Introduction

WIP. This lesson is being written

Structure of this Lesson

In the first two chapters, we will introduce the concept of subsurface scattering in non technical terms. These chapters are ideal for readers looking to understand what subsurface scattering is, and what's causing it. In the second chapter, we will be looking at one of the models which is being used in CG to reproduce this effect without getting into the mathematical details. Understanding what the parameters to this model mean is critical to control the look of subsurface materials. The full explanations about the various techniques (including an implementation of each one of these algorithms) that can be used to simulate subsurface scattering will be given in the remaining chapters.

What is Subsurface Scattering ?

If it's easy for you to identify around you objects which appear translucent, then you are not too far from understanding what subsurface scattering is. Because in essence, translucent materials and subsurface scattering are about the same thing. Although not as noticeable maybe than the other types of materials we can observe in our daily lives, translucent materials are still quite common. Human skin might be the most obvious of them all, but a wild variety of other materials exhibit translucent properties from the hardest rocks (marble, jade) to plants (leaves), plastic and malleable materials (wax), liquids (milk), etc. Take a moment to look around you and see how many translucent objects you can identify (starting with your own skin). Translucent objects are sometimes easier to spot when they are back lit (looking at the palm of your hand against the sun on a bright day light. Do not burn your eyes!).

What makes translucent object different from the other objects ? Other objects you might say, in comparison to translucent objects, are either transparent (the glass of a window) or completely opaque (wood, brick wall, concrete, metal). So an intuitive definition of translucency is that of an object which is not completely opaque nor completely transparent. Something in between these two states. This isn't a bad way of looking at it. Even if you haven't read the various lessons that treat of the topic of the interaction between light and objects, you might be familiar with the idea that transparent objects (such as glass or water) are objects which let light pass through them and opaque objects are objects that don't let light pass through them at all. What happens with an opaque object is that the light bounces off right away when it hits its surface, no light passes through; the light is like a tennis ball bouncing off the hard surface of a tennis court. Transparent objects on the other hand do not block any of the light that hits their surface like opaque objects do. Instead, they let the light passing through them freely.

Translucent objects are somewhere in between. What does that mean? It's slightly more complicated than just saying that some of the light passes through and some of the light bounces off. When light hits the surface of a translucent object, it will penetrate the opaque and continue traveling through it until its trajectory is deviated (when a light hits the surface of a translucent object, it travels through the object more or less in the same direction but we will see later that it is in fact slightly bended due to the phenomenon of refraction).

What can cause a light ray to change direction while it traverses an object ? Lets start to say in the first place that the light ray goes through the object because the substance it travels through offers no or very little resistance to it (like glass or water). For instance the main constituent of milk is water. As long as a light ray traversing milk is traveling through water, nothing will happen to it. However milk is a mixture of water and many other complex and fairly large particles which are floating in the water. If our light ray hits one of these particles, its direction is likely to change. 

Heterogeneous materials (such as for instance marble) are more likely to cause a light ray to change direction while it travels through them (the light ray is more likely to collide with the various elements that these materials are made of). But even materials that appear homogeneous such as milk for instance are in fact a suspension of tiny particles or various molecules mixed in some other liquid or homogeneous substance (such as water). When light hits one of these particles, it could be deflected to another direction. We say that the light is scattered (scatter: to cause to separate and go in different directions). And since the phenomenon takes place under the surface of the object, we call the effect subsurface scattering (literally scattering under the surface). If a light ray changes direction because it collided with a particle, it will either continue traversing the object or it will leave it (if it makes its way to the surface). When the light leaves the surface, the most important thing to notice is that the point where the ray exits the surface is different from the point where the ray entered it. Light that penetrated the object at point x leaves the object at point x0 (figure xx). This observation is crucial in understanding the phenomenon of subsurface scattering (both from a visual and technical point of view).

The main difference between opaque objects and translucent objects, is that in the case of opaque objects, light bounces off from the surface where it hit the surface (like a ball bouncing off from a hard surface) while for translucent materials, the light is more likely to exit at a certain distance away from the point of illumination. In the case of opaque objects, x and x0 are the same. In the case of translucent material x and x0 are separated by some distance. How far x0 is from x defines how translucent the material is. For materials which are not so translucent, this distance is small (they can even almost appear like opaque. The light source may have to be really strong for them to appear even slightly translucent). As the materials get more translucent, the distance between x and x0 increases.

