Boundary Correct Real-Time Soft Shadows

Transcription

1 Boundary Corret Real-Time Soft Shadows Bjarke Jakobsen Niels J. Christensen Bent D. Larsen Kim S. Petersen Informatis and Mathematial Modelling Tehnial University of Denmark {bj, nj, Abstrat This paper desribes a method to determine orret shadow boundaries from an area light soure using umbra and penumbra volumes. The light soure is approximated by a irular disk as this gives a fast way to extrude the volumes. The method also gives a rude estimate of the visibility of the area light soure as seen from a point in the shadow region. Instead of rendering the volumes to the stenil buffer, we use an extended shadow map - a so-alled D-buffer, whih among other things stores distanes from the enter of the light soure to the umbra and penumbra volumes. The method is suited for implementation on most programmable hardware. Though some rude approximations are used in the visibility funtion, the method an be used to produe soft shadows with orret boundaries in real time. 1. Introdution Soft shadows provide information whih is important to give the viewer of a sene an impression of spatial relationships between objets. They are able do so beause the softness of a shadow provides information of the relative distanes between the light soure, the oluding objets and the objets reeiving the shadow. When hard shadows are used, this information is not available, whih an lead to misinterpretations of the sene. Non-real time global illumination methods, suh as radiosity and distributed ray traing, an handle various types of area light soures and thus render soft shadows by approximating the mathematis of light transportation. Real-time rendering of shadows is dominated by two hard shadow methods - shadow mapping ([23]) and shadow volumes ([9], [15]). The urrent methods for generating soft shadows in real time an be divided into: Image based methods, whih use one or more shadow maps. Objet based methods, where the atual geometry in the sene is used Image Based Methods In a number of methods the area light soure is sampled by a number of point light soures, and regular shadow maps are rendered from eah sample. These samples are then ombined into either a Radiane Texture([14]) or a Layered Depth Map([1]). To redue the number of samples needed, other methods render shadow maps for only the verties of a linear ([16]) or polygonal ([25]) light soure and interpolate these for eah reeiving pixel. In [22], a onvolution filter is used on a traditional shadow map to produe a soft shadow texture. The onvolution filter is dependent on the light soure and the distanes between the light soure, the oluder and the reeiver. [18] uses both a regular shadow map and a shadowwidth map and performs shadow alulation by applying a 2D funtion from a look-up of eah map Objet Based Methods Several methods ([12], [6], [24], [8]) add information of objet geometry to a shadow map in order to generate objet driven penumbrae. [19] uses Penumbra wedges as an extension of the shadow volume method, and visibility is approximated using a 16-bit stenilbuffer. The wedges are generated by extruding silhouette edges, whih are determined from the enter of the light soure. In more reent work ([2], [3]), the authors use programmable graphis hardware to projet the silhouette edges onto the light soure to alulate the visibility Problem statement In survey [13] a number of general issues onerning soft shadows are presented. It is laimed that physially exat soft shadows are impossible to inlude in a real time rendering method for general senes. Where most global illumination methods approximate the visibility using hundreds of samples of the light soure, the presented real time methods all try to overome this sampling through various simplifiations. However, in none of the urrent methods, true soft shadows an be gener-

2 (a) Point determined silhouette (b) Area determined silhouette Fig. 1. The shadow regions are not ast orretly if silhouettes are found using a single point when generating penumbra wedges instead of using the umbra and penumbra volumes from an area light soure. ated in real time for any given sene. Among the presented methods, the one from [3] is superior to the others, as it produes very onvining soft shadows. The two main approximations used are that the silhouettes are found using a single point of the light soure, and that overlapping oluders are not handled orretly in the penumbra region. An example of the onsequenes of the first approximation is shown in Figure 1. The ase in Figure 1 is quite extreme, sine the light soure is very large ompared to the geometry of the sene. It is noted