Lighting Series Part 3 – Radiance

In this post from Lighting Series, we are going to talk about Radiance. The previous post in this series is Lighting Series Part 2 – Radiant Energy and Radiant Power. In the next post, we are going to talk about irradiance.

The SI System (Système International d’unites) defines six radiometric units used for describing the radiation coupling between a light source and an optical system.  The most commonly used units are: Radiant Flux, Irradiance, and Radiance.

What is Solid Angle?

A direction is indicated by a vector. Since this is a unit vector, it can be represented by a point on the unit sphere.

Positions on the Earth’s surface can be defined by two angles:

• The zenith angle θ, between the zenith (point on the celestial sphere located on the observer’s ascending vertical) and the direction observed.
• The azimuth angle φ between the North (on the local meridian) and the projection of the line on the Earth’s surface.

The height (altitude or elevation) is sometimes used instead of θ: h = (π / 2) – θ. θ varies along the vertical plane from 0 to π / 2 (0 to 90 degrees). φ varies along the horizontal plane from 0 to (0 to 360 degrees).

Directions ω and spherical coordinates (θ, φ) can be used interchangeably. A big advantage of thinking of directions as points on a sphere comes when considering differential distributions of directions. A differential distribution of directions can be represented by a small region on the unit sphere.

The area of a small differential surface element on a sphere of radius r is

dA = (r * dθ) * (r * sin φ * dφ) = r^2 * sin θ * dθ * dφ

Here r * dθ is the length of the longitudinal area generated as θ goes to θ + dθ. Similarly, r * sin θ * dφ is the length of the latitudinal area generated as φ goes to φ + . The product of these two lengths is the differential area of that patch on the sphere dA.

In 2D one can think of directions (unit vectors) as being points on the unit circle, and in 3D one can think of directions as being points on the unit sphere.

An angle is a section of the unit circle and its magnitude is its arc length. The angle subtended by a circular arc of length l is equal to l / r. The circle itself subtends an angle of radians because the circumference of the circle is 2π * r.

By using a similar idea we can define a solid angle. A solid angle is a section of the unit sphere and it describes angular spans in three-dimensional space, analogous to the way in which the radian describes angles in a two-dimensional plane. The solid angle subtended by a spherical area a is equal to a / r^2.

The steradian (sr, radiant squared) is the SI unit for measuring solid angles, defined by the solid angle Ω that projects on the surface of a sphere with a radius of r, having an area A equal to r^2:

Ω = A / (r^2) = (r^2) / (r^2) = 1 sr

A differential solid angle, indicated as dΩ, delimits a cone in space:

dΩ = dA / r^2 sr = sin θ * dθ * dφ

where dA is the area cut by the cone over a sphere of radius r the center of which is at the apex of the cone.

The solid angle corresponding to all the space around a point equals 4π sr. For an observer on Earth, the half-space formed by the celestial arch (in other words an hemisphere) therefore corresponds to 2π sr.

In the previous article, we described radiant power as the radiant energy per unit time. We can also consider the radiant energy per unit volume (instead per unit time) as the photon volume density times the energy of a single photon h * c /  λ, where h is Planck’s constant, c is the speed of light and λ is the photon’s wavelength. The radiometric term for this quantity is radiance.

L(x, ω) = ∫ p(x, ω, λ) * (h * c / λ) * dλ

Radiance is power dW per unit area dA perpendicular to a particular direction ω per unit solid angle in the direction ω at a particular point x on a surface or in space.

Another way to describe it: For a point x and a direction ω, L(x, ω) is a measure of the density of photons passing near x and traveling in directions near ω. Radiance measures the radiant flux that would be collected if we look at photons whose directions lie within a solid angle around ω and land on an area of surface dA at x that is perpendicular to the direction ω.

The SI unit of radiance is watts per square meter per steradian [W / (m^2 * sr)].

What happens if the surface is not perpendicular to the ray?

The definition of radiance that we shown is for a surface area perpendicular to the direction in question. We can see that a surface that is not perpendicular, that is, a surface that is inclined at an angle θ to ω, has the same particles passing through it in the direction ω as a perpendicular surface whose area is smaller by a factor of cos θ.

We can call the inclined surface area as the projected area.

Example of projected area:

You can think that power arriving at a grazing angle is “smeared out” over a larger surface.

Since we explicitly want to express power per (unit) projected area and per (unit) direction, we have to take the larger area into account, and that is where the cosine term comes from.

So the radiance formula can be written:

Another option is to group and cos θ and call them the projected solid angle.

