We already mentioned that good visual approximations of trees should display the fringed character of the foliage. If only a few large-sized polygons are used, the desired look is not achieved, not even if partially transparent textures are used.
It is more efficient to represent the foliage deliberately using unconnected prim- point-based representation ^ itives. Since leaves mostly correspond to small discrete objects, points can be
used instead of polygons. That way the already-mentioned geometric overspecification can be decreased. For example, such an overspecification occurs when for one million pixels (image size is approximately 1000×1000), 100 million polygons are used for rendering... >
The appearance of trees is mostly determined by the foliage, so that methods have to achieve an optimal approximation for the many isolated surfaces of the leaves. The problem here is the special characteristic of the foliage, since, in contrast to smooth surfaces, the approximation should not reveal a smoothness but rather the fringed outline of the crown.
In an earlier study, Gardner  approximated trees and terrains with a few geometric primitives. He used quadrics, i. e., surfaces that can be described with square functions such as paraboloidal or hyperboloidal functions. Here procedural textures control the color and the transparency of the surfaces. Similar methods are used in a number of computer games in order to illustrate foliage... >
The successive simplification of closed, smooth surfaces is the subject of many approaches. Some methods work directly with geometric data; others transform the data into a new representation in which the reduction is executed. A common example is the simplification in wavelet space [53, 209].
The classic method, which works directly with geometric data, is described by Hoppe et al. [93, 94]. An initial surface is successively simplified by removing triangle edges. If an edge has to be deleted, the end points are moved to the middle point of the edge, and all adjacent triangles are modified accordingly. This is called an “edge collapse”. The deletion of the edges is repeated until a sufficiently simple base geometry is yielded... >
The procedures discussed in the last chapter, and particularly the efficient raytracing implementations, allow for a relatively fast, though sufficiently exact calculation of the light distribution in complex botanical scenes. The performance of the procedures is here mostly only restricted through the quantities of the data that have to be processed.
However, it is not always necessary to show the plant models in their full complexity. For example, if the visual size of a plant only consists of a few pixels on screen because it is part of the background, it is pointless to use thousands of polygons for its production. The geometric resolution thus should possibly be adjusted to the visual size of the plant... >
In concluding this chapter, some questions remain when viewing images, such as those on the following pages. Why is it actually possible to reproduce using relatively simple methods a sometimes astonishing degree of realism? One reason could possibly be the relatively simple lighting conditions within the plants that are easy to approximate using traditional computer graphics procedures.
Another reason lies in the complexity of the data. The eye is used to process large amounts of visual information at the viewing of images. If this information is missing, an impression of flatness is created.
Although the images shown so far were produced with standard computer – graphics algorithms that produce an almost synthetic impression with other data, the images shown appear relatively “natur... >
In the rendering of landscapes, radiosity has so far rarely been used. The enormous amount of geometric information causes extreme computing times and needs a huge amount of memory. Soler and Sillion  decreased this by integrating hierarchical instancing in radiosity procedures (see Sect. 8.5).
hierarchical radiosity ^ With the hierarchical version of the algorithms, Eqn. (9.7) is not solved in one
step, but rather this is accomplished through a recursive mechanism that first calculates directly the energy exchange between the larger part of the scene, and only then includes the smaller units when a given error threshold is crossed ... >
The method described here takes advantage of the fact that in plant scenes very seldomly must reflection rays be traced. A great part of the ray query thus consists of the so called primary rays, and the number of secondary rays caused by reflection is minimal.
All primary rays start at the viewing point of the viewer using the virtual camera model, and penetrate through the pixels of the image to be produced. The ray query thus represents the question: Which object can be viewed in each pixel of the image to be produced?
At this point we take advantage of the fact that the local lighting simulation can be computed very quickly via the graphics hardware, which helps to answer the question just posed... >
As already mentioned, raytracing is the most widely used method for the simulation of global illumination. With the described spatial division methods and bounding objects, the time-consuming process of the ray query can be optimized. Additional optimizations can be reached with special versions of raytracing that are described in the following.
First two aspects for the implementation of efficient raytracing algorithms are explained, which allow for a better ray query. The first method affects the optimal processing of rays using an efficient memory management. This is especially useful when complex scenes are rendered. The second process takes advantage of the fact that in natural scenes reflection rays are rare, since they are not needed in combination with plant materials... >
The term “occlusion culling” defines methods that exclude all the objects hidden by other objects before the actual image computation. In connection with local rendering methods, these objects are not even transferred to the graphics hardware. In connection with raytracing, these procedures are used to decrease the total complexity of the scene description, and to avoid unnecessary shadow computation.
In this case, the visibility of the light source is to be computed for the object points, and we must find out in the most efficient way whether or not one or more objects are in-between. If there are very many objects to be considered, occlusion culling can be helpful in decreasing this effort.
To execute the computation of the complete visibility information for every single point in ... >
In raytracing, a great amount of computing time must be spent on the so-called ray query. Here, with each tracing ray, we have to check which object or polygon of the scene is hit first. On the other hand, in local illumination models a lot of work is due to the visibility calculation, which defines for all polygons of each object whether or not they are actually visible. In both cases the efforts can be simplified and the computing time decreased, if complex objects are approximated using simple bounding geometries such as cubes or spheres, which include these, so-called bounding objects or bounding boxes [108,180]. If such objects were produced, then for the ray request as well as for the computation of the visibility the bounding object can be used for a first test... >