Hemi-Cube Ray-Tracing: A Method for Generating Soft Shadows

Transcription

1 EUROGRAPHICS 90 / C.E. Vandoni and D.A Duce (Editors) Elsevier Science Publishers B.V. (North-Holland) Eurographics Association, Hemi-Cube Ray-Tracing: A Method for Generating Soft Shadows Urs Meyer Department of Computer Science Multimedia Laboratory University of Zurich, CH-8057 Zurich Irchel meyer@ifi.unizh.ch Abstract This paper presents a new ray-tracing technique for generating soft shadows. The technique treats scenes consisting of light sources and opaque objects which are polygons or polyhedra of arbitrary shape and size. To determine the intensity at a point on a surface, the hemisphere is sampled extensively through the use of hemi-cubes. So-called item-buffer boxes, a combination of itembuffers and buffer boxes, are used to calculate ray intersections as well as to suppress image aliasing. Several methods for reducing aliasing caused by hemi-cubes are discussed. The uniform treatment of rays allows for a straightforward extension of the algorithm to produce fuzzy reflections. The success of the new technique depends on a fast implementation of a visible surface algorithm as provided by today s high-end graphics workstations. The results are images of realistically illuminated synthetic environments. 1. Introduction One of the goals in computer graphics is the synthesis of photorealistic images. The research of the eighties indicates that simulating the global propagation of light leads to the best and most impressive results. Currently known methods include ray-tracing and radiosity. One important aspect of realistic image synthesis is the capability of creating shadows. The type of light source determines the type of the shadow. Point light sources as well as parallel light lead to shadows with sharp edges, whereas finite size light sources lead to soft shadows with fuzzy boundaries. Soft shadows consist of umbrae and penumbrae. An umbra is the part of a surface which receives no light from the source and a penumbra is the region which receives only part of the light emitted by the source. Several techniques have been proposed to compute shadows. Most of them fall into the following categories: shadow volume algorithms, ray-tracing methods, and radiosity methods. An overview over these techniques with concentration on shadowing will be given in the following. Crow [7] describes a shadow algorithm for point light sources and polygonal objects that involves calculating a shadow volume, which is the surface enclosing the volume of space swept out by the shadow of an object. The shadow surfaces are then added to the data and treated as invisible objects. Brotman [3] and Nishita [14] extend Crow s technique to cast soft shadows. Brotman approximates extended light sources using a modified depth buffer algorithm. The light sources are modeled as a set of randomly placed point light sources. The shadows are generated by superposition of the shadow volumes of each point light source, The modified depth buffer stores the information necessary to accomplish the task. Nishita on the other hand calculates the volumes for umbra and

2 366 U. Meyer penumbra and uses contour integrals to determine intensity values. Both methods are restricted to convex light sources and convex objects. In ray-tracing rays are traced from the observer s eye through the scene to the light sources [19]. If a ray intersects a surface, a probe ray is sent to each light source to determine if the intersection point is in shadow. The point is in shadow if another object intersects the line segment between that point and the light source. Since a single ray is used per light source only point light sources can be simulated. Verbeck introduced a comprehensive light source description for computer graphics [ 15]. Concerning the geometry of light sources he distinguished point sources, linear sources, and area sources. Soft shadows are created by approximating the last two types with a series of point sources. Their individual contributions are calculated using probe rays and summed up to find the intensity. This approach depends on the number of point sources used and becomes computationally expensive for large area sources. Another approach to soft shadows is distributed ray-tracing by Cook [6]. Extended light sources are stochastically sampled by perturbing the rays around the main direction to the source. Complex scenes need many rays to achieve accurate results. Amanatides uses cones instead of single rays to sample light sources in order to produce fuzzy shadows [1]. When a ray is sent to an area light source for shadow testing, the ray is broadened to the size of the source. The visible part of the light source gives enough information to generate fuzzy shadows. The radiosity method introduced by Goral simulates the interaction of light between diffusely reflecting surfaces [11]. Although it deals with a different model - every surface is treated as a light source - it is mentioned because of its capability to produce soft shadows. Cohen extended the radiosity method to handle complex environments containing occluded surfaces and shadows [4]. He introduced the hemi-cube algorithm which samples the half-space above a surface. It is used in radiosity methods to determine form-factors which describe the energy exchange between pairs of surfaces. An object space smoothing algorithm creates a continuous shading across a surface. This paper presents an improved ray-tracing algorithm to calculate soft shadows and fuzzy reflections. When rays are traced through the scene to the light sources, a visible surface processor determines quickly the surfaces hit by the rays for all rays at once. The light intensity at an intersection point of a ray with a surface is determined using the hemi-cube algorithm which efficiently samples the half-space above the surface. If the surface is specular, the intensity arriving from the reflected direction is calculated. The algorithm deals with polygonal and polyhedral light sources and objects which are not restricted to be convex. This publication will first describe a general formulation for the propagation of light and then discuss interpretations of the intensity and reflectance functions used in standard ray-tracing and hemi-cube ray tracing. Thereafter the aliasing problem is discussed. 2. Illumination Models An equation describing the propagation of light is introduced. It will be shown that standard raytracing in general and the algorithm presented can be derived from this equation by approximations and simplifications.