Example of projected solid angle:

Then, we can see that the definition of radiance is

• “power dW per unit projected area dA’ per unit solid angle in the direction ω at a particular point x on a surface or in space”
• “power dW per unit area dA perpendicular to the direction ω per unit projected solid angle dΩ’ at a particular point on a surface or in space”.

The “projected” word is saying that the area for radiance is measured on a surface perpendicular to the direction of propagation.

Note that it is a function of five independent variables, three that specify position and two that specify direction.

Consider a point p on the surface of an object. There is some distribution of radiance arriving at the point that can be described mathematically by a function of position and direction. This function is denoted by Li(p, ω).

The function that describes the outgoing reflected radiance from the surface at that point is denoted by Lo(p, ω). It is important to see that, in general Li(p, ω) != Lo(p, ω).

Another property to keep in mind is that at a point in space where there is no surface and where there is no participating media causing scattering, these functions are related by Lo(p, ω) = Li(p, -ω)

Intuitively, radiance expresses how much power arrives at (or leaves from) a certain point on a surface, per unit solid angle, and per unit projected area.

Property 1: Radiance is invariant along straight paths.

The radiance in the direction of a light ray remains constant as it propagates along the ray . This law follows from the conservation of energy within a thin pencil of light between two differential surfaces. This property of radiance is only valid in the absence of participating media, which can absorb and scatter energy between the two surfaces.

If we look at the radiance at two points x1 and x2, in the direction ω that points from one to the other, we will measure the same radiance at both points: L(x1, ω) = L(x2, ω)

It follows that once incident or exitant radiance at all surface points is known, the radiance distribution for all points in a three-dimensional scene is also known.

Radiance at surface points is referred to as surface radiance by some authors, whereas radiance for general points in three-dimensional space is sometimes called field radiance.

Property 2: Sensors, such as cameras and the human eye, are sensitive to radiance.

The response of sensors (for example, a camera or the human eye) is proportional to the radiance incident upon them, where the constant of proportionality depends on the geometry of the sensor. This can be explained intuitively. Consider a simple exposure meter.

Each sensor element sees that part of the environment inside the beam defined by the aperture and the receptive area of the sensor. If a surface is far away from the sensor, the sensor sees more of it. One might conclude that the surface appears brighter because more energy arrives on the sensor, but the sensor is also far from the surface, which means that the sensor subtends a smaller angle with respect to the surface. The increase in energy resulting from integrating over a larger surface area is exactly counterbalanced by the decrease in percentage of light that makes it to the sensor. This property of radiance explains why a large uniformly illuminated painted wall appears equally bright over a wide range of viewing distances.

These two properties explain why the perceived color or brightness of an object does not change with distance.

The radiance of a source is increased/decreased by:

• Increasing/Decreasing its emitted power
• Making the emitting area of the source smaller/bigger
• Emitting the radiation into a smaller/bigger solid angle.

Example of different lamps wattage:

Example of different emitting areas:

Light Absorption and Scattering

Radiance is a conserved quantity in an optical system so that radiance measured as watts per unit area per unit solid angle incident on a detector will not exceed the radiance at the emitter. For any bunch of rays mapping an emitter to a detector, the radiance seen at the detector could be diminished by the light which is absorbed along the way or scattered out of the solid angle of the bunch of rays reaching the detector.

Example of light being absorbed by the fog:

Example of light scattering:

Fluorescent and Xenon Arc Lamp

Suppose we observe with the eye a 35W Xenon short-arc lamp, and then a 60W straight tube fluorescent lamp, both at a similar distance of a few meters.

Example 60W fluorescent tube and 35W Xe arc-lamp:

We want to know which light source is perceived to be brighter (higher radiance). The Xenon short-arc lamp is perceived to be much brigher, although it emits less power than the tube. This is a result of the much smaller emitting area A of the short-arc lamp compared to the very large emitting area of the fluorescent lamp, while the eye is receiving the radiation at more or less the same solid angle Ω when the distance between the eye and the source is the same.

The eye’s lens forms a bright image of the Xenon arc on a very small area of the retina and the eye does not feel comfortable. The larger area of the fluorescent lamp will form an image over a much large area on the retina, which the eye can tolerate more comfortably. The arc-lamp has a much higher radiance than the fluorescent lamp, even though it emits less power.

Example of light impacting the retina:

References

Real-Time Rendering, 3rd Edition

Physically Based Rendering, 3rd Edition: From theory to implementation

Fundamentals of Computer Graphics, 4th Edition

Essential Mathematics for Games and Interactive Applications, 3rd Edition

Introduction to Computer Graphics: A Practical Learning Approach, 1st Edition

Computer Graphics: Principles and Practices, 3rd Edition

Foundations of 3D Computer Graphics