Physically-Based Rendering Shih-Chin Weng shihchin.weng@gmail.com - - PowerPoint PPT Presentation

Physically-Based Rendering

Shih-Chin Weng

shihchin.weng@gmail.com

What is PBR?

The Chemical Brothers - Wide Open, The Mill

Physically Based Rendering

Simulate materials and lights based on physical laws

• r observations of real world more accurately.

Stages of Photorealistic Rendering

• 1. Measurement and acquisition of scene data

– BRDF, BSSRDF, BTF, etc.

• 2. Light transport simulation

– Ray tracing, photon-mapping, radiosity, etc.

• 3. Visual display

– Tone mapping

What Is Light?

Video: What Is Light? by Kurzgesagt

Video: What Is Light? by Kurzgesagt

Geometric Optics

• Assumption: the wavelength of light is much smaller

than the scale of interacted object

• Light travels

– in straight lines – instantaneously through a medium

• Light is not influenced by gravity or magnetic fields

– No diffraction, dispersion – But the movie “Interstellar” does simulate the light bent by gravity!!

Light Matter Interaction

specular diffuse diffuse scattering particles

Photo by Gabriel Gurrola

Snell’s Law

Snell’s Law

𝑡𝑗𝑜 𝜄𝑗 𝜃𝑗 = 𝑡𝑗𝑜 𝜄𝑢 𝜃𝑢

Index of Refraction (IOR): 𝜃

𝜄𝑗 𝜄𝑗 𝜄𝑢

𝜃𝑗 𝜃𝑢

https://en.wikipedia.org/wiki/Snell%27s_law#/media/File:Snells_law_wavefronts.gif

Snell’s Law

𝑡𝑗𝑜 𝜄𝑗 𝜃𝑗 = 𝑡𝑗𝑜 𝜄𝑢 𝜃𝑢

Index of Refraction (IOR): 𝜃

𝜄𝑗 𝜄𝑗 𝜄𝑢

𝜃𝑗 𝜃𝑢

https://en.wikipedia.org/wiki/Snell%27s_law#/media/File:Snells_law_wavefronts.gif

Photo by Ales Krivec

Fresnel Effect

refraction reflection

Photo by Ashes Sitoula

Fresnel Effect

F0

reflectance at normal more and more reflective as the angle of view approaches a grazing angle

Fresnel

• Fresnel reflectance

– the amount of reflected light w.r.t. the viewing angle

• Relates the ratio of reflected and transmitted energy

as a function of

– Incident direction – Polarization – Materials’ properties

Material Properties

Non-metal (dielectrics)

• Only reflect 4~10% of

incoming light in average

• The reflection intensity is

independent on the wavelength

• No energy is absorbed

during reflection

– but might be absorbed during subsurface scattering

Metal

• IOR strongly depends on the

wavelength

• Immediately absorbs

refracted lights (i.e. no refraction)

– The reflected lights would change their color

Fresnel Reflectance

[Real-time Rendering, 3/e, A K Peters 2008]

Fresnel Reflectance

[Real-time Rendering, 3/e, A K Peters 2008]

Fresnel Reflectance

[Real-time Rendering, 3/e, A K Peters 2008]

Fresnel Reflectance

[Real-time Rendering, 3/e, A K Peters 2008]

Fresnel Reflectance

[Real-time Rendering, 3/e, A K Peters 2008]

Reflection goes to 100% at grazing angle!

Fresnel

𝜄𝑗 𝜄𝑗 𝜄𝑢

𝜃𝑗 𝜃𝑢

Fr = 1 2 𝑠

∥ 2 + 𝑠 ⊥ 2 Dielectric

r∥ = ηt cos 𝜄𝑗 − 𝜃𝑗 cos 𝜄𝑢 ηt cos 𝜄𝑗 + 𝜃𝑗 cos 𝜄𝑢 r⊥ = ηi cos 𝜄𝑗 − 𝜃𝑢 cos 𝜄𝑢 ηi cos 𝜄𝑗 + 𝜃𝑢 cos 𝜄𝑢

Conductor

r∥

2 = η2 + k2 cos2 θ𝑗 − 2η cos 𝜄𝑗 + 1

η2 + k2 cos2 θ𝑗 + 2η cos 𝜄𝑗 + 1 r⊥

2 = η2 + k2 − 2η cos 𝜄𝑗 + cos2 𝜄𝑗

η2 + k2 + 2η cos 𝜄𝑗 + cos2 𝜄𝑗 for unpolarized light

Radiometry

Radiant flux Φ =

dQ dt (J/sec) The total amount of energy

passing through a region of surface per unit time

Irradiance 𝐹 =

𝑒𝛸 𝑒𝐵 Pre area incoming flux at a surface

Radiant Exitance or Radiosity

𝑁 = 𝐶 = 𝑒𝛸 𝑒𝐵 𝐹 = Φ 4𝜌𝑠2

the total amount 𝛸 measured at inner and outer sphere is the same (equals to the radiant flux of the point light) r

