**Photorealistic Rendering **

**Wanho Choi **

**fxwano@gmail.com **

**Radiometry **

The science of measuring light

• Time / position / direction

**Photometry is also the science of measuring light, but it **
is based on sensitivity of human eyes rather than physical
model. PBR only deals with radiometry exclusively.

However, the visual response can be added as a post-process. This is referred to as tone mapping.

Radiometry uses **solid angle **in the various radiometry
equations.

**Solid Angle **

𝝎 𝑠𝑡𝑒𝑟𝑎𝑑𝑖𝑎𝑛𝑠
The **projected area** on a unit sphere
Max. value: 4𝜋 (for hemisphere: 2𝜋)

If you are at the origin, and I look out, The solid angle tells you

**Why Solid Angle? **

We need to consider light propagation in a cone.

light source sensor (eye or camera) Infinitesimal area of a surface surface normal infinitesimal hemisphere

**Infinitesimal Solid Angle **

Differential solid angle(𝑑𝝎) in spherical coordtes

𝑑𝝎 = 𝑠𝑖𝑛𝜃𝑑𝜃𝑑𝜙 r𝑑𝜃 𝑟𝑠𝑖𝑛𝜃𝑑𝜙 𝑠𝑖𝑛𝜃𝑑𝜃 𝜋 0 2𝜋 0 𝑑𝜙 = −𝑐𝑜𝑠𝜋 − 𝑐𝑜𝑠0 2𝜋 0 𝑑𝜙 = 2 2𝜋 0 𝑑𝜙 = 4𝜋

Finite solid angle

**Frequently Asked Integral Calculus 2 **

Integrals over a hemisphere
𝑐𝑜𝑠𝑛𝜃 2𝜋 𝑑𝜔 = 𝑐𝑜𝑠𝑛𝜃𝑠𝑖𝑛𝜃𝑑𝜃 𝜋/2 0 2𝜋 0 𝑑𝜙 = 𝑑𝜙 2𝜋 0 𝑐𝑜𝑠𝑛𝜃𝑠𝑖𝑛𝜃𝑑𝜃 𝜋/2 0 = 2𝜋 𝑢𝑛𝑑𝑢 1 0 = 2𝜋 𝑢 𝑛+1 𝑛 + 1 0 1 = 2𝜋 𝑛 + 1

**Radiometric Quantities **

𝑄: energy [J]
• A count of photons

𝛷: flux [𝑊]

• Radiant energy per time

𝐸, 𝐵: flux density [𝑊𝑚−2]

• 𝐸 : arriving density (=irradiance) • 𝐵 : radiant density (=radiosity)

𝐿: radiance [𝑊𝑚−2𝑠𝑟−1]

• 𝑑𝝎⊥ _{= 𝑐𝑜𝑠𝜃𝑑𝝎 = 𝒏 ⋅ 𝝎 𝑑𝝎}

(projected solid angle)

### 𝑄 = 𝑄

_{𝜆}

### 𝑑𝜆

∞ 0### 𝛷 =

### 𝑑𝑄

### 𝑑𝑡

### 𝐸 =

### 𝑑𝛷

### 𝑑𝑨

⊥### 𝐿 =

### 𝑑𝐸

### 𝑑𝝎

**differentiation**

**integration**

**differentiation**

**integration**

**differentiation**

**integration**

Uniformly spread flux

**Point Light Source **

𝐸 = Φ 4𝜋𝑟2 Φ[𝑊] 𝜃 𝐸′ = Φ𝑐𝑜𝑠𝜃 4𝜋𝑟2 𝐸′ = 𝐸𝑐𝑜𝑠𝜃

**Radiance **

Radiance is the most important quantity for ray tracing. The response of a sensor (cameras, human eyes)

is proportional to radiance.

• In this moment, we are seeing the radiance of the world.

The flux per unit solid angle per unit projected area

**Radiance **

𝒙 𝒅𝐴
𝒅𝐴⊥
𝜽
𝝎
𝒏
𝐿 𝒙, 𝝎 = 𝑑𝐸
𝑑𝝎 =
𝑑
𝑑𝝎
𝑑𝛷
𝑑𝐴⊥ =
𝑑2𝛷
𝑑𝐴⊥_{𝑑𝝎}= 𝑑2𝛷 𝑑𝐴𝑐𝑜𝑠𝜃𝑑𝝎 = 𝑑2𝛷 𝑑𝐴(𝒏 ⋅ 𝝎)𝑑𝝎

Sometimes, radiance can be thought as the intensity of light at a given point in a given direction.

**Radiance **

𝒙

𝝎

**Radiance **

Radiance is constant as it propagates along a ray.

• Radiance doesn’t change with distance.

