Radiometry

Radiometry is the measurement of electromagnetic radiation in physical units. In rendering, radiometry gives the language used to describe how light energy moves through space, across surfaces, and along directions.

Geometric Quantities

Surface

In radiometry, a surface is usually modeled as a two-dimensional manifold in Euclidean space, typically

M \subseteq ℝ^{3}

. This means that near each ordinary point, the surface can be described by two coordinates. When edges are relevant, the surface is treated as a manifold with boundary. Rendering systems may approximate surfaces by polygon meshes, but the radiometric definitions below apply pointwise on smooth pieces.

Surface Patch

A surface patch around a point

p

is a small region of a surface containing

p

. Area densities in radiometry are defined by shrinking such patches around the point.

Surface Normal

A surface normal at a point

p

on a surface

M \subseteq ℝ^{3}

is a unit direction perpendicular to the tangent space

M_{p}

Solid Angle

Let

p \in ℝ^{3}

. The direction projection from $p$ is the function

π_{p} : ℝ^{3} ∖ {p} \to 𝕊^{2}, π_{p} (x) = \frac{x - p}{‖ x - p ‖} .

X \subseteq ℝ^{3} ∖ {p}

and

π_{p} (X)

is surface-measurable as a subset of the unit sphere

𝕊^{2}

, then the solid angle subtended by $X$ from $p$ is

σ_{p} (X) = {area}_{𝕊^{2}} (π_{p} (X)) .

For a measurable direction set

A \subseteq 𝕊^{2}

, we write

σ (A) = {area}_{𝕊^{2}} (A) .

This is the special case of measuring the surface area of a set of directions.

The projection $π_{p}$ forgets distance and remembers only direction from $p$ . If many points of $X$ lie on the same ray from $p$ , they contribute the same direction to $π_{p} (X)$ . Thus solid angle measures angular size as seen from $p$ , not physical area or distance.

Solid Angle is a Measure

Let

𝒮_{𝕊^{2}}

be the

σ

-algebra of surface-measurable subsets of

𝕊^{2}

. The solid angle function

σ : 𝒮_{𝕊^{2}} \to ℝ^{\geq 0}

is a measure on

(𝕊^{2}, 𝒮_{𝕊^{2}})

The empty set has surface area

0

, so

σ (\emptyset) = {area}_{𝕊^{2}} (\emptyset) = 0.

Now let

A_{1}, A_{2}, \dots \in 𝒮_{𝕊^{2}}

be pairwise disjoint. By the definition of induced surface area, the surface area of a countable disjoint union is obtained by adding the surface areas of the pieces. Hence

σ (⋃_{j = 1}^{\infty} A_{j}) = {area}_{𝕊^{2}} (⋃_{j = 1}^{\infty} A_{j}) = \sum_{j = 1}^{\infty} {area}_{𝕊^{2}} (A_{j}) = \sum_{j = 1}^{\infty} σ (A_{j}) .

Therefore

σ

satisfies countable additivity and is a measure.

Solid Angle of a Single Direction

Let

ω \in 𝕊^{2}

. Compute the solid angle of the one-point set

{ω}

By the definition of solid angle,

σ ({ω}) = {area}_{𝕊^{2}} ({ω}) .

A single point on a surface has surface area

0

. Therefore

σ ({ω}) = 0.

Thus a single point of

𝕊^{2}

has solid angle

0

, even though small neighborhoods around that point can have positive solid angle.

Countable Direction Sets Have Zero Solid Angle

A \subseteq 𝕊^{2}

is countable, then

σ (A) = 0.

Since

A

is countable, write

A = {ω_{1}, ω_{2}, \dots},

allowing repetitions if needed. For each

k

, the one-point set

{ω_{k}}

has solid angle

0

by solid angle of a single direction. By countable subadditivity of surface area on

𝕊^{2}

σ (A) = σ (⋃_{k = 1}^{\infty} {ω_{k}}) \leq \sum_{k = 1}^{\infty} σ ({ω_{k}}) = 0.

Since

σ (A) \geq 0

, this gives

σ (A) = 0

Projected Unit Disk Has Solid Angle

2 π

Let

W = {(u, v) \in ℝ^{2} : u^{2} + v^{2} < 1}

and define

f (u, v) = (u, v, \sqrt{1 - u^{2} - v^{2}}) .

Compute the solid angle of

f (W) \subseteq 𝕊^{2}

By the definition of solid angle,

σ (f (W)) = {area}_{𝕊^{2}} (f (W)) .

The projected unit disk area computation gives

{area}_{𝕊^{2}} (f (W)) = 2 π .

Therefore

σ (f (W)) = 2 π .

Steradian

A steradian, denoted

s r

, is the unit used to record values of solid angle. If

σ (A) = 1

, then

A

has solid angle

1 s r

Orthogonal Projection onto a View Plane

Let

ω \in 𝕊^{2}

. The view plane perpendicular to $ω$ is the plane through the origin with normal

ω

ω^{⟂} = {x \in ℝ^{3} : x \cdot ω = 0} .

The orthogonal projection onto the view plane is the function

{proj}_{ω^{⟂}} : ℝ^{3} \to ω^{⟂}

defined by

{proj}_{ω^{⟂}} (x) = x - (x \cdot ω) ω .

Projected Area

Let

P \subseteq ℝ^{3}

be a surface patch, and let

ω \in 𝕊^{2}

. The projected area of

P

in direction

ω

is the surface area of its image under orthogonal projection onto the view plane:

{area}_{ω}^{⟂} (P) = {area}_{ω^{⟂}} ({proj}_{ω^{⟂}} (P)),

when this projected image has a well-defined two-dimensional area.

Projected Area of a Flat Patch

Let

P

be a surface patch contained in a plane with unit surface normal

n

. For every

ω \in 𝕊^{2}

{area}_{ω}^{⟂} (P) = | n \cdot ω | area (P) .

First consider a parallelogram

P

spanned by tangent vectors

u, v

in the plane. Its area is

‖ u \times v ‖

, and because

u

and

v

lie in the plane,

u \times v

is parallel to

n

. The projected parallelogram is spanned by

u^{'} = {proj}_{ω^{⟂}} (u), v^{'} = {proj}_{ω^{⟂}} (v) .

The area of the projected parallelogram is

‖ u^{'} \times v^{'} ‖ .

The vectors

u^{'}

and

v^{'}

lie in

ω^{⟂}

, so

u^{'} \times v^{'}

is perpendicular to

ω^{⟂}

, and therefore is a scalar multiple of

ω

. Since

ω

is a unit vector, norm equals absolute dot product with a parallel unit vector gives

‖ u^{'} \times v^{'} ‖ = | (u^{'} \times v^{'}) \cdot ω | .

By the oriented projected area identity,

(u^{'} \times v^{'}) \cdot ω = (u \times v) \cdot ω .

Hence

{area}_{ω}^{⟂} (P) = | (u \times v) \cdot ω | .

Since

u \times v

is a scalar multiple of the unit normal

n

, the dot product of a vector parallel to a unit vector gives

| (u \times v) \cdot ω | = | n \cdot ω | ‖ u \times v ‖ = | n \cdot ω | area (P) .

Thus the formula holds for parallelograms. Since

{proj}_{ω^{⟂}}

is a linear map on the plane containing

P

, the same constant area-scaling factor

| n \cdot ω |

applies to every measurable patch in that plane. Therefore

{area}_{ω}^{⟂} (P) = | n \cdot ω | area (P) .

Radiometric Quantities

Radiant Energy

Radiant energy is energy carried by electromagnetic radiation. Radiant energy is a scalar physical quantity: it records an amount of energy, not a direction. Radiant energy is measured in joules.

For example, if a short pulse of light carries $2 J$ of energy, then its radiant energy is $2 J$ , where $J$ denotes the joule unit of energy.

Watt

A watt, denoted

W

, is the unit of power. One watt is one joule per second:

1 W = 1 J \cdot s^{- 1} .

Radiometric Particle State

A radiometric particle state is the struct type

Particle_State : : {\begin{matrix} position & : & ℝ^{3} \\ direction & : & 𝕊^{2} \\ wavelength & : & ℝ^{> 0} \end{matrix}

A state

s \in Particle_State

has fields

s . position

s . direction

, and

s . wavelength

Travelling Particle

A travelling particle is a continuous function

TP : ℝ \to Particle_State .

The value

TP (t) . position

is the particle's position at time

t

, and

TP (t) . direction

is its direction at time

t

Travelling Particle Hits a Surface Patch

Let

P \subseteq ℝ^{3}

be a surface patch. A travelling particle

TP

hits

P

during the time interval

[t, t + τ]

if there exists

u \in [t, t + τ]

such that

TP (u) . position \in P .

Travelling Particle Locally Leaves a Surface Patch

Let

P \subseteq ℝ^{3}

be a surface patch. A travelling particle

TP

locally leaves

P

at time

u

TP (u) . position \in P

and there exists

ϵ \in ℝ^{> 0}

such that

TP (u + s) . position \notin P for every s \in (0, ϵ) .

