Eurographics Workshop on Natural Phenomena (2006), pp. 18 E. Galin, N.

Chiba (Editors)

Realistic Water Volumes in Real-Time

Figure 1: Examples of rendering of water volumes. All images are 800 600 and are generated at about 30Hz.

Abstract We present a real-time technique to render realistic water volumes. Water volumes are represented as the space enclosed between a ground heighteld and an animable water surface heighteld. This representation allows the application of recent GPU-based heighteld rendering algorithms. Our method is a simplied raytracing approach which correctly handles reections and refractions and allows us to render complex effects such as light absorption, refracted shadows and refracted caustics. It runs at high framerates by exploiting the power of the latest graphic cards, and could be used in real-time applications like video games, or interactive simulation. Categories and Subject Descriptors (according to ACM CCS): I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism

1. Introduction Realistic rendering of water is one of those effects that can highly increase the perceived realism of virtual scenes. However, the appearance of water is caused by many physical phenomena that were typically not affordable in realtime until recently. Indeed, the increasing functionalities and power offered by graphics hardware allow the adaptation of well-known off-line techniques to real-time applications. In particular, ray-tracing is well adapted to the SIMD nature of fragment processors. However, porting generic ray-tracers to the GPU is quite involved [PBMH02]. Hierarchical acceleration data structures, or generic mesh-ray intersection routines, must be carrefuly rewritten to t the GPU organisation. Fortunately, in specic situations, the computations can be greatly simplied and efciently mapped to the GPU. In particular, this is the case for the ray-tracing of heightelds.
submitted to Eurographics Workshop on Natural Phenomena (2006)

The question is then to identify the class of objects that can be adequately represented with these simpler models. The contribution of this paper is to show that water basins and similar structures can be quite realisically modelled by two heightelds, one for the ground, and one for the water surface. Using adaptations of recent techniques for heighteld rendering, complex effects like reections, refractions, light absorption, refracted shadows and caustics can be efciently simulated in real-time. Except for a part of the caustics computations, it runs entirely on the GPU. The amount of displayed geometry is negligible and the rendering time is proportional to the number of pixels occupied by the water on screen. The different features are mostly independent and can be enabled separately, allowing a tradeoff between realism and performance.

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2. Previous work Displaying water surfaces involves two orthogonal problems : simulating the movement and rendering it accurately. Our work focuses only on this latter part, and any animation method can be used. We briey review existing techniques. Water Simulation Fluids are governed by the NavierStokes equations which have been used at different levels of simplication. Applying these equations to a volumetric representation of water leads to physically accurate but computationally intensive simulations [FF01, EMF02]. They are required for complex movements like breaking waves or splashing water. The study of Navier-Stokes equations in the context of ocean surfaces modeling has led to the Gerstner and Biesel swell model which decomposes ocean waves into a sum of trochoidal wave trains. Realistic results [FR86, HNC02] have been obtained by combining it with models expressing waves amplitude, frequency and direction spectra derived from measurements [PM64, HDE80], and waves propagation properties like refraction near coasts [Pea86, TB87, Tes99, GM02]. These approaches generally allow one to compute a heighteld representation of the water surface in real-time. In the case of bounded water surfaces, the surface heighteld can be expressed as the solution of simple partial differential equations [KM90] which can be solved numerically by efcient local diffusion schemes [Gom00, Sta03]. The spectral approaches [MWM87, Tes99, PA00] rst compute a Fourier spectrum of the water surface heighteld, using physically measured distributions, before transforming it into a heighteld with a Fast Fourier Transform. The resulting heighteld is bounded but periodic. Water Rendering Rendering of water surfaces is difcult due to complex interactions with light. As explained by [PA00], the physical study of light-water interactions is a full-edged research eld with a vast litterature. Several works have focused on the transcription of these phenomena for computer generated images [Wat90, PA00, IDN03]. Although realistic, these approaches are computationally intensive. Realtime solutions exist, but they generally make important simplifying assumptions [NN94, HVT 04]. Reection and refraction are important effects for the realism of virtual water. Until recently, reection approaches would only address special congurations such as planar reectors [FvDFH90] or innitely far reected objects, or would be only interactive [HNC02] or based on precomputations [HLCS99, YYM05]. Recently, real-time approaches using GPU-based ray-tracing and image based approximations of the reected objects have been proposed [SKALP05, PMDS06]. Refraction algorithms usually consider one transparent object surrounded by an innitely far environment represented by an environment map (see [GLD06] for a recent review of these methods). This assumption can not be used in the case of the refraction of a ground surface through

