Displacement Mapping

Transcription

1 HELSINKI UNIVERSITY OF TECHNOLOGY Telecommunications Software and Multimedia Laboratory Tik Seminar on computer graphics Spring 2002: Rendering of High-Quality 3-D Graphics Displacement Mapping Andrei Räisänen 40402k

2 Displacement Mapping Andrei Räisänen Abstract This paper describes the basics of displacement mapping, a technique used for enhancing the visual quality of any 3-D model by inserting more detail into it. Displacement mapping creates new geometry by first tessellating (dividing) existing triangles into smaller ones and then perturbs the new geometry by displacing the created vertices according to a displacement map. The objective of this paper is to present an overview of the process, not to delve too deep into the complex mathematics involved. A generic algorithm for tessellation is explained, as well as some discussion of the present state of displacement mapping and its applications. 1

3 1 INTRODUCTION The increasing hunger for more and more detailed 3-D implementations drives developers to model even more complex objects. However, the processing power of computers rendering the objects sets limits to the complexity of the models. One way of increasing the detail of surfaces is to apply another map to the surface, beside the texture map. The idea of perturbing a surface s geometry to add more detail is not new, Mr. Blinn introduced some ideas back in 1978 [1]. Until recently, the most used technique for visually enhancing a surface s geometric detail has been bump mapping, particularly so with real-time rendering applications. With bump mapping (also referred to as normal mapping), the surface of a triangle is textured with a map which effectively alters the normal of the surface at every unit of the bump map. When the surface is shaded (most often with the Phong shading model), the changed surface normals at different points of the surface polygon make the visual outcome look like the surface had, indeed, bumps. Figure 1.1 demonstrates this. Figure 1.1: Bump-mapped objects. 2

4 Bump mapping is indeed an effective way of rendering high-frequency detail on a given 3-D object. However, problems arise when viewing the surface very closely or from a steep angle, thus exposing the object s silhouette. When examined closely, the object s silhouette looks flat although the bumping suggests otherwise (see Figure 1.1). This is due to the fact that bump-mapping does not alter the surface geometry, but only the normal of the surface. Another artifact with bump mapping is that shadows might get rendered totally wrong, as Figure 1.2 demonstrates. Figure 1.2: Left: Shadows with traditional bump-mapping. Right: Correct shadows. 3

5 2 DISPLACEMENT MAPPING - OVERVIEW To avoid the problems of bump mapping, a new concept was introduced: displacement mapping first by Cook [2]. The idea behind displacement mapping is to actually change the geometry of the 3-D model, not just play with the surface normals. This is achieved by first tessellating (dividing) the existing triangles into smaller ones (Section 3) and then applying a displacement map on the surface which transforms the geometry by displacing the vertexes of the new, smaller triangles along the normal of the original surface (Section 4). Figure 2.1 summarizes the whole process. The use of displacement mapping induces much overhead to the processing, but eliminates some of the artifacts which plague bump mapping, incorrect shadowing for example (see Figure 1.2). Figure 2.1: A displacement mapped triangle (big, bottom grey triangle). 4

6 3 TESSELLATION The first step in displacement mapping is to tessellate the existing, big source triangles into smaller ones. There are many algorithms for this, one of which will be presented here. The best result is achieved by creating sufficiently small new triangles, preferably smaller in average size than the pixels of the target surface (i.e. screen) to which they will be drawn. The most common technique of subdividing a triangle is shown in Figure 3.1. The triangles are then iteratively subdivided further, if necessary, to produce geometry small enough to suit the purpose of creating new, finer detail. Using this algorithm, the number of resulting triangles for one source triangle is 4 n, where n is the number of iterations. For example, applying this tessellation algorithm four times to a source triangle results in 4 4 = 256 triangles. Figure 3.1: Tessellating a triangle. The most straightforward way of subdividing a big, original triangle is to uniformly divide it into a web of smaller triangles, throughout the whole mesh. The advantage of this method is that it will look very good and the new geometry can be generated easily, regardless the amount and type of detail in the displacement map. However, this method also creates very much overhead to the rendering process. What if the displacement map being applied consists of large flat areas with little or no detail? In such cases tessellating the whole source triangle into hundreds or thousands of small triangles is totally unnecessary. To lessen the overhead of the rendering process, schemes for preventing the unnecessary creation of too many triangles while tessellating can be used. One method for this is to process the displacement map to examine how much detail it has and make a height variation map for every texel [4]. This is done by calculating the second derivative of the height variation to the adjacent eight texels of every texel, and storing the result into another map. A displacement map and its second derivative are presented in Figure 3.2. In this example, the measure of the second derivative is based on the greatest absolute angle between the normals of triangles constructed between the texels. 5

7 Figure 3.2: A displacement map (left) and its second derivative (right). As can be clearly seen from Figure 3.2, the second derivative of the displacement map highlights the parts that have detail or slope. When tessellating a mesh, every triangle is scan-converted in texture space and the texels inside are examined for detail from the second derivative map. If not enough detail occurs, the triangle is left untouched and the rest of the mesh is processed further. Figure 3.3 shows a 3-D mesh both in its original geometry and after being tessellated using the technique described above. The displacement map used is that of Figure 3.2. Note that in Figure 3.3, displacement of the vertices has not yet taken place. Figure 3.3: A 3-D mesh in its original form (left) and after tessellation (right). 6

8 When tessellating a triangle, it is important to avoid creating edges that are split unevenly, also called T-junctions (see Figure 3.4). These problematic edges occur when an edge is split from the other side, but not from the other. In other words, a single edge spans the same line as two or more other edges. Applying displacement mapping on a mesh containing T-junctions easily produces artifacts like cracks or jaggedness when rendered. Figure 3.4: An edge containing a T-junction (at the vertex highlighted in red). When a mesh with a T-junction is displaced, the renderer processes the problematic edge two (or even more) times differently, depending which adjacent triangle it is working on at the time. This may produce especially nasty artifacting when the vertex in question is to be displaced relatively far from its original position, as has happened in Figure 3.5. Of course, the viewpoint of the scene displayed is also a factor in the gravity of the artifacting - sometimes artifacts are obscured by other geometry and may be thus left unnoticed. Figure 3.5: A crack artifact in a mesh containing a T-junction. 7

9 To prevent such artifacts from occurring, we must eliminate all T-junctions. Neulander[4] suggests a generic algorithm for this. When tessellating, we must keep a table of shared edges, so that these are not be split twice. In the table, the edges are uniquely identified by the ordered pair of its original two vertices. The mesh must be searched for T-junctions, and when a problematic triangle is found, it must be subdivided so that its edges are not split further. We first consider whether all of the triangle s edges have been split. If all edges are split, we recursively subdivide the triangle into four subtriangles until we find a triangle that has at least one edge that has not been split. After that we form a zig-zag triangulation along the split edges until we ve reached the second-last vertex of either edge (see Figure 3.6). From this vertex, we create a triangle-fan to the remaining vertices of the opposing edge. The rest of the triangles are processed in the same way. Figure 3.6: Re-tessellating a problematic triangle. When the T-junctions are processed this way, the result is a 3-D mesh that does not produce artifacts when rendering. It is worth noting that T-junctions only appear in meshes that are subdivided unequally. If all triangles are equally subdivided, for example four times recursively throughout the whole mesh, all edges are split evenly and we do not have to worry about T-junctions. The price to pay with even subdivision, however, is the vast amount of mathematical calculations to cope with. 8

10 4 DISPLACEMENT After the tessellation, it is a matter of displacing the newly created vertices according to the displacement map. The vertices are moved along the normal of the original surface; thus resulting in bumpy new geometry. It is also possible to re-interpolate the directions of the displacement vectors according to the three vertex normals of the original triangle. Usually this is unnecessary, as the original mesh should be constructed finely enough to take into account the detailed curvature. Sometimes, however, interpolation of the direction of the displacement vectors yields better results. Formally, we define the displacement as follows [3]: - The function P(u,v) defines the 3-D points on the surface. - The function D(u,v) defines the scalar amount of displacement from the surface. - The function N (u,v) defines the direction of the displacement vectors (these may be all identical, depending on whether if normal interpolation is being used or not). The displacement function P defining the new, perturbed surface is thus as follows: P (u,v) = P(u,v) + D(u,v)N (u,v) where N (u,v) is a normalized unit vector. See Figure 4.1. Figure 4.1: Displacement of the vertices. 9

11 5 DISCUSSION Displacement mapping has been used mainly in ray-tracing, as the mathematical overhead has been too heavy for real-time rendering, mainly because of the sheer volume of geometry created by the tessellation. The problem is that the overall process of displacement mapping is very complex to implement in hardware. Things are about to change, however; hardware implementations have been introduced [5] and it seems like displacement mapping is making its way to applications that operate at interactive frame rates. Furthermore, schemes for tackling overtessellation have been developed, most important of these is view dependent tessellation [3], in which the tessellation algorithm varies the intensity of tessellation according to the distance between the camera and the area to be tessellated. REFERENCES [1] James F. Blinn, "Simulation of Wrinkled Surfaces", Computer Graphics, (Proc. Siggraph), Vol. 12, No. 3, August 1978, pp [2] Robert L. Cook, Shade Trees, Computer Graphics, (Proceedings of SIGGRAPH 84), 18(3): , July [3] Doggett & Hirsche, Adaptive View Dependent Tessellation of Displacement Maps, WSI/GRIS, University of Tübingen, Germany [4] Ivan Neulander, Displacement Mapping, [5] Gumhold & Huttner, Multiresolution Rendering with Displacement Mapping, WSI/GRIS, University of Tübingen, Germany, ndex.htm [6] Becker & Max, Smooth Transitions between Bump Rendering Algorithms, University of California, Davis and Lawrence Livermore National Laboratory. 10