in [3], that few subdivisions of a large light soure will make the artifat skethed in Figure 1 beome pratially indistinguishable. This is, however, not a general solution, and the problem of generating soft shadows in real time thus still needs work. The situation in Figure 1(b) ould be obtained if umbra volumes and penumbra volumes were used as an extension to the shadow volume method. These volumes are defined in [20] as respetively the intersetion of and the minimum onvex hull enlosing all shadow volumes generated from every point on the light soure. The different types of volumes are illustrated in Figure 2. The first task in generating these volumes is to determine the silhouettes of the oluding objets. This step is not shown for the simple oluding triangle in Figure 2, sine all edges are silhouette edges. When using more omplex oluders, the silhouette determination must be performed prior to the volume extrusion. The determination proedure depends on the shape of the partiular light soure. The silhouette extration for shadow volumes is desribed in [11]. For losed polygonal, two-manifold meshes, the method marks eah polygon as being either front or bak faing with regards to the light soure. An edge belongs to the silhouette if it onnets a front and a bak faing polygon. The shadow volume is generated by extruding eah silhouette edge toward infinity in the diretion defined by the light soure and the edge itself. The umbra and penumbra volumes are desribed elaborately in [7]. Here, as in Figure 2, the volumes are generated separately for eah oluding triangle, sine (a) Shadow volume (b) Umbra volume () Penumbra volume Fig. 2. Different types of volumes for shadows no silhouette is alulated. In the following, we desribe a method that overomes the problem of using the wrong silhouette. This is done beause the shape of the soft shadow probably is an important fator in the pereption of spatial relations between objets. The method is hardware aelerable with ability to run in real-time. 2. Our Method The key part of the algorithm is to determine the boundary of the soft shadow orretly. This means, that we shall be able to determine the umbra and penumbra regions of the shadow reeiver orretly. Seondly, we will make an estimate of the visibility of the area light soure as seen from every point x in the shadow region. For these purposes, we will introdue some simplifiations. First, a planar n-polygonal light soure is approximated by a irular disk in order to determine the umbra and penumbra regions fast and to be able to use a single parameter, t, for the estimation of the visibility funtion. For the hardware implementation of the visibility alulation, we will introdue an extension of the shadow map onept - a so-alled D-buffer, whih among other things stores distanes from the enter of the light soure to the umbra and penumbra volume, respetively. The method thus ontains elements from both the shadow volume and shadow map methods. The individual parts of the method are desribed in the following setions: Setion 2.1 desribes the proedure for generating the umbra and penumbra volumes. Setions 2.2 and 2.3 give an introdution to the rendering proedure. Setions 3 and 4 desribe eah of the two render passes. Setions 5 and 6 present the results and disus these along with suggestions for future work. Finally, we onlude in setion Umbra and Penumbra Volumes For this partiular method, a fast proedure for generating umbra and penumbra volumes for a irular light soure is required. As with shadow volumes, the proedure onsists of: a) finding the silhouettes and b) extruding the atual volumes. Silhouette Determination To determine the silhouettes, we test the front faing relationship between the

3 partiular fae Φ and the light soure for eah fae. Sine the light soure is a irular disk, the previously used method annot be used diretly. Instead we use the fat, that - in onstant time - we are able to find a single point p near on the disk whih is used to determine whether no points on the disk are front faing to the given fae. Similar, a point p far determines whether all points are front faing. This is illustrated in Figure 3(a). Assuming the normals are normalized, we find p near = + r(( n n ) n ) (1a) p far = r(( n n ) n ) (1b) where r, and n are radius, enter and normal of the light soure, respetively. If p near is in the negative half-spae of Φ, so is every other point on the disk. Similarly if p far is in the positive half-spae, then every other point is too. Using this property, the polygons are marked as being front faing to either the whole light soure (allfrontfaing), or to no part of the light soure (nonefrontfaing). An