a water volume. Computing the refraction of nearby objects can be done with memory costly light-eld precomputations [HLCS99]. Caustics are caused by the convergence of light due to reections and refractions. They are computationally expensive to render and are traditionally handled by global illumination algorithms like photon mapping [Jen01]. Interactive solutions exist based on different photon emitting and gathering strategies [EAMJ05,WS03,WD06,DS06] but they need to reduce the number of emitted photons to run in realtime, thereby producing noisy or blurred caustics. We use a method conceptually very similar to backward beam tracing with illumination maps [Arv86]. Heighteld rendering Our technique is based on efcient heighteld rendering. Efcient GPU-based implementations [POC05, Tat06, BD06] have made it an advantageous way to render geometric details with small computation and memory costs. These algorithms sample the heighteld texture along each viewing ray using different sampling strategies. [POC05] intersect the viewing ray with xed horizontal planes before computing a precise intersection with few iterations of a binary search. This amounts to an even sampling in the vertical direction, which can cause stair-stepping artifacts at grazing angles. This dependence of the sampling on the view angle is reduced by [Tat06] which chooses the number of samples between xed bounds according to the viewing ray direction. The horizontal sampling method described by [BD06] uses precomputed information to sample the ray optimally. This allows the accurate display of heightelds potentially containing ne features. 3. Our approacch 3.1. Modelling hypothesis Our observation is that, except for big waves, water surfaces are mostly 2.5D. Similarly, because of gravity, most terrain can be represented as a displacement over a at surface. Therefore, we choose to represent a water volume with two heightelds, the ground surface and the water surface, both encoded as 2D textures (Fig. 2). The texture for the water surface is the output of a simulation procedure which can be procedural (e.g. a sum of basis functions) or numerical (e.g. a physical simulation). To render the water volume, we ren-

(a)

(b)

(c)

Figure 2: The ground (a) and water (b) surfaces heightmaps and the corresponding water volume (c). der its bounding box and use a fragment shader to perform
submitted to Eurographics Workshop on Natural Phenomena (2006)

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ray-tracing. During this, we only consider one level of recursivity : for each viewing ray, we nd the intersection with the water surface, then the intersection of the refracted ray with the ground. Avoiding recursion is important because current fragment shaders do not support recursive function calls. This single level recursion assumption is correct if the

environment

ma

Figure 4: Four possible situations are handled by the algorithm. Figure 3: The single level recursivity assumption generally holds, even for large waves. ground is non-reective, and if no light-ray crosses the water surface more than once. Due to the leaning effect induced by refraction, even in presence of big waves, this hypothesis generally holds (Fig.3). When it is not the case, the visual difference induced by this simplication is hard to notice for the human eye. The rays we consider are shown on Figure 4. The algorithm outline is the following :
v viewpoint
for each screen pixel d direction of the corresponding viewing ray pg intersection (ground, ray ( v , d ) ) pw intersection (water , ray ( v , d ) ) i f ( pg undefined and pw undefined) do nothing ( discard fragment) else i f ( pg before pw ) pixel lightedGroundColor ( pg , d ) else dt refractedDirection ( d , n(pw ) , ) dr reflectedDirection ( d , n(pw ) ) pg intersection (ground, ray ( pw , dt ) ) pe intersection (ground, ray ( pw , dr ) ) Ct lightedGroundColor( pg , dt ) Cr i f ( pe undefined) envMap( dr ) else lightedGroundColor ( pe , dr ) F fresnelReflectivity ( d , n(pw ) , ) pixel (1 F) Ct + F Cr

arriving at pw after refractions. Instead, we make a simplication. We use a simple Phong model involving the ray from pw to the light, the normal, and the material at that point (the latter two being specied by texture maps). Only the amount of light is computed, taking into account the many possible light rays (see Section 4.3 on caustics). Note that we implicitly supposed that each ray entering the water hits the ground further on. This condition is satised if the water is correctly enclosed by the ground surface, i.e. if on the borders the ground height values are higher than the water ones. This holds for a basin or a pond, but not for an aquarium (Fig.5). In that latter case, a dynamic environment map can be used to approximate what the ray sees when exiting the water.

Figure 5: Outgoing light rays from an unclosed water volume. We look up an environment map to nd out what they see.