It locally leaves

P

during

[t, t + τ]

if it locally leaves

P

at some

u \in [t, t + τ]

The word locally matters. A reflected particle can leave a patch at one interaction and hit the same patch again later; that later hit is a separate event. The outgoing condition only says that immediately after the event, the particle is no longer on the patch.

Travelling Particle in a Direction Set

Let

A \subseteq 𝕊^{2}

. A travelling particle

TP

is travelling in the direction set

A

at time

t

TP (t) . direction \in A .

A direction set $A \subseteq 𝕊^{2}$ is the same data as a cone of directions from the origin. Saying that a particle is travelling in $A$ means its direction field points into that cone at the time being measured.

Radiant Energy of a Particle Collection

Let

𝒯

be the set of travelling particles under consideration. A radiant energy function is a function

radiant_energy : ℰ \to ℝ^{\geq 0},

where

ℰ

is a collection of subsets of

𝒯

, such that

radiant_energy (C)

is the total radiant energy carried by the particle collection

C

. In the continuous radiometric model, individual particles have zero radiant energy:

radiant_energy ({TP}) = 0

for every

TP \in 𝒯

This follows the usual continuum idealization: instead of counting a finite list of photons, radiometry treats photon events as so numerous that their energy distribution is modeled continuously. A single exact travelling particle has zero energy, while a measurable bundle of travelling particles can have positive radiant energy.

Radiant Flux

Let

𝒯

be the set of travelling particles under consideration. A time-windowed particle collection is a function

C : ℝ \times ℝ^{> 0} \to ℰ,

where

C (t, τ) \subseteq 𝒯

is the collection of travelling particles selected during the time interval from

t

t + τ

. The radiant flux, also called radiant power, of

C

at time

t

Φ (C, t) = \lim_{τ \to 0^{+}} \frac{radiant_energy (C (t, τ))}{τ},

whenever this limit exists. Radiant flux has units of joules per second, more commonly called watts.

If $C$ and $D$ are time-windowed particle collections, then $C \cap D$ means the time-windowed collection defined by $(C \cap D) (t, τ) = C (t, τ) \cap D (t, τ) .$ This lets us restrict radiant flux by several conditions at once, such as hitting a surface patch and travelling in a chosen set of directions.

For example, if $C_{P}^{hit} (t, τ)$ is the collection of particles that hit a surface patch $P$ during $[t, t + τ]$ , then $Φ (C_{P}^{hit}, t)$ is the radiant flux incident on $P$ at time $t$ . If no such particles hit $P$ during a small time interval, then $radiant_energy (C_{P}^{hit} (t, τ)) = 0 J$ , so $Φ (C_{P}^{hit}, t) = 0 W$ . If particles hitting $P$ carry $3$ joules every $1$ second, then $radiant_energy (C_{P}^{hit} (t, τ)) = 3 τ$ joules, so $Φ (C_{P}^{hit}, t) = 3 W$ .

Radiant Energy from Flux

Let

C

be a time-windowed particle collection, let

t_{0}, t_{1} \in ℝ

with

t_{0} < t_{1}

, and suppose that there is a differentiable accumulated radiant energy function

Q_{C} : [t_{0}, t_{1}] \to ℝ

such that

Q_{C} (t_{1}) - Q_{C} (t_{0}) = radiant_energy (C (t_{0}, t_{1} - t_{0})) .

If the radiant flux of

C

Φ (C, t) = Q_{C}^{'} (t)

for every

t \in (t_{0}, t_{1})

, and

t \mapsto Φ (C, t)

is continuous on

[t_{0}, t_{1}]

, then the radiant energy selected by

C

from

t_{0}

t_{1}

radiant_energy (C (t_{0}, t_{1} - t_{0})) = \int_{t_{0}}^{t_{1}} Φ (C, t) d t .

Since

Φ (C, t) = Q_{C}^{'} (t)

(t_{0}, t_{1})

, the Fundamental Theorem of Calculus II gives

\int_{t_{0}}^{t_{1}} Φ (C, t) d t = Q_{C} (t_{1}) - Q_{C} (t_{0}) .

By the defining property of the accumulated radiant energy function on this interval,

Q_{C} (t_{1}) - Q_{C} (t_{0}) = radiant_energy (C (t_{0}, t_{1} - t_{0})) .

Therefore

radiant_energy (C (t_{0}, t_{1} - t_{0})) = \int_{t_{0}}^{t_{1}} Φ (C, t) d t .

As a more physical example, aim a flashlight at a small circular patch $P$ on a wall, and let $C_{P}^{hit}$ select the particles that hit $P$ during each time interval. If the flashlight output pulses in time, then the radiant flux $Φ (C_{P}^{hit}, t)$ through that circular patch changes with time. For instance, a simple model might be $Φ (C_{P}^{hit}, t) = 2 W + W \sin (t),$ meaning the patch receives $2 W$ on average, with the instantaneous radiant flux rising and falling between $1 W$ and $3 W$ .

Irradiance

Let

(P_{k})

be a sequence of surface patches that shrinks to

p

. Let

C_{P_{k}}^{hit}

be the time-windowed particle collection that selects particles that hit

P_{k}

. The irradiance at

p

and time

t

is the area density of incoming flux:

irradiance (p, t) = \lim_{k \to \infty} \frac{Φ (C_{P_{k}}^{hit}, t)}{area (P_{k})},

when this limit exists and has the same value for the shrinking patches under consideration. Irradiance is measured in

W \cdot m^{- 2}

, using the watt and the metre.

A helpful picture is rain passing through a small window. If the window is twice as large, it catches about twice as many drops, so dividing by the window area gives a rate per area. For light, the situation is smoother: a light source is modeled as producing a continuous flow of radiant flux, not separate drops that must be counted one by one. That is why shrinking the patch area and taking a limit can approach a nonzero density, even though an actual rain window made small enough might have no drops pass through during a short measurement.

For example, if $20 W$ of radiant flux arrives uniformly across a flat surface patch of area $4 m^{2}$ , then the irradiance on that patch is $20 W / 4 m^{2} = 5 W \cdot m^{- 2}$ , where $m$ denotes the metre unit of length.

Radiant Flux from Irradiance

Let

P

be a surface patch on a surface, fix a time

t

, and let

C_{P}^{hit}

select particles that hit

P

. Suppose the incoming irradiance

irradiance (q, t)

is defined at each point

q \in P

, is measurable and integrable with respect to surface area on

P

, and represents the area density of incoming radiant flux with respect to the surface area on

P

. Then the incoming radiant flux through

P

Φ (C_{P}^{hit}, t) = \int_{P} irradiance (q, t) d A,

where

d A

is the surface area element on

P

Partition

P

into small surface patches

P_{1}, \dots, P_{N}

, and choose a representative point

q_{j} \in P_{j}

. Since

irradiance (q, t)

is the area density of incoming flux, the flux through

P_{j}

is approximated by

irradiance (q_{j}, t) area (P_{j}) .

Adding the approximations gives

Φ (C_{P}^{hit}, t) \approx \sum_{j = 1}^{N} irradiance (q_{j}, t) area (P_{j}) .

In the limit as the partition is refined, these sums converge to the integral of

irradiance

over the subset

P

with respect to surface area. Therefore

Φ (C_{P}^{hit}, t) = \int_{P} irradiance (q, t) d A .

The measurability of $q \mapsto irradiance (q, t)$ is not automatic from the pointwise limit notation in the definition of irradiance. To use the integral over $P$ , one either assumes this surface-measurability as part of the radiometric model or proves it from more primitive data. A common sufficient route is to express $irradiance$ in each coordinate patch of the surface and prove that the coordinate expression is Borel measurable; in particular, a continuous coordinate expression is Borel measurable by continuous functions are Borel measurable.

Irradiance from Shrinking Circular Patches

Let

M = ℝ^{2} \times {0} \subseteq ℝ^{3}

be a flat surface, and let

p = (0,0,0)

. For each

k \in ℕ_{1}

, let

P_{k} = {(x, y, 0) \in M : x^{2} + y^{2} \leq \frac{1}{k^{2}}} .

Let

C_{P_{k}}^{hit}

select particles that hit

P_{k}

. Suppose that, at a fixed time

t

, the incoming radiant flux through every patch

P_{k}

Φ (C_{P_{k}}^{hit}, t) = 5 W \cdot m^{- 2} area (P_{k}) .

