# Realtime Shading of Folded Surfaces

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3 Example images can be found in the appendix. 2 Algorithm outline This section gives a brief outline of our algorithm: Preprocessing For each vertex v of the mesh we compute the visibility map by rendering the scene using v as eye point and the normal n of the mesh in v as viewing direction. Pixels of the visibility map not covered by the mesh encode their corresponding direction in the hemisphere. The visibility maps are computed and stored for each vertex. At runtime The radiance with respect to the observers position is calculated for each vertex using a standard reflection model. The vertex visibility map is used to determine whether the radiance from a certain direction contributes to its radiance value. The positions of point light sources are transformed into the coordinate system of the visibility map. It is subsequently decided whether the point light source contributes to the illumination of the vertex or not. This can easily be performed using the visibility map. Finally, we assign the calculated radiance values to their vertices and perform a Gouraud interpolation. 3 Preprocessing During the preprocessing phase we generate the visibility maps for each vertex of the mesh. 3.1 Computation of visibility maps We compare three different ways to encode a visibility map: the hemicube [2], a single plane [3] and the hemisphere discretized into a rectangular grid. To obtain the visibility map in a vertex v the mesh is projected by a central projection with center v to one of these models (see figure 2). The hemicube, the hemisphere and the single plane are centered around v s surface normal n. The single plane is oriented perpendicularly to n. If the environment cannot be seen in a certain direction, the visibility of this direction in the visibility map is set to the RGB value (0, 0, 0), otherwise the direction itself is encoded in an RGB value. As we restrict ourselves to a cube model of the environment map, the RGB value (x, y, n) is given by the x and y coordinate

4 Figure No 1: Visibility test. The top image shows the mesh with a vertex (blue dot) inside a fold. The vertex is marked by its red normal. The bottom image shows the visibility map of this vertex. The model used for the visibility map is a hemicube with a resolution of pixels for the top side of the cube and for the sides. The figure shows the unfolded hemicube. For simplicity in this picture the directions in which the environment can be seen are coded by black and not by the direction itself. The directions where the environment cannot be seen are drawn in green. The red dot shows a projected point light source. In this example the light source can be seen by the specified vertex. Therefore the vertex is lit.

5 in the picture of the n-th side of the cube of the environment map. The numbering of the sides is as follows: The top face is numbered 0, the bottom face 5, the front face 1, the right face 2, the back face 3 and the left face 4. Figure 1 shows a sample mesh and the corresponding visibility map for one vertex. A hemicube model is used with a resolution of for the top and for the sides. The top image shows the mesh with a vertex (blue dot) inside a fold. The vertex is marked by its red normal. The bottom image shows the unfolded hemicube for the above mentioned vertex. For simplicity in this picture the directions in which the environment can be seen are coded by black and not by the direction itself. The directions where the environment cannot be seen are drawn in green. The red dot shows a projected point light source. In this example the light source can be seen by the specified vertex. Therefore the vertex is lit. The resolution of the visibility maps is n m. We experimented with values n, m [4, 256]. The resolution can be defined by the user. For good results we recommend n m 64, as the possible resolution of the outgoing radiance in a vertex is limited by this number in case of diffuse lighting of the environment. For a resolution less than 64 this results in blotchy images. The central projection for all three models can be computed using standard ray tracing. For the hemicube and the single plane standard OpenGL rendering can be applied. However, using standard OpenGL rendering to project the mesh onto the sphere is complicated and involves specific hard to implement clipping steps. Details on this can be found in [10]. For a static mesh the visibility map is fixed and is stored together with the coordinates and the material parameters (reflectance parameters) of a point. 3.2 Comparison of Methods Let v be the vertex of the mesh we determine the visibility map for. We evaluated the following methods for generating the visibility maps. 1. OpenGL-Rendering: We rendered the mesh using a triangle stripped display list of all triangles in the mesh. 2. OpenGL-Rendering with triangle pre-selection: In a first step, all triangles of the mesh are sorted into a three-dimensional grid. Then, triangle strips are generated for all triangles contained in a certain cell using a straightforward stripping algorithm. Finally, the triangle strips are stored in display lists. During the rendering of the mesh, we call the display lists of only those voxels that are within a given distance from the vertex v. This distance is given in units of the maximum dimension of the mesh. See table Raytracer: For every discretized direction of the visibility map we test all triangles of the mesh for an intersection with the ray leaving the point in that direction.