edge belongs to the umbra volume silhouette if it onnets a none-frontfaing fae with one that is not. Similarly, an edge belongs to the penumbra volume silhouette if it onnets an all-frontfaing fae with one that is not. It must be noted that some speial faes are neither all-frontfaing or none-frontfaing. These faes are loated between the two silhouettes, and we denote these silhouette faes. The silhouette faes are desribed further in setion 3.3. Volume Extrusion One the edges defining the silhouettes are found, the volumes an be extruded. This is done similarly to the way a shadow volume is extruded. Here, the point light soure is used as the extrusion point, that defines the diretion, in whih the silhouette edge is extruded. When an area light soure is used, we alulate separate extrusion points for eah umbra and penumbra volume quad. These are denoted p u and p p, respetively. In the following, we desribe how to alulate these points for an edge e, whih is assumed to be part of the silhouette for both volumes. This is not a general ase, sine an edge might be part of only one silhouette. p u and p p must be the two points eah ontaining a plane through e that is tangent to the irle defining the light soure. If e, defined by the points v 1 and v 2, and the light soure are not parallel, we find the intersetion a between the light soure plane and the line defined by e. v 1 n a = v 1 v 2 + v 1 v 2 v 1 n If a is outside the irle, p u and p p are found using regular 2D geometry, whih an easily be dedued from p far n Φ n p near (a) Silhouette determination e v 1 v 2 p u p p (b) Quad extrusion Fig. 3. The points used to determine silhouettes and extrude the volume quads Figure 3(b). If a is inside the irle, the extrusion points are undefined. Fortunately, this situation never ours for a silhouette edge in a losed two-manifold mesh. The speial ase where e is parallel to the light soure require speial treatment, see [17]. As seen in Figure 2, the extruded quads of the umbra volume need to be trimmed as they will otherwise interset. Similarly, the penumbra volume requires extrusion of triangles between the quads in order to lose the volume. The way this is done is ompletely dependent on the shape of the light soure, but a solution is to form a onneting triangle between the extruded quads. This is orret for the triangular light soure shown in Figure 2, and the approah an be used as an approximation for other light soure shapes. In the ase of a irular light soure, it is preferable to extrude more triangles using interpolation between p p for eah of the two adjaent edges Rendering Proedure The proposed rendering method is a multi-pass algorithm. The first render pass renders the objets of the sene and the umbra and penumbra volumes as seen from the enter of the light soure. This is done using a vertex program where distanes to the rendered entities are written to the individual olor hannels of the framebuffer. The result of this render-pass is stored in a D-buffer, similar to the way the shadow map method uses the Z-buffer. The main differene between the two buffers is, that the shadow map ontains only one depth value, whereas the D-Buffer ontains several distanes. This render pass is where the method differs from other shadow volume based methods, where the volumes normally are rendered in sreen spae to the stenil buffer. The seond render-pass renders the objets in sreen spae using a fragment program to alulate visibility. This fragment program uses the distanes from the D- buffer by performing a texture look-up and alulates the visibility using the retrieved values. In order to make the generation of the umbra and a

4 replaements non-oluded area oluded area t Penumbra volume x o x s x uv z z o x o x (a) Light soure intersetion x (b) 2D view Umbra x volume (a) D-Buffer x (b) Z-Buffer Fig. 4. The 2D projetion of the problem. penumbra volumes fast, we have limited the sene to ontain only one irular light soure Parameterization To alulate the visibility, V, we introdue a rude simplifiation: Assumption 1 For a given x in the penumbra region, the light soure is divided between oluded and nonoluded area by one and only one straight line. The ases where x is outside the penumbra region are trivial, sine the light soure seen from suh a point is either ompletely oluded or ompletely non-oluded, hene V = 0 or V = 1. This enables a projetion of the problem into 2D, whih simplifies the problem into finding the parameter t, as illustrated in Figure 4. In general, the partition of the light soure an be any urve resulting in both onvex or onave areas. One t is found for the given x, the problem is projeted bak to 3D using a parameter funtion, that desribes the visible area of the light soure. We have thus redefined the visibility funtion to a parameter funtion V (t). 