Prove that the irradiance at

p

and time

t

5 W \cdot m^{- 2}

The patches form a decreasing sequence, since

P_{k + 1} \subseteq P_{k}

for every

k

. Also,

⋂_{k = 1}^{\infty} P_{k} = {p},

because the only point whose distance from

p

is at most

1 / k

for every

k

p

itself. Therefore

P_{k}

shrinks to

p

The area of $P_{k}$ is $π / k^{2}$ . By the given flux formula, $\frac{Φ (C_{P_{k}}^{hit}, t)}{area (P_{k})} = \frac{5 W \cdot m^{- 2} area (P_{k})}{area (P_{k})} = 5 W \cdot m^{- 2} .$ Hence $irradiance (p, t) = \lim_{k \to \infty} \frac{Φ (C_{P_{k}}^{hit}, t)}{area (P_{k})} = 5 W \cdot m^{- 2} .$

Radiant Exitance

Let

(P_{k})

be a sequence of surface patches that shrinks to

p

. Let

C_{P_{k}}^{leave}

be the time-windowed particle collection that selects particles that locally leave

P_{k}

. The radiant exitance at

p

and time

t

is the area density of outgoing flux:

M (p, t) = \lim_{k \to \infty} \frac{Φ (C_{P_{k}}^{leave}, t)}{area (P_{k})},

when this limit exists and has the same value for the shrinking patches under consideration. Radiant exitance is measured in

W \cdot m^{- 2}

, using the watt and the metre.

For example, if a glowing panel emits $12 W$ uniformly from a surface area of $3 m^{2}$ , then its radiant exitance is $12 W / 3 m^{2} = 4 W \cdot m^{- 2}$ .

Radiant Intensity

Suppose

ω \in 𝕊^{2}

, and let

(A_{k})

be a sequence of subsets of

𝕊^{2}

that shrinks to

ω

. Let

C_{A_{k}}

be the time-windowed particle collection that selects particles travelling in the direction set

A_{k}

. The radiant intensity at

ω

and time

t

is the solid-angle density of emitted flux:

I (ω, t) = \lim_{k \to \infty} \frac{Φ (C_{A_{k}}, t)}{σ (A_{k})},

when this limit exists and has the same value for the shrinking direction sets under consideration. Radiant intensity is measured in

W \cdot s r^{- 1}

, using the watt and steradian.

In this definition, $C_{A_{k}}$ selects the travelling particles whose directions lie in $A_{k}$ . Equivalently, if $A_{k}$ is pictured as an imaginary surface patch on the unit sphere centered at the light source, those are the same particles that cross that empty-space patch. The patch is not a material surface; it is a way to visualize a bundle of directions.

Usually, radiant intensity is pictured with a small light source at the center of a sphere. Each point of the unit sphere represents one outgoing direction, and radiant intensity asks how much emitted power is concentrated near that direction. In the special case of the unit sphere, the numerical value of radiant intensity in a direction $ω$ matches the numerical value of the irradiance measured at the corresponding point $ω \in 𝕊^{2}$ , using surface patches on $𝕊^{2}$ that shrink to $ω$ . The limits have the same denominator because, on the unit sphere, solid angle is surface area. The quantities are still different: radiant intensity has units $W \cdot s r^{- 1}$ , while irradiance has units $W \cdot m^{- 2}$ . If the receiving sphere has radius $2$ , then the same bundle of directions lands on four times as much physical area, so the irradiance at the corresponding point $2 ω$ is one fourth as large. Radiant intensity does not change in this comparison because the bundle of directions has the same solid angle.

Radiant intensity uses the same density idea as irradiance, but with direction sets instead of physical surface patches. If we accept twice as large a cone of outgoing directions, we expect about twice as much emitted radiant flux to lie in that cone. Dividing by the cone's solid angle, then shrinking the cone around one direction, gives a directional density of emitted power.

For example, if a small source emits $10 W$ uniformly into a cone whose solid angle is $2 s r$ , then its average radiant intensity over that cone is $10 W / 2 s r = 5 W \cdot s r^{- 1}$ .

Radiant Flux from Radiant Intensity

Let

A \subseteq 𝕊^{2}

be a measurable set of outgoing directions, fix a time

t

, and let

C_{A}

be the time-windowed particle collection that selects particles travelling in

A

. Suppose the radiant intensity

I (ω, t)

is defined for

ω \in A

, is measurable and integrable with respect to solid angle measure on

A

, and represents the solid-angle density of emitted radiant flux. Then the radiant flux emitted into

A

Φ (C_{A}, t) = \int_{A} I (ω, t) d σ (ω),

where

d σ

means integration with respect to solid angle on

𝕊^{2}

Partition

A

into small direction sets

A_{1}, \dots, A_{N}

, and choose a representative direction

ω_{j} \in A_{j}

. Since

I (ω, t)

is the solid-angle density of emitted flux, the flux emitted into

A_{j}

is approximated by

I (ω_{j}, t) σ (A_{j}) .

Adding the approximations gives

Φ (C_{A}, t) \approx \sum_{j = 1}^{N} I (ω_{j}, t) σ (A_{j}) .

In the limit as the direction partition is refined, these sums converge to the integral of

I

over the subset

A

with respect to solid angle. Therefore

Φ (C_{A}, t) = \int_{A} I (ω, t) d σ (ω) .

Radiant Intensity from Shrinking Direction Sets

Fix a direction

ω \in 𝕊^{2}

and a time

t_{0}

. Let

(A_{k})

be direction sets that shrink to

ω

, and let

C_{A_{k}}

select particles travelling in

A_{k}

. Suppose

σ (A_{k}) = \frac{1}{k} .

Suppose the emitted radiant flux into

A_{k}

at time

t_{0}

Φ (C_{A_{k}}, t_{0}) = \frac{6}{k} + \frac{2}{k^{2}}

watts. Compute the radiant intensity at

ω

and

t_{0}

By the definition of radiant intensity,

I (ω, t_{0}) = \lim_{k \to \infty} \frac{Φ (C_{A_{k}}, t_{0})}{σ (A_{k})} .

Substituting the given values,

\frac{Φ (C_{A_{k}}, t_{0})}{σ (A_{k})} = \frac{\frac{6}{k} + \frac{2}{k^{2}}}{\frac{1}{k}} = 6 + \frac{2}{k} .

Therefore

I (ω, t_{0}) = \lim_{k \to \infty} (6 + \frac{2}{k}) = 6.

Thus the radiant intensity at

ω

and

t_{0}

6 W \cdot s r^{- 1}

Radiance

Radiance measures radiant flux per unit projected area and per unit solid angle. Let

(P_{k})

be a sequence of surface patches that shrinks to

p

, and let

(A_{k})

be a sequence of direction sets that shrinks to

ω

. Let

C_{P_{k}}

be the time-windowed particle collection that selects particles whose positions lie in

P_{k}

, and let

C_{A_{k}}

be the time-windowed particle collection that selects particles travelling in

A_{k}

. The radiance at

p

, in direction

ω

, at time

t

, is

radiance (p, ω, t) = \lim_{k \to \infty} \frac{Φ (C_{P_{k}} \cap C_{A_{k}}, t)}{{area}_{ω}^{⟂} (P_{k}) σ (A_{k})},

when this limit exists and has the same value for the shrinking patches and direction sets under consideration. Here

{area}_{ω}^{⟂} (P_{k})

is the projected area of

P_{k}

in direction

ω

. Equivalently, when

{area}_{ω}^{⟂} (P_{k}) = | n \cdot ω | area (P_{k})

radiance (p, ω, t) = \lim_{k \to \infty} \frac{Φ (C_{P_{k}} \cap C_{A_{k}}, t)}{| n \cdot ω | area (P_{k}) σ (A_{k})} .

Radiance is measured in

W \cdot m^{- 2} \cdot s r^{- 1}

, using the watt, metre, and steradian.

Radiance combines the two localizations above. It measures how much radiant flux passes through an imaginary tiny window around $p$ , restricted to a tiny cone of directions around $ω$ , after dividing out both the window's projected area and the cone's solid angle. The window is just a measuring device around the point $p$ ; it does not have to be a physical surface.

For example, if $6 W$ of flux is associated with $2 m^{2}$ of projected area and $3 s r$ of directions, then the average radiance over that area-direction set is $6 W / (2 m^{2} \cdot 3 s r) = 1 W \cdot m^{- 2} \cdot s r^{- 1}$ .

Radiance Contribution to Irradiance Density

Let

p

be a point on a surface patch with unit surface normal

n

. For a direction set

A \subseteq 𝕊^{2}

, let

{irradiance}_{p} (A)

be the irradiance contribution at

p

from incoming directions in

A

. If

(A_{k})

is a sequence of direction sets that shrinks to

ω_{i}

, and

radiance_in

is locally stable near

ω_{i}

, then the solid-angle density of

{irradiance}_{p}

ω_{i}

\lim_{k \to \infty} \frac{{irradiance}_{p} (A_{k})}{σ (A_{k})} = radiance_in (p, ω_{i}) | n \cdot ω_{i} | .

By the definition of radiance, flux from directions in a small direction set around

ω_{i}

is approximated by

radiance_in (p, ω_{i}) {area}_{ω_{i}}^{⟂} (P) σ (A),

where

P

is a small surface patch around

p

A

is the small direction set, and

σ (A)

is its solid angle. By the projected area formula,

{area}_{ω_{i}}^{⟂} (P) = | n \cdot ω_{i} | area (P) .

