Photorealistic Rendering 2 – Radiometry

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

(∵ 𝑑𝝎₁ = 𝑑𝝎₂ & 𝑑𝐴₁ = 𝑑𝐴₂) ⟺ 𝐿₁𝑑𝝎₁𝑑𝐴₁ = 𝐿₂𝑑𝝎₂𝑑𝐴₂

∴ 𝐿₁ = 𝐿₂ 𝑑Φ₁ = 𝑑Φ₂

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

Bidirectional Reflectance Distribution Function

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

Bidirectional Reflectance Distribution Function

• 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