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Next: Ray-Tracing Inversion Up: Forward Simulation Previous: Light Source

Ray Representation

Having developed representations of the surface and of the light entering the surface, the next step is mapping the refracted rays from the surface to the pool floor. At this stage, a ray is simply a normalized vector $ \vec r$ at a surface position pointing downward. This vector controls the contribution to the light intensity field. The contribution can be a simple binned ray or a beam.

It is a simple exercise of trigonometry to project the refracted ray $ \vec r$ to a point $ \vec p$ the bottom of the pool.

$\displaystyle \vec p=\frac{\mbox{depth}}{r_z}\vec r$ (5)

The point $ \vec p$ is unlikely to fall on a grid point, and therefore the ray amplitude distribution can be binned. If each ray is given a unit amplitude, the sum of amplitudes is the ray count for each bin. Figure 4 shows the result obtained using binning. The observable caustics have sharp, though not continuous boundaries. Zones of low light intensity can be confused with shadow zones, simply because the light is not distributed continuously. Some of the image degradation might be attributed to the nearest-neighbor interpolation; however, even with the best interpolation algorithms, the result will not improve significantly. In Figure 3, the aliasing is due to an insufficient number of rays and binning. de Ridder (2008) suggests using a Gaussian distribution as a function of the lateral distance from the ray projection point on the pool floor. The proposed method yields more stable results. Nevertheless, the symmetry of the distribution neglects the tilting of rays.

fig-binning3d
Figure 4.
The result of binning is a good approximation but not realistic. [ER]
fig-binning3d
[pdf] [png]

So far I have used the specular-transport mechanism within water and ignored light diffusion, which is a shortcoming of the ray-tracing. It would take orders of magnitude more rays for ray tracing to simulate diffusion, which is prohibitively expensive. Also, it will be inefficient, because diffusion does not change much from one pixel to the next (Watt, 1990). Figure 5 shows the diffusion of light incident from a point on a disordered medium (ICMM, 2008). It is obvious that light intensity within a beam is dependent on the angle of refraction. It can be modeled with a normal distribution as a function of the angle, with a decay factor that is a function of distance from the refraction point. In vector notation, the contribution of the diffusion to a point is

fig-diffusion
Figure 5.
Diffusion of a light beam in an isotropically disordered medium (ICMM, 2008). [NR]
fig-diffusion
[pdf] [png]

$\displaystyle <tex2html_comment_mark>49 <tex2html_file> ... (6)

where $ \vec p$ is the position vector from the refraction point $ (x_0, y_0, z_0)$ on the surface that is defined as

$\displaystyle \vec p(x,y,z)= (x-x_0) {\bf\hat i} + (y-y_0) {\bf\hat j} + (z-z_0) {\bf\hat k }\quad,$ (7)

$ \sigma$ determines how narrow the distribution is, and $ \alpha$ is the rate of exponential decay.

Beams[*] demonstrate some of the characteristics of both specular and diffusive transport mechanisms. One way of handling diffusive and specular transport is to model them separately and stack the results. However, I use beams to model the two simultaneously.

In designing the beam shape, one can choose to favor either the diffusive or specular distribution. Favoring diffusion by using equation 6 can negatively affect the resolution of the light patterns. The following equation can be used to describe the distribution of light within the beam:


$\displaystyle %\input{eqn_gaussian}\quad ,\\
\rho(\vec p)$ $\displaystyle =$ $\displaystyle {1 \over b}\exp\left\{-{{\vert \vec r \times \vec p \vert} \over {2b}}-\alpha \vert\vec p\vert\right\}
\quad ,$ (8)

where $ b$ controls the width of the beam, and $ \alpha$ is the rate of exponential decay. Figures 6 and 7 show the beam coming from a single point on the surface and the section obtained by stacking all the beams coming from the surface respectively.

fig-beam
Figure 6.
A single beam refracting into the pool. [ER]
fig-beam
[pdf] [png]

fig-section2d
Figure 7.
2D forward modeling using beam tracing. The refracted light focuses into caustics. [ER]
fig-section2d
[pdf] [png]

The disadvantage of using beam tracing is that every beam contributes to a very large number of points, which can be computationally more expensive -$ O(N^2)$- than binning-$ O(N)$, where N is the number of samples on the surface function. As mentioned earlier, the boundaries of the caustics are not as sharp. However, they become more realistic, as in Figure 1, where the details of caustics are not usually seen with sharp boundaries. Figure 8 is the result of the 3D beam tracing of the surface in Figure 2. Figure 9 shows the light intensity field of a surface distorted by many ripples. Although it is more expensive, beam tracing works better than ray tracing. We can distinguish shadow zones and the elongated caustics that usually connect two point caustics.

Beam tracing has many of the limitations of ray tracing with binning. The prominent limitation is that beam tracing needs a large number of beams to model light intensity under a very narrow trough on the surface. Also, for depths where many rays do not project on the pool floor, there are too few beams to construct the correct intensity field. Moreover, the quality of the forward modeling results is sensitive to the beam width chosen; too wide beam hinders the resolution, and too narrow beams behave like rays and can produce aliased results.

fig-finitdiffproj
Figure 8.
Intensity field under the surface at Figure 2. [CR]
fig-finitdiffproj
[pdf] [png]

fig-exploding3d
Figure 9.
A result from the exploding surface model. [CR]
fig-exploding3d
[pdf] [png]


next up previous [pdf]

Next: Ray-Tracing Inversion Up: Forward Simulation Previous: Light Source

2009-04-13