The fresnelReectivity procedure computes the ratio F between reected and incident radiance with the Fresnel formula. For a xed refraction index , it is only a function of the incident angle, which can be efciently computed with the approximation given by [Sch93] or simply sampled in a precomputed 1D texture. The lightedGroundColor procedure computes the color of a point on the ground from its coordinates, the viewing direction and the light position. True computations should integrate all rays coming from the light and
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3.2. Optimized rendering Our rendering algorithm is a tailored version of [BD06]. We refer the viewer to that paper for detailed description. In brief, it walks along the ray until a binary search can be safely run, i.e. it is guaranteed that the interval of that search contains at most one intersection with the heighteld. For that, it uses a pre-computed safety radius texture indicating for each texel the maximum neighborood in which a ray

e nv

iro n

m e nt m

ap

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above the texel can intersect the heighteld no more than once. The paper also describes a slower but still exact method that does not use the safety radius texture. In the rendering of water surface, we perform intersections with two heightelds, the ground and the water surface. As the ground is static, precomputing the safety texture can be done off-line. This is not the case for the water texture which is dynamic. Fortunately, compared to the ground, it generally has a small amplitude. We thus estimate the average amplitude and size of the features, and compute a constant safety radius accordingly. In the worst case, it gives a radius of 1, which boils down to using the exact intersection described in [BD06]. To further optimize the number of steps required to nd the intersection, we intersect the viewing ray with the bounding box of the water surface instead of that of the ground and water altogether. Conceptually, we run the ray intersection algorithm several times (see pseudo-code in Section 3). However, in practice, we can perform the search along the ray for the two heightelds at the same time, thereby saving the computations of the successive samples taken on the light ray. It allows to stop as soon as the rst of the two surfaces is hit. Moreover, if the two heightelds are packed in a single texture, this divides by two the number of texture lookups performed. Packing the ground heights with the water ones can be done during the precomputing of the water texture at no additional cost. We want to emphasize that various strategies can be adopted here. Although we use the artifact-free approach of [BD06], any other method such as [POC05, Tat06] can be used. One can also choose to use different methods (binary search, horizontal or vertical xed steps, Amanatides traversal, constant or texture-based safety radius, etc.) depending on the desired tradeoff between speed and quality, and depending on possible constraints on particular viewing conditions (distant or nearby water, grazing angles, large or small amplitudes of the heightelds, etc.). The only limitation is that the joint intersection described in previous paragraph is applicable only if the same method is used for intersecting with the ground and the water. Our goal in this paper was to show that the availability of these techniques and the heighteld representation for water basin and alike makes the rendering of complex effects achievable in real-time. We provide images and timings of what the programmers can expect, but the tuning for a particular application remains their responsibility. Finally, we would like to say a few words about the precision used for heighteld values. For most situations, using one 8 bit channel per heighteld is sufcient. But if the ground or water surfaces are to be viewed from very close, or if they have large amplitudes, it will not be sufcient. Even if the ground is smooth and is correctly sampled, the bilinear interpolations performed on 8 bits numbers will produce stairs artifacts. This can be avoided by using 16 bits textures

(the two height maps can be packed in a 32 bits texture with two 16 bits channels). Note also that as water is likely to have a small amplitude, it is more appropriate to store it in a normalized form by passing two additional parameters to the shaders : a height offset (the mean water level) and an amplitude. Doing so allows us to fully use the precision offered by the texture. 3.3. Integration with other objects For our method to be integrated within a classically modelled scene, we must handle correctly the interactions with other objects. The rst thing to notice is that for each fragment, the exact 3D intersection point of the viewing ray with the rst encountered surface is known. Transforming it into the camera frame, we can trivially compute an appropriate zvalue (instead of that corresponding to the bounding box we are actually drawing). Furthermore, for rays that do not intersect any of the two surfaces, the corresponding fragment is discarded, so it does not interfere with any past or future value in the framebuffer. Finally, the current depth in the zbuffer can easily be taken into account during the intersection procedure. We just stop the walk along the ray when this value is exceeded (if it corresponds to a point vz between the segment [v0 , v1 ], the only modication to do is to restrict this segment to [v0 , vz ]). For all these reasons, our water volumes seamlessly integrate with the z-buffer, yielding correct occlusions with other rendered primitives. This statement is however true only for objects that do not enter totally (like shes) or partially (like the legs of a character) the water volume. Treating these cases is difcult because after undergoing the refraction, the viewing rays can be scrambled in a way which is difcult to predict (Fig.6). Having objects occluding viewing rays between the water surface and the ground breaks the organization into heightelds of this specic volume which made the raytracing feasible. Note that this effect is typically faked with adhoc hacks in games.

part viewed after correct refraction part viewed after simply rendering from viewpoint

Figure 6: Objects in water are difcult to correctly display because of refraction.

