GPU-based rendering of point-sampled water surfaces

Visual Comput (2008) 24: 77–84DOI 10.1007/s00371-007-0186-8 O R I G I N A L A R T I C L E

Kei IwasakiYoshinori DobashiFujiichi YoshimotoTomoyuki Nishita

GPU-based rendering of point-sampledwater surfaces

Published online: 7 December 2007© Springer-Verlag 2007

K. Iwasaki (�) · F. YoshimotoDepartment of Computer andCommunication Sciences,Wakayama University, Wakayama, Japan{iwasaki, fuji}@sys.wakayama-u.ac.jp

Y. DobashiGraduate School of Information Scienceand Technology, Hokkaido University,Hokkaido, [email protected]

T. NishitaGraduate School of Frontier Sciences, TheUniversity of Tokyo, Tokyo, [email protected]

Abstract Particle-based simulationsare widely used to simulate fluids. Wepresent a real-time rendering methodfor the results of particle-basedsimulations of water. Traditionalapproaches to visualize the results ofparticle-based simulations constructwater surfaces that are usuallyrepresented by polygons. To constructwater surfaces from the results ofparticle-based simulations, a densityfunction is assigned to each particleand a density field is computedby accumulating the values of thedensity functions of all particles.However, the computation of thedensity field is time consuming. Toaddress this problem, we propose

an efficient calculation of densityfield using a graphics processingunit (GPU). We present a renderingmethod for water surfaces sampled bypoints. The use of the GPU permitsefficient simulation of optical effects,such as refraction, reflection, andcaustics.

Keywords Real-time rendering ·GPU · Point-sampled geometry ·Caustics

1 Introduction

The research into fluid simulation is one of the mostimportant research topics in computer graphics. Manymethods have been developed for the simulation of fluidssuch as water, smoke, and fire [4, 5, 18, 19]. Most of thesemethods subdivide the simulation space into grids andsolve the Navier–Stokes equations by discretizing theequations, using the grids to simulate the fluid dynamics.These methods are based on the Eulerian method. On theother hand, particle-based fluid simulations have been de-veloped that represent the fluid as particles and calculatethe fluid dynamics by solving the particles dynamics [8,20]. Particle-based fluid simulation has received attentionsince this simulation method is free from the numericaldiffusions in the convection terms, suffered by the Eu-lerian method, and the surface transformation is easy tohandle.

One of the methods of visualizing particle-based simu-lation is to construct the water surface by polygons and torender these polygons. The water surface is constructed asfollows. Initially, a density function (or smoothing kernel)is defined with the distance from the center of the par-ticle as parameter. The simulation space is subdivided intoa grid, and the summation of the densities of the particlesis calculated at each grid point. Then the water surface isextracted as an iso-surface by using either the marchingcube [10] or the level set method [4, 15]. To render highquality images of the water surfaces, the simulation spacemust be subdivided into numerous small cells. This indi-cates that the computational cost of the density calculationat each cell also increases; thus, the cost of the construc-tion of the water surface becomes quite high. Moreover,many small polygons are generated from a fully subdi-vided grid. For the animation of the particle-based fluidsimulation, the processing of enormous numbers of small

78 K. Iwasaki et al.

polygons compared to the number of screen pixels in eachframe results in bandwidth bottlenecks. Therefore, theseproblems prevent the particle-based fluid simulation frombeing applied to interactive applications such as the pre-view of the simulation, video games and virtual reality.

In recent years, point-based rendering methods havebeen developed, using the points as primitives instead ofthe polygons [14, 22]. Several methods that are acceler-ated by the GPU have been presented [2, 6]. Moreover,a point-based method has been developed for visualizingiso-surfaces [3]. This method demonstrates that the pointbased visualization method for iso-surfaces can obtainstorage and rendering efficiency compared with standardpolygon-based methods.

Particle-based fluid simulation represents the fluid asparticles and calculates the dynamics. Therefore, visual-izing the particle-based fluid simulation by using pointprimitives is straightforward, since both the result data ofthe simulation and the data from the rendering are unifiedinto points.

This paper presents a fast rendering method, resultingin the particle-based fluid simulation without explicitlyconstructing polygons. In this paper, we deal with thewater as a fluid and describe a rendering method for thewater, represented by point primitives. To render the wa-ter surface, we have to take into account optical effects dueto water surfaces such as reflection, refraction, and caus-tics. Rendering these optical effects is essential to increaserealism. We present a fast rendering method for these ef-fects from water surfaces, represented by points.

