The effect of diffuse reflections on noise maps using phonon mapping

Constructing noise maps for large urban areas is a tedious and CPU intensive task. Engineering approximations to outdoor sound propagation are commonly introduced to reduce the computational complexity. In particular, diffuse reflections are often neglected because of their great computational expense. This paper reports on an effort to approximate the diffusely reflected field by using a technique called phonon mapping. In noise mapping, many techniques can and are adapted from the field of computer graphics where various techniques to approximate indirect lighting exist. A popular technique in computer graphics is photon mapping. It was introduced in 1995 by Jensen and is known as one of the fastest algorithms for simulating indirect lighting. In this paper, it will be shown how this photon mapping algorithm can be adapted to the field of noise mapping to simulate diffuse reflections. The phonon mapping algorithm exists of two passes. In the first pass, a phonon map is build by bouncing phonons, abstract sound particles, through the environment. On each interaction with a (partially) diffusely reflecting surface, it is stored in the phonon map. The Russian roulette technique is used to determine if the phonon is absorbed, specular or diffusely reflected. Upon reflection, the phonon is traced through the environment again. This process is continued until the phonon map is sufficiently populated. The second pass is a more traditional backward ray tracer. Rays are emitted from the receivers. Specular reflections are handled as usual but for diffuse reflections, no secondary, reflected rays are cast. Instead, the phonon map is queried for nearby phonons. Their presence is used to estimate the energy that is available from this diffuse reflector. DEFINITIONS Directional sound power D: sound power per unit solid angle dω. Its units are W sr . B. de Greve, T. Willems, T. De Muer, D. Botteldooren Acoustic radiance L: sound power per unit projected area dA and per unit solid angle dω, with dA = dA cos θ the projected area of dA on the hypothecial surface orthogonal to dω. Its units are W msr . INTRODUCTION Computational methods to construct noise maps of large areas are tedious and computational complex, even when engineering methods are applied. Classic ray/beam tracers [3, 4, 6] often only implement specular reflections using the image sources. Diffuse reflections are often neglected because of their great computational cost and complexity, despite the fact that the diffusely reflected field can be a significant contribution to urban soundscapes. This paper reports on an effort to approximate the diffusely reflected field by using a technique called phonon mapping. Phonon mapping belongs to the family of geometrical acoustics, together with methods like ray tracing and beam tracing, making the same high frequency approximation. Phonon mapping is a particle tracing technique in which packets of acoustic power called phonons are bounced through the geometrical scene. On diffuse surfaces, incoming phonons are stored in a phonon map. Later, this phonon map is used to estimate the incoming power per area which is used to find the outgoing acoustic radiance L available from this diffuse reflector. This technique is adopted from the field of computer graphics where it is known as photon mapping. The latter is introduced in 1995 by Jensen [8, 9] as an efficient alternative to radiosity [5] and Monte Carlo ray tracing [10] for simulating global illumination. The goal of noise mapping is to determine how much acoustic power W is reaching a receiver located at r. It is an integral of the incoming directional power D over all directions: W (r) = ∫ Ω D (r, ω) dω (1) In geometrical acoustics, this integral is written as the sum over all sound paths that terminate at r: W (r) = ∑ k Dk (r, ωk) (2) This sum includes the set of all sound paths from a source to the receiver via any number of specular or diffuse reflections. It contains an infinite number of sound paths, therefore for practical applications an approximation of this sum must be made. A traditional ray or beam tracer can only handle direct paths or paths with specular reflections only. The sum of all their contributions is Ws. Such a ray tracer misses all sound paths that have at least one diffuse reflection. The goal of the phonon mapper in this paper is exactly to estimate the sum Wd of the contributions of the latter. Finally, both results must be summed to get W : W (r) = Ws (r) +Wd (r) (3) ICSV13, July 2-6, 2006, Vienna, Austria FIRST PASS – PHONON TRACING Emitting Phonons From A Single Source A number of phonons is emitted from the source with power P until the phonon map is populated with Nmap phonons. Nemit is the total number of phonons emitted and is not known beforehand because some phonons may never end up in the phonon map, and others may be stored several times. All phonons emitted have the same power. This is important for two reasons: Firstly computational resources are not spent on low power phonons that would contribute only a little to the result, and secondly it will reduce the variance of the radiance estimate in the gathering pass. As a consequence, more phonons must be emitted in directions with higher power contribution. For omnidirectional sources, directions of phonon emission ω must be uniformly distributed over the unit sphere. This can be accomplished by transforming a sample (ξ1, ξ2) from the uniform distribution over [0, 1]× [0, 1) using (4).