3 Hemi-Cube Ray-Tracing The general equation A general formulation for the propagation of light has been provided by several authors. Kajiya established the rendering equation and Immel presented a general radiosity method for non-diffuse environments [13, 12]. They both described a complete equation accounting for global illumination. The formulation given by Wallace [16] will be used in the following: (1) where intensity leaving the surface in the outgoing direction incident intensity arriving at the surface from the incoming direction surface emission sphere of incoming directions angle between incoming direction and the surface normal bidirectional reflectance (and transmittance) function of the surface Figure 1. Definition of terms. Solving the integral of equation (1) is very difficult since it involves a one-dimensional and a twodimensional continuous function which are not analytically available. Nevertheless, there exist a number of pragmatic solutions for illumination models in computer graphics which are approximations of the reference formulation expressed in equation (1). The main reason for simplifying equation (1) is to obtain a model which is computationally tractable in a reasonable amount of time. The disadvantage of simplifications is that one looses the capability to simulate the propagation of light accurately. One traditional approach is treating the bidirectional reflectance (transmittance) function as composed of a diffuse non-directional term and a specular (transmissive) directional term: Another approach is to sample the integral at a finite number of points. Ray-racing is such a method and will be further explained in the next section. (2)

4 368 U. Meyer 2.2. Standard Ray-Tracing Standard ray-tracing simulates point light sources, shadows, multiple specular reflections and refractions [19]: (3) The reflectance function has been reduced to constants for the diffuse term and to and for the specular term and the transmissive term respectively. The diffuse term is approximated by a sum of n terms where each term calculates the contribution of a point light source. This is equivalent to interpreting the function of equation (1) as a delta-function, which is non-zero only for a finite number of directions, specifically the directions to the point light sources. Concerning the specular and transmissive terms. a single ray is cast to sat-ole the associated direction, i.e., and are non-zero in one direction only. These rays are traced further until they either eventually hit another surface or nail off into space. The integral of equation (1) has been thus reduced to a small number of evaluation points. However, the drawbacks of such an approximation are quite substantial. Standard ray-tracing is not capable of correctly accounting for global illumination effects such as soft shadows, indirect illumination, and more Hemi-cube ray-tracing The method proposed improves the standard ray-tracing algorithm using better approximations of and than expressed in equation (3). To calculate the intensity leaving a surface at a given point p, the diffuse and specular parts are considered separately. Figure 2. Principle of hem-cube ray tracing.

5 Hemi-Cube Ray-Tracing 369 For calculating the diffuse component at point p, the half-space above the surface is extensively sampled independent of light source directions. For that purpose a hemi-cube is placed on the surface such that its center coincides with p (figure 2). The hemi-cube sides are discretized into pixels which divide the half-space into small solid angles. The equation describing the diffuse component is as follows: (4) where set of all pixels on the hemi-cube incoming intensity from direction of pixel q (only light sources contribute) delta form-factor for pixel q on the hemi-cube The delta form-factor describes the fraction of light arriving through pixel q [4]. The total intensity at the surface point p is obtained by summing up the delta form-factors of all hemi-cube pixels multiplied with the intensity arriving through pixels q. The cosine term for diffuse reflections is incorporated in the delta form-factors. In case of a mirror-like surface the specular component at p is calculated by sampling the reflected direction. A reflection frustum with a small opening angle and a small resolution is placed at the surface point [16]. Each pixel of the frustum defines a ray which is traced to get an intensity value. These values are then weighted according to the reflectance function p using an array This leads to the following equation for the specular component (5) where set of all pixels of the reflection frustum incident intensity from direction of pixel r weight of pixel r (simulates the bidirectional reflectance function) solid angle of pixel r In summary, the sides of the hemi-cube as well as the reflection frustum each define a viewing volume which is used by a visible-surface algorithm to create a projection of the scene. This process is then used to calculate the diffuse and the specular components of the intensity leaving a surface as described by the equations (4) and (5). Since the method presented considers the whole half-space above a point on a surface and the neighborhood of the reflected direction, it is capable of simulating soft shadows and fuzzy reflections. To process the large number of sample points involved with the hemi-cube ray-tracing approach in a reasonable time it is inevitable to rely on fast hardware.