The contributions of our method are as follows.

– Fast generation of point primitives, representing watersurfaces by using the GPU

– Fast rendering of the water surface, represented bypoints to obtain optical effects, such as refraction, re-flection, and caustics

The rest of our paper is organized as follows. Sec-tion 2 describes the related work. In Sect. 3, the overviewof our method is presented. Section 4 describes the calcu-lation of the density at each grid point by using the GPU.The method of rendering water surfaces, represented bypoints, is described in Sect. 5. The rendering results ofpoint-based fluid simulation are shown in Sect. 6. Finally,conclusions and future work are summarized in Sect. 7.

2 Related work

There have been many methods for visualizing the re-sults of the fluid simulation. These are categorized intotwo types. One is to polygonize the iso-surfaces, repre-sented by implicit functions, and then to render the poly-gons. Another is to directly render the implicit surface,without creating polygons. One of the methods to createthe implicit surface using polygons is the marching cube

method [10]. Many methods have employed this march-ing cube approach to render the water surface [9, 12, 19].Moreover, a GPU accelerated iso-surface polygonizationmethod has been proposed in recent years. Matsumuraet al. proposed a fast method of iso-surface polygoniza-tion using programmable graphics hardware [11]. Recket al. developed a hardware accelerated method to extractiso-surfaces from unstructured tetrahedral grids [16]. Al-though the marching cube method is efficient, represent-ing iso-surfaces by creating polygons requires the memoryfor the connectivity information and two different datastructures are required for points and polygons.

Another visualization method for fluid simulation ofwater involves the rendering of the iso-surface directly.Enright et al. [4] and Premoze et al. [15] employed a levelset method to represent the water surface. Their methodsrender the water surface by using Monte Carlo path tracingmethods. Whilst these methods can render realistic im-ages, the computational cost for the rendering is high.

Although not for the rendering of the results of thefluid simulation, a visualization method has been de-veloped for iso-surfaces using point primitives. Co et al.proposed a new algorithm called iso-splatting for render-ing iso-surfaces using point primitives [3]. This methodshows that the point based rendering of iso-surfaces canexceed the traditional polygon-based approach, such asa marching cube method in time and space efficiency.This method, however, does not describe the calculationmethod of the scalar(density) field, whose computationalcost is high.

To solve these issues, we present a novel approach torender the water surface in a particle-based fluid simula-tion. In our method, the iso-surface, representing the watersurface, is calculated efficiently by using fluid particles.Then the water surface is sampled, point by point, and ren-dered by surfels [14]. This makes it possible to unify thedata structure into points in the simulation and then ren-der, without the construction of polygons. Moreover, ourmethod presents a fast rendering method for reflection, re-fraction, and caustics by use of the point-sampled watersurface.

3 Overview

Figure 1 shows the overview of our method. This methoddeals with the results of the particle-based fluid simu-lation (Fig. 1a), calculated by particle-based simulationmethods, such as moving particle semi-implicit (MPS)and smoothed particle hydrodynamics (SPH). Then thewater surfaces, including caustics, are rendered as shownin Fig. 1d. To render the water surfaces, including caus-tics, particles that represent the surfaces must be extracted.Directly rendering the particles representing surfaces isone solution to visualize the result of the particle-basedfluid simulation. However, the number of particles used

GPU-based rendering of point-sampled water surfaces 79

Fig. 1a–d. Overview of our method: a Our method renders water surfaces from the results of the particle-based simulation. b We firstassign the density function to each particle and calculate the density field. c Then the iso-surfaces that represent the water surfaces areextracted and sampled by points. d Our method renders water surfaces represented by points and renders caustics taking into accountrefractions

in the simulation is usually between about 1000 and100 000 so that the number of particles representing a sur-face is, at most, several ten thousands. As Muller pointedout, this is not sufficient number to render high qualityimages [12]. On the other hand, point-based renderingmethods [2, 6, 14, 22] are designed to render huge numberof points measured by range scanners. Thus, it is diffi-cult to create high quality images of water surfaces byrendering only the particles used in the simulation.

Therefore, our method generates dense sampled surfels(Fig. 1c), representing water surfaces from all the particlesused in the simulation (Fig. 1a). We create a temporarygrid in the simulation space, where the densities of the par-ticles are accumulated in each grid point (Fig. 1b). Thedensity at each grid point is calculated as a density func-tion.