4. Simulated effects We now discuss a list of effects that can be straightforwardly simulated with our approach. Each effect can be turned on
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or off in the shaders, allowing for a tradeoff between realism and speed. 4.1. Underwater light absorption Careful attention has to be paid to lighting phenomena occuring in water. As explained in [PA00], some of these phenomena, like light scattering, are difcult to handle. Thus lighting equations have to be simplied. We use the lighting model given by [PA00] : for a given wavelength , the radiance transmitted from a point pg under water to a point pw on the surface is given by : L (pw , ) = (d, 0)L (pg , ) + (1 (d, z))Ld (1) where is the direction from pg to pw , L (p, ) is the outgoing radiance from the point p in the direction , d and z are respectiveley the distance and the depth difference between pg and pw , Ld is a constant diffuse radiance computed from scattering measurements involving sky and sun colors, and (d, z) describes an exponential attenuation depending on traveled distance and depth : (d, z) = ea db z (2) where a and b are attenuations coefcients depending on water properties. The rst term of the equation equates radiance coming from pg attenuated with traveled distance, the second one equates the contribution of diffuse scattering, depending also on depth. Remember that during the rendering process we compute the precise coordinates of pg and pw so traveled depth and distance are easy to compute. This makes the implementation of this lighting equation costless relative to other computations. Light attenuation can vary with wavelength, producing visually compelling chromatic scales. Ideally the color spectrum has to be discretized and computations must be done per wavelength. Using only the three components of the RGB color space gives an appropriate approximation. This is done by computing the attenuation color : (d, z) = (R (d, z), G (d, z), B (d, z)) (3) and combining it component-wise with the incoming and diffuse colors. Precisely desired color variations (e.g. physically measured) can be obtained by using a precomputed 2D color texture containing sampled values for (d, z). In practice, the inuence of depth (the b z term) is subtle and considering only traveled distance can be sufcient. Doing so simplies the attenuation coefcient to : (d) = ea d (4) which can now be sampled in a 1D color texture (Fig.7). With this simplication, L(pw , ) is the simple linear blending between L(pg , ) and Ld with respect to (d). Notice that a can be seen as a distance scaling factor, so
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Figure 7: An example of color variations induced by the exponential attenuation.

it can be scaled according to the scale of the water volume. If this value is large, it means that light will be quickly absorbed so the deep parts of the ground surface will not be visible. In that case it is useless to keep running on the refracted ray after a certain limit distance dmax . This distance can be dened by xing an attenuation threshold under which the incoming ground color is considered to have a negligible contribution : , ea dmax < log() a log() min a (5) hence simply computable as : dmax = max

(6)

4.2. Self-reections Reections on the water surface can be separated into three classes : reections of distant objects, reections of near objects and reections of the parts of the ground emerging from water (Fig. 8). The rst class can be easily handled using an environment map whereas only heuristics exist for the second one. We call the third class self-reection. This case is more specic : emerging parts of the ground are represented in the ground heighteld, which allows us to perform a rapid raytracing of the reected ray, as we do for refracted rays. So to take these reections into account, the reected ray is intersected with the ground surface. If an intersection exists, the corresponding lighted ground color replaces the one that would be otherwise looked up in the environment map.

pe pw

Figure 8: Reected ray hitting an emerging relief. One can notice that, as with refracted rays, we simplied the exact raytracing algorithm : the reected ray could actually hit the water surface before reaching the ground or exiting the box, thus splitting in two new secondary reected and refracted rays. But it is an indirect, difcult to witness, effect which doesnt affect realism, making this simplication reasonable.

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4.3. Caustics and shadows Computing caustics is hard to carry out in a forward raytracing approach because it requires us to count the amount of incoming light at each rendered point. That is why most existing caustics algorithms are based on backward raytracing : rays are cast from the light source instead of the viewpoint. We use a two-pass photon-mapping-like algorithm, involving GPU/CPU transfers of textures. Fortunately, we only need small textures and this does not impede the performance too much. The rst pass renders the water surface from the light source into a photon texture. The texels of this texture record the coordinates of where the corresponding light rays hits the ground. Since the ground is a heighteld, we need only recording the (x, y) coordinates, which leaves the third channel of the texture available to store the photon contribution, based on the Fresnel transmittance, the traveled distance, the incident angle and a ray sampling weight. We apply a random jittering to light rays to reduce aliasing artifacts (see Figure 9). We then gather the photons recorded in
light frustum mapped photons

(a)

(b)

(c)

(d)

Figure 10: To compute caustics and shadows (a), we render a photon texture from the light (b) and invert it into an illumination texture (c) which is bilaterally ltered (d).