6 370 U. Meyer 3. The Aliasing Problem Point sampling a continuous function causes aliasing if the maximum frequency component of that function is too high. The Nyquist theorem states that the sampling rate must be at least twice the maximum frequency of the function to be sampled in order to avoid aliasing. Several techniques have been proposed to reduce aliasing or to make it less conspicuous: filter out or bandlimit the high frequencies before sampling, increasing the sampling density, or perform non-uniform sampling [8, 19, 20]. The following sections discuss two kinds of aliasing: image plane aliasing and hemi-cube aliasing. Methods to reduce aliasing implemented in a experimental version of the algorithm are presented and discussed afterwards Image Plane Aliasing The image plane consists of a set of pixels arranged in a regular, rectangular grid. The scene is sampled by casting one ray per pixel. This sampling rate is too low to accurately represent object silhouettes or sharp shadow boundaries leading to aliasing appearing as jaggies. They can be suppressed by using a sufficient number of intensity levels [9], i.e., more rays are needed per pixel to generate more intensity levels. The relatively simple and popular method of adaptive super-sampling [19, 10] permits to reduce aliasing for most cases by using just a few more rays. The general idea is to cast more rays and average the resulting intensities if the scene gets complex within the pixel. A scene is determined to be complex if the intensities taken at the four corners of a pixel vary too much as in the case of object silhouettes. Then the pixel is subdivided into subpixels and the process is repeated for each subpixel. It is assumed that the subdivision process is limited to a maximum depth. The set of the samples used in adaptive supersampling is a subset of the samples of the same image at a resolution corresponding to the maximum depth. Brotman used so-called buffer boxes to produce an anti-aliased image [3]. The screen is divided into buffer boxes, which when enlarged fit into the screen. This allows one to super-sample the image space. By averaging the resulting intensities, a partial anti-aliased image corresponding to the buffer box is produced. If that process is repeated for each buffer box the final anti-aliased image can be composed step by step. Furthermore, in ray-tracing the process of determining the intensity at a given sample point consists of two parts: 1) intersect the ray with the scene and determine the nearest intersection and 2) compute the intensity value at the point of intersection. The first part is equivalent to determine first the visible surface and afterwards intersect the ray with the surface. Weghorst introduced the itembuffer to speed up the first part using a visible surface algorithm [18]. The item-buffer is a table which stores the identifications of the visible surfaces for each pixel. The observations made above are used to introduce an extension called item-buffer boxes (IBB s), a combination of the known item-buffers and buffer boxes, to be used for adaptive supersampling. A visible-surface processor (VSP) creates the IBB s. The buffer-boxes principle cancels the memory limitations imposed by the VSP. If a unique color is assigned to each surface, the VSP directly creates an item-buffer, since the color value obtained for each pixel identifies the visible surface. The creation of IBB s is a preprocess for adaptive super-sampling. Whenever the intensity for a sample point has to be evaluated, the IBB serves as a lookup table for the ray-surface intersection procedure (figure 3). Experiments show that the overhead incurred by the multiple rendering of the scene for the IBB s is negligible compared to the time needed to render hemi-cubes.

7 Hemi-Cube Ray-Tracing 371 Figure 3. a) original image consisting of 16 item-buffer boxes (i.e., max. 2 subdivisions for anti-aliasing). b) individual item-buffer box: line crossings represent lookup values for visible surfaces (thick lines indicate resolution of original image, thin lines indicate resolution of IBB) Hemi-Cube Aliasing The limited resolution and the regular arrangement of the pixels of the hemi-cube lead to hemi-cube aliasing. A visible surface algorithm projects the scene onto the hemi-cube. The pixel values are used to gather incoming intensities from light sources in order to calculate the diffuse part of the surface s intensity. In other words, the solid angle of the visible part of the light source is estimated using a finite number of regularly spaced sample values. Errors in the estimation lead to regular low-frequency intensity patterns on a surface (plate 1.a).* Baum [2] investigates the hemi-cube algorithm very carefully and establishes the necessary assumptions for the hemi-cube to get exact results. The problems resulting from violations of the assumptions are then discussed. In the following, these assumptions are presented briefly and are related to hemi-cube ray-tracing Proximity Assumption The proximity assumption requires that the distance between two surfaces A, and A, is great compared to the effective diameter of A, (figure 4). In hemi-cube ray-tracing this assumption is violated in the case of two adjacent surfaces, if one is a light source and the other one is opaque. Errors will occur at the boundaries of the two surfaces. The intensity in these regions will be very high; actually too high for the display process in most cases and, therefore, will be clipped to the permitted range of values for the frame buffer. Thus, errors created due to the violation of the assumption vanish when using intensity clipping * See page 557 for Plate 1 a

