Technical Reference

Bulk Scattering Standard Expert

When a ray enters an object that has bulk scattering, the ray propagates a random distance, then the direction of the ray is deviated, similar to the way rays are diffracted by an aperture in TracePro. Importance sampled rays may also be generated at your option. The ray deviation and importance sampled flux are governed by a probability distribution similar to a BSDF, but slightly different. Whereas the BSDF is defined as the scattered radiance per unit incident irradiance, the scattering distribution function or SDF is defined as the scattered intensity per unit incident flux,

ˆ
∂I(r)	(7.39)
SDF = ----------- .
∂Φ

In addition, the SDF is independent of the direction of incidence.

Henyey-Greenstein Phase Function

One SDF that has been implemented in TracePro is after a paper by Jacques and Wang1 that describes scattering in biological tissue and has the form

SDF = p(θ) =	1 – g2
	4----------------------------------------------------------π(1 + g2 – 2g cosθ)3 ⁄ 2 ,	(7.40)

where g is called the anisotropy factor, and g can take on values between –1 and 1. When g is positive, rays are scattered more in the forward direction, and when g is negative, they are scattered more in the backward direction. When g is zero, the scattering is isotropic, i.e. the same in all directions. Figure 7.17 shows an example scattering distribution function for g = 0.5.

FIGURE 7.17 - Example scattering distribution with g = 0.5. Zero is the forward direction

1. S. L. Jacques and L.-H. Wang, “Monte Carlo modeling of light transport in tissues,” in Optical Thermal Response of Laser Irradiated Tissue, edited by A. J. Welch and M. J. C. van Gemert (Plenum Press, New York, 1995), pp. 73-100.

7.54	TracePro 5.0 User’s Manual

Bulk Scattering

Gegenbauer Phase Function

Another bulk scattering phase function offered in TracePro is the Gegenbauer2,3 phase function. The Henyey-Greenstein function is actually a special case of the Gegenbauer phase function. The Gegenbauer function is

SDF = f(θ)	= K[1 + g2 – 2g cosθ]–(α + 1)								(7.41)
where
K = αg	(1	– g				2	)	2α	(7.42)
	π[----------------------------------------------------------(1 + g)2α – (1 – g2α)]
and
α > –1 ⁄ 2 ,			g		≤ 1				(7.43)

The Henyey-Greenstein phase function is a special case of the Gegenbauer phase function when α = 1/2.

Scattering Coefficient

When a ray enters a scattering medium, it propagates a random distance x governed by the probability distribution

P(x)dx = e–μsxdx ,

(7.44)

where μs is called the scattering coefficient. The inverse of the scattering

coefficient is the mean free path of the ray in the material. When a ray enters a piece of material that is thin compared to the mean free path, it is likely to pass through the material without being scattered. Conversely, if the material is thick compared to the mean free path, the ray is almost certain to scatter within the material. When a strong scattering coefficient is combined with a strong absorption coefficient, rays will be only weakly transmitted through the material.

Using Bulk Scattering in TracePro

In order to simulate bulk scattering in TracePro, select an object in which you wish scattering to occur and apply a material property using the Apply Properties dialog box. Then apply a bulk scattering model to the object. Finally, you can apply importance sampling to the object optionally in a way that is like surface importance sampling.

To define new bulk scattering models, open the bulk scattering model editor. Create a new scattering property and add a new wavelength entry at the desired wavelength. Enter values for g (the anisotropy factor) and μs, the scattering

coefficient. The units for μs are 1/mm, (i.e. inverse millimeters). Most bulk scattering materials also have some bulk absorption, so you need to define a

2.L.O. Reynolds and N.J. McCormick, “Approximate two-parameter phase function for light scattering,” J.O.S.A. 70, 1206 (1980).

3.A.N. Yaroslavsky, I.V. Yaroslavsky, T. Goldbach, and H.-J. Schwarzmaier, “Influence of the Scattering Phase Function Approximation on the Optical Properties of Blood Determined from the Integrating Sphere Measurements,” J. Biomed. Opt., 4, 47 (1999).

TracePro 5.0 User’s Manual

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