Propagation Paths in 2.5D Environments

This paper describes the implementation of an algorithm capable of enumerating a large number of propagation paths in interactive time for the special case of 2.5D environments. A two-dimensional beam tracing algorithm is used to compute the projection of the propagation paths in the plane and the heights lost in this projection, allowing the reconstruction of the actual propagation paths in three-dimensional space. The propagation paths computed are comprised of specular reflections and diffractions. As the propagation beams are created in a preprocessing stage, the system treats only sources with fixed position and moving receivers. For a long time, the computational simulation of acoustic phenomena has been used mainly in the design and study of the acoustic properties of concert and lecture halls. Recently, however, there has been a growing interest in the use of such simulations in order to enhance virtual reality applications, as computer games for example. To achieve the interactive rates necessary for these applications, an approximation known as geometrical room acoustics [1] is used. As Kuttruff describes it, in geometrical room acoustics the concept of a sound wave is replaced by the concept of a sound ray. The use of sound rays to simulate the propagation of sound in an environment makes the algorithms created for this purpose very similar to the ones used in the analysis of wireless communication networks [2] and in visualization (hidden surface removal), such as ray tracing [3] and beam tracing [4]. This means that all these applications can benefit from the same techniques used to accelerate visualization applications, as was shown by Funkhouser et al. [5] for acoustic simulations. In this paper we present an algorithm capable of enumerating propagation paths comprised of specular reflections and diffractions in 2.5D environments (environments defined by the vertical sweeping of twodimensional shapes). By using the acceleration techniques mentioned above, the algorithm is capable of enumerating a large number of propagation paths in interactive time. Figure 1 contains a dataflow diagram that illustrates how the algorithm works. The first stage of the algorithm, represented by the gray boxes in the diagram, is a preprocessing stage. This stage is responsible for the construction of the beam tree data structure and of the height paths. The first allows the fast computation of propagation paths in the plane and the second allows us to unproject these paths, transforming them into 3D propagation paths. In the diagram, one can see that the environment, the position of the source and the z coordinate of the receiver are inputs to the preprocessing stage. If any of these inputs is modified, the preprocessing stage must be repeated. The second stage, represented by the white boxes in the dataflow diagram is responsible for the creation of the propagation paths in the plane and for the reconstruction of the 3D propagation paths, represented by the Path Merger box in the diagram. The following sections present each step of the algorithm in more detail. We begin by reviewing some previous work in Section 2 and by making a brief introduction to the beam tracing technique in Section 3. This introduction covers the Cell Decomposition, Beam Tracer and Path Tracer steps of the algorithm. Then, the construction of the height paths and the reconstruction of the 3D paths are explained, respectively, in Section 4 and in Section 5. Finally, we present some results (Section 6) and our conclusions and future work (Section 7).