3. D-Buffer The parameter t an, as it will be shown in setion 4, be found from various distanes along the line between the enter of the light soure and the arbitrary point x on the reeiver. These distanes are stored in the D-Buffer, whih an be thought of as a modified (and augmented) shadow map Definition The differenes between the ordinary shadow map and the D-Buffer is shown on Figure 5. The shadow map is generated by rendering the sene as seen from the light soure and storing the ontest of the Z-Buffer as shown in Figure 5(b). Instead of z-values, the D-Buffer stores distanes from to various geometry in the sene as illustrated in Figure 5(a). Fig. 5. Differene between shadow map and D-buffer. In the shadow map, only z o is stored, whereas the D-Buffer stores the distanes x o, x uv ( x pv ) and x s The distane x o is the distane from to the oluder. In the ase where x is not plaed in the umbra region x o and x will be the same. The distanes x pv and x uv are used to get information of distanes from the light soure to the penumbra and umbra volumes, respetively. x s is used to normalize the volume distanes and desribes the distane between the light soure and the silhouette edge. The distane x s an be alulated when the volumes needed to determine x uv and x pv are rendered. In the ase where the line x intersets the umbra volume (see Figure 5(a)), the alulation is performed using the following formula for the partiular silhouette edge e. x s = dir(,x uv) n dist(,e) (2) dir(,e) n where n is the normal of the light soure, dir(,e) and dir(,x uv ) are normalized diretion vetors from to e and x uv, respetively. dist(,e) is the distane between and e whih is uniquely defined beause of assumption 1. If the line x intersets the penumbra volume, x s is alulated using x pv instead of x uv Rendering to the D-Buffer The values x o, x uv and x pv are generated by rendering the oluder and the volume geometry as seen from. Instead of using standard matrix multipliation to transform the verties from world spae to light soure view spae, the z-oordinate of the verties is alulated as the distane between the vertex and using the eulidian distane formula. This an be done by programmable vertex engines. The distane x s is generated simultaneously with x uv and x pv by implementing equation (2) in the same vertex shader, that performs the speialized view transformation. This way, x s is generated without onstruting additional geometry. This vertex shader must have the silhouette edge e passed as a parameter.

5 Penumbra volume Gap Silhouette fae Umbra volume Fig. 6. Silhouette faes an ause undefined values in the D-buffer Furthermore, olor masking an be used to ensure that the distanes are rendered to the appropriate olor hannels of the texture, that stores the D-Buffer. During rasterization, the distanes are interpolated. This yields inaurate values, sine the interpolation sheme is based on linearly distributed values instead of spherially distributed values whih is the ase for the D-Buffer. This auses problems if the oluding objets onsists of very large polygons without subdivisions. In pratie, this artifat is virtually unnotieable Silhouette Faes When the volumes are rendered into the D-buffer, a partiular speial ase often arises, when a given edge is part of only one of the two silhouettes. When rendering only the volume quads into the D-buffer, gaps appear in the D-Buffer beause of the presene of silhouette faes, as desribed in setion 2.1. This is shown in Figure 6. In the gap, neither x uv values nor x pv values are rendered as part of either the umbra or penumbra volume, respetively. The straight-forward solution to this problem is to render the silhouette faes as a part of the umbra volume. There are two types of silhouette faes. The one shown in Figure 6 is front faing to less than half of the light soure area. The other type is front faing to more than half of the light soure area. To determine whih ategory a given silhouette fae belongs to, we hek whih half-spae of the faes plane the enter of the light soure lies in. If the enter is in the negative half-spae, it belongs to the ategory shown in Figure Visibility Approximation The visibility V is alulated by a parameter t, whih an be found from values in the D-Buffer. It an be shown using the 2D-projetion that two ongruent triangles exist and that these share an edge. This property yields t = { xs ( x x uv ) 2 x uv ( x uv x s ), x < x o 1 xs ( x xpv ) 2 x pv ( x pv x s ), x x o (3) The seond branh of equation (3) is