What is a BRDF ?

Introduction

There is many aspect of the CG lighting pipeline which are mysterious to people using it, such as how the brightness of an object relates to the intensity of a light. This chapter explains this relationship and introduces the concept of BRDF which you may have heard often without really understanding what it means and how it works. In general, this lesson should lift the veil on a few similar rather (what appears to be) mysterious topics.

Radiometry: defining relations between light and matter

The techniques we use to simulate how light in the real world interacts with objects are based on a series of measures defined in radiometry which is the field that studies the measurements of electromagnetic radiation (including visible light).

To understand where these techniques come from and how they work, we will first look at these radiometric terms and explain how they related to the physical world. From there we will introduce the concepts of BRDF and albedo, and give an example of a BRDF for diffuse surfaces. 

Radiometry terms: flux, irradiance and radiance

Light can be seen as a collection of particles (photons) travelling in space, carrying energy. We model the photons direction as a straight line. When we talk of the amount of light emitted by an object, the term actually relates to a quantity of photons that is emitted by an object. In CG what interests us the most is the amount of light arriving at and leaving the surface of an object.

The first important notion is the light power, called flux, expressed in watts (W). The flux is a quantity of luminous energy (number of photons) passing through a surface per unit of time. Intuitively one can see that the properties of a material, such as its color and brightness, have some form of relation with the ratio of the reflected flux (the amount of light that is reflected off the surface) to the incoming flux (the amount of light arrived at the surface). A black surface should reflect no flux (no light), while a pure reflector should reflect all the incoming flux. In the real world most object's behaviour lies somewhere in the middle.

$$material \quad brightness = \frac{light \quad reflected}{light \quad received}$$

Figure 1: flux measures the quantity of luminous energy passing through a surface per unit of time.

In the real world the appearance of objects often varies across their surface. We say that their appearance vary spatially. A good example of such material is marble. If we try to define what is the flux that is reflected by a slate of marble we need to refine the definition of flux by introducing a concept of locality. Instead of just being interested in the flux that is received or reflected by a surface, we will look at the flux that is received and reflected at a particular point on the surface (lets call this point x). However as a point cannot be physically defined in the real world, instead we consider a very very small area around a point (so small that the surface of this area is guaranteed to have a constant color), which we call the differential area (figure 2).

When expressed with regards to this differential area we now treat the flux as a density, and the flux density arriving at a surface is named irradiance. The flux density emitted at a point can be called radiant exitance or radiosity if it is the sum of the emitted (but we will be ignoring emission for now) and reflected flux density. Irradiance is expressed in watts per unit square meter (W.m-2). While a textured object (an object made of marble for instance), may receive the same irradiance all over its surface, the reflected irradiance will change spatially.

Figure 2: to measure the flux at a point we need to take a very small region around the point called the differential area. Because this region is very small that there's not flux variation over its area.

Finally, most objects also exhibit view-dependent properties. For a given point on the surface the reflected light changes with the view angle. The definition of irradiance doesn't take into consideration any view dependent behaviour, therefore we need to come up with a new quantity called radiance, which is measured in watts per unit square meter per unit solid angle (W.m-2.sr-1). Radiance is the quantity of light incoming at (or emitted from) a given point, for a given direction in space. The ratio of reflected radiance at a point into a given direction allows us to describe almost any material.

Radiance along a direction can be seen as a flux of photons going through a cylinder of infinitesimal radius. Note that the human eye is a surface itself and consequently we actually use radiance in computer graphics to describe how light reflected by objects interact with the surface that is responsible for creating an image.

Flux, irradiance and radiance allows us to describe almost every lighting phenomenon. Radiance being the basic quantity, irradiance and flux can be computed from it.

Short introduction to the radiometry equations

Flux is noted ϕ and is expressed in W.

Irradiance is noted E and is expressed in W.m-2.

Radiance is noted L and is expressed in W.m-2.sr-1.