• Exception: participating media

(∵ 𝑑𝝎_{1} = 𝑑𝝎_{2} & 𝑑𝐴_{1} = 𝑑𝐴_{2})
⟺ 𝐿_{1}𝑑𝝎_{1}𝑑𝐴_{1} = 𝐿_{2}𝑑𝝎_{2}𝑑𝐴_{2}

∴ 𝐿_{1} = 𝐿_{2}
𝑑Φ_{1} = 𝑑Φ_{2}

Total incoming radiance at 𝒙 = the irradiance at 𝒙

**Radiance **

𝐸(𝒙) = 𝑑𝐸
𝛺
= 𝐿(𝒙, 𝝎)(𝒏 ⋅ 𝝎)𝑑𝝎
𝛺
𝜴: hemisphere
from other light sources (direct illumination) from other objects (indirect illumination)

𝒙

**B**idirectional **R**eflectance **D**istribution **F**unction

How much light coming in from direction 𝝎_{𝑖} is reflected
out in direction 𝝎_{𝑜} at surface point 𝒙.

**Reflection Equation **

𝐿_{𝑜}𝒙, 𝝎

_{𝑜}= 𝑓

_{𝑟}𝒙, 𝝎

_{𝑖}, 𝝎

_{𝑜}𝐿

_{𝑖}(𝒙, 𝝎

_{𝑖})(𝒏 ∙ 𝝎

_{𝑖})𝒅𝝎

_{𝑖}Ω

**BRDF**𝒙 𝒏 𝑑𝝎

_{𝑜}𝜴: hemisphere

**BRDF **

**B**idirectional **R**eflectance **D**istribution **F**unction

• The amount of light that is scattered in each outgoing angle for

each incoming angle

• How much light coming in from direction 𝝎_{𝑖} is reflected out in

direction 𝝎_{𝑜} at surface point 𝒙.

• The relative distribution of the surface reflection of light from 𝝎_{𝑖} at

point 𝒙.
𝑓_{𝑟} 𝒙, 𝝎_{𝑖}, 𝝎_{𝑜} = 𝑑𝐿𝑜(𝒙,𝝎𝑜)
𝑑𝐸_{𝑖}(𝒙) =
𝑑𝐿_{𝑜}(𝒙,𝝎_{𝑜})
𝐿_{𝑖}(𝒙,𝝎_{𝑖})(𝒏∙𝝎_{𝑖})𝒅𝝎_{𝑖} 𝑠𝑟
−1

• 𝒙: the interesting point

• 𝒏: the unit surface normal at 𝒙

• 𝝎_{𝑖}: solid angle from 𝒙 to the incoming direction

**BRDF Visualization **

Lobe
**Properties of BRDF **

**Reciprocity **(bidirectionality, symmetry)

• 𝑓_{𝑟} 𝒙, 𝝎_{𝑖}, 𝝎_{𝑜} = 𝑓_{𝑟} 𝒙, 𝝎_{𝑜}, 𝝎_{𝑖}

• Light paths can be traced in the opposite direction that light would

normally travel.

**Linearity **

• When using multiple BRDFs, the total reflected radiance at a

surface point is sum of the reflected radiance from each BRDF.

**Energy conservation **

• 𝑓_{𝛺} _{𝑟} 𝒙, 𝝎_{𝑖}, 𝝎_{𝑜} (𝒏 ∙ 𝝎_{𝑖})𝒅𝝎_{𝑖} ≤ 1

**Reflectance (𝝆) **

Albedo: 𝜌 𝒙 = 𝐵 𝒙
𝐸 𝒙 =
𝐿_{𝛺}

_{𝑜}𝒏⋅𝝎 𝑑𝝎 𝐸 𝒙 0 ≤ 𝜌 𝒙 ≤ 1

**Ideal diffuse reflection (Lambertian) **

• 𝐿_{𝑜} 𝒙 is identical in all directions. (constant)

• 𝜌_{𝑑} 𝒙 = 𝐿𝑜 𝒙

𝐸 𝒙 𝝅 ⇔ 𝑓𝑟,𝑑 𝒙 =

𝜌_{𝑑} 𝒙
𝝅

**Ideal specular reflection (mirrorlike) **

• 𝐿_{𝑜} 𝒙 is non-zero only in the mirror direction.

• 𝜌_{𝑠} 𝒙 = 𝐿𝑜 𝒙

𝐸 𝒙

1

**Rendering Equation **

BRDF does not consider fluoresce or emitting surfaces.
𝐿 𝒙, 𝝎_{𝑜} = 𝐿_{𝑒} 𝒙, 𝝎_{𝑜} + 𝑓_{𝑟} 𝒙, 𝝎_{𝑖}, 𝝎_{𝑜} 𝐿 𝒙, 𝝎_{𝑖} 𝒏 ⋅ 𝝎_{𝑖} 𝑑𝝎_{𝑖}

Ω

No analytic solution Very difficult to solve

radiance at 𝒙 in 𝝎_{𝑜}

radiance emitted from surface radiance reflected by surface