6 Method Model Resolution Running time Distance (sec) (ums) Triangle Rasterizer HC 8x HC 16x SP 8x8 4.3 SP 16x HS 16x HS 32x Triangle Rasterizer HC 8x /6 with triangle HC 16x preselection SP 8x8 3.1 SP 16x HS 16x HS 32x Raytracer with HC 8x /6 triangle preselection HC 16x SP 8x SP 16x HS 16x HS 32x Raytracer with grid HC 8x traversal HC 16x SP 8x8 9.3 SP 16x HS 16x4 7.2 HS 32x Table No 1: Overview of the running times of the visibility calculation routines. Abbreviations used: HC= hemicube, SP= single plane, HS= hemisphere. UMS=units of mesh size.

7 Environment Map T i L i I N Surface P i Hemicube V Figure No 2: The mapping between the binary visibility map of vertex v and the global environment map T i is encoded in each pixel P i of the visibility map itself. The environment patch T i emits L i in the direction I. θ is the angle between the surface normal N in v and the direction I of the incoming radiance. For simplicity, only a side view is shown. 4. Raytracer with triangle preselection: Similarly to OpenGL rendering with triangle pre-selection, we sort all triangles of the mesh into a three-dimensional grid and test only triangles in gris cells close to v. 5. Raytracer with grid traversal: We test only triangles lying in grid cells passed by the ray. Table 1 summarizes the run times for all combinations of methods and models. The computations were performed on an Intel Celeron 800 MHz machine with a NVIDIA TNT2 graphics card. To evaluate the core speed of the visibility calculation routines, we do not include the rendering of the environment map into the measurements, i.e. we do not encode the directions. The table demonstrates quite clearly that OpenGL rendering with triangle preselection in singleplane mode and raytracer with grid traversal in hemisphere mode are the fastest techniques for the preprocessing step. The singleplane model is efficient, however it has the disadvantage that the aperture angle must be less than 180 degrees, so only part of the half space is evaluated.

9 da : differential area at x (x x): radiance leaving x in the direction towards x S: set of all surface points { V (x, x 1 : x and x ) = are mutually visible 0 : otherwise (2) G(x, x ) = ( ω n )( ω n) x x 2 (3) For the evaluation of the hemicubes we assume, that every surface in the model is Lambertian, so that the reflected radiance is constant in all directions. This reduces equation (8) to: B(x) = B e (x) + f r,d (x)b(x )V (x, x )G(x, x ) da S = B e (x) + ρ d(x) π B: radiosity (outgoing) [W m 2 ] ρ d : diffuse reflectance for a Lambertian surface (ρ d = πf r,d (x)) S B(x )V (x, x )G(x, x ) da (4) Discretizing: B i = B e,i + ρ i N j=1 B j F ij (5) The form factor F ij from differential area da i to differential area da j is F ij = 1 V (x, x )G(x, x ) da j da i (6) A i A j π A i Delta form factors for the hemicube pixels: (A hemicube pixel covers the area A, the visibility information ist encoded into the pixels): F F = G(x, x ) π A (7) For the hemicube model the -form factors for top and side faces are computed analog to [11]. The contribution of one pixel in the visibility map of x is computed using the radiance stored in the corresponding texel of the environment map. This radiance is weighted by the -form factor of the pixel and then used as input of a local illumination model, which is Lambertian reflection in our algorithm. Note that for real-world environment maps no -form factor has to be applied to the outgoing radiance stored in the environment map, since it is already encoded in the corresponding real-world picture. Due

