# Point-Based Approximate Color Bleeding

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4 are not needed. Figure 3 shows a few surfels and their rasterization onto the six faces of a cube raster. The solid angle associated with each raster pixel differs slightly depending on where on the cube face the pixel is. Figure 3: Coarse rasterization onto six cube faces. We use three different strategies to rasterize the color contributions; the appropriate rasterization strategy depends on the distance between the receiving point and the surfel Near-field: overlapping surfels For surfels that overlap (or nearly overlap) the receiving point we use ray tracing for full precision. We trace a ray through each raster pixel and check for intersection with the surfel disk. Although ray tracing in general is relatively time consuming, there are only a few surfels that overlap any given receiving point, and ray casting against a few disks without checking distances and without evaluating shaders at the hit points is actually very fast. Visible artifacts appear near sharp edges if overlapping surfels are not handled carefully like this Mid-field: other nearby surfels Surfels that are close to the receiving point, but do not overlap it, are rasterized individually. We compute the projected area of the surfel as seen from the receiving point. The correct, projected shape of a disk is an ellipse. However, as a speed optimization we replace the ellipse with an axis-aligned square with the same area as the projected disk. This is of course a very coarse approximation, but it is consistent and stable and seems to work well in practice. Duff 1984] could be applied without first checking which color goes over which (painter s algorithm). 3.5 Integrating over the hemisphere After all this, we have arrived at a coarse rasterization of the scene as seen from the receiving point. In order to compute the color bleeding we need to convolve the incident illumination with the BRDF. We do this by looping over all raster pixels and multiplying the rasterized colors with the BRDF for the direction corresponding to that pixel, and with the solid angle corresponding to that pixel. For diffuse reflection we can disregard all pixels below the horizon, and pixels above the horizon are weighted by the cosine of the pixel direction and the surface normal. 4 Caches For scenes of low to medium complexity (point clouds containing a few million surfels), the entire point cloud and octree can be stored in memory. However, for more complex scenes it is necessary to read octree nodes and surfels on demand and cache them. We use two caches: one for groups of surfels, and one for small groups of octree nodes. We observe very high cache hit rates in both caches when the receiving points are coherent. The caching approach means that point cloud files much larger than the available memory can be handled efficiently. 5 Parameter tweaking The point-based algorithm has two accuracy parameters: maximum solid angle σ max and the pixel resolution of the cube face rasters. Figures 4(a) (c) are rendered with decreasing values of the maximum solid angle (for raster resolution pixels). The render times are 4 seconds, 9 seconds, and 22 seconds, respectively Far-field: clustering for distant surfels For distant surfels we evaluate the spherical harmonic representation of the power and area of the cluster. The rgb power P of a cluster in direction (θ, φ) is: P(θ, φ) = 2 l=0 m= l l p lm Y lm (θ, φ), Figure 4: Varying maximum solid angle σ max: (a) σ max = 1. (b) σ max = 0.1. (c) σ max = Figure 5 shows the image quality for various rasterization resolutions. (The maximum solid angle is 0.03 for all images.) The render time is 20 seconds for the first image and 21 seconds for the second and third. and similar for the cluster area A(θ, φ). Then we rasterize an axisaligned square with the appropriate color and area Optimization In the description above, the octree traversal and rasterization are interleaved. This requires a z buffer for each of the six faces, and works just fine. However, we have found it slightly more efficient in practice to traverse the octree first (storing all directions, distances, areas, and colors), sort these contributions according to distance, and then finally rasterize them. Knowing that the contributions have already been sorted according to distance means that the rasterizer can be simpler. We have chosen to sort front-to-back; when a pixel is fully covered no further color is added to it. Alternatively, the sorting could be back-to-front, and the over operator [Porter and Figure 5: Varying the pixel resolution of the raster cube faces: (a) 2 2 pixels. (b) 4 4 pixels. (c) 8 8 pixels. 6 Results Figure 6 shows color bleeding in a scene with a car consisting of 2155 NURBS patches (many of which have trimming curves). The