Multiple vs Single Scattering

So far, we said that when the light ray was deflected it was either continuing to travel through the substance the object is made of or exiting the object. We suggested that only one single deflection event could could cause the ray to leave the object. However for some materials, the light ray will continue being deflected by other particles many times until it potentially leaves the object (following some sort of random walk). In some translucent materials the light direction direction will change only once before it leaves the object (one deflection event). The phenomenon is called single scattering. In some other translucent materials, light is deflected many times before it leaves the object (it is scattered multiple times). In that case, we speak of multiple scattering. Materials are rarely one or the other but more often a combination of the two where one phenomenon potentially dominates the other. In human skin for instance, multiple scattering is more important than single scattering (in skin, light bounces around many times before it leaves the surface).

There is one important thing to notice about single and multiple scattering. In the case of multiple scattering light entering a surface is deflected many times before it eventually leaves the object. That means that the direction of the light ray also changes many times. It changes so many times, that the position (x0) where the light ray exits the surface and its direction are random and do not depend of the incoming light direction. In the case of multiple scattering there's no relation between the direction of the light ray when it hits the surface in x, and its direction when it leaves the surface in x0. For materials for which multiple scattering is predominant, the light direction doesn't influence the subsurface effect. To get a photographic reference of translucent materials, we can take an image of such materials illuminated with a small light beam (laser). If you get an image where light seems to diffuse away from the light beam in concentric circles (and fade out as the distance to the light beam increases) you can most certainly deduct that for this material, the multiple scattering component is more important than the single scattering one.

You can also change the direction of the laser beam and you are very likely to see that changing this direction do not affect what you see. Light still scatters around the point where the light beam enters the surface of the object in concentric circles (because the ray is being deflected many times there is equal probability that it will exist the medium anywhere around x). In reality it's not completely true so you might see some dependency to the light direction but we will look at these subtleties later. For now it's only important that you understand the difference between single and multiple scattering and to understand that in the case of multiple scattering, there's not dependency between the scattering effect and the light direction and that light leaves the medium around the point of illumination (x) in concentric circles.

It is also important to point out, that the pixels closer to the light in the picture of the potato are brighter than the pixels away from the point of illumination (this is clearly visible when the photograph is displayed with false colors). This is due to the fact that light exiting far away from x has been scattered more often and has traveled (in average) a greater distance before leaving the medium than light leaving the medium in the proximity of x. Traveling a greater distance means losing energy along the way due to absorption. The probability of a photon leaving the medium also decreases as we walk away from x. So naturally we will also find more photons leaving the medium around x than away from it. We will explain these effects in detail later in the lesson.

Another way of looking at multiple scattering in a intuitive way, is to imaging that the light entering the object at point x is being blurred due to the effect of being deflected many times under the surface of that object. We have illustrated this process in the following figure.

In the top image image we have only rendered a few photons. But if we were to let the simulation run for a longer period of time we would get a result very similar to the bottom image which is a blurred version of the top image (a blur was applied in Photoshop). We can look at the second image as what we would typically get if we were to illuminate a real translucent material (where the multiple scattering component is more important than the single scattering one) with a small light (notice the similarities between this image and the photograph of the potato illuminated with a small light). Pay attention to how the light fades away from the point of illumination (the yellow dot in the image). Light diffuses all around x and fades out in a way that is very characteristic of translucent materials. The way light fades out away from x can be plotted on a graph and we can use various mathematical model to fit the data. This observation is at the heart of the mathematical models which are used to simulate subsurface in CG and will be explained in detail in the second chapter.

Single scattering on the other hand has a strong dependency to the light direction. From a technical point of view this component is not as complicated to simulate than multiple scattering. Single scattering plays an important role in the appearance of the sky, clouds and ocean water to just name a few common things from our daily lives.