The cost of the density computation at each grid pointis quite high, since it depends on the number of grid pointsand the number of particles. We present a fast methodfor accumulating densities of particles by using the GPU.Our density calculation method can be applied not onlyto the particle-based simulation, but also to the grid-basedsimulation, since the marching cube method requires thedensity at each grid.

The points (surfels) on the iso-surface representing thewater surface are then extracted. The calculation of surfelson the water surface is explained in Sect. 4.

The water surfaces are rendered by splatting surfels(Fig. 1d). Refraction and reflection of light is calculatedper pixel. The rendering method of caustics from watersurfaces represented by surfels is described in Sect. 5.

4 Fast density calculation using the GPU

This section describes the calculation method of the dens-ity field from particles used in the fluid simulations. Wecreate a grid in the simulation space and calculate thedensity at each grid point by using particles. The simula-tion space is subdivided into nx ×ny ×nz grid points.

The density function F(r, h) in this paper is calculatedfrom the following equation [21].

F(r, h) =⎧⎨

⎩

405748πh

(− 49 a6 + 17

9 a4 − 229 a2 +1

)

(0 ≤ r ≤ h),

0 (r > h),

(1)

where a = r/h, and where r is the distance from the cen-ter of particle to a calculation point, and h the effectiveradius of the particle. Although we have used this smooth-ing function as a density function for the prototype, othersmoothing functions, such as the smoothing kernel of theSPH, could also be used as the density function.

The simulation space is located as shown in Fig. 2 andthe z-axis is set to be the vertical direction. A virtual cam-era is set along the z-axis and the reference point of thevirtual camera is set to be the center of the simulationspace. A virtual screen is then set to be perpendicular tothe z-axis. The virtual screen consists of nx ×ny pixels.Each pixel corresponds to a grid point on the grid planesperpendicular to the z-axis, as shown in Fig. 2. The pix-els in the screen frame buffer consist of R, G, B, and αcomponents. To calculate the density of each grid point in-fluenced by a particle, we use a metaball whose center isthe position of the particle. The disk of intersection be-tween the grid plane and the metaball, with an effectiveradius h, is calculated. The densities of pixels within thedisk of intersection are calculated. By drawing the disks

Fig. 2. Calculation of densities at each grid point by using splatting


of intersection with the densities and accumulating thedensities in the frame buffer, the density of each pixel,corresponding to each grid point of the grid plane, is cal-culated by using the GPU.

4.1 Density calculation

To calculate the density at each grid point, texture-mappeddisks are projected onto the screen corresponding to thegrid planes (see Fig. 2). The intersection disk between thegrid plane and the metaball whose center is the particleand the effective radius of h. The texture mapped onto thedisk represents the density function F on the disk. Thedensities on the disk are calculated from the distance fromthe center of the particle to the grid point using Eq. 1. Byprojecting the disks of the particles, intersecting the gridplane, onto the screen, and accumulating the densities, thedensities of the grid points on each grid plane are calcu-lated by using the GPU. For the computation accuracy inthe accumulation of densities, our method uses floating-point buffers1.

The disks are rendered by using point sprites. Thismakes it possible to accelerate the rendering process bythe GPU. The point sprites are hardware functions thatrender a point by drawing a square, consisting of four ver-tices, instead of drawing a single vertex. The point spritesare automatically assigned texture coordinates for eachvertex corner of the square. This indicates that each pixelinside the point sprite is automatically parameterized inthe square. Therefore, the distance, d, from the center ofthe particle to each pixel of the point sprites can be cal-culated by using the fragment program. By comparing thedistance, d, with the effective radius h, we can determinewhether the pixel is within the disk or not. The density ofthe grid point corresponding to the pixel is calculated byinserting the distance, d, into the density function F. Forthe density calculation, we prepare a texture whose par-ameter is the distance from the calculation point to thecenter of the particle. The density of the pixel correspond-ing to the grid point is efficiently calculated by mappingthis texture.

The density is scalar and the pixel of the frame bufferobject consists of four components. Therefore, our methodcalculates disks of intersection between the particle andfour grid planes at once, and renders four disks by stor-ing four densities in the RGB and α components. Afterdrawing all the disks intersecting the four grid planes, theRGBα components are read from the frame buffer objectinto the main memory.

