, and an more easil be ombined ith arping for better results. Ever perspetive arping tehnique fails hen the light and vie diretions are near parallel. In this situation, the parameteriation is usuall fored to onverge to the standard shado mapping b imposing n [,5]. This diffiult an be overome b dividing the vie frustum aording to its faes, as seen b the light soure, and generating a different shado map for eah division ith its on transformation [4,5]. This fae partitioning approah, hoever, an be too expensive, as it requires at least four shado maps to be generated eah frame. Besides, despite its exellent results in the near-parallel ase, the tehnique does not improve shado qualit hen the light is perpendiular to the vie diretion. As a different approah, -partitioning attempts to ahieve higher shado qualit in all ases b dividing the vie frustum along its axis, as shon in Figure 5. A shado map, either arped or not, is then generated for eah partition. This an be seen as a pieeise approximation of the optimal logarithmi parameteriation of the shado map [5], hih gives onstant aliasing error in both x and diretions. V a V b

5 0 n a φ a a / tan( light n f V b P (-nsin( -n f map φ a n P V b f 0 n light f map (a φ π/ (b 0 < < φ Figure 6. Shado Map generalied perspetive parameteriation With -partitioning, the maximum aliasing error in eah partition is proportional to its f / n ratio. As a result, the optimal -partitioning sheme uses self-similar partitions, that is, hih have the same f / n ratio [4,5], as in Figure 5a. For a total of k partitions, the i th partition satisfies: ( i k i k f f ni = n, f i = ni n + =, i {,,..., k} n n (5 As shon b Llod et al. [5], -partitioning an provide more benefit than fae-partitioning ith the same number of subdivisions. Moreover, the use of onl or 3 partitions ith LiSPSM arping suffies to greatl redue aliasing errors hen the light is either perpendiular or parallel to the vie diretion. This is thus the reommended setting for most sstems. 3. Proposed improvements In this setion, e propose to improvements to existing tehniques. First, e build on [,3,5] to derive a LiSPSM parameter generalied to different angles beteen the light and vie diretions. Seond, e propose an adaptive -partitioning sheme based on the ork of Llod et al. [4,5]. 3.. Generalied LiSPSM parameter Reall that Figure 4 shos the optimal onfiguration for the LiSPSM arp, hen the light and vie diretions are perpendiular. Based on the figure and on our aliasing error metri (Equation 3, it an be shon [5] that, for an image of (res x res pixels and a shado map of (res ls res lt texels, the maximum error in the diretion is given b: (6 The derivative in Equation (6 is alulated from the standard OpenGL perspetive transformation: Figure 4 also gives: (7 (8 (9 Equations (8 and (9 are onl valid for the situation depited in Figure 4, here the light and vie diretions are perpendiular. If the light is tilted toards or aa from the amera, e must onsider the to general ases shon in Figure 6, here the smallest angle beteen the light and vie diretions is. Aided b the auxiliar triangles shon in Figure 6a, e an see that: n + a os = n + b os n + f = n + t m ( = l = + res = res ( + ( n sin (, ( + ( f sin (, tan φ π 0 < < φ (0 ( In these equations, a and b are half the height of the vie frustum s near and far planes, respetivel. Substituting (0 into (7, differentiating, and using the lt ( φ dt d ( f + n f n ( f n ( f n = n + n f = n + f n ( a + b os ( + ( f n sin (, b os ( 0 < < φ φ π

6 result in (6, e find the generalied error in the diretion: m (, n, res ξ ( n, = res l = = ξ ( n, tan n + a os ψ (, n, = n + b os lt ( φ ( ψ (, n, ( f n n f ( + ( n sin (, ( + ( f sin (, ( Differentiation of Equation ( shos that the error funtion has onl one loal minimum in [n,f]. The maximum error must then our at the amera s near and far planes, and an be minimied b making its value equal on those extremes: m ( n, n, = m ( f, n, n n = n (3 Equation (3 represents a generalied LiSPSM parameter that keeps the maximum error minimied hen < π/. Unfortunatel, hoever, n assumes negative values hen the light and vie diretions are near parallel. This means that our onstraint is impossible to satisf, hih orroborates the fat that ever perspetive arp must onverge to the standard shado mapping in this ase [,3,5]. In order to obtain a smooth transition in the shado qualit, e modif Equation (3 using the folloing ne onstraints: (4 In Equation (4, is the it angle here e stop using Equation (3. To avoid numerial instabilit and the generation of overl ide arp frustums, it should be kept higher than φ. We reommend using the angle hih makes the near and far faes of the vie sin ( ( = ( n + f n sin ( a os (, ( = ( f + f n sin ( b os (, n 0 n ( = n ( = φ π 0 < < φ φ π 0 < < φ frustum superpose in the light s vie: n n = n a + b = artan (5 f n Our modified parameter an no be ritten as: ( = ( n + f n sin ( a os (, ( n = r ( (, π 0 < < (6 In Equation (6, the funtion r( desends from r( = to r(0 = 0, hih satisfies the onstraints of Equation (4. In their paper, Zhang et al. propose the folloing funtion for the PSM ase [3]: r ( = sin (7 For our LiSPSM ase, hoever, e required r( to desend faster to ero, one again to avoid exessivel ide arping frustums. We have ahieved good results ith the folloing funtion: r ( sin = + (8 Figure 7 shos the form of the final generalied LiSPSM parameter. The arp at first beomes stronger as the light is tilted toards or aa from the vie diretion. Belo the it angle, the arp quikl onverges to the standard shado mapping. In general, e have ahieved better shado qualit ith our generalied LiSPSM than ith other parameteriations for most light diretions. Figure 8 ompares the standard and generalied LiSPSM arps for a lattie shado lose to the near plane, here the improvement is most notieable. Nevertheless, one must keep in mind that minimiing the maximum error over the entire vie frustum does not guarantee a minimal error everhere in the sene. A simpler tehnique like the original LiSPSM or PSM might give better results for speifi ases. π π Figure 7. Generalied LiSPSM parameter (n =, f = 00, φ = π/6. Figure 8. Left: Standard LiSPSM. Right: Generalied LiSPSM.

7 3.. Adaptive -partitioning Reall the -partitioning sheme desribed in Setion.4. When the light and vie diretions are perpendiular, the use of self-similar partitions and of LiSPSM arping guarantees that the maximum aliasing error in the diretion is minimal and equal in the near and far planes of ever partition. In the x diretion, the error is small and does not hange drastiall over the vie frustum. These are important properties beause otherise a sudden hange in the shado qualit ould be notieable as a seam at the interfae beteen partitions. Hoever, hen the light and vie diretions are not perpendiular, the LiSPSM arp onverges to the standard shado mapping. As a result, the aliasing error in both diretions has its maximum value at the near plane and minimum value at the far plane of eah partition. In this ase, the algorithm produes visible seams beteen eah pair of partitions, as shon b the detail on the left in Figure 9. Moreover, the seams move together ith the amera, hih athes the user s attention and beomes ver distrating. In order to minimie seams, the ratio f / n of all partitions, exept the one losest to the amera, should be kept small. This redues the differene beteen the maximum error at the near plane and the minimum error at the far plane. The disadvantage is that the error lose to the amera (near plane of the first partition beomes higher than ith the optimal partitioning. In this paper, e propose an adaptive partitioning sheme. When the light and vie diretions are perpendiular, the optimal -partitioning should be used to minimie the maximum error over the vie frustum. In the near-parallel ase, hoever, ever i th partition but the first should have its (f i / n i ratio redued to a small value α i. A smooth transition an be ahieved b Figure 9. Adaptive -partitioning ith detail on the right. The detail on the left shos the result of optimal -partitioning ith visible seam linearl interpolating the distane of eah subdivision beteen the optimal and it ases, using the angle beteen the light and vie diretions as the eight: λ =, π f k = f ni = f i n = n 0 λ i (9 f i i f i = λ n + ( λ, i { k,...,3,} n α i Note that Equation (9 is alulated iterativel, starting ith the last partition (i=k and bakard until the seond (i=, alas using the values of f i previousl alulated b the linear interpolation. The first partition has n =n and f =n. It an be shon that, hen = π/, Equations (5 and (9 are equivalent. Figure 9 ompares the shados generated b the optimal -partitioning tehnique and our adaptive one ith partitions. The interfae beteen the partitions is indiated ith a hite arro. In the left detail, optimal partitioning produes the best shado qualit lose to the far plane of the first partition. Hoever, the tehnique produes a visible seam beteen this region and the beginning of the next partition. When using the proposed adaptive sheme ith α=, aliasing is not redued so dramatiall in the first partition, but the seam beomes almost unnotieable. Empiriall, e reommend a value α no higher than 4 ithout PCF filtering and no higher than ith PCF, sine the filter in texel spae as desribed in Setion. makes seams more notieable (note the blur in the seond partition in the left of figure Results and additional remarks We have implemented all the tehniques disussed thoroughl in this paper on a mahine ith a dual ore.4gh proessor and a GeFore 8800 GTS graphis ard. Tests ere performed on an oil platform CAD model ith 50k triangles, using sreen resolution and shado maps. For our performane tests, e used onstant bias, generalied LiSPSM arping, adaptive partitions along the vie frustum s axis and 5 5 texel-spae PCF ith jittered kernel samples and multiple nonaligned virtual sample generation. Despite the extra ost assoiated ith the generation of to shado maps and the use of PCF in the final rendering pass, these tehniques an run faster than the standard shado mapping for some amera and light vies, as shon in Table, sine frustum ulling an be opti-