Wavelength Dependency

An introduction on this phenomenon wouldn't be complete without mentioning that the scattering effect is wavelength dependent effect. White light is made of photons which wavelength is in the range of what we call the visible light spectrum (roughly from 390 to 750 nm). The interaction between the photons from a beam of light and the various particles or molecules a medium is made off depends on these photons' wavelength. Various molecular-photon interactions come into play which are responsible for filtering out or deflecting photons differently depending on their wavelength. As a consequence of these interactions some photons in certain ranges of wavelengths will be preferentially absorbed by the material while photons in the remaining ranges of wavelengths will be left alone.

Similarly for scattering, these interactions will cause photons in some ranges of wavelengths to be deflected differently than photons from some other wavelengths (not so much differently in fact. It's more the frequency of a scatter event that changes based on the photon's wavelength). When put together, these two phenomenons cause a white beam of light that is being scattered to leave the surface of the object, not white, but colored (and this is true for almost all translucent materials even those which appear white under normal lightning conditions such as milk). The wavelength dependent absorption effect will cause some of the photons from the white light to be sucked in.

As for the wavelength dependent scattering effect: imagine that the substance the material is made of, scatters the green and red photons more often than the blue photons. When a red and green photon have been scattered 100 times, a blue photon has only been scattered 10 times. This has a few consequences. If multiple scattering dominates, green and red photons have a greater probability to leave the surface than blue photons. There is less probability for a blue photon to be deflected and therefore find it's way back to the surface. Furthermore a blue photon will travel longer distances (since it's not scattered as often) which means that, if it finds its way back to the surface, it will leave the object further away from the point of illumination (x) than the green and red photons. In that particular case the material would appear yellow around x and slightly blue farther away from x. As you can see, when all these phenomenons come into play and interact with each other (wavelength dependent absorption and scattering), the result of subsurface scattering is not necessarily a plain constant color but can be a complex combination of colors.

Multi-Layered Translucent Materials

It is also worth mentioning that many translucent objects are made of several thin translucent layers (thin translucent slabs) which is the case of a lot of organic materials (such as plant and skin tissues). Each one of the layers absorb and scatter light in their own way. Light is being filtered and scattered as it travels from layer to layer before it can eventually leave the object. It is important to note that each layer can either reflect or transmit light. To differentiate the two possible cases we speak of reflectance and transmittance. When a light source illuminates one side of a thin slab of a translucent material, some of the light will leave this slab on the same side (reflectance) and some of the light will traverse the object and exit on the other side (transmittance).

Any light entering the slab will be absorbed as it travels through the slab. We know that some of the light is absorbed but why is the remaining light either leaving the object on the front or on the back of the slab ? The remaining light is made of photons which haven't been absorbed. However they still can be scattered. As they travel through the slab in a straight line, some of them might be deflected in another direction due to scattering. In they are scattered many times they might end up leaving the object on the front side. If they have rarely been scattered or not scattered at all, they will leave the slab on the back side.

What's important to remember here is that light leaving the slab from the front is more likely to have been scattered than light leaving the slab from the back. Because scattering is a wavelength effect, the color of the light leaving the object from the front will be influenced by the absorption and scattering properties of the material while the color of the light leaving the slab from the back will be mainly changed due to absorption. The result of this difference is that light reflected (reflectance) and light transmitted (transmittance) by a slab of a translucent material have different colors.

This is of particular importance when we try to simulate the appearance of a multi layered material. Imagine a material made of two layers with different absorption and scattering properties, and illuminated with a white light source. The upper layer will reflect some of that light (L1R) and will also transmit (L1T) some of it to the under layer. Light transmitted by the upper layer (L1T) will be transmitted by the second layer (L1T->L2T) and some of it will also be reflected (L1T->L2R). Light reflected by the second layer (L1T->L2R) will be reflected (L1T->L2R->L1R) and transmitted (L1T->L2R->L1T) by the upper layer. Light being reflected by the first layer will again be reflected and transmitted by the second layer etc. As you can see this process goes on and on. There is always a fraction of the light coming from layer A illuminating layer B that is reflected back to layer A.

