# 8.2: Path Tracing

- Page ID
- 13753

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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We have seen how ray tracing can be extended to approximate a variety of effects that are not handled by the basic algorithm. We look next at an algorithm that accounts for all those effects and more in a fairly straightforward and unified way: * path tracing*. Like ray tracing, path tracing computes colors for points in an image by tracing the paths of light rays backwards from the viewer through points on the image and into the scene. But in path tracing, the idea is to account for

**all**possible paths that the light could have followed. Of course, that is not literally possible, but following a large number of paths can give a good approximation — one that gets better as the number of paths is increased (“Forward path tracing,” where paths of light rays emitted from light sources are traced forward in time, is also sometimes used.)

## BSDF’s

In order to model a wide variety of physical phenomena, path tracing uses a generalization of the idea of material property. In OpenGL, a material is a combination of ambient, diffuse, specular, and emission colors, plus shininess. These properties, except for emission color, model how the surface interacts with light. Material properties can vary from point to point on a surface; that’s an example of a texture.

OpenGL material is only a rough approximation of reality. In path tracing, a more general notion is used that is capable of more accurately representing the properties of almost any real physical surface or volume. The replacement for materials is called a * BSDF*, or Bidirectional Scattering Distribution Function.

Think about how light that arrives at some point can be affected by the physical properties of whatever substance exists at that point. Some of the light might be absorbed. Some might pass through the point without being affected at all. And some might be “scattered,” that is, sent off in another direction. In fact, we consider passing through the point as a special case of scattering. A BSDF describes how light is scattered from each point on a surface or in a volume.

Think of a single ray, or photon, of light that arrives at some point. What happens to it can depend on the direction from which it arrives. In general, assuming that it is not absorbed, the light is more likely to be scattered in some directions than in others. (As in specular reflection, for example.) The BSDF at the point gives the probability that the ray will leave the point heading in a given direction. It is a “bidirectional” function because the answer is a function of two directions, the direction from which the light arrives and the outgoing direction that you are asking about. (It is a “distribution function” in the sense of the mathematical theory of continuous probability distributions, but you don’t need to understand that to get the general idea. For us, it’s enough to understand that the function says how light coming in from a given direction is distributed among possible outgoing directions.) Note that a BSDF is also a function of the point that you are talking about, and it can be a function of the wavelength of the light as well.

Any point in space can be assigned a BSDF. For empty space, the BSDF is trivial: It simply says that light arriving at a point has a 100% probability of continuing in the same direction. But light passing through fog or dusty air or dirty water has some probability of being absorbed and some probability of being scattered to a random direction. Similar remarks apply to light passing through the interior of a translucent solid object.

Traditionally, though, computer graphics has been mostly concerned with what happens to light at the surface of an object. Light can be absorbed or reflected or, if the object is translucent, transmitted through the surface. The function that describes the reflection of light from a surface is sometimes called a BRDF (Bidirectional Reflectance Distribution Function), and the formula for transmission of light is a BTDF (Bidirectional Transmission Distribution function). The BSDF for a surface is a combination of the two.

Let’s consider OpenGL materials in terms of BSDFs. In basic OpenGL, light can only be reflected or absorbed. For diffuse reflection, light has an equal probability of being reflected in every direction that makes an angle of less than 90 degrees with the normal vector to the surface, and there is no dependence on the direction from which the light arrives. For specular reflection, the incoming light direction matters. In OpenGL, the possible outgoing directions for specularly reflected light form a cone, where the angle between the axis of the cone and the normal vector is equal to the angle between the normal vector and the incoming light direction. The axis of the cone is the most likely direction for outgoing light, and the probability falls

off as the angle between the outgoing direction and the direction of the axis increases. The rate of falloff is specified by the shininess property of the material. The BRFD for the surface combines the diffuse and specular reflection. (The ambient material property doesn’t fit well into the BSDF framework, since physically there is no such thing as an “ambient light” that is somehow different from regular light.)

Ray tracing adds two new possibilities to the interaction of light with a surface: perfect, mirror-like reflection, where the outgoing light makes exactly the same angle with the normal vector as the incoming light, and transmission of light into a translucent object, where the outgoing angle is determined by the indices of refraction outside and inside the object.

But BSDFs can provide even more realistic models of the interaction of light with surfaces. For example, the distinction between mirror-like reflection of an object and specular reflection of a light source is artificial. A perfect mirror should reflect both light sources and objects in a mirror-like way. For a shiny but rough surface, all specular reflection would send the light in a cone of directions, giving fuzzy images of objects and lights alike. A BSFD should handle both cases, and it shouldn’t distinguish between light from light sources and light reflected off other objects.