found by observing symmetry around where t = 1 2. This symmetry is only valid in the ases where the given edge is a silhouette edge in both the umbra and penumbra volume. However, when rendering the silhouette faes as part of the umbra volume, equation (3) is valid in general. In equation (3) only values from the D-Buffer and x appear. This is essential for the possibility to alulate t (and thereby V ) for eah pixel in the framebuffer. When using a irular light soure, the funtion V (t) an be found by integrating the equation of a plane irle. The funtion obtained by this is, however, quite omplex and needs to be approximated by a simpler funtion. As noted in [21], a reasonable approximation would be to use V (t) 3t 2 2t 3 (4) The desribed visibility alulation works for many senes where assumption 1 holds. The ases where the assumption does not hold are: 1. Conave and/or overlapping oluders 2. No umbra region 3. Aute silhouette orners In the first ase, the problem is to determine from what oluding geometry the values stored in the D- Buffer should ome from. This hoie must be made, sine the visibility funtion is only dependant of one parameter. When the desire is to be able to handle onave oluders, the best solution is to hoose the geometry with the greatest distane from the light soure. This an be done using depth test when rendering to the D- Buffer. In this ase, the visibility funtion should still be equation (4). The seond ase ours when the light soure is muh larger than the oluder. In this ase, a given point on the reeiver an see the light soure from more than one side of the oluder. The problem beomes to render appropriate geometry for the umbra volume to the D-Buffer beause of self-intersetion. This an be handled by geometrial trimming of the umbra volume, but this is not trivial. It is muh easier to use depth test to ensure storing of proper distanes in the D-Buffer. The visibility alulated using equation (4) will in this ase yield values that are too low. Depending on the sene, an alternative funtion might be preferable. The third ase is hard to improve, beause of the way visibility is alulated. The artifats due to this violation are fortunately less visible than the other two ases. 5. Results The proposed method for approximating soft shadows was implemented using OpenGL, and the Cg language was used for the GPU-programming. The test system was a 3Ghz PC equipped with a QuadroFX 3000 Graphis ard. The fragment program needed to alulate the

6 (a) Our method (b) 1024 samples (a) Varying distane (b) Varying size light soure Fig. 7. Sene as in Figure 1 where the umbra region is determined orretly. 7(a) is rendered with our method, and 7(b) is ray traed using 1024 samples of the light soure visibility ontains 21 lines in the urrent implementation. Due to this simpliity, it is also possible to implement the method with the older NV20-arhiteture by using Register Combiners, as desribed in [17]. Figure 7 shows how the method handles the situation skethed in Figure 1. The images in Figure 7(a) are made by our method, and Figure 7(b) are ray traed referene pitures, where visibility is alulated using 1024 samples on the light soure. These sequenes show that the method alulates the shadow boundaries orretly. Figure 8 shows the apabilities of the method. In Figure 8(b) it is notied that the penumbra region expands orretly as the light soure size inreases. Similarly, it is notied in Figure 8(a) that the penumbra region diminishes as the oluder approahes the reeiver. Figure 8() shows a sene with a fairly omplex oluder. This image shows that the method is apable of asting shadows onto arbitrary geometry, whih is the ase beause () Complex oluder. Rendered using our method Table 1. Performane of different senes tri. FPS at different resolutions Sene ount 300* *600 1k*1k Fig. 8(a) Fig. 8(b) Fig. 8() Fig. 8() * * Simplified oluder geometry (d) Same as above ray traed using 1024 samples Fig. 8. Different senes showing the apabilities of the method visibility is alulated per pixel. The referene piture is shown in Figure 8(d). The overall performane of the