Thus the incoming flux is approximated by

radiance_in (p, ω_{i}) | n \cdot ω_{i} | area (P) σ (A) .

Dividing by

area (P)

gives the contribution per unit surface area,

radiance_in (p, ω_{i}) | n \cdot ω_{i} | σ (A) .

Thus, for small direction sets

A

near

ω_{i}

{irradiance}_{p} (A) \approx radiance_in (p, ω_{i}) | n \cdot ω_{i} | σ (A) .

Therefore, along any shrinking sequence

A_{k} \to ω_{i}

for which the density limit exists,

\lim_{k \to \infty} \frac{{irradiance}_{p} (A_{k})}{σ (A_{k})} = radiance_in (p, ω_{i}) | n \cdot ω_{i} | .

Equivalently, if

B

is a small direction set near

ω_{i}

, then

{irradiance}_{p} (B) = radiance_in (p, ω_{i}) | n \cdot ω_{i} | σ (B) + r (B),

where the error

r (B)

is smaller than linear in solid angle:

\frac{r (B)}{σ (B)} \to 0

B

shrinks to

ω_{i}

This is the same logic as a one-variable derivative. When a derivative exists, the function has a best linear approximation and the error divided by the input size tends to $0$ . Here the input size is the solid angle $σ (B)$ , and the linear term is $radiance_in (p, ω_{i}) | n \cdot ω_{i} | σ (B)$ . This is why, later, a small direction set $B_{j}$ with representative direction $η_{j}$ contributes approximately $radiance_in (p, η_{j}) | n \cdot η_{j} | σ (B_{j})$ to irradiance.

Radiance Functions

Incident Radiance

The incident radiance

radiance_in (p, ω)

is the radiance arriving at a surface point

p

from direction

ω

For example, sunlight arriving at a point on a wall from the direction of the sun contributes to $radiance_in (p, ω)$ for that incoming direction.

Rendering Scene

A rendering scene is the collection of surfaces, light sources, and material scattering functions used to determine the radiance transported through the scene.

No-Hit Sentinel

The no-hit sentinel, denoted

⊥

, is a special value used to record that a ray did not intersect any surface in a scene. It is not a point of

ℝ^{3}

Raycast Function

Given a rendering scene

S

, the raycast function

{raycast}_{S} : ℝ^{3} \times 𝕊^{2} \to ℝ^{3} \cup {⊥}

sends a point

p \in ℝ^{3}

and a direction

ω \in 𝕊^{2}

to the first surface point hit by the ray starting at

p

and traveling in direction

ω

. If the ray hits no surface in

S

, then

{raycast}_{S} (p, ω) = ⊥

Incident Radiance from Raycast

Let

S

be a rendering scene, let

p \in ℝ^{3}

, let

ω \in 𝕊^{2}

, and suppose

p^{'} = {raycast}_{S} (p, ω) \neq ⊥ .

If radiance is transported without absorption or emission along the open segment from

p^{'}

p

, then

radiance_in (p, ω) = radiance_out (p^{'}, - ω) .

The value

{raycast}_{S} (p, ω) = p^{'}

says that the ray starting at

p

in direction

ω

first meets the scene at

p^{'}

. Thus light arriving at

p

from direction

ω

is the same bundle of light that left

p^{'}

in direction

- ω

. Since there is no absorption or emission along the segment between the two points, radiance is unchanged during this straight-line transport. Therefore

radiance_in (p, ω) = radiance_out (p^{'}, - ω) .

Another way to read the same statement is with path notation. If $p$ and $p^{'}$ are consecutive points on a light path, then the direction $p \to p^{'}$ is the same direction as $ω$ , and the opposite direction $p^{'} \to p$ is $- ω$ . So the same transport identity can be read as $radiance_in (p, p \to p^{'}) = radiance_out (p^{'}, p^{'} \to p) .$

Exitant Radiance

The exitant radiance

radiance_out (p, ω)

is the radiance leaving a surface point

p

in direction

ω

For example, light reflected from a painted wall toward a camera contributes to $radiance_out (p, ω)$ for the direction from the wall to the camera.

Surface Interaction

Hemisphere of Directions

Let

p

be a point on a surface

S

with surface normal

n (p)

. The positive hemisphere of directions at

p

H_{p}^{+} : = {ω \in 𝕊^{2} : n (p) \cdot ω > 0} .

The opposite hemisphere is

H_{p}^{-} : = {ω \in 𝕊^{2} : n (p) \cdot ω < 0} .

BRDF

The bidirectional reflectance distribution function, abbreviated BRDF, describes how much incoming radiance from one direction is reflected into another direction at a surface point. For a surface

S

, write

brdf : {(p, ω_{i}, ω_{o}) : p \in S, ω_{i}, ω_{o} \in H_{p}^{+} or ω_{i}, ω_{o} \in H_{p}^{-}} \to [0, \infty) .

The value

brdf (p, ω_{i}, ω_{o})

describes reflected scattering at

p

when the incoming unit vector

ω_{i} \in 𝕊^{2}

and outgoing unit vector

ω_{o} \in 𝕊^{2}

lie in the same hemisphere. Informally,

brdf

is outgoing radiance per incoming irradiance, so its unit is

s r^{- 1}

For example, an ideal matte white surface has nearly the same value of $brdf (p, ω_{i}, ω_{o})$ for every outgoing direction $ω_{o}$ , while a shiny surface has a value of $brdf (p, ω_{i}, ω_{o})$ that is much larger near the mirror-reflection direction.

Lambertian BRDF

A Lambertian BRDF with reflectance

ρ \in [0,1]

is a BRDF whose value is constant over incoming and outgoing directions on the same side of the surface:

brdf (p, ω_{i}, ω_{o}) = \frac{ρ}{π} .

The number

ρ

is the fraction of incident surface power density reflected by the surface.

BTDF

The bidirectional transmittance distribution function, abbreviated BTDF, describes how much incoming radiance from one side of a surface is transmitted into an outgoing direction on the other side. For a surface

S

, write

btdf : {(p, ω_{i}, ω_{o}) : p \in S, ω_{i} \in H_{p}^{+}, ω_{o} \in H_{p}^{-} or ω_{i} \in H_{p}^{-}, ω_{o} \in H_{p}^{+}} \to [0, \infty) .

The value

btdf (p, ω_{i}, ω_{o})

describes transmitted scattering at

p

when the incoming unit direction

ω_{i}

and outgoing unit direction

ω_{o}

lie in opposite hemispheres.

The difference from a BRDF is which side of the surface contains the outgoing light. A BRDF describes reflection: the incoming and outgoing directions lie on the same side of the surface. A BTDF describes transmission: the incoming light passes through the surface and leaves on the opposite side.

For example, a clear glass pane has a BTDF that is concentrated around directions determined by refraction.

BSDF

The bidirectional scattering distribution function, abbreviated BSDF, combines the BRDF and BTDF. For a surface

S

, write

bsdf : S \times 𝕊^{2} \times 𝕊^{2} \to [0, \infty) .

Away from tangent directions, it is the piecewise function

bsdf (p, ω_{i}, ω_{o}) = {\begin{matrix} brdf (p, ω_{i}, ω_{o}), & ω_{i}, ω_{o} \in H_{p}^{+} or ω_{i}, ω_{o} \in H_{p}^{-}, \\ btdf (p, ω_{i}, ω_{o}), & ω_{i} \in H_{p}^{+}, ω_{o} \in H_{p}^{-} or ω_{i} \in H_{p}^{-}, ω_{o} \in H_{p}^{+} . \end{matrix}

Thus

bsdf (p, ω_{i}, ω_{o})

describes surface scattering at

p \in S

from the incoming unit direction

ω_{i}

to the outgoing unit direction

ω_{o}

, whether the light is reflected or transmitted.

The codomain $[0, \infty)$ does not mean that a BSDF value is a fraction of light, and it does not mean that $bsdf \leq 1$ . A BSDF value is a conversion density from incoming irradiance, measured in $W \cdot m^{- 2}$ , to outgoing radiance, measured in $W \cdot m^{- 2} \cdot s r^{- 1}$ . Therefore the raw output of $bsdf$ is measured in $\frac{outgoing radiance}{incoming irradiance} = \frac{W \cdot m^{- 2} \cdot s r^{- 1}}{W \cdot m^{- 2}} = s r^{- 1} .$ Thus, by the linear approximation coming from irradiance density, if a small direction set $B_{j}$ has representative incoming direction $η_{j}$ and solid angle $σ_{j}$ , then $radiance_in (p, η_{j}) | n \cdot η_{j} | σ_{j}$ is an approximate incoming irradiance contribution, not a radiant flux. Multiplying by $bsdf (p, η_{j}, ω_{o})$ gives the corresponding outgoing radiance contribution in direction $ω_{o}$ . A BSDF can be larger than $1$ at a particular pair of directions when scattering is concentrated into a small set of outgoing directions; the energy-conserving condition is an integral condition over outgoing directions, not a pointwise bound.