4.2 Acceleration of density calculation using clustering

As shown in Eq. 1, the density contribution from the par-ticle at the grid point is zero, when the distance betweenthe particle and the grid point is larger than the effective

1 Our method uses framebuffer objects as floating point buffers.

radius h. To reduce the computational time of the densitycalculation, the particles whose density contributions arezero are eliminated. The particles are classified into clus-ters by using the z coordinates of the particles. Let zi1 , zi2 ,zi3 , and zi4 be the z coordinates of four successive gridplanes i1, i2, i3, and i4. Cluster Ci (i is the cluster number)includes the particles pi whose z coordinate pi

z satisfieszi1 −h ≤ pi

z ≤ zi4 +h. To compute the densities on fourgrid planes i1, i2, i3, and i4, particles in the cluster Ci areprojected.

4.3 Generation of surfels

After the density field in the simulation space is calcu-lated, the iso-surfaces are extracted. The density of theiso-surface is specified by the user. The iso-surfaces aresampled by points. The positions of the surfels, si , are setto the positions of these sample points. Normal vector, ni ,of surfel si is calculated by using the gradient of the den-sities. To render the water surfaces, a disk is assigned tosurfel si . The radius of the disk is Ri and the disk is per-pendicular to normal ni of the surfel. The radius, Ri , of thesurfel, si , is assigned and is determined so that there areno gaps between the surfels. If the distance between thesampled point and neighbor point is larger than a thresh-old, we add points on the iso-surface to fill gaps betweenthe surfels.

5 Rendering point-sampled water surface

This section describes the rendering method for watersurfaces represented by surfels. In this section, we firstexplain the rendering method of caustics due to water sur-faces represented by surfels. Then the rendering method ofwater surfaces is described.

5.1 Rendering caustics for point-sampled water surface

Our rendering method for caustics is based on Nishita’smethod [13] and Iwasaki’s method [7]. In these methods,the water surface is represented by a triangular mesh. Ateach vertex, the refracted direction of the incident light iscalculated. Then the volumes are created by sweeping thevectors refracted from the triangle mesh. These volumesare called illumination volumes [13]. The intersection tri-angles between illumination volumes and the object sur-faces are called caustics triangles. However, illuminationvolumes and caustics triangles cannot be created directlyfrom point-sampled water surfaces, since the surfels re-presenting water surfaces have no connectivity. To addressthis problem, we propose a rendering method of causticstriangles for water surfaces represented by surfels. More-over, our rendering method are fully implemented on theGPU, whereas the previous method [7] calculated illu-mination volumes and intensities of caustics triangles onthe CPU.


Fig. 3. Virtual screen for rendering caustics

To render caustics for point-sampled water surfaces,we set a virtual screen horizontally as shown in Fig. 3.Then point s(x,y) on the water surface corresponding topixel P(x, y) of the virtual screen, and the refracted ray ofthe incident light at s(x,y) are calculated. An illuminationvolume is created by sweeping the refracted vectors frompoints s(x,y), s(x+1,y), and s(x,y+1) (or s(x+1,y+1), s(x+1,y),and s(x,y+1)) that correspond to neighboring pixels. Thevertex c(x,y) of the caustics triangle corresponding to pixelP(x, y) is obtained by calculating the intersection pointbetween the refracted ray and the object surface. By relat-ing s(x,y) and c(x,y) to pixel P(x, y), s(x,y) and c(x,y) can becalculated on the fragment program.

The rendering algorithm of caustics on the GPU is fol-lows:

1. calculate normal and depth of each point correspondingto each pixel to obtain the point and the refracted ray,

2. calculate the intersection point (the vertex of the caus-tics triangle) between the refracted ray and the objectsurface,

3. calculate intensity at the vertex of the caustics triangle,and

4. render caustics triangles.

5.1.1 Calculation of points on the water surfaceand normals

The position of s(x,y) is calculated by using the depthd(x,y) of the water surface from the virtual screen. Normal,n(x,y), and depth, d(x,y), at point s(x,y) on the water surfaceare calculated from the following equations,

n(x,y) =∑

i g(ri(s)

Ri

)ni

∑i g

(ri(s)Ri

) , d(x,y) =∑

i g( ri(s)

Ri

)di

∑i g

(ri(s)Ri

) , (2)

where g is a Gaussian function whose parameter is dis-tance, ri(s), between each surfel, si and s(x,y), and re-turns 0 if ri(s) is larger than radius Ri . The calculation of

the normal, n(x,y), and depth, d(x,y), is accelerated by usingthe GPU as the previous method [1]. The normal n(x,y) andthe depth d(x,y) are stored as two textures, normal map anddepth map, respectively.