Radiometry is very similar to the science of photometry, but photometry only measures the brightness of objects (lights) as perceived by the human eye while radiometry deals with absolute power values. Photometry is limited to the light from the visible spectrum while radiometry is not.

As we mentioned before, points and directions are concepts which do not exist in the physical world. A point on a surface usually means a very small area of the surface around this point. Similarly, a direction usually describes a very small angle around this direction (which is called a differential solid angle). As we said, irradiance at a point describes the flux going through a very small element of the surface around the point. Radiance is the flux going through a very small area on the surface around x, into a very small cylinder (which is displayed as a cone in figure because of perspective) centred around a given direction. This concept of a small cylinder is similar to the concept of a small area around x. We usually speak of differential solid angle and we use the letter omega to represent it: dω or dΩ.

Figure 3: radiance (L) measures the outgoing radiant flux per solid angle (dω here represents a differential solid angle) per area projected (dA cosθ) on to the surface in the direction of radiation (the eye in our case).

Irradiance is the quantity of light incoming at a given point. Radiance is the quantity of light incoming at a given point from a given direction. This means that the total irradiance at point x can be computed from the radiance by simply summing up the radiance for all directions in the hemisphere oriented around the normal at x. If we consider radiance for only one incoming direction (as opposed to the entire hemisphere), we can compute the corresponding fraction of irradiance, which we will call differential irradiance (denoted dE).

The following figures shows that for a given radiance (photon flux through a small cylinder) the quantity of photons going through a small element of surface (that is our differential irradiance) depends on the angle of incidence of the cylinder. When the cylinder is perpendicular to the surface, all the photons travelling through the cylinder hit the surface element. As we rotate the cylinder the intersection between the surface and the cylinder becomes larger which means that less photons go through our original surface element. Ultimately when the cylinder is parallel to the surface, no photons hit the surface element.

Figure 4: as the beam of light makes an angle with the plane the surface covered by the beam increases. It means that the differential area (the blue square in this figure) receives less and less photons (the same number of photons is spread across a larger region) as the light angle of incidence increases. The light intensity can be multiplied by the cosine of the angle subtended by the light and the surface normal. This relation is known as the Lambert cosine law.

This relationship is known as the Lambert cosine law. To account for the reduced amount of photons reaching the surface element as the angle between the normal at the surface and the light increases, we multiply the light energy by the cosine of this angle.
$$ dE=cos(\theta)Ld\omega $$ where θ is the incidence angle, L is the radiance incoming along a direction, and dω represents the small cylinder centred along this direction (which we express with a differential solid angle).

And finally to compute the total irradiance, we use a mathematical construction called integration to compute the sum of all the fractional irradiance for all directions in the hemisphere. In mathematical form it looks like this:

$$ E=\int\limits_\Omega dE=\int \limits_\Omega cos(\theta)Ld\omega $$

Properties of a surface

Given the quantity we have introduced so far we can know define the properties of a surface as ratios between incoming and reflected flux.

If this ratio is 0, it means that our surface does not reflect any light and appears totally black. This ratio is 1 for a pure reflective surface (meaning all the incoming irradiance is reflected). This ratio cannot be greater than 1 because it would mean that the surface would create light (we ignore the case of emissive surfaces for now). A surface may reflect less light than the amount it receives because some of the incoming light is absorbed by the material. Our ratio somehow defines how much of that incoming light is absorbed by an object.

Lets see now how we can express this ratio in a mathematical form for a beam of light (single direction) arriving at x on the surface. Remember that when we consider a single direction instead of all the directions in the hemisphere we are interested in differential quanties. In the case of a beam of light, this ratio is therefore between the differential incoming radiance (light coming in) with direction wi and the differential radiance in the direction of the eye (with direction wo). This ratio is called reflectance (denoted Fr) and is expressed as:

$$ fr=\frac{dL(\omega_o)}{dE(\omega_i)}=\frac{dL(\omega_o)}{cos(\theta_i)L(\omega_i)d\omega_i} $$

Since Fr depends on an incoming and outgoing direction, it is 4-dimensional (because the directions are usually expressed in spherical coordinates). The reflectance is also known as the BRDF (Bidirectional Reflectance Distribution Function). A BRDF is dimensionless but is expressed in terms of 1/sr (or sr-1).