For example, a pane of slightly tinted glass can have a BSDF with a reflected part for mirror-like glare on the front surface and a transmitted part for light that passes through the glass with some absorption. Geometrically, after fixing the surface normal, the BRDF describes scattering into the same hemisphere of directions as reflection, the BTDF describes scattering into the opposite hemisphere, and the BSDF describes both hemispheres together.

Reflected Radiance

Reflected radiance is the part of the exitant radiance at a surface point that comes from incident light scattering at the surface rather than from the surface emitting light by itself.

Emitted Radiance

The emitted radiance

emitted_radiance (p, ω)

is the radiance produced by a surface point

p

itself and leaving in direction

ω

For example, a glowing screen has nonzero emitted radiance, while a wall lit only by another lamp has zero emitted radiance and only reflected radiance.

Reflection Equation

The reflection equation is the local bookkeeping rule for outgoing radiance caused by emission and reflection from incoming directions on the same side of the surface:

radiance_out (p, ω_{o}) = emitted_radiance (p, ω_{o}) + \int_{Ω^{+}} brdf (p, ω_{i}, ω_{o}) radiance_in (p, ω_{i}) | n \cdot ω_{i} | d ω_{i} .

Here

radiance_out

is exitant radiance,

radiance_in

is incident radiance,

emitted_radiance

is emitted radiance,

brdf

is a BRDF,

n

is the surface normal, and

Ω^{+}

is the hemisphere of directions above the surface.

The factor $| n \cdot ω_{i} |$ converts incoming radiance from a direction into its contribution per unit surface area through projected area.

Energy-Conserving BSDF

A BSDF is energy-conserving if, for every surface point

p

and incoming direction

ω_{i}

\int_{𝕊^{2}} bsdf (p, ω_{i}, ω_{o}) | n \cdot ω_{o} | d ω_{o} \leq 1,

where

n

is the surface normal at

p

. The integral is the total fraction of incoming irradiance from

ω_{i}

that is scattered into all outgoing directions. The inequality says that the surface may absorb energy, but it does not scatter out more energy than arrived from that incoming direction.

Scattered Radiance

Suppose the local BSDF scattering approximation is being used. Let

p

be a surface point with surface normal

n

, let

ω_{o} \in 𝕊^{2}

be an outgoing direction, and let

A \subseteq 𝕊^{2}

be a measurable incoming direction set. Suppose the function

ω_{i} \mapsto bsdf (p, ω_{i}, ω_{o}) radiance_in (p, ω_{i}) | n \cdot ω_{i} |

is measurable and integrable on

A

with respect to solid angle measure. The scattered radiance from

A

into

ω_{o}

scattered_radiance (p, ω_{o}, A) : = \int_{A} bsdf (p, ω_{i}, ω_{o}) radiance_in (p, ω_{i}) | n \cdot ω_{i} | d σ (ω_{i}) .

This is the part of the exitant radiance leaving

p

in direction

ω_{o}

that comes from incident light arriving from directions in

A

and scattering at

p

The integral in the definition is a Lebesgue integral with respect to the solid angle measure $σ$ . If $g (ω_{i}) = bsdf (p, ω_{i}, ω_{o}) radiance_in (p, ω_{i}) | n \cdot ω_{i} |,$ then formally $\int_{A} g d σ$ is obtained from the supremum of lower Lebesgue sums. For a finite measurable partition $𝒫 = (B_{1}, \dots, B_{m})$ of $A$ , the lower sum with respect to the solid angle measure is $L (g, 𝒫, σ) = \sum_{j = 1}^{m} σ (B_{j}) \inf_{B_{j}} g .$ When the pieces $B_{j}$ are small and $g$ changes little on each piece, this lower sum is close to the representative-direction sum $\sum_{j = 1}^{m} σ (B_{j}) g (η_{j}),$ where $η_{j} \in B_{j}$ . Expanding the definition of $g$ , this is $\sum_{j = 1}^{m} σ (B_{j}) bsdf (p, η_{j}, ω_{o}) radiance_in (p, η_{j}) | n \cdot η_{j} | .$ The factor $radiance_in (p, η_{j}) | n \cdot η_{j} | σ (B_{j})$ is the approximate irradiance contribution from $B_{j}$ , by the linear approximation coming from irradiance density. Multiplying by $bsdf (p, η_{j}, ω_{o})$ converts that incoming irradiance contribution into an outgoing scattered radiance contribution in direction $ω_{o}$ . The integral is the rigorous version of taking all of those direction pieces at once.

Local Linear BSDF Scattering Model

The local linear BSDF scattering model is a rendering approximation that assumes the exitant radiance at each surface point

p

and outgoing direction

ω_{o}

is modeled using two local contributions:

emitted radiance $emitted_radiance (p, ω_{o})$ , produced at $p$ itself, and
scattered radiance from incident light arriving at $p$ .

It also assumes the scattered contribution is linear and additive over incoming direction sets. With these assumptions,

radiance_out (p, ω_{o}) = emitted_radiance (p, ω_{o}) + scattered_radiance (p, ω_{o}, 𝕊^{2}) .

BSDF Scattering Equation

Suppose the local linear BSDF scattering model is used for a surface, and suppose scattered radiance is defined for

A = 𝕊^{2}

. Then the exitant radiance in direction

ω_{o}

radiance_out (p, ω_{o}) = emitted_radiance (p, ω_{o}) + \int_{𝕊^{2}} bsdf (p, ω_{i}, ω_{o}) radiance_in (p, ω_{i}) | n \cdot ω_{i} | d σ (ω_{i}) .

If one is only describing the scattered outgoing radiance, or if

emitted_radiance (p, ω_{o}) = 0

, this reduces to the fundamental scattering equation

radiance_out (p, ω_{o}) = \int_{𝕊^{2}} bsdf (p, ω_{i}, ω_{o}) radiance_in (p, ω_{i}) | n \cdot ω_{i} | d σ (ω_{i}) .

By the assumptions in the local linear BSDF scattering model, radiance leaving

p

in direction

ω_{o}

is modeled as an emitted part plus a scattered part:

radiance_out (p, ω_{o}) = emitted_radiance (p, ω_{o}) + scattered_radiance (p, ω_{o}, 𝕊^{2}) .

Applying the definition of scattered radiance with

A = 𝕊^{2}

gives

scattered_radiance (p, ω_{o}, 𝕊^{2}) = \int_{𝕊^{2}} bsdf (p, ω_{i}, ω_{o}) radiance_in (p, ω_{i}) | n \cdot ω_{i} | d σ (ω_{i}) .

Substituting this expression for the scattered part gives the BSDF scattering equation.

Raycast Form of the BSDF Scattering Equation

Let

S

be a rendering scene, let

p

be a surface point with surface normal

n

, and let

ω_{o} \in 𝕊^{2}

be an outgoing direction. Suppose the local linear BSDF scattering model is used at

p

, and suppose that for almost every incoming direction

ω_{i} \in 𝕊^{2}

p_{i} = {raycast}_{S} (p, ω_{i}) \neq ⊥,

with straight-line radiance transport from

p_{i}

p

. Then

radiance_out (p, ω_{o}) = emitted_radiance (p, ω_{o}) + \int_{𝕊^{2}} bsdf (p, ω_{i}, ω_{o}) radiance_out ({raycast}_{S} (p, ω_{i}), - ω_{i}) | n \cdot ω_{i} | d σ (ω_{i}) .

The BSDF scattering equation gives

radiance_out (p, ω_{o}) = emitted_radiance (p, ω_{o}) + \int_{𝕊^{2}} bsdf (p, ω_{i}, ω_{o}) radiance_in (p, ω_{i}) | n \cdot ω_{i} | d σ (ω_{i}) .

For almost every

ω_{i}

, write

p_{i} = {raycast}_{S} (p, ω_{i})

. By incident radiance from raycast,

radiance_in (p, ω_{i}) = radiance_out (p_{i}, - ω_{i}) .

Substituting this expression into the integral gives the raycast form.

In path language, this says that the outgoing radiance at $p$ in direction $ω_{o}$ is emitted radiance at $p$ , plus the scattered contribution from all possible previous path vertices. For each incoming direction $ω_{i}$ , the raycast finds the previous surface point $p_{i}$ , and the incoming radiance at $p$ is the outgoing radiance from $p_{i}$ back toward $p$ .

Point-to-Point Direction

p, q \in ℝ^{3}

and

p \neq q

, then

ω_{p \to q} : = \frac{q - p}{‖ q - p ‖}

is the unit direction from

p

toward

q

Visibility Between Surface Points

Let

S

be a rendering scene, and let

p, q

be surface points in

S

. The visibility factor

V_{S} (p, q)

1

when the open line segment from

p

q

intersects no blocking surface in

S

, and is

0

otherwise.