5.1.2 Calculation of vertices of caustics triangles

By using the depth map, the position of s(x,y) on the watersurface is calculated on the fragment program. The re-fracted vector at s(x,y) is calculated by using the normaln(x,y). The vertex c(x,y) of the caustics triangle is obtainedby the intersection calculation between the refracted rayfrom s(x,y) and the object surface. The vertex c(x,y) is it-eratively calculated on the fragment program, which issimilar to [17].

Let us explain the algorithm of the intersection calcula-tion using Fig. 4. We first calculate the point on the objectsurface corresponding to each pixel P(x, y) by renderingthe object surface to a texture. We call the texture, geom-etry map. Then we calculate the intersection point itera-tively as follows. First, the point P1 on the object surfacecorresponding to P(x, y) is calculated by referring to thegeometry map. Then the distance between s(x,y) and P1is calculated. If the distance is smaller than a thresholdspecified by the user, point P1 is regarded as the intersec-tion point. Otherwise P1 is projected onto the vector ofthe refracted ray at s(x,y). The projected point is referredto as P2. Then the point P3 on the object surface corres-ponding to P2 is obtained by using the geometry map.The distance P2 P3 is calculated to determine whetherP2 is regarded as the intersection point. Our method re-peats the above processes until the distance between theprojected point on the refracted ray and the correspond-ing point on the object surface becomes smaller than thethreshold.

Fig. 4. Calculation of the vertex of the caustics triangle

The vertices of the caustics triangles are stored as a tex-ture to calculate the intensities of the caustics triangles.The texture is referred to as intersection map. Moreover,the vertices of the caustics triangles are required to rendercaustics triangles. To do this, the vertices of the causticstriangles are rendered to a vertex buffer object that makesthe vertices of the caustics triangles be stored in the videomemory.


Fig. 5. Rendering the result of the MPS simulation of making waves

5.1.3 Calculation of intensities of caustics triangles

The intensity Lc at c(x,y) of the caustics triangle is calcu-lated from the following equation [13]:

Lc = Li cos θi T(θi, θt) exp(−σtlc)Fc fr + La, (3)

where Li cos θi is the intensity of the incident light ontothe water surface, θi is the incident angle between the nor-mal n(x,y) and the incident light, T(θi, θt) is the Fresneltransmittance, σt is the extinction coefficient of the lightin the water, lc is the length between s(x,y) and c(x,y). Fc isthe flux ratio and is calculated from Fc = S/Sc, where S isthe area of the triangle consisting of the three points on thewater surface corresponding to three pixels of the virtualscreen, and Sc is the area of the caustics triangle. fr is thediffuse reflectance of the object surface and La is the in-tensity of the ambient light. To calculate the flux ratio, thearea of the caustics triangle must be calculated. The areaof the caustics triangle is calculated by using the intersec-tion map that stores the vertices of the caustics triangles.The intensity Lc is stored in the video memory2.

5.1.4 Rendering of caustics using caustics triangles

Caustics are rendered by drawing caustics triangles andaccumulating the intensities of the caustics triangles.Since the vertices of the caustics triangles and the intensi-ties of the vertices are stored in the GPU, caustics trianglesare rendered efficiently.

5.2 Rendering water surfaces represented by surfels

Water surfaces are rendered through the use of a splattingtechnique. Our rendering method extends the method pro-

2 In our implementation, the intensity is stored in the vertex buffer object.To render the intensity to the vertex buffer object, our method uses a pixelbuffer object.

posed by Botsch et al. [1] to take into account refraction,reflection and caustics.

Before rendering, we eliminate invisible surfels byusing a backface-culling method. First, the surfels are ren-dered only to the z-buffer with all z values having anε offset added and the update of the z-buffer is enabled. εis specified by the user.

Then, the update of the z-buffer is turned off so thatthe overlapping surfels are blended if and only if the dif-ference of their depth values is less than ε. The positionof the water surface and its normal corresponding to eachpixel of the frame buffer are calculated by accumulatingthe Gaussian weighted normals and depths as described inSect. 5.1.1. By using the position and the normal at eachpixel, the reflection and refraction rays of the viewing rayare calculated per each pixel. The images of the object sur-face and caustics viewed from the viewpoint without thewater surface are rendered to textures. Then the intersec-tion point between the refracted ray and the object surfaceis calculated by using the method described in Sect. 5.1.2.The refraction of an object, with caustics through the wa-ter surface is rendered by refraction mapping of the im-ages of the object and caustics.