8 372 U. Meyer Figure 4. Hemi-cube aliasing showing the true projection and the pixels used to approximate the projection calculation Visibility Assumption The visibility assumption requires that the visibility of surface does not change when observed from where is that part of A which is projected to a given pixel on the hemi-cube. A third interposing surface may hide part of visibility without changing the projection on the hemicube. The delta form-factor and, therefore, the intensity will be over- or underestimated. If each pixel is evaluated using a single sample, significant errors may result. Small changes in the hemi-cube s position may cause significant sudden changes of the projection of a surface. This will generate aliasing artifacts at shadow boundaries. The magnitude of the error for a given configuration depends primarily on the size of A j, which itself depends on the sampling density of the corresponding surface: the higher the sampling density the smaller the size of A,. Thus, increasing that sampling density reduces the error. The solutions presented in a subsequent section will reduce such aliasing. Since the intensity varies heavily at shadow boundaries, adaptive methods will most likely proceed to the maximum sampling rate Resolution or Aliasing Assumption The resolution or aliasing assumption states that the true projection of each visible surface onto the hemi-cube must be accurately accounted for using a finite resolution hemi-cube. Only surfaces projecting exactly on entire hemi-cube pixels meet this criterion. However, the projection typically uses only fractions of pixels at the boundary. This kind of aliasing causes the projected surface area and, thus, the delta form-factors to be over- or underestimated (figure 5). If the scene is uniformly sampled, low frequency aliasing occurs along the track of a scanline caused by abrupt changes of projections on the hemi-cube (figure 6).

9 Hemi-Cube Ray-Tracing 373 Figure 5. The same triangle sampled at different positions leading to different estimations of its size. Figure 6. Hemi-cube aliasing: while the hemi-cube moves along a scanline the projection of a light source changes abruptly. Low-frequency regular aliasing patterns appear on the surface. The magnitude of the error depends on the size of the projection, i.e., on the number of pixel it covers, and on the circumference per area ratio of the projection. Projections of thin rectangular surfaces lead to large errors Anti-Aliasing Solutions The violation of the resolution assumption remains the principal source of computational errors. Anti-aliasing methods should, therefore, try to improve the precision of projection calculations or to distribute the error locally using non-uniform sampling methods such that aliasing is less conspicuous. Increasing the hemi-cube resolution raises the accuracy of the intensity calculation. Theoretically an infinite resolution hemi-cube would solve this aliasing problem perfectly. But implementations can only deal with finite resolutions. The computational costs increase with O(n²) if n by n is the hemicube resolution. Supersampling the image space and average reduces aliasing as well. If each image plane pixel is subdivided the complexity of this approach is O(m²), where m is the subpixel resolution. Adaptive