Point-to-Point Exitant Radiance

Let

S

be a rendering scene, and let

p, q

be distinct surface points. The exitant radiance from $p$ to $q$ is

radiance_out (p \to q) : = {\begin{matrix} radiance_out (p, ω_{p \to q}), & V_{S} (p, q) = 1, \\ 0, & V_{S} (p, q) = 0. \end{matrix}

Three-Point BSDF

Let

p_{i}, p, p_{o}

be distinct surface points. The three-point BSDF at

p

, from

p_{i}

through

p

toward

p_{o}

, is

bsdf (p_{i} \to p \to p_{o}) : = bsdf (p, ω_{p \to p_{i}}, ω_{p \to p_{o}}) .

The incoming direction at

p

is the direction from

p

toward the previous point

p_{i}

, and the outgoing direction at

p

is the direction from

p

toward the next point

p_{o}

Geometric Coupling

Let

S

be a rendering scene, and let

p, q

be distinct surface points with surface normals

n (p)

and

n (q)

. The geometric coupling between

p

and

q

{geometric_coupling}_{S} (p, q) : = V_{S} (p, q) \frac{| n (p) \cdot ω_{p \to q} | | n (q) \cdot ω_{q \to p} |}{{‖ p - q ‖}^{2}} .

The factor ${geometric_coupling}_{S} (p, q)$ packages three things that otherwise appear separately: whether $p$ and $q$ can see each other, the projected-area cosine at $p$ , and the solid-angle-to-surface-area conversion at $q$ . This is useful when evaluating the transport equation by sampling surface points instead of sampling directions.

Radial Surface Patch from a Point

Let

p \in ℝ^{3}

, and let

Q

be a surface patch not containing

p

. We say that

Q

is radial from $p$ if every half-line starting at

p

intersects

Q

in at most one point. Equivalently, for every

ω \in 𝕊^{2}

, there is at most one

r \in ℝ^{> 0}

such that

p + r ω \in Q .

Intuitively, a radial surface patch from $p$ is like a patch of the unit sphere around $p$ , except the radius is allowed to vary smoothly with direction. One way to picture it is $Q = {p + ρ (ω) ω : ω \in A},$ where $A \subseteq 𝕊^{2}$ is a direction set and $ρ : A \to ℝ^{> 0}$ gives one distance from $p$ for each direction. A genuine subset of the sphere centered at $p$ is the special case $ρ (ω) = 1$ . A filled disk containing whole radial line segments from $p$ is not radial from $p$ , because one direction would correspond to many points of the patch.

Direction Map is Injective on a Radial Patch

Let

p \in ℝ^{3}

, and let

Q

be a radial surface patch from

p

. The direction map

g : Q \to 𝕊^{2}, g (q) = ω_{p \to q}

is injective.

Suppose

q_{1}, q_{2} \in Q

and

g (q_{1}) = g (q_{2}) = ω

. By the definition of point-to-point direction,

q_{1} = p + ‖ q_{1} - p ‖ ω, q_{2} = p + ‖ q_{2} - p ‖ ω .

Since

p \notin Q

, both

‖ q_{1} - p ‖

and

‖ q_{2} - p ‖

are positive. Thus both

q_{1}

and

q_{2}

lie on the same half-line

{p + r ω : r \in ℝ^{> 0}} .

Because

Q

is radial from

p

, that half-line intersects

Q

in at most one point. Therefore

q_{1} = q_{2}

, so

g

is injective.

Surface Patches are Locally Coordinate Images

Let

Q

be a smooth surface patch on a surface

M \subseteq ℝ^{3}

. For each ordinary point

q_{0} \in Q

, there is a smaller patch

Q_{0} \subseteq Q

containing

q_{0}

and a coordinate system

x : U \to ℝ^{3}, U \subseteq ℝ^{2},

such that

x (U) = Q_{0}

By the definition of surface, the smooth pieces of

M

are two-dimensional manifolds in Euclidean space. By the definition of coordinate system on a manifold, every point of such a smooth piece has a neighborhood of the form

x (U) = M \cap O

, where

O \subseteq ℝ^{3}

is open and

U \subseteq ℝ^{2}

is open. Restricting

O

if necessary so that

M \cap O \subseteq Q

, set

Q_{0} = M \cap O

. Then

x (U) = Q_{0}

Derivative of the Normalization Map

Let

N : ℝ^{3} ∖ {0} \to 𝕊^{2}, N (z) = \frac{z}{‖ z ‖} .

z \in ℝ^{3} ∖ {0}

ω = N (z)

, and

v \in ℝ^{3}

, then

D N (z) (v) = \frac{1}{‖ z ‖} (v - (v \cdot ω) ω) = \frac{1}{‖ z ‖} {proj}_{ω^{⟂}} (v) .

Here

D N (z)

is the derivative of

N

z

. It is a linear transformation

ℝ^{3} \to ℝ^{3}

, and

D N (z) (v)

means that this linear transformation is applied to the vector

v

Define the length function

L (z) = ‖ z ‖ .

First compute

D L (z) (v)

. Define the squared-length function

S : ℝ^{3} \to ℝ, S (z) = L {(z)}^{2} = z \cdot z .

By the derivative of squared norm exercise, for

v \in ℝ^{3}

D S (z) (v) = 2 z \cdot v .

Now define the one-variable square function

m : ℝ \to ℝ, m (t) = t^{2} .

Since

S (z) = L {(z)}^{2}

, this means

S = m \circ L .

Therefore, by the chain rule,

D S (z) (v) = D m (L (z)) (D L (z) (v)) .

The one-variable derivative of

m (t) = t^{2}

L (z)

is the linear map

s \mapsto 2 L (z) s

. Hence

D S (z) (v) = 2 L (z) D L (z) (v) .

Combining this with the earlier value

D S (z) (v) = 2 z \cdot v

gives

2 z \cdot v = 2 L (z) D L (z) (v) .

Because

z \neq 0

L (z) = ‖ z ‖ > 0

, so

2 L (z) \neq 0

. Dividing both sides by

2 L (z)

gives

D L (z) (v) = \frac{z \cdot v}{‖ z ‖} .

Now define

Q : ℝ^{3} \times (ℝ ∖ {0}) \to ℝ^{3}, Q (y, t) = \frac{y}{t},

and

P : ℝ^{3} ∖ {0} \to ℝ^{3} \times (ℝ ∖ {0}), P (z) = (z, L (z)) .

Then

N = Q \circ P

. By the derivative of the identity function,

D P (z) (v) = (v, D L (z) (v)) .

Therefore the chain rule and the derivative of the vector scalar division map give

D N (z) (v) = D Q (P (z)) (D P (z) (v)) = D Q (z, L (z)) (v, D L (z) (v)) = \frac{1}{L (z)} v - \frac{D L (z) (v)}{L {(z)}^{2}} z .

Substitute the formula for

D L (z) (v)

D N (z) (v) = \frac{1}{‖ z ‖} v - \frac{z \cdot v}{{‖ z ‖}^{3}} z .

Since

ω = z / ‖ z ‖

, we have

z = ‖ z ‖ ω

and

z \cdot v = ‖ z ‖ (ω \cdot v) .

Thus

\frac{z \cdot v}{{‖ z ‖}^{3}} z = \frac{‖ z ‖ (ω \cdot v)}{{‖ z ‖}^{3}} ‖ z ‖ ω = \frac{(ω \cdot v) ω}{‖ z ‖} .

Therefore

D N (z) (v) = \frac{1}{‖ z ‖} (v - (ω \cdot v) ω) .

Since

ω \cdot v = v \cdot ω

, and by the definition of orthogonal projection onto

ω^{⟂}

{proj}_{ω^{⟂}} (v) = v - (v \cdot ω) ω,

the desired formula follows.

Surface Area to Solid Angle Change of Variables

Let

p \in ℝ^{3}

, let

Q

be a smooth radial surface patch from

p

, let

A_{Q}

be the surface area measure on

Q

, let

n : Q \to 𝕊^{2}

assign a unit surface normal

n (q)

to each point

q \in Q

, and let

g : Q \to 𝕊^{2}, g (q) = ω_{p \to q}

be the direction map. If

F : g (Q) \to ℝ

is integrable, then

\int_{g (Q)} F (ω) d σ (ω) = \int_{Q} F (ω_{p \to q}) \frac{| n (q) \cdot ω_{q \to p} |}{{‖ p - q ‖}^{2}} d A_{Q} (q) .

The notation

d A_{Q} (q)

means integration with respect to the surface area measure

A_{Q}

; it is not evaluation of an

A

-function at

q

By direction map is injective on a radial patch,

g

is one-to-one on

Q

. By surface patches are locally coordinate images, it is enough to do the calculation on a smaller patch described by one coordinate system and then add the pieces.

So let $x : U \to Q$ be a coordinate system, with $U \subseteq ℝ^{2}$ . For each $u \in U$ , define

q (u) = x (u), r (u) = ‖ q (u) - p ‖, ω (u) = ω_{p \to q (u)} .

Thus the coordinate expression of the direction map is

g (x (u)) = \frac{x (u) - p}{‖ x (u) - p ‖} = ω (u) .