Figure 9: Photons are emitted from jittered pixel positions and mapped on the ground surface. the texture. This boils down to computing a two-dimensional histogram of this texture. Unfortunately, it can not be done efciently on the GPU because of the lack of forward mapping, so the photon texture is transfered to CPU where it is processed to construct an illumination texture (by analogy to illumination maps of [Arv86]). We traverse the texels of the photon texture, retrieve the position and intensity of the corresponding photon, and add this intensity to that stored at that position in the illumination texture. The illumination texture is then transferred to the GPU where it is used for lighting in the nal render pass. The illumination texture can be very noisy due to the crude sampling of our approach. Applying a simple gaussian blur would remove noise but would also blur caustics patterns which, by nature, present high frequencies. That is why edge preserving lters like anisotropic diffusion or bilateral ltering [TM98, DD02, WMM 04] are traditionally used to lter density maps in photon mapping techniques. We apply a GPU-based bilateral lter which gives good results once correctly tuned. The whole process is summarized in Figure 10.

The resolutions for the two intermediate textures must be adjusted to balance the desired quality and framerate. The resolution of the photon texture directly determines the number of photons. The more photons are used, the more accurate the caustics will be. The illumination map can be seen as a lighting texture directly mapped on the ground, so its resolution affects the neness of the reproduced caustic patterns. Note however that these two resolutions are linked : if too few photons are emitted, a large caustics map will be noisy. The resolutions are limited by the cost of their transfer between GPU and CPU memory. Using a 256 256 photon texture and a 128 128 illumination texture gives good results in practice. This approach for computing caustics also accounts for shadows cast by the ground on itself, with refractions handled. Shadows of other objects in the scene onto the ground can not be cast that easily. Once again, the difculty is to handle the refraction of rays. However, one could use a shadow map approach as a coarse approximation. It would integrate directly with our algorithm by performing a projective texture lookup into the shadow map, using the world position where the ray hits the ground. 5. Results and discussion We implemented the proposed algorithm using GLSL shading language, and ran the example on a GeForce 7800FX. We implemented animation using [Gom00] with forces specied by random rain-drops or by mouse interactions
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(see accompanying videos). Table 1 indicates performance measurments. The cost is highly dependent on the number coverage 100% 50% 25% refraction 26 Hz 45 Hz 87 Hz + auto-reection 20 Hz 36 Hz 69 Hz + caustics 15 Hz 23 Hz 34 Hz

Table 1: Framerates obtained at 800 600 for different screen coverages and with various effects turned on successively (rightmost column combines all effects). of fragments rasterized. For our tests, the displayed volume covers a large part of a 800 600 window, yet it renders in real-time. Another interest of the method is that its memory requirements are low: only a couple of textures are required. In all examples shown, we used 128 128 textures for the ground and water surfaces, yet they look appealing. Figure 1 and Figure 11 show the kind of effects we can simulate. We believe the accompanying videos should also convince that a high level of realism is achieved with this technique. In particular, relevant effects are simulated, and the illusion of water is convincing. 6. Conclusion and future work In this paper, we show how the encoding of water volumes with two heightelds allows for real-time rendering of realistic water. Using an efcient ray-tracing approach for such a representation, we show that many effects can be combined : single bound refractions and reections, fresnel effect, light absorption, caustics and shadows. None of these effects are new; the contribution of this paper is to validate the amenability and suitability of heightelds in this context. Each effect can be turned on or off depending on the desired tradeoff between speed and quality. The method is based on simplications of the underlying physical phenomena, yet gives very appealing results. The rendering cost is directly proportional to the screen occupancy of the displayed water. Thus, we believe it can be of great interest for rendering elements of a virtual world such as aquariums, ponds, muddy puddles, fountains, etc. in a much more realistic way that previously. Yet, even at large resolutions, it is still fast and may also prove useful for sea and lake rendering. In the future, we would like to investigate a couple of optimizations (we plan to release the shaders and a detailed description of the formulae used). We also want to generalize the approach to other objects that combine a transparent layer in front of an opaque one, such as windows in a city walkthrough. We also want to tackle the problem of geometry-based objects that penetrate the water, for which we investigate approaches that dynamically project the geometry onto the heightelds.
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Figure 11: Examples of renderings obtained with our technique.

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