11 6 Conclusions and Future Work The algorithm presented is able of illuminating folded surfaces with extended light sources in realtime. The illumination conditions can be changed at runtime. So far our meshes cannot be deformed in realtime. This would require a complete new set of visibility maps at every frame. At the moment, our preprocessing step needs at least about 3 seconds of runtime, using the hemiplane model, which is not enough to achieve interactive frame rates. A possible optimization could be to update only the parts of the mesh which have changed. Furthermore, it might be possible to further exploit graphics hardware acceleration for the preprocessing step. Future versions of our method should use high dynamic range images for the environment maps, which, at this state, consist only of RGB images. This will increase the realism of effects simulated by the environment illumination. Moreover, a local illumination model described by Stewart and Langer [9] can be applied to estimate secondary irradiance. The use of this model yields perceptually acceptable shading without resorting to an expensive global illumination step. 7 Acknowledgements We would like to thank Markus Wacker from the University of Tuebingen for the mesh we used in our examples and our colleague Marcin Novotni for useful hints and corrections. Some of the used models were provided by References [1] A. James Stewart. Computing Visibility from Folded Surfaces, Elsevier Preprint, [2] J. D. Foley, A. van Dam, S. K. Feiner, and J. F. Hughes. Fundamentals of Interactive Computer Graphics, Addison Wesley, second edition [3] Micheal F. Cohen, John R. Wallace. Radiosity and Realistic Image Syynthesis, Morgan Kaufmann Publishers, Inc [4] Peter Shirley. Realistic Ray Tracing, A K Peters [5] T. Whitted. An Improved Illumination Model for Shaded Display, Communications of the ACM, vol.23, no [6] F. C. Crow. Shadow Algortihms for Computer Graphics, SIGGRAPH [7] L. Williams. Casting Curve Shadows on Curved Surfaces, SIGGRAPH

12 [8] J. T. Kajiya. The rendering equation, Computer Graphics, vol. 20, no [9] A. James Stewart. and M. S. Langer Towards accurate recovery of shape from shading under diffuse lighting, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 19, no.9, [10] B. Ganster Efficient cloth shading, Diploma Thesis, University of Bonn, [11] Michael F. Cohen and Donald P. Greenberg The Hemi-Cube: A radiosity solution for complex environments, SIGGRAPH 85 Proc., vol. 19, no.3, [12] H.W. Jensen Realistic Image Synthesis Using Photon Mapping, A K Peters, 2001.

13 8 Appendix Figure No 3: These three images show a folded dress consisting of 3120 vertices. The left image shows a frontal view of the dress. Due to the pre-computed shadowing the folds in the lower part of the dress are clearly visible. In the center image, the back of the dress is shown, slightly rotated against the front side of the environment cube. The right image is rendered using another point of view, showing the folds in more detail. Figure No 4: Environment cube map used for the rendering of the above images, with only the front side white.

14 Figure No 5: These images show the effects of the usage of the environment faces as colored light sources. The mesh used, is the same as in figure 3. The corresponding environment maps are shown in figure 6. The reflection of the different light sources can be distinguished from each other in the folds. Figure No 6: Environment cube maps used for the rendering of the above images. For the two left images top and bottom are black. Sides are red, green, blue and yellow. For the two right images left and right are blue and red, the top is white and the rest is colored black.

15 Figure No 7: These images show a textured mesh of parts of the Grand Canyon based on satellite altitude data with vertices. In the left image pre-computed shadows calculated with a hemicube model are used. In the right image no illumination is used. The visual impression of the left images reveals a greater depth impression due to the self-shadowing in the faults. Figure No 8: The Utah teapot. The mesh consist out of 3907 vertices and is illuminated with a point light source (yellow dot) in front of the teapot (left image). The light source visibility is calculated on a per vertex basis, as described in the paper. The right image shows the teapot with the light source moved above it. Self shadowing is also visible in both images.

16 Figure No 9: These images show a spaceship model with vertices, provided by A hemicube model with a resolution of pixels for the top of the cube and no additional point light sources were used. Self-shadowing is visible in the propulsion units, at the cockpit and under the wings.

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