7 method is shown in table 1. It is notied, that the method is fill-limited for senes with oluders of low omplexity, and CPU-limited for senes with high omplexity. 6. Disussion The presented method is based on onserving the properties mentioned in the introdution. Here, it was noted to whih extent the shadows depend on having orret shadow boundaries. Figure 7 shows how the orret shadow boundaries give more information of the shape of the oluder than the method proposed in [3] where a single silhouette is used. The method from [3] would render exatly the same shadow for the two different oluders. Another very important benefit of our method is that it is implementable on a broader set of hardware arhitetures inluding the X-Box and all Nvidia GPUs sine the GeFore3. Sine the urrent implementation of the fragment program uses about 35% fewer instrutions than the one used in [3], a performane inrease is ahieved ompared to their method Optimizations The performane of the method ould be improved if some optimizations were made in the implementation. The urrent implementation uses no oherene between eah rendered frame, whih means that everything is alulated from srath for eah frame. This is done to ensure omplete interativity as everything in the sene an be hanged in real time. However, if the volumes only were updated when the relation between oluders and light soure is hanged, it ould be possible to improve the overall performane of the method. Another point where the urrent implementation ould be improved is the way the volumes are generated, sine the method beomes very CPU-limited when the omplexity of the oluder yields many geometry alulations done by the CPU. There is a possibility that a hardware aelerating method for volume generation ould be used to improve this. It is, however, not trivial to aelerate the volume generation in urrent vertex engines, as the number of verties needed for the various volumes is not onstant. If future vertex engines were to allow sene geometry to be stored in the memory of the graphis hardware, this might be exploitable in order to ahieve hardware aelerated volume generation. Furthermore, the method presented in [5] for generating shadow volumes entirely on the GPU ould possibly be enhaned to generate umbra and penumbra volumes Artifats The primary drawbak of the method is the limitations aused by the assumption. The fat that the light soure for a given x only an be oluded by one onvex objet should diminish the number of senes where the method is appliable. However, The implementation shows that in situations where the assumption is violated, the method still provides renderings that are reasonably satisfying, and in the ases where the assumption is valid, the results are quite onvining. An example of this is seen in Figure 7(a) where the onave oluder violates the assumption. As noted earlier, the D-Buffer stores only the distanes of the furthest of the overlapping oluding faes. This yields artifats in the penumbra region of the oluder s self shadow, but this problem is not very notieable in ontrast to an inorret penumbra region on the reeiver objet. This is beause self shadows generally have small penumbra regions sine distanes between typial oluders and typial reeivers are greater than distanes within onave oluders. Furthermore, the stenil buffer is used to avoid overwriting umbra regions. Another ase where the assumption is not valid is when the oluder is very small as shown in the top image of Figure 8(a). In this ase x an see the light soure from more than one side of the oluder. The approximate visibility beomes too low, whih makes small oluders ast more signifiant shadows than they physially ought to. This artifat an be redued as shown in Figure 7(a), where a more suited visibility funtion is used (V = t). In the ase where the angle between two onneted silhouette edges is very aute, the penumbra region will have some artifats due to the simplifiation of only dealing with olusion of one of the two edges. Beause the method uses projetive texturing, sampling artifats are bound to our beause of the limited resolution of the texture. This artifat is usually handled by implementing the light soure as a narrow spot light, thus using as muh as possible of the resolution in the projetion texture. This issue is disussed in [4] Future Work As mentioned earlier, the primary limitation is the restrition of only handling one intersetion of the light soure. This topi is where future expansions should fous. When dealing with more intersetions, the D-Buffer must ontain more values than the urrent method. This expansion ould be performed by applying the Depth Peeling tehnique desribed in [10]. This would enable storing more intersetions. The main problem lies in determining the area funtion when more intersetions are present. Figure 9 shows this problem for two intersetions of a irular light soure. As seen in the Figure, the angle α must be taken into aount when the area funtion V is defined. The angle between some vetor in the plane of the irle and the

