Rendering equation

1. Physically
Based
Rendering
RENDERING EQUATION

2. Empirical Model
Phong Shading
Ambient + Diffuse + Specular = Phong
Reflection
Parameters are chosen arbitrary by the user

3. Realistic Model
Radiosity
Radiosity (radiometry), the total
radiation (emitted plus reflected)
leaving a surface, certainly
including the reflected radiation
and the emitted radiation.
BRDF Equation
A BRDF, bi-directional reflectance
distribution function, is a tool for
describing the distribution of
reflected light at a surface.
Light Transport Equation
The light transport equation (LTE)
is the governing equation that
describes the equilibrium
distribution of radiance in a
scene.

4. Spectrum and Spectral
Power Distribution
For each surface, it reflects lights with different
wavelengths. The power of different lights can be plot as
Spectral Power Distribution graph.
A remarkable property of the human visual system makes
it possible to represent colors for human perception with
just three floating-point numbers. The tristimulus theory
of color perception says that all visible SPDs can be
accurately represented for human observers with three
values.

5. Spectrum and Spectral
Power Distribution
When we display an RGB color on a display, the spectrum that is
displayed is basically determined by the weighted sum of three
spectral response curves, one for each of red, green, and blue, as
emitted by the display’s phosphors, LED or LCD elements, or plasma
cells. Figure plots the red, green, and blue distributions emitted by a
LED display and an LCD display; note that they are remarkably
different.
Given (X, Y, Z) representation of an SPD, we can convert it to
corresponding RGB coefficients, given the choice of a set of SPDs
that define red, green, and blue for a display of interest.

6. Angle and Solid Angle
In order to define light intensity, we first need to define the notion
of a solid angle. Solid angles are just the extension of 2D angles in
a plane to an angle on a sphere.
2D angles are defined by the length of the arc on the unit circle.
Solid Angle (3D Angle) are defined by the area on the unit sphere.
So for a 2D angle s, if we calculate 360 degree, then it is equal to
! 𝑑𝑠 = 2𝜋
While for solid angle 𝜔
! 𝑑𝜔 = 4𝜋

7. Radiometry
Quantities Definition
Energy (Single Photon) 𝑄 =
ℎ𝑐
𝜆
𝑐 is the speed of light, ℎ is the Planck’s
constant, and 𝜆 is the wavelength.
Flux (Power) 𝜙 =
𝑑𝑄
𝑑𝑡
, Q = + ∅𝑑𝑡 Energy per second
Irradiance and Radiant
Exitance
𝐸 =
𝑑𝜙
𝑑𝐴
, ∅ = + 𝐸𝑑𝐴 Flux (Power) per area
Intensity 𝐼 =
𝑑𝜙
𝑑𝜔
, 𝜙 = +
!
𝐼𝑑𝜔 Flux (Power) per solid angle
Radiance (Brightness) 𝐿 𝑝, 𝜔 =
𝑑𝜙
𝑑𝜔 ⋅ 𝑑𝐴 ⋅ 𝑐𝑜𝑠𝜃
Flux (Power) per area per solid angle

8. Radiometry
By knowing Radiance, then all other quantities can be
calculated by integrating Radiance over solid angle or over
area.
Consider point P, then the received Irradiance is
𝐸! 𝑝, 𝑛 = !
"
𝐿! 𝑝, 𝜔 |𝑐𝑜𝑠𝜃|𝑑𝜔
Most importantly, we need to know 𝐿# 𝑝, 𝑤# , as it is the
power our eye receives.

9. Linearity Assumption
If the direction 𝜔! is considered as a differential cone of directions, the
differential irradiance at 𝑝 is
𝑑𝐸 𝑝, 𝜔! = 𝐿! 𝑝, 𝜔! |𝑐𝑜𝑠𝜃!|𝑑𝜔!
A differential amount of radiance will be reflected in the direction 𝜔" due to this
irradiance. Because of the linearity assumption from geometric optics, the
reflected differential radiance is proportional to the irradiance
𝑑𝐿"(𝑝, 𝜔") ∝ 𝑑𝐸(𝑝, 𝑤!)
A good way to understand is that the outgoing radiance has a constant ratio
against with the incoming irradiance. When the irradiance increase the outgoing
radiance increase as well. Considering different directions, when the irradiance
increases, it results in increasing different amounts of the outgoing radiance.
The ratio between the outgoing radiance and incoming irradiance is like the
probability density function between probability and random variables.

10. BRDF Equation
Because we have
𝑑𝐿!(𝑝, 𝜔!) ∝ 𝑑𝐸(𝑝, 𝑤")
The constant of proportionality defines the surface’s BRDF for the pair of directions 𝜔" and 𝜔!:
𝑓# 𝑝, 𝜔", 𝜔! =
𝑑𝐿!(𝑝, 𝜔!)
𝑑𝐸(𝑝, 𝜔")
=
𝑑𝐿!(𝑝, 𝜔!)
𝐿" 𝑝, 𝜔" |𝑐𝑜𝑠𝜃"|𝑑𝜔"
So we have
𝐿! 𝑝, 𝜔! = 2
$
𝑓# 𝑝, 𝜔", 𝜔! 𝐿" 𝑝, 𝜔" |𝑐𝑜𝑠𝜃"|𝑑𝜔"
Physically based BRDFs have two important qualities:
1. Reciprocity: For all pairs of directions:
𝑓# 𝑝, 𝜔", 𝜔! = 𝑓# 𝑝, 𝜔!, 𝜔"
2. Energy conservation: The total energy of light reflected is less than or equal to the energy of incident
light. For all directions:
2
$
𝑓# 𝑝, 𝜔, 𝜔! 𝑐𝑜𝑠𝜃𝑑𝜔 ≤ 1

11. Rendering
Equation
𝐿!$ = 𝐿#% = ! 𝐿!% …