III.2.2 Image-source Method in a Rectangular Room
The principle of mirror image sources, presented above for a single surface, can be applied in room geometries consisting of multiple planar surfaces as well. All the first order reflections in a room are obtained by reflecting the original sound source against each surface in the room geometry. Higher order reflections can be obtained by recursively reflecting the new image sources such that reflection of the first order image sources provides the second order reflections, and those can be further used to reconstruct the third-order reflections, and so on.

The result of this image-source computation can be seen as an image-source tree. The original sound source is the root of that tree, first order reflections are seen as the first branches from the root, and in the end, the the leaf nodes of the tree represent the highest order reflections.

- You can select the reflection orders to see the specular reflections in a 2-D rectangular room.
- The 'Wavefronts' visualizes the circular wavefronts whereas 'Full rays' draws the actual reflection paths between the source and the receiver.
- The 'Ray extensions' extends the reflection paths to their corresponding image sources to show their origin.
- The 'Ray sections' visualizes short segments of ray propagation when the 'Time' slider is moved.
- Move both the sound source and the listener and see how they affect the image sources and the reflection paths.
The 'Show image-source tree' opens a new window and illustrates the tree formed of image sources such that the ray-paths shown with 'Full ray paths' originate from the red image sources.


Time:
Show outside Zoom: (or SHIFT + mousewheel)
Reflection orders: Direct sound 1st 2nd 3rd 4th 5th and 6th 7th to 10th
Visualization: Wavefronts Full ray paths Ray extensions Raylets

It is essential to note that the image sources and, thus, the wavefronts are invariant against the receiver location, whereas the actual reflection paths depend on the listener location. The image sources move only when the sound source moves (or the geometry changes). This means that it is possible to pre-compute all the image source locations if both the geometry and the source location remain static, and there are either several listener positions or the listener can move. In such cases, it is sufficient to perform only the listener-dependent operations (covered later in this section) separately for each listener position.

The main challenge in the image-source method is the exponential growth in the number of image sources. The fast growth in the width of the image-source tree, i.e. the number of leaf nodes in the tree, illustrates this well. In the most basic case the number of image-sources upto the $K^{th}$-order reflections is given by the expression $\sum\limits_{k=1}^{K}N(N−1)^{k-1}$ for $N$ surfaces. That is obtained by assuming that in each reflection all the image-sources of the previous order are reflected against all the other $N-1$ surfaces with the exception of the surface that created the image source that is being reflected.

In the case of the rectangular room presented above, the number of image sources seen is lower than predicted by the expression above as some of the image sources overlap and those are actually double, or even higher order duplicate, image sources as there are several ways to reach the same location. For example, reflection against top and right wall in that order results in exactly the same location (one step right and one step up in the illustration) as performing the reflection first against the right wall and only after that against top wall. By moving the listener around it is possible to see that this transition from one image source, and thus from one order of reflecting surfaces, to another one is smooth while the ray-path goes via the corner of the room (you can turn on the 'Ray extensions' option to see better this transition). In addition, the principle of having only single-sided reflectors also limits the growth speed. This is seen in the image source tree such that all the nodes purely in the axial directions (up-down, or left-right) have three children whereas all the others have only two children.

III.2 Image source technique (previous)III.2 Image source technique (up)III.2.3 Implementation of basic image-source algorithm (next)
© 2016 Lauri Savioja