Fix

u \in U

. The direction map is the composition of the translation

q \mapsto q - p

with the normalization map

N (z) = z / ‖ z ‖

. For a tangent vector

v \in T_{q (u)} Q

, apply the derivative of the normalization map with

z = q (u) - p, ‖ z ‖ = r (u), N (z) = ω (u) .

Since the derivative of

q \mapsto q - p

sends

v

v

, this gives

D g (q (u)) (v) = \frac{1}{r (u)} (v - (v \cdot ω (u)) ω (u)) = \frac{1}{r (u)} {proj}_{ω {(u)}^{⟂}} (v) .

Let

x_{1} = \frac{\partial x}{\partial u_{1}} (u), x_{2} = \frac{\partial x}{\partial u_{2}} (u) .

Then the two coordinate tangent vectors on

𝕊^{2}

induced by

g \circ x

are

D g (q (u)) (x_{1}) = \frac{1}{r (u)} {proj}_{ω {(u)}^{⟂}} (x_{1}), D g (q (u)) (x_{2}) = \frac{1}{r (u)} {proj}_{ω {(u)}^{⟂}} (x_{2}) .

By the surface area density cross product formula, the surface-area density of

g (Q)

in the coordinates

g \circ x

‖ D g (q (u)) (x_{1}) \times D g (q (u)) (x_{2}) ‖ = \frac{1}{r {(u)}^{2}} ‖ {proj}_{ω {(u)}^{⟂}} (x_{1}) \times {proj}_{ω {(u)}^{⟂}} (x_{2}) ‖ .

The projected vectors lie in

ω {(u)}^{⟂}

, so their cross product is parallel to

ω (u)

. Therefore, using norm equals absolute dot product with a parallel unit vector and the oriented projected area identity,

‖ {proj}_{ω {(u)}^{⟂}} (x_{1}) \times {proj}_{ω {(u)}^{⟂}} (x_{2}) ‖ = | (x_{1} \times x_{2}) \cdot ω (u) | .

Since

n

is the chosen unit normal field on

Q

, the vector

x_{1} \times x_{2}

is parallel to

n (q (u))

. Hence

| (x_{1} \times x_{2}) \cdot ω (u) | = | n (q (u)) \cdot ω (u) | ‖ x_{1} \times x_{2} ‖ .

Because

ω_{q (u) \to p} = - ω_{p \to q (u)} = - ω (u)

, this becomes

‖ D g (q (u)) (x_{1}) \times D g (q (u)) (x_{2}) ‖ = \frac{| n (q (u)) \cdot ω_{q (u) \to p} |}{{‖ p - q (u) ‖}^{2}} ‖ x_{1} \times x_{2} ‖ .

Now express the integral with coordinates on the sphere. Let

y : V \to 𝕊^{2}

be a coordinate system on the sphere with

y (V) = g (Q)

, and define

h = y^{- 1} \circ g \circ x : U \to V,

so that

y (h (u)) = g (x (u))

. The map

h

is a coordinate expression of the direction map. The absolute Jacobian determinant

| \det {[D h (u)]}_{E} |

, together with the sphere area density from

y

, is exactly

\frac{| n (x (u)) \cdot ω_{x (u) \to p} |}{{‖ p - x (u) ‖}^{2}} ‖ x_{1} \times x_{2} ‖ .

Since

σ

is the solid angle measure, which is surface area measure on

𝕊^{2}

, the sphere-coordinate expression of the left side is

\int_{g (Q)} F (ω) d σ (ω) = \int_{V} F (y (v)) ρ_{y} (v) d λ (v),

where

ρ_{y}

is the surface area density of the sphere in the coordinates

y

. Applying change of variables for Lebesgue integrals to

h : U \to V

gives

\int_{g (Q)} F (ω) d σ (ω) = \int_{U} F (g (x (u))) \frac{| n (x (u)) \cdot ω_{x (u) \to p} |}{{‖ p - x (u) ‖}^{2}} ‖ x_{1} \times x_{2} ‖ d u .

By the definition of surface area measure and the surface area density cross product formula, the measure

A_{Q}

has local density

d A_{Q} (q (u)) = ‖ x_{1} \times x_{2} ‖ d u .

Therefore the local integral is

\int_{Q} F (ω_{p \to q}) \frac{| n (q) \cdot ω_{q \to p} |}{{‖ p - q ‖}^{2}} d A_{Q} (q) .

For a patch covered by several coordinate systems, apply the same argument on each piece and add the resulting integrals.

Surface Form of the BSDF Scattering Equation

Let

S

be a rendering scene, let

Σ_{S}

be the union of the scene's surfaces, and let

p, p_{o} \in Σ_{S}

be distinct surface points. Suppose the raycast form of the BSDF scattering equation applies at

p

. Then the point-to-point exitant radiance satisfies

radiance_out (p \to p_{o}) = emitted_radiance (p, ω_{p \to p_{o}}) + \int_{Σ_{S}} bsdf (q \to p \to p_{o}) radiance_out (q \to p) {geometric_coupling}_{S} (p, q) d A_{Σ_{S}} (q) .

Apply the BSDF scattering equation with outgoing direction

ω_{p \to p_{o}}

radiance_out (p, ω_{p \to p_{o}}) = emitted_radiance (p, ω_{p \to p_{o}}) + \int_{𝕊^{2}} bsdf (p, ω_{i}, ω_{p \to p_{o}}) radiance_in (p, ω_{i}) | n (p) \cdot ω_{i} | d σ (ω_{i}) .

For a visible surface point

q

, the corresponding incoming direction at

p

ω_{i} = ω_{p \to q}

. By incident radiance from raycast,

radiance_in (p, ω_{p \to q}) = radiance_out (q \to p) .

On each smooth visible patch

Q \subseteq Σ_{S}

, apply surface area to solid angle change of variables to the direction map

g (q) = ω_{p \to q}

. The determinant calculation in that proposition gives the Jacobian factor

d σ (ω_{p \to q}) = \frac{| n (q) \cdot ω_{q \to p} |}{{‖ p - q ‖}^{2}} d A_{Q} (q) .

Directions that do not hit a surface contribute nothing to the surface integral, and blocked point pairs are recorded by the visibility factor

V_{S} (p, q)

. Substituting the radiance identity and the change-of-variables factor into the direction integral gives

\int_{Σ_{S}} bsdf (q \to p \to p_{o}) radiance_out (q \to p) V_{S} (p, q) \frac{| n (p) \cdot ω_{p \to q} | | n (q) \cdot ω_{q \to p} |}{{‖ p - q ‖}^{2}} d A_{Σ_{S}} (q) .

The factor multiplying the BSDF and radiance is exactly

{geometric_coupling}_{S} (p, q)

, so the claimed surface form follows.

The direction form suggests sampling directions $ω_{i}$ and then raycasting to find $q$ . The surface form suggests sampling points $q$ on scene surfaces and then using ${geometric_coupling}_{S} (p, q)$ to account for visibility, orientation, and distance. Both equations describe the same local light transport; they organize the computation differently.

Analytic LTE Solution for an Emitting Lambertian Sphere

Consider the interior surface of a sphere. Suppose every point on the sphere has the same Lambertian BRDF with reflectance

ρ

, where

0 \leq ρ < 1

, and suppose every point emits the same constant emitted radiance

emitted_radiance (p, ω) = E_{0}

in every direction. Then the solution of the BSDF scattering equation is constant over all surface points and directions, and its value is

radiance_out (p, ω) = \frac{E_{0}}{1 - ρ} .

By the rotational symmetry of the sphere interior and the constant material and emission assumptions, the outgoing radiance has the same value at every surface point and direction. Write this constant value as

L

. The corresponding incident radiance is also

L

, because every incoming ray last left another point on the same sphere with the same outgoing radiance. Using the BSDF scattering equation on the inward hemisphere gives

L = E_{0} + \int_{H_{p}^{+}} \frac{ρ}{π} L | n \cdot ω_{i} | d σ (ω_{i}) .

The hemisphere cosine integral is

π

. Indeed, choosing coordinates so that

n = (0,0,1)

, write

ω_{i}

in spherical coordinates with polar angle

θ \in [0, π / 2]

and azimuthal angle

φ \in [0,2 π]

. Then

| n \cdot ω_{i} | = \cos θ

and

d σ (ω_{i}) = \sin θ d θ d φ

, so

\int_{H_{p}^{+}} | n \cdot ω_{i} | d σ (ω_{i}) = \int_{0}^{2 π} \int_{0}^{π / 2} \cos θ \sin θ d θ d φ = π .

Therefore the scattering equation reduces to

L = E_{0} + ρ L .

Solving gives

L = \frac{E_{0}}{1 - ρ} .

The same result appears by successive substitution. From

L = E_{0} + ρ L

L = E_{0} + ρ (E_{0} + ρ L) = E_{0} + ρ E_{0} + ρ^{2} L .

Repeating this

N

times gives

L = \sum_{j = 0}^{N} ρ^{j} E_{0} + ρ^{N + 1} L .