Not only this leads in theory to an infinite series of light exchanges between two adjacent layers which is complicated to simulate with a mathematical model but to make an accurate simulation, we would also need to consider some other optical phenomenons that take place when light goes from one layer to the other. The main one being refraction of the light rays which are bent and which intensity are attenuated according to the Fresnel equation as they travel through the different layers. All to say that many complex things are happening when light illuminates a multi-layered translucent object and that simulating them in CG is still a challenge today. Research is made in this area because it is thought that ultimate perfection in the photorealism of CG images can only be obtained if we simulate nature as closely as possible. Some models have been developed which are using a wavelength representation of light (we speak of spectral rendering) but they are still very experimental, not fast and not always very robust (quite a lot of freedom must be taken with the mathematical models to get a practical implementation working). However they lead to make the idea more common that the appearance of translucent materials (especially skin) can be misleading, because while a material such as skin appears red for instance, it is actually made of various layers which taken individually may have a color very different from the overall appearance of the material. Human skin to only mention this example is made of three main layers (the epidermis, the upper dermis and the bloody layer). Taken individually, the epidermis and upper dermis have absorption and scattering properties that make their appearance blue (their average reflectance color is bluish). The bloody layer is on the other hand very red. It's only when you combine these layers together that we get the final appearance of the caucasian human skin which overall looks more red than blue (the epidermis and upper dermis layers are very thin and they transmit quite a lot of light to the bloody layer. Light reflected by the bloody layer is then transmitted back through the two upper layers. While still noticeable, being quite thin, the appearance of these layers on the overall appearance of the material is not as important as the contribution of the thicker bloody layer which explains the overall red tint).

Terminology

Lets finish this introduction on the topic by introducing a few technical terms and making an observation that might be of more importance to readers with an artistic background. The first term we will introduce is the concept of mean free path. You will come across this term in the literature on subsurface scattering (and anything related to diffusion phenomenons in general) and it is therefore important to understand its meaning. It describes the average distance a photon travels through a particular material before it gets scattered (before its trajectory gets changed). Looking at what we said earlier on the wavelength dependency of subsurface scattering and the fact that materials can be more or less translucent, you can easily understand that this value relates to how translucent a particular material appears. A material (A) that appears more translucent than another (B), will have a greater mean free path than the one of the material it is compared to. In other words, in A, light will travel a greater distance in the object before it gets scattered compared to B. When light travels a greater distance before it leaves the object, the material appears more translucent than a material in which light travels a shorter distance before it goes out of the object. Furthermore, this mean free path is a wavelength dependent value. If we simplify our visible spectrum to three colors (red, green and blue), the mean free path will take a different value for each one of these colors. As mentioned earlier, this explained why subsurface scattering is a colored effect. Finally before we need to get into more technical explanation lets just say for now that this mean free path value relates to two other fundamental properties of the material: its absorption and scattering coefficients. Without getting into the detail of how they relate to each other, lets just say that the absorption and scattering coefficients are also wavelength dependant. The absorption coefficients relates  to the fraction of light that is being absorbed by the material per unit distance (millimetre, centimetre, etc. You have to be careful about the unit numbers are expressed in). Large coefficients means that light is quickly attenuated as it travels through the object. Light traveling through the medium is absorbed but is also scattered. Scattering coefficient relates to the fraction of light that is being scattered per unit length. The effect of scattering on the final result is harder to predict because some of the light that is traveling through the medium is "lost" due to outward scattering and some of it is "gained" through in-scattering. These concepts are better explained in the lesson on "Atmospheric Scattering".

But overall one might question how the loss and the gain relate to each other and what is the net result of this "exchange" of light due to scattering ? Lets not think of scattering in these terms for now. Lets just remember one thing. A light beam passing through a material (A) with a large scattering coefficient, will be scattered more often than when it passes through a medium (B) with a smaller scattering coefficient. In A, statistically, light will travel a greater distance than in B before it leaves the object. That has an important consequence. Light will potentially leave the medium further away from x (the point of illumination) than in B (making the object to appear more translucent than B if we assume that they have the same absorption coefficients). The higher the scattering coefficient the more "random" is the path followed by the light before it leaves the object. In A, light is more "blurred" than in B which means than more of the light will leave the object than in A where more of the light will continue on a straight line until it gets potentially absorbed. There is usually a direct relationship between the overall color of an object (we speak of its albedo, or diffuse reflectance) and its scattering coefficient. The "whiter" the object the higher the scattering coefficients (more light is reflected of its surface due to scattering being very high). Very "white" objects such as spectralon for instance have very high scattering coefficient, while their absorption coefficients are close to 0. Scattering is also a wavelength dependent effect. If the "red" component of the coefficient is greater than the two other components (green and blue), it is more likely than your object will appear reddish. Obviously this is a hard assumption to make because the final result is a mixture of scattering and absorption. If the material also absorbs a lot of red it could cancel out the red color due to scattering. This inter-relationess between the two effects (absorption and scattering) explains why it is often difficult for an artist to tweak the coefficients and come up with the right color for the object and the right amount of translucency.