Finally, to compute the final color of a point as seen from from a particular direction (wo), we have to compute the total reflected radiance by summing all the differential radiance.

$$\begin{array}{l} dL(\omega_o)=frdE(\omega_i)\\ L=\int \limits_\Omega frdE(\omega_i)=\int \limits_\Omega fr cos(\theta_i)L(\omega_i)d\omega_i \end{array}$$

A BRDF for diffuse surface

To simplify the case let's consider only a purely diffuse material. Purely diffuse means that the material reflects light in the same way for all directions in the hemisphere, its reflectance is constant (and because it is a constant, we can take fr out of the integral). The previous equations can then be simplified into:

$$\begin{array}{l} L(\omega_o)=\int \limits_\Omega fr cos(\theta_i)L(\omega_i)d\omega_i=fr \int \limits_\Omega cos(\theta_i)L(\omega_i)d\omega_i=frE \end{array}$$

Remember that E is the total irradiance. Lets consider that the incoming radiance (irradiance) is constant over the hemisphere, that is L(ωi) = 1. Reflected radiance in any direction can then be expressed as:

$$\begin{array}{l} L(\omega_o)=fr \int \limits_\Omega cos(\theta_i)d\omega_i=fr\pi \end{array}$$

This integral resolves to π in that particular case. Readers interested to know how we got to π, can find the math derivation in the following note (this note itself takes some shortcuts with explaining how we get to this result. Check the lesson on BRDFs in the basic section for a full introduction on the topic).

Equation 1 represents how we have defined the integral of the function cos(θ) so far. This equation can be re-written in a more classical form (2) where the integral over the solid angle dω can be re-written as two integrals: dω is replaced by sin(θ)dθdΦ, θ (theta) is in the range [0, π/2] and Φ (phi) is in the range [0 2π]. The integration of Φ gives 2π. The remaining integral which includes the original function cos(θ) can be re-written using what's known in calculus as the definite integral. In short if F(x) is the primitive function of f(x) (or if f(x) is the derivative of F(x)), then you can write that the integral of f(x) in the range [a, b] is equal to F(b) - F(a) (equation 5). Equation 4 shows what is the primitive function of cos(θ)sin(θ). We then apply equation 5 using this primitive function (-1/2cos^2(x)) to find the value 1/2 (a=0 and b=π/2). Finally we multiply 2π by 1/2 which gives us π.

Therefore if we're considering a diffuse material with a constant BRDF fr, illuminated with a constant radiance of 1, the reflected radiance in all the directions is πfr. In the case where the amount of incident light is 1 and fr is 1 the amount of reflected light is π which means that the surface creates energy. In order to have a physically plausible result (energy conserving), the reflected light in this case should not be greater than 1. Therefore fr, the BRDF for a diffuse surface needs to be:

$$BRDF \quad diffuse = \frac{k}{\pi} $$

with k in the range 0 to 1 and k is usually what we call albedo.

From a lighting perspective, it means that if k is 1 for an object that is illuminated by a distant light which is perpendicular to the surface (so that there is not attenuation due to the lambert law), the reflected value should be 1 over π. 

Inversely, if you illuminate the same object with a dome light which has a constant radiance of 1 we will have to compute the total incoming irradiance over the hemisphere by integrating the differential incoming irradiance. Which is solving for this equation that we have introduced further up:

$$E=\int \limits_\Omega dE= \int \limits_\Omega cos(\theta)Ld\omega $$

This equation is very similar to the one we have just solved. Assuming L equals 1 we can integrate and find that E equals π. Therefore when you multiply π by a diffuse BRDF which constant k equals 1 you get 1. When k equals 1 and the surface is illuminated by a dome light of constant radiance 1, the brightness of the surface is 1.

Chapter 1 of 3