Since

0 \leq ρ < 1

, the remainder

ρ^{N + 1} L

tends to

0

, and the geometric series converges to

L = \sum_{j = 0}^{\infty} ρ^{j} E_{0} = \frac{E_{0}}{1 - ρ} .

The successive-substitution series above is a Neumann series. In rendering language, the first term is radiance emitted directly by the surface, the next term is light that has scattered once after emission, the next has scattered twice, and so on. Direct illumination methods keep only the first scattering event from light sources; path tracing estimates the longer sequence by randomly sampling paths.

Energy-Conserving BSDF Does Not Amplify Incident Energy

Suppose that

bsdf

is an energy-conserving BSDF and

emitted_radiance (p, ω_{o}) = 0

. Then the scattered part of the BSDF scattering equation does not send out more total surface power density than the incoming radiance delivers to the surface.

The outgoing surface power density due to scattering is obtained by integrating outgoing radiance against the projected-area factor:

\int_{𝕊^{2}} radiance_out (p, ω_{o}) | n \cdot ω_{o} | d ω_{o} .

Since

emitted_radiance (p, ω_{o}) = 0

, substitute the scattered part of the BSDF scattering equation:

\int_{𝕊^{2}} \int_{𝕊^{2}} bsdf (p, ω_{i}, ω_{o}) radiance_in (p, ω_{i}) | n \cdot ω_{i} | | n \cdot ω_{o} | d ω_{i} d ω_{o} .

Reordering the integrations gives

\int_{𝕊^{2}} radiance_in (p, ω_{i}) | n \cdot ω_{i} | (\int_{𝕊^{2}} bsdf (p, ω_{i}, ω_{o}) | n \cdot ω_{o} | d ω_{o}) d ω_{i} .

By the energy-conserving BSDF condition, the parenthesized integral is at most

1

. Therefore the outgoing scattered power density is at most

\int_{𝕊^{2}} radiance_in (p, ω_{i}) | n \cdot ω_{i} | d ω_{i},

which is the incoming surface power density. Thus the scattering term redistributes or absorbs incident energy; it does not create extra energy.

BSSRDF

The bidirectional surface scattering reflectance distribution function, abbreviated BSSRDF, generalizes the BRDF by allowing light to enter a surface at one point and leave at another point. It depends on an incident surface point

p_{i}

, an incoming direction

ω_{i}

, an exitant surface point

p_{o}

, and an outgoing direction

ω_{o}

For example, wax and skin often need a BSSRDF because light can enter at one nearby point, scatter under the surface, and exit elsewhere.

Spectral Quantities

Wavelength

A wavelength is the spatial period of a wave. For light, wavelength is commonly denoted by

λ

Visible Wavelength Range

The visible wavelength range is the range of wavelengths that humans can usually see. A common approximate range is

380 n m

780 n m

Photon

A photon is a discrete packet of light or other electromagnetic radiation. A photon with wavelength

λ

has energy

E = \frac{h c}{λ},

where

h

is the Planck constant and

c

is the speed of light.

Spectral Distribution

A spectral distribution is a function whose input is a wavelength and whose output is the amount of some physical quantity at that wavelength.

Spectral Radiometric Quantity

A spectral radiometric quantity describes how a radiometric quantity is distributed over wavelength. Spectral radiance

L_{λ}

satisfies

L = \lim_{\max_{i} Δ λ_{i} \to 0} \sum_{i} L_{λ} (λ_{i}) Δ λ_{i},

whenever this limit exists over the wavelengths being considered.

Spectral Power Distribution

A spectral power distribution is a spectral distribution for radiant flux.

For example, two lamps may emit the same total radiant flux but have different spectral power distributions if one emits more short-wavelength light and the other emits more long-wavelength light.

Sampled Spectrum

A sampled spectrum represents a spectral distribution by storing values at finitely many wavelength samples or wavelength intervals.

For example, instead of storing a value for every wavelength from $380 n m$ to $780 n m$ , a renderer might store values at a fixed set of sample wavelengths and interpolate or average between them.

Spectral Rendering

Spectral rendering is rendering that transports and scatters light using spectral distributions rather than only three color coordinates.

Light Emission

Light Source

A light source is an object or region with nonzero emitted radiance.

Area Light

An area light is a light source whose emission is distributed over a surface region.

For example, a glowing rectangular panel is naturally modeled as an area light.

Blackbody Emitter

A blackbody emitter is an idealized light source whose spectral power distribution is determined only by its temperature.

For example, as the temperature increases, an ideal blackbody shifts from a reddish appearance toward white and then bluish-white.

Standard Illuminant

A standard illuminant is a specified spectral power distribution used as a reference light source.

For example, a daylight standard illuminant gives a repeatable reference spectrum for describing colors under daylight-like illumination.

Color

Color is the perceptual response associated with a light spectrum and an observer. It is not the same thing as a spectral distribution: different spectra can produce the same perceived color for a given observer.

Color Matching Functions

Color matching functions are three weighting functions over visible wavelengths used to convert a spectral distribution into three color coordinates.

XYZ Color

An XYZ color is a triple

(X, Y, Z)

obtained from a spectral distribution using color matching functions. The

Y

coordinate is chosen to correspond to perceived brightness under the standard observer model.

RGB Color

An RGB color is a triple

(R, G, B)

of coordinates relative to chosen red, green, and blue primaries. RGB is a coordinate representation of color, not a complete description of the physical light spectrum.

RGB Color Space

An RGB color space specifies how RGB triples correspond to colors. It includes the red, green, and blue primaries, a white point, and a transfer rule for encoding numerical channel values.

For example, the same triple $(1,0,0)$ means the red primary of whichever RGB color space is being used, so changing the color space can change the actual displayed color.

Metamer

Two spectral distributions are metamers for an observer if they produce the same perceived color for that observer.

Exercises

Photons from a Monochromatic Lightbulb

How many photons would a

50 W

lightbulb emit in

1 s

, assuming all of its emitted light has the single wavelength

λ = 600 n m

50 W

bulb emits

50 J

of radiant energy in

1 s

, using the joule and second. A photon with wavelength

λ

has energy

E_{γ} = \frac{h c}{λ} .

Therefore the number of photons is

N = \frac{50 J}{E_{γ}} = \frac{50 λ}{h c} .

Using

λ = 600 \times 10^{- 9} m

h = 6.62607015 \times 10^{- 34} J \cdot s

, and

c = 299792458 m \cdot s^{- 1}

N \approx 1.51 \times 10^{20} .

Thus the bulb emits about

1.51 \times 10^{20}

photons in

1 s

Irradiance from a Unit-Radius Disk

Compute the irradiance at a point due to a unit-radius disk

h

units above the point along the surface normal. Assume the disk has constant outgoing radiance

L = 10 W \cdot m^{- 2} \cdot s r^{- 1} .

Do the computation once as an integral over solid angle and once as an integral over area.

Let

θ

be the angle from the receiver normal. The disk subtends a cone with maximum angle

θ_{0}

, where

\tan θ_{0} = \frac{1}{h}, \sin^{2} θ_{0} = \frac{1}{h^{2} + 1} .

Using solid angle,

irradiance = \int L \cos θ d ω = L \int_{0}^{2 π} \int_{0}^{θ_{0}} \cos θ \sin θ d θ d φ = L π \sin^{2} θ_{0} = \frac{10 π}{h^{2} + 1} .

Now compute over disk area. Let $ρ$ be the radial coordinate on the disk. The distance from the point to a disk point is $r = \sqrt{h^{2} + ρ^{2}}$ , and the cosine factor at both the receiver and the disk is $h / r$ . Since $d ω = \frac{h}{r^{3}} d A,$ the irradiance is $irradiance = L \int_{0}^{2 π} \int_{0}^{1} \frac{h^{2}}{{(h^{2} + ρ^{2})}^{2}} ρ d ρ d φ = \frac{10 π}{h^{2} + 1} .$ The two computations agree.

Irradiance from a Unit Square

Compute the irradiance at a point due to a square with side length

1

, centered

1

unit above the point in the direction of its surface normal, with constant outgoing radiance

L = 10 W \cdot m^{- 2} \cdot s r^{- 1} .

Place the receiving point at the origin and the square in the plane

z = 1

, with

- \frac{1}{2} \leq x \leq \frac{1}{2}, - \frac{1}{2} \leq y \leq \frac{1}{2} .

For a point

(x, y, 1)

on the square, the squared distance to the receiver is

r^{2} = 1 + x^{2} + y^{2} .

The receiver cosine and emitter cosine are both

1 / r

. For a small area element represented by

d x d y

, the corresponding irradiance contribution is approximated by

L \frac{1}{r} \frac{1}{r^{3}} d A = L \frac{1}{{(1 + x^{2} + y^{2})}^{2}} d x d y .

Therefore

irradiance = 10 \int_{- 1 / 2}^{1 / 2} \int_{- 1 / 2}^{1 / 2} \frac{1}{{(1 + x^{2} + y^{2})}^{2}} d y d x \approx 7.5227468845 W \cdot m^{- 2} .