6 Results

Figure 5 shows the result of MPS simulation of makingwaves. The numbers of the points representing the wa-ter surface are from 55 000 to 120 000 in this animation.The average rendering frame rate of these figures is about43fps. That is, our method can render the water surfaces,represented by points, including caustics, refraction, andreflection in real-time. These images are created on a lap-top PC (CPU : Centrino Duo 2.16GHz, 2GB memory)with a NVIDIA GeForce Go 7900 GTX. The image size


of these figures is 512×512. The size of the virtual screenfor creating illumination volumes is 256×64.

The number of water particles used for the simula-tion is about 150 000. The temporary grid is subdividedinto 2563. The computational time of density calculationfrom the particles using the GPU is 0.14 s. The mem-ory for calculating the density in the GPU is only 1MB.For the software calculation, the computational time isabout 8.4 s. That is, our method using the GPU can cal-culate the densities about 60 times faster than the methodusing the software. The computational time for extract-ing the surfels on the iso-surface is about 0.16 s. OurGPU-based method extremely reduces the time of con-structing the water surface from the particles comparedto the software-based method. The relative difference be-tween the densities calculated by the GPU and those by thesoftware is about 1.9%. To verify the quantization errordue to the GPU-based density calculation, Fig. 6 shows the

Fig. 6a,b. Comparisons of the water surface generated by using theGPU-based density calculation (a) and the water surface calculatedby the software (b). These images are rendered from the particlesof the simulation of dropping a parallelepiped into the water. Thewater surfaces with caustics viewed from above the water surfacesare rendered

comparisons of the water surface that is extracted from thedensities calculated by the GPU and that by the software.The image quality of the water surface (Fig. 6a) calculatedby using the GPU-based density calculation is indistin-guishable from that by using the software-based densitycalculation (Fig. 6b).

7 Conclusions and future work

In this paper, we have presented a fast rendering methodfor the particle-based simulation. To calculate the watersurface from the result of the particle-based simulation,a temporary grid is created and the densities at each gridpoint by using the particles are calculated. We acceler-ate this density calculation by using a GPU-based splattingmethod. Then the iso-surface is extracted and representedby surfels. The rendering method has been developed fora water surface, which is represented by the surfels. More-over, our method can render the reflection, refraction, andcaustics due to the point-sampled water surface in real-time. Our method drastically reduces the times of the sur-face construction and rendering. This makes it possible toeasily preview the result of the particle-based simulation.

In future work, we plan to accelerate the extraction ofsurfels by using the GPU. Moreover, we would like to de-velop a method for rendering splashes and foams usingsurfels.

Acknowledgement This research was partly supported by theOkawa Foundation.

References1. Botsch, M., Hornung, A., Zwicker, M.,

Kobbelt, L.: High-quality surface splattingon today’s gpus. In: Proc. EurographicsSymposium on Point-Based Graphics 2005,pp. 17–24 (2005)

2. Botsch, M., Kobbelt, L.: High-qualitypoint-based rendering on modern GPUs. In:Proc. Pacific Graphics 2003, pp. 335–343(2003)

3. Co, C., Hamann, B., Joy, K.: Iso-splatting:A point-based alternative isosurfacevisualization. In: Proc. Pacific Graphics2003, pp. 325–334 (2003)

4. Enright, D., Marschner, S., Fedkiw, R.:Animation and rendering of complex watersurfaces. In: Proc. SIGGRAPH 2002,pp. 736–744 (2002)

5. Foster, N., Fedkiw, R.: Practical animationof liquids. In: Proc. SIGGRAPH 2001,pp. 23–30 (2001)

6. Guennebaud, G., Barthe, L., Paulin, M.:Deferred splatting. Comput. Graph. Forum23(3), 653–660 (2004)

7. Iwasaki, K., Dobashi, Y., Nishita, T.: A fastrendering method for refractive and

reflective caustics due to water surfaces.Comput. Graph. Forum 22(3), 601–609(2003)

8. Koshizuka, S., Tamako, H., Oka, Y.:A particle method for incompressibleviscous flow with fluid fragmentation. Int.J. Comput. Fluid Dyn. 29(4), 29–46(1995)