Lambert’s Cosine Law

𝐹 = 𝑒𝛸 𝑒𝐵

𝑒𝐵 = 𝑒𝐵′ cos 𝜄 𝑒𝐵

𝐹1 = 𝑒𝛸 𝑒𝐵 𝐹2 = 𝑒𝛸 𝑒𝐵′ = cos 𝜄 𝑒𝛸 𝑒𝐵 = 𝐹1 cos 𝜄

𝜄 𝑒𝐵

Solidangle

– The total area on a unit sphere subtended by the object – A set of directions – Measured in steradians (sr) – Often denoted as 𝜕 Ω = A r2

Radiance

𝑀 = 𝑒2𝛸 𝑒𝜕𝑒𝐵⊥ = 𝑒2𝛸 𝑒𝜕𝑒𝐵 𝑑𝑝𝑡 𝜄

flux projected area solidangle

𝑒𝐵 𝑒𝜕 𝑒𝐵⊥ The density of photons passing near x and traveling in directions near ω

Bidirectional Reflection Distribution Function

𝑐 Ԧ 𝑢 𝑜

𝜕𝑗 𝜕𝑝 𝜄𝑗 𝜄𝑝 𝜚𝑗 𝜚𝑝

𝑔(𝜄𝑗, 𝜚𝑗, 𝜄𝑝, 𝜚𝑝) = 𝑔(𝜕𝑗, 𝜕𝑝)

BRDF Definition

𝑔 𝜕𝑗, 𝜕𝑝 = 𝑒𝑀𝑠 𝜕𝑝 𝑒𝐹𝑗(𝜕𝑗) = 𝑒𝑀𝑠 𝜕𝑝 𝑀𝑗 𝜕𝑗 𝑑𝑝𝑡 𝜄𝑗 𝑒𝜕𝑗

• utgoing radiance

incoming irradiance

BRDF Definition

𝑔 𝜕𝑗, 𝜕𝑝 = 𝑒𝑀𝑠 𝜕𝑝 𝑒𝐹𝑗(𝜕𝑗) = 𝑒𝑀𝑠 𝜕𝑝 𝑀𝑗 𝜕𝑗 𝑑𝑝𝑡 𝜄𝑗 𝑒𝜕𝑗

• utgoing radiance

incoming irradiance

𝒕𝒒𝒇𝒐𝒆𝒋𝒐𝒉 𝒋𝒐𝒅𝒑𝒏𝒇

Properties of BRDFs

• Helmholtz reciprocity

– symmetric surface reflectance 𝑔 𝜕𝑗, 𝜕𝑝 = 𝑔(𝜕𝑝, 𝜕𝑗)

• Positivity

𝑔 𝜕𝑗, 𝜕𝑝 ≥ 0

• Energy conservation

– Total amount of outgoing energy must be less than or equal to the incoming energy

http://www.disneyanimation.com/technology/brdf.html

[Image courtesy of Disney.]

from Disney Animation

Isotropic vs. Anisotropic

• Isotropic BRDFs are independent of incident azimuth

angle 𝜚

isotropic anisotropic

BRDF Acquisition

[White et al, JAO 98] [Marschner et al. 1999]

MERL 100

http://www.merl.com/brdf/

“A Data-Driven Reflectance Model”, Matusik et al., SIG’03

BRDF Data Fitting

[Ngan et al., 2005]

Microfacet Model

Microfacet Model

ℎ = Ԧ 𝑚 + Ԧ 𝑤 Ԧ 𝑚 + Ԧ 𝑤 macrogeometry

Microfacet Model

ℎ = Ԧ 𝑚 + Ԧ 𝑤 Ԧ 𝑚 + Ԧ 𝑤 macrogeometry

Microfacet Model

𝜄𝑛 𝜄𝑛 𝑛 Ԧ 𝑚 Ԧ 𝑤 ℎ = Ԧ 𝑚 + Ԧ 𝑤 Ԧ 𝑚 + Ԧ 𝑤 microfacet: ideal mirror macrogeometry

General Microfacet BRDF

𝑔

𝑠 Ԧ

𝑚, Ԧ 𝑤 = 𝑒𝑗𝑔𝑔𝑣𝑡𝑓 + 𝐸 𝜄ℎ 𝐺 𝜄𝑒 𝐻(𝜄𝑚, 𝜄𝑤) 4 𝑑𝑝𝑡 𝜄𝑚 𝑑𝑝𝑡 𝜄𝑤

Normal Distribution Function (NDF) Fresnel reflectance Geometric Term 𝜄𝑚, 𝜄𝑤: angle between Ԧ 𝑚, Ԧ 𝑤 and normal 𝜄ℎ: angle between normal and ℎ 𝜄𝑒: difference between Ԧ 𝑚 (𝑝𝑠 Ԧ 𝑤) and ℎ

The ratio of micro-surface area visible to the light, viewer

Fresnel

• Schlick’s approximation

𝐺

𝑇𝑑ℎ𝑚𝑗𝑑𝑙 = 𝐺 0 + 1 − 𝐺

1 − 𝑑𝑝𝑡 𝜄𝑗 5 – Where F0 =

η2−𝜃1 η2+𝜃1 2

• a.k.a. reflectance at normal, normal reflectance, etc.