10 374 U. Meyer supersampling concentrates on regions with high intensity gradients and regions containing object silhouettes. Its complexity depends on the length of the object contours and the maximum subdivision level. Jittering is an attractive method for non-uniform sampling. There are several possibilities for applying a jitter: the x-, y-position of the hemi-cube s center on a surface (x-y-jitter), the rotation of the hemi-cube about its z-axis (orientation-jitter), the shape of the hemi-cube (it would then be a hemi-box), and the resolution of the hemi-cube, although the last could only be done for a finite number of discrete values. In this paper jittering x-, and y-coordinates, and jittering the rotation angle, as well as their combination have been investigated, since the other ones involve recomputation of the delta-form factors. X-y-jitter of the hemi-cube is transformed to jitter primary rays within their pixel of the image plane. X-y-jitter and orientation jitter have a complexity of O(1). The methods described above can be combined arbitrarily. Jittering a regular grid of m by m subpixels is a combination of supersampling and x-y-jitter. The complexity is O(m²). Other combinations are discussed in the following section. 4. Results Plates 1*and 2*show sequences of a simple test scene. The white polygon above is an area light source and the remaining two polygons are opaque surfaces. The red polygon casts a shadow on the grey polygon. Each pair of images in plate 2 show the scene with a different hemi-cube resolution. The left one has been rendered using a 32 by 32 hemi-cube and the right one using a 64 by 64 hemicube. The intensity has been measured at the indicated scanline and plotted for comparisons. In plates 2.a)-g) a thin light source has been chosen to amplify the aliasing artifacts while plate 2.h) shows the same configuration with a larger light source. Plate 1 shows enlargements of the left images of plates 2.a), 2.b), and 2.c). In plate 2.a) the hemi-cubes used for a given surface have the same orientation. Aliasing artifacts appear on the bottom surface as regular patterns indicating the hemi-cube movements during the surface sampling process. In plate 2.b) the hemi-cubes have been randomly rotated for the intensity calculation (orientation-jitter). Artifacts appear as strong noise. The intensity changes from pixel to pixel are larger than in plate 2.a). Plates 2.c) and d) show images 2.a) and 2.b) after x-y-jitter has been applied. Aliasing artifacts are substantially reduced in 2.c) whereas in 2.d) the noise introduced by orientation-jitter is still dominant. Plates 2.e) and 2.f) show images 2.c) and 2.d) after adaptive supersampling has been applied. A maximum of 2 subdivisions has been chosen. Image quality could further be enhanced. The right image in plate 2.e) is almost perfect. Plate 2.g) is an example of jittering a regular grid of 2 by 2 subpixels together with adaptive supersampling. The image quality could not further be enhanced, since it is less likely that the projection of the scene onto the hemi-cube changes significantly for subpixel movements, especially if the hemi-cube resolution is low. Plate 2.h) shows the same configuration as in 2.a) but with a larger light source. Although no antialiasing methods have been used, the image quality is much better than in plate 2.a). The projections of the light source onto the hemi-cube are large enough for not to create aliasing artifacts * See page 557 for Plates 1 a-c and Plates 2a-h

11 Hemi-Cube Ray-Tracing 375 Finally, plates 3*and 4*show realistically illuminated environments. The hemi-cube resolutions have been 32 by 32 for the room with table and chairs and 64 by 64 for the highly reflective room with the Sierpinsky pyramid. In both cases, x-y-jittering and orientation-jittering as well as adaptive super-sampling up to a maximum of two subdivisions have been applied. The rendering times were 4.7 and 21 hours, respectively, on a Silicon Graphics 4D/20G. 5. Conclusion Generating soft shadows within complex polygonal environments is a time consuming operation. The new ray-tracing method presented simulates area light sources employing hemi-cubes, which permit an extensive sampling of the intensities needed to produce realistically illuminated environments. Compared to other shadow algorithms the scenes that can be processed are not restricted to convex objects [3,7, 14]. The complexity of the algorithm is dependent on the number of polygons in a scene, whether they are light sources or not. Rays are treated uniformly by utilizing viewing volumes and visible surface processing. This is true for primary rays which are generated using item-buffer boxes, as well as secondary rays which are generated using reflection frusta, as well as light feeler rays which are generated using hemi-cubes. Several anti-aliasing techniques have been analyzed. Experiments with test scenes indicate that nonuniform sampling leads to better results. Specifically, low frequency aliasing patterns resulting from uniform sampling can be reduced or suppressed using nonuniform sampling. Hemi-cube raytracing produces acceptable results for scenes not containing very thin light sources. It is very well suited for scenes with very large light sources. Ray-tracing allows the geometry of objects to be different for various purposes. Wallace divides spheres and cylinders into polygonal elements for shading and display and uses the exact geometry during the form-factor computation [17]. In the method discussed here, spheres or cylinders or other geometric primitives may be modeled by dividing them into polygonal facets. These are used for visible surface processing and illumination sampling including shadow calculation. The exact geometry is used to find the true intersection point of ray-object intersections and to calculate the true surface normals. Acknowledgements Thanks go to Peter Stucki, Hartwig Thomas and Charles Wüthrich for many inspiring discussions and for their help in preparation of this text. References [1] Amanatides, John: Ray Tracing with Cones. Siggraph 84 Proceedings, ACM Computer Graphics 18(3), pp , July [2] [3] Baum, Daniel R., Rushmeier, Holly E., and Winget, James M.: Improving Radiosity Solutions Through the Use of Analytically Determined Form-Factors, Siggraph 89 Proceedings, ACM Computer Graphics 23(3), pp , July Brotman, Lynne Shapiro and Badler, Norman I.: Generating Soft Shadows with a Depth Buffer Algorithm, IEEE Computer Graphics & Applications 4(10), pp. 5-12, October * See page 558 for Plates 3 and 4

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