8 PSfrag t 1 α Fig. 9. Two silhouette edges ause a irular light t2 soure to be interseted by two lines. axis of t ould be stored in the D-Buffer by projeting the normal of the quads of volumes onto the irle. α ould then be found as the differene between the angle orresponding to t 1 and t 2 respetively. The problem remains to determine a funtion V (t 1,t 2,α) that takes into aount the overlap between the two areas. One possibility would be to use a look-up texture, as done in [3], instead of alulating V (t 1,t 2,α) on the fly. If more than two intersetions must be taken into aount, the problem beomes even more omplex. 7. Conlusion We have presented a new method for produing soft shadows with orret boundaries in real time. The method is implementable on a wider range of existing graphis hardware than other existing methods produing similar (or better) quality shadows. The limitations disussed do of ourse narrow the range of senes where the method an give aeptable results, but as the image in Figure 8() shows, quite omplex geometry an be used with deent results. Furthermore, we have sueeded in in resolving the problem from Figure 1, sine our method determines the shadow boundaries better than other methods. This is mainly beause the hoie of using umbra and penumbra volumes has proven suessful. The artifats are all related to the visibility alulation, whih implies that this is the weaker part of the method. Even though the images from the urrent implementation are limited to a irular light soure, the method in itself may, as it has been noted, be used with other shapes of light soures. The modifiations needed for this are relatively small. Referenes [1] M. Agrawala, R. Ramamoorthi, A. Heirih, and L. Moll. Effiient image-based methods for rendering soft shadows. In Pro. of Siggraph, pages , [2] U. Assarsson and T. Akenine-Moller. A geometry-based soft shadow volume algorithm using graphis hardware. ACM Trans. on Graphis (TOG), 22(3): , [3] U. Assarsson, M. Dougherty, M. Mounier, and T. Akenine-Moller. An optimized soft shadow volume algorithm with real-time performane. In Pro. of the ACM SIGGRAPH/EUROGRAPHICS onf. on Graphis hardware, pages 33 40, [4] S. Brabe, T. Annen, and H.-P. Seidel. Pratial shadow mapping. Journal of Graphis Tools, 7(4):9 18, [5] S. Brabe and H.-P. Seidel. Shadow volumes on programmable graphis hardware. Computer Graphis Forum, 22(3): , [6] S. Brabe and H.-P. Seidel. Single sample soft shadows using depth maps. Graphis Interfae, pages , [7] A. T. Campbell. Modeling Global Diffuse Illumination for Image Synthesis. PhD thesis, Dept. of Computer Sienes, University of Texas, [8] E. Chan and F. Durand. Rendering fake soft shadows with smoothies. In Pro. of the 13th Eurographis workshop on Rendering, pages , [9] F. C. Crow. Shadow algorithms for omputer graphis. Computer Graphis, 11(2), [10] C. Everitt. Interative order-independant transpareny, Available at [11] C. Everitt and M. J. Kilgard. Pratial and robust shadow volumes, Available at rlwww.nvidia.om. [12] E. Haines. Soft planar shadows using plateaus. Journal of Graphis Tools, 6(1):19 27, [13] J.-M. Hasenfratz, M. Lapierre, N. Holzshuh, and F. ois Sillion. A survey of real-time soft shadows algorithms. In Eurographis, State-of-the-Art Report. [14] P. S. Hekbert and M. Herf. Simulating soft shadows with graphis hardware. Tehnial Report CMU-CS , CS Dept., Carnegie Mellon U., Jan [15] T. Heidmann. Real shadows, real time. Iris Universe, 18:28 31, [16] W. Heidrih, S. Brabe, and H.-P. Seidel. Soft shadow maps for linear lights. Eurographis Workshop on Rendering, pages , [17] B. Jakobsen, K. S. Petersen, N. J. Christensen, and B. D. Larsen. Implementing boundary orret soft shadows. Tehnial report, Informatis and Mathematial Modelling, Tehnial University of Denmark, DTU, [18] F. Kirsh and J. Doellner. Real-time soft shadows using a single light sample. Journal of WSCG, 11(2): , [19] T. A. Moller and U. Assarsson. Approximate soft shadows on arbitrary surfaes using penumbra wedges. Eurographis Workshop on Rendering, pages 1 9, [20] T. Nishita and E. Nakamae. Continuous tone representation of three-dimensional objets taking aount of shadows and interrefletion. In Pro. of Siggraph, pages 23 30, [21] S. Parker, P. Shirley, and B. Smits. Single sample soft shadows. Tehnial Report UUCS , Computer Siene Department, University of Utah, Otober [22] C. Soler and F. ois X. Sillion. Fast alulation of soft shadow textures using onvolution. In Pro. of Siggraph, pages , [23] L. Williams. Casting urved shadows on urved surfaes. In Pro. of Siggraph, pages , [24] C. Wyman and C. Hansen. Penumbra maps: approximate soft shadows in real-time. In Eurographis workshop on Rendering, pages , [25] Z. Ying, M. Tang, and J. Dong. Soft shadow maps for area light by area approximation. 10th Paifi Conf. on Computer Graphis and Appl., pages , 2002.