8 mied for eah partition. The results in the table ere obtained for a lose-up vie here about 0% of the sene s geometr as visible to the amera, hile the light soures irled around the sene at a far distane. From our results, it should also be noted that, due to the inreasing fragment proessing poer of graphis ards, PCF an be applied ith nearl negligible ost. As a final remark, it is important to notie that, hen aliasing is not ompletel einated, viedependent arping tehniques an ause shado borders to ral hile the amera or light moves around. Fortunatel, blurring the shado borders ith PCF makes this effet muh less notieable. Hoever, the situation of near-parallel light and vie diretions must still be handled ith are to avoid disturbing temporal hanges in shado qualit. This emphasies the importane of a good partitioning sheme. Table. Performane results Tehnique Standard Reommended Lights Shado Map Tehniques dnami light 45 fps 9 to 59 fps dnami lights 5 fps 8 to 30 fps 5. Conlusion We have presented a surve of shado mapping tehniques that attempt to redue self-shadoing and aliasing artifats. In partiular, e have investigated the effetiveness of depth bias, sample alignment, filtering, perspetive arping and partitioning tehniques hen applied to CAD model visualiation. For this kind of appliation, e reommend the use of a onstant bias or the dual-depth tehnique hen generating the shado map, as ell as LiSPSM arping and or 3 partitions along the vie frustum s axis. In the final rendering pass, e reommend the use of texel-spae PCF ith a 3 3 to 5 5 jittered kernel and multiple non-aligned virtual samples. We have onluded that this set of tehniques an produe high qualit shados in real-time for CAD models. Additionall, e have presented improvements to existing tehniques: a generalied parameter for LiSPSMs, hih keeps the maximum error in the diretion minimied for different angles beteen the light and vie diretions; and an adaptive - partitioning sheme that minimies seams, that is, avoids sudden hanges in shado qualit beteen eah pair of partitions. Aknoledgements During this researh, the first author as finaniall supported b Brailian agen CNPq. Tegraf is a laborator at PUC-Rio that is mainl funded b the Brailian oil ompan, Petrobras. Referenes [] L. Williams, Casting urved shados on urved surfaes, Proeedings of ACM SIGGRAPH 78, ACM Press, 978, vol., pp [] M. Wimmer, D. Sherer, and W. Purgathofer, Light spae perspetive shado maps, Proeedings of the Eurographis Smposium on Rendering 004, Eurographis Assoiation, 004, pp [3] F. Zhang et al., Generalied Linear Perspetive Shado Map Reparameteriation, Proeedings of the ACM international onferene on virtual realit ontinuum and its appliations, 006. pp [4] B. Llod et al., Subdivided Shado Maps, Universit of North Carolina at Chapel Hill, 005, Tehnial Report TR [5] B. Llod et al., Warping and Partitioning for Lo Error Shado Maps, Proeedings of the Eurographis Smposium on Rendering 006, Eurographis Assoiation, 006, pp [6] Y. Wang, and S. Molnar, Seond-Depth Shado Mapping, The Universit of North Carolina at Chapel Hill, 994, Tehnial Report TR [7] A. Woo, The Shado Map Revisited. D. Kirk (ed., Graphis Gems III, AP Professional, Boston, 99, pp [8] D. Weiskopf, and T. Ertl, Shado Mapping Based on Dual Depth Laers, Proeedings of the Eurographis 003 Short Papers, Eurographis Assoiation, 003, pp [9] W. Reeves, D. Salesin, and R. Cook, Rendering antialiased shados ith depth maps, Proeedings of ACM SIG- GRAPH 87, ACM Press, 987, vol., pp [0] W. Donnell, and A. Lauriten, Variane shado maps, Proeedings of the 006 smposium on Interative 3D graphis and games, ACM Press, Ne York, NY, USA, 006, pp [] Internet forum, last aess on 07/0/007. [] Y. Krahko, Effiient Soft-Edged Shados Using Pixel Shader Branhing, M. Phar, and R. Fernando (Ed., GPU Gems II, h. 7, NVidia Corp., 005, pp [3] M. Stamminger, and G. Drettakis, Perspetive shado maps, Proeedings of ACM SIGGRAPH 0, ACM Press, 00, pp [4] J. Arvo, Tiled shado maps, Proeedings of Computer Graphis International 004, IEEE Computer Soiet, 004, pp [5] R. Fernando et al., Adaptive shado maps, Proeedings of ACM SIGGRAPH, ACM Press, 00, pp