BSDFs can also correctly handle a phenomenon called * subsurface scattering*, which can be an important visual effect for materials that are just a bit translucent, such as milk, jade, and skin. In sub-surface scattering, light that hits a surface can be transmitted into the object, be scattered a few times internally inside the object, and then emerge from the surface at another point. How the light behaves inside the object is determined by the BSDF of the material in the interior of the object. The BSDF in this case would be similar to the one for fog, except that the probability of scattering would be larger.

The point is that just about any physically realistic material can be modeled by a correctly chosen BSDF.

## The Path Tracing Algorithm

Path tracing is based on a formula known as the “rendering equation.” The formula says that the amount of light energy leaving a given point in a given direction is equal to the amount of light energy emitted by the point in that direction plus the amount of light energy arriving at the point from other sources that is then scattered in that direction.

Here, emitted light means light that is created, as by a light source. In the rendering equation, any object can be an emitter of light. In OpenGL terms, it’s as if an object with an emission color actually emits light that can illuminate other objects. An area light is just an extended object that emits light from every point, and it is common to illuminate scenes with large light-emitting objects. (In fact, in a typical path tracing setup, point lights and directional lights have to be assigned some area to make them work correctly in the algorithm.)

As for scattered light, the BSDF at a point determines how light arriving at that point is scattered. Light can, in general, arrive from any direction and can originate from any other point in the scene. The rendering equation holds at **every** point. It relates the light arriving at and departing from each point to the light arriving at and departing from every other point. It describes, in other words, an immensely complicated system, one for which you are unlikely to be able to find an exact solution. A rendering algorithm can be thought of as an attempt to find a good approximate solution to the rendering equation.

Path tracing is a probabilistic rendering algorithm. It looks at possible paths that might have been followed by light arriving at the position of the viewer. Each possible path has a certain probability. Path tracing generates a random sample of possible paths, choosing paths in the sample according to their probabilities. It uses those paths to create an image that approximates a solution to the rendering equation. It can be shown that as the size of the random sample increases, the image that is generated will approach the true solution. To get a good quality image, the algorithm will have to trace thousands of paths for each pixel in the image, but the result can be an almost shocking level of realism.

Let’s think about how it should work. First, consider the case where light is only emitted and reflected by surfaces. As with ray tracing, we start at the position of the viewer and cast a ray in the direction of a point on the image, into the scene. (See Subsection 8.1.1.) We find the first intersection of that ray with an object in the scene. Our goal to trace one possible path that the ray could have followed from its point of origin until it arrives at the viewer, and we want the probability that we select a given path to be the probability that the light actually followed that path. This means that each time the light is scattered from a surface, we should choose the direction of the next segment of the path based on the BSDF for the surface. That is, the direction is chosen at random, using the probability distribution that is encoded in the BSDF. We construct the next segment of the path by casting a ray in the selected direction.

We continue to trace the path, backwards in time, possibly through multiple reflections, until it encounters an object that emits light. That object serves as the original source of the light. The color that the path contributes to the image is determined by the color and intensity of the emitter, by the colors of surfaces that the light hits along the way, and by the angles at which the light hits each surface. If the path escapes from the scene before it hits a light emitting object, then it does not contribute any color to the image. (It might be desirable to have a light-emitting background, like a sky, that emits light over a large area.) Note that it is possible for an object to be both an emitter and a reflector of light. In that case, a path can continue even after it gets to a light source.

Of course, we have to trace many such paths. The color for a pixel in the image is computed as an average of the colors obtained for all the paths that pass through that pixel.

The algorithm can be extended to handle the case where light can be scattered at arbitrary points in space, and not just at surfaces. For light traveling in a medium in 3D space, the question is, how far will the light travel before it is scattered? The BSDF for the medium will determine a probability distribution on possible travel distances between scatterings. When light enters a medium, that probability distribution is used to select a random distance that the light will travel before it is scattered (unless it hits a surface or enters a new medium before it has traveled that distance). When it scatters from a point in the medium, a new direction and length are chosen at random for the next segment of the path, according to the BSDF of the medium. For a light fog, the average distance between scatterings would be quite large; for a dense medium like milk, it would be quite short.

A great deal of computation is required to trace enough light paths to get a high-quality image. Although path tracing was invented in the 1980s, it is only recently that it has become practical for general use, and it can still take many hours to get acceptable quality. In fact, you can do path tracing on your desktop computer using the 3D modeling program Blender, which is discussed in Appendix B. Blender has an alternative rendering engine, called the Cycles renderer, that uses path tracing. Cycles is not discussed in the appendix, but you can look up some tutorials on it, if you are interested in seeing what path tracing can do.