9. Kunimatsu, A., Watanabe, Y., Fujii, H.,Saito, T., Hiwada, K., Takahashi, T.,Ueki, H.: Fast simulation and renderingtechniques for fluid objects. Comput.Graph. Forum 20(3), 57–66 (2001)

10. Lorensen, W., Cline, H.: Marching cubes:A high resolution 3D surface constructionalgorithm. In: Proc. SIGGRAPH’87,pp. 163–169 (1987)

11. Matsumura, M., Anjo, K.: Acceleratedisosurface polygonization for dynamicvolume data using programmable graphicshardware. In: Proc. ElectronicImaging2003, pp. 145–152 (2003)

12. Muller, M., Charypar, D., Gross, M.:Particle-based fluid simulation forinteractive applications. In: Proc.

Symposium on Computer Animation 2003,pp. 154–159 (2003)

13. Nishita, T., Nakamae, E.: Method ofdisplaying optical effects within waterusing accumulation-buffer. In: Proc.SIGGRAPH’94, pp. 373–380(1994)

14. Pfister, H., Zwicker, M., Baar, J., Gross, M.:Surfels: Surface elements as renderingprimitives. In: Proc.SIGGRAPH 2000,pp. 335–342 (2000)

15. Premoze, S., Tasdizen, T., Bigler, J.,Lefohn, A., Whitaker, R.: Particle basedsimulation of fluids. Comput. Graph.Forum 22(3), 335–343 (2003)

16. Reck, F., Dachsbacher, C., Grosso, R.,Greiner, G., Stamminger, M.: Realtimeisosurface extraction with graphicshardware. In: Proc. Eurographics 2004Short Presentation (2004)

17. Stah, M., Konttinen, J., Pattanaik, S.:Caustics mapping: An image-spacetechnique for real-time caustics. IEEETrans. Vis. Comput. Graph. 13(2), 272–280(2007)


18. Stam, J.: Stable fluids. In: Proc.SIGGRAPH’99, pp. 121–128 (1999)

19. Takahashi, T., Fujii, H., Kunimatsu, A.,Hiwada, K., Saito, T., Tanaka, K., Ueki, H.:Realistic animation of fluid with splash andfoam. Comput. Graph. Forum 22(3),391–400 (2003)

20. Thurey, N., Keiser, R., Pauly, M., Rude, U.:Detail-preserving fluid control. In: Proc.Symposium on Computer Animation 2006,pp. 7–12 (2006)

21. Wyvill, G., Trotman, A.: Ray-tracing softobjects. In: Proc. Computer GraphicsInternational, pp. 439–475 (1990)

22. Zwicker, M., Pfister, H., Baar, J., Gross, M.:Surface splatting. In: Proc. SIGGRAPH2001, pp. 371–378 (2001)

KEI IWASAKI received the B.S., M.S., andPh.D. degrees from the University of Tokyoin 1999, 2001, and 2004, respectively. He ispresently an associate professor in the Depart-ment of Computer and Communication Sciences,at Wakayama University, Wakayama, Japan. Hisresearch interests are mainly for computer graph-ics.

YOSHINORI DOBASHI received the B.E., M.E.,and Ph.D. in engineering in 1992, 1994, and1997, respectively, from Hiroshima University.He worked at Hiroshima City University from1997 to 2000 as a research associate. He is

presently an assistant professor at Hokkaido Uni-versity in the Graduate School of Engineering,Japan since 2000. His research interests are com-puter graphics, including lighting models.

TOMOYUKI NISHITA received the B.E., M.E.,and Ph.D. degrees in electrical engineering fromthe Hiroshima University, Japan, in 1971, 1973,and 1985, respectively. He worked for MazdaMotor Corp. from 1973 to 1979. He has beena lecturer at the Fukuyama University since1979, then became an associate professor in1984, and later became a professor in 1990. Hemoved to the Department of Information Sci-

ence of the University of Tokyo as a professor in1998. He has been a professor at the Departmentof Complexity Science and Engineering of theUniversity of Tokyo since 1999. His researchinterests are mainly for computer graphics.

FUJIICHI YOSHIMOTO received a Ph.D. in com-puter science from Kyoto University, Japan in1977. He is presently a professor in the Depart-ment of Computer and Communication Sciences,at Wakayama University, Wakayama, Japan. Hisresearch interests are in entertainment comput-ing, computer graphics and CAD.

Documents

GPU-based rendering of point-sampled water surfaces