6 7.3 Environment illumination Environment illumination from a standard or high-dynamic range image is easily added to the method. First a filtered environment map lookup is done for each direction and cone angle corresponding to a raster pixel. With the axis-aligned cube rasterization approach, this can be done once and for all for each environment map, so comes at neglible run-time cost. Then, during the rasterization for each receiving point, the rasterization simply renders disk colors over the environment map colors. The result is that the environment illuminates the receiving points from the directions that are unblocked by geometry. Figure 9(a) shows an HDRI environment map with four areas of bright illumination: two white, one green, and one blue), and figure 9(b) shows a dragon scene illuminated by the environment map. The image took 2.2 minutes to compute. 7.5 Ambient occlusion Ambient occlusion [Zhukov et al. 1998; Landis 2002] is a representation of how much of the hemisphere above each point is covered by geometry. Ambient occlusion is widely used in movie production since it is relatively fast to compute and gives an intuitive indication of curvature and spatial relationships. It is usually computed with ray tracing: a number of rays are shot from each receiving point, sampling the coverage above that point. Point-based ambient occlusion is a simplified version of the pointbased color bleeding algorithm. There is no need to store radiosity in the point cloud; only the point areas are needed. In the rasterization phase, only the alpha channel is used. Also, there is no need to sort according to distance since near-vs-far does not matter for occlusion. It is common to compute occlusion from both sides of each surfel. In this case the 9 spherical harmonics coefficients should be computed using the absolute value of d ni : alm = Z 2π φ=0 Z π Ai d ni Ylm (θ, φ) sin θ dθ dφ. θ=0 (The three coefficients with l = 1 will always be 0 due to the symmetry of the absolute dot product and the asymmetry of those spherical harmonics.) Figure 11 shows point-based ambient occlusion on the car from figure 6. The point cloud used for the computation contains 6.6 million surfels. The image took 3.4 minutes to compute. Figure 9: Point-based environment illumination: (a) HDRI environment map (dimmed for display). (b) Dragon illuminated by environment map. Note that this method will blur high frequencies in the environment map. With our raster resolution of pixel per cube face, the finest details in the environment map that can be represented are approximately 4π/( ) steradian. 7.4 Multiple bounces A simple extension allows the method to compute multiple bounces of diffuse reflection. For this variation, it is necessary to store surface reflection coefficients (diffuse color) along with the radiosity and area of each surfel. Then we can simply iterate the algorithm, updating the radiosity of the surfels in the point cloud in each iteration. Figure 10 shows a comparison of 0, 1, and 2 bounces in a Cornell box scene. The indirect illumination gets brighter with more bounces. Figure 10: Cornell box with spheres: (a) Direct illumination. (b) 1 bounce. (c) 2 bounces. Figure 11: Point-based ambient occlusion 7.6 Glossy reflection The point-based approach can also be used to compute glossy reflections. For this variation, we collect illumination from a cone of directions centered around the main reflection direction (instead of a hemisphere of directions centered around the surface normal). When traversing the octree, clusters and points outside the cone can be rejected. After rasterization, we multiply the glossy BRDF with the color of the rasterized pixels within the cone. Figure 12 shows an example of point-based glossy reflection. We use the same six raster faces as for diffuse reflection and ignore the raster pixels outside the cone. This means that the glossy coneangle can not be too narrow: for the resolution of the raster cube faces, we have found a coneangle of 0.2 radians to be the smallest that gives artifact-free results. For narrower reflection we would recommend using ray tracing. Alternatively, one could use a rasterization onto a single raster covering the directions within the cone. This would avoid wasting raster pixels for directions outside the cone, but is slower since it involves more trigonometric functions.

9 WILLMOTT, A. J., HECKBERT, P. S., AND GARLAND, M Face cluster radiosity. In Rendering Techniques 99 (Proc. 10th Eurographics Workshop on Rendering), ZHUKOV, S., IONES, A., AND KRONIN, G An ambient light illumination model. In Rendering Techniques 98 (Proc. 9th Eurographics Workshop on Rendering),

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