? What if 𝜃2 = 𝜃1

• 𝐺 should be zero but 𝐺

𝑇𝑑ℎ𝑚𝑗𝑑𝑙 = 1 − 𝑑𝑝𝑡 𝜄𝑗 5 ≠ 0

NDF (Normal Distribution Function)

• Half vector ℎ =

Ԧ 𝑚+𝑤 Ԧ 𝑚+𝑤

• As for perfect mirror microfacets, we can only see

those facets whose normal vector 𝑛 = ℎ

Photo by Liu Zai Hou

Highlights at Grazing Angles

Data Fitting of Acquired Data

[Ngan et al., SIG’04]

Highlights at Grazing Angles

[Ngan et al., SIG’04]

Data Fitting of Acquired Data (Cont’d)

[Ngan et al., SIG’04]

• Measures area density of microsurface with respect

to microsurface normal

𝐸 𝜕 = න

ℳ

𝜀𝜕 𝜕𝑛 𝑞𝑛 𝑒𝑞𝑛

NDF (Cont’d)

microsurface

• Measures area density of microsurface with respect

to microsurface normal

𝐸 𝜕 = න

ℳ

𝜀𝜕 𝜕𝑛 𝑞𝑛 𝑒𝑞𝑛

NDF (Cont’d)

microsurface

• Measures area density of microsurface with respect

to microsurface normal

𝐸 𝜕 = න

ℳ

𝜀𝜕 𝜕𝑛 𝑞𝑛 𝑒𝑞𝑛

NDF (Cont’d)

microsurface Ω

• Measures area density of microsurface with respect

to microsurface normal

𝐸 𝜕 = න

ℳ

𝜀𝜕 𝜕𝑛 𝑞𝑛 𝑒𝑞𝑛

NDF (Cont’d)

microsurface Ω 𝜕𝑛: ℳ → Ω

NDF (Cont’d)

𝑛𝑗𝑑𝑠𝑝𝑡𝑣𝑠𝑔𝑏𝑑𝑓 𝑏𝑠𝑓𝑏 = න

ℳ

𝑒𝑞𝑛 = න

Ω

𝐸 𝜕𝑛 𝑒𝜕𝑛

𝜕𝑕: normal of macrosurface

𝑞𝑠𝑝𝑘𝑓𝑑𝑢𝑓𝑒 𝑛𝑗𝑑𝑠𝑝𝑡𝑣𝑠𝑔𝑏𝑑𝑓 𝑏𝑠𝑓𝑏 = න

Ω

𝜕𝑛 ⋅ 𝜕𝑕 𝐸 𝜕𝑛 𝑒𝜕𝑛

projection

Masking/Shadowing

masking shadowing

Conservation of Projected Area

[Heitz ‘14]

cos 𝜄𝑝 = න

Ω

𝐻1 𝜕𝑝, ω𝑛 𝜕𝑝, ω𝑛 𝐸 𝜕𝑛 𝑒𝜕𝑛

masking function

Conservation of Projected Area

[Heitz ‘14]

cos 𝜄𝑝 = න

Ω

𝐻1 𝜕𝑝, ω𝑛 𝜕𝑝, ω𝑛 𝐸 𝜕𝑛 𝑒𝜕𝑛

masking function

BRDF Validation

• What makes it physically-based?
• 1. Reciprocity: f l, v = f v, l
• 2. Positivity: f l, v > 0
• 3. Energy conservation: ׬

Ω 𝑔 𝑚, 𝑤 cos 𝜄𝑗 𝑒𝜕𝑗 ≤ 1

What do we miss?

Multiple Surface Bounces?

[Heitz, 2015]

References

• Physically-based Rendering. SIGGRAPH Course Notes 2011~15.
• Ngan et al., Experimental Analysis of BRDF Models. Technical Report 2005.
• Eric Heitz, Understanding the Masking-Shadowing Function. SIG’14.
• Brent Burley, Physically Based Shading at Disney. SIG’12 Course Note.
• Steve Marschner, Microfacet models for refection and refraction, Cornell

University, CS 6630, Fall 2015.