The final term we will introduce is the extinction coefficient. It corresponds to the sum of the scattering and coefficient scattering. For material where the single scattering component is more important than multiple scattering, the mean free path is is the inverse of the extinction coefficient. For materials where the multiple scattering component dominates the mean free path is the inverse of the reduced extinction coefficient (which is also computed using the scattering and absorption coefficients). All you want to remember at that stage is that, the mean free path is more of less the inverse of the extinction coefficient which is the sum of the scattering and absorption coefficients. Keeping this in mind, you can make try to mentally imagine what happens to the mean free path for various combinations of scattering and absorption coefficients. We mentioned the case of spectralon which has a very high scattering coefficient (compared to skin for example). Its absorption coefficient is 0. Its mean free path is therefore quite small, which means that the material is not very translucent.

In the literature these coefficients can be expressed with the greek lower case letter mu (μ) or sigma (σ) or tau (τ). We will be using sigma for the rest of this lesson.

$$\begin{array}{ll} scattering \quad coefficient & \sigma_{s}\\ absorption \quad coefficient & \sigma_{a}\\ extinction \quad coefficient &\sigma_{t}=\sigma_{a}+\sigma_{s}\\ mean \quad free \quad path & \mu = \frac{1}{\sigma_{t}} \end{array}$$

In the following table, you will the coefficients of a few common materials (source "A Practical Model for Subsurface Light Transport" Jensen et Al. and "Acquiring Scattering Properties of Participating Media by Dilution" Narasimhan et Al.).

XX we are missing the scale of these values (unit) XX

Material	Absorption coefficient σa			Scattering coefficient σs
	Red	Green	Blue	Red	Green	Blue
Milk	0.0014	0.0025	0.0142	0.70	1.22	1.90
Ketchup	0.0610	0.9700	1.4500	0.18	0.07	0.03
Skin	0.0320	0.1700	0.4800	0.74	0.88	1.01
Marble	0.0210	0.0041	0.0071	2.19	2.62	3.00
Spectralon	0.0000	0.0000	0.0000	11.6	20.4	14.9

Color Bleeding

XX light bleeding within materials and diffusion of light across shadow boundaries XX We are waiting for approval to publish images from papers XX

Summary

In this chapter we have learned what makes translucent materials different from opaque or transparent objects. For translucent material, light enters and leaves the surface of an object at different locations. The greater the distance between the point of illumination and the point where light leaves the medium, the more translucent the material is. Subsurface can be split into a single and multiple scattering component. When light enters an object it will be either scattered or absorbed. The color of a material is the result of these two phenomenons which are wavelength dependent. We have also learned that when the object is thin enough, some of the light might be transmitted on the other side of the material sample. Light reflected and transmitted by a translucent material doesn't necessarily have the same color. Finally many translucent materials are made of thin translucent layers and the overall color of an object is the result of light between reflected and transmitted by these layers. We have learned some terms which will be used throught the rest of this lesson: absorption, scattering, extinction coefficients and mean free path which is the average distance a photon travels in the medium before it gets scattered.

What's Next?

In the following chapter we will give a historical background about the development of different techniques used to simulate subsurface scattering. We will also give a simple way of simulating subsurface scattering without getting into any mathematical details yet. This simple and practical exercise will make the study of the algorithms more straightforward and is aimed at readers who want to understand the principle of the technique without learning too much about the maths. Finally in the following chapter we will study various algorithms to simulate subsurface scattering, including photon mapping, the dipole model (used to simulate multiple scattering), the sum of gaussian (which is popular in video game and can easily be implemented on the GPU) and a simple implementation of single scattering (using raytracing).

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