Lecture 6 - Light Transport Welcome! , = (, ) , - - PowerPoint PPT Presentation

𝑱 𝒚, 𝒚′ = 𝒉(𝒚, 𝒚′) 𝝑 𝒚, 𝒚′ + න

𝑻

𝝇 𝒚, 𝒚′, 𝒚′′ 𝑱 𝒚′, 𝒚′′ 𝒆𝒚′′

INFOMAGR – Advanced Graphics

Jacco Bikker - November 2017 - February 2018

Lecture 6 - “Light Transport”

Welcome!

Today’s Agenda:

• Introduction
• The Rendering Equation
• Light Transport

Introduction

Advanced Graphics – Light Transport 3

Whitted

Introduction

Advanced Graphics – Light Transport 4

Whitted

Introduction

Advanced Graphics – Light Transport 5

Whitted

Missing:

• Area lights
• Glossy reflections
• Caustics
• Diffuse interreflections
• Diffraction
• Polarization
• Phosphorescence
• Temporal effects
• Motion blur
• Depth of field
• Anti-aliasing

Introduction

Advanced Graphics – Light Transport 6

Anti-aliasing

Adding anti-aliasing to a Whitted-style ray tracer: Send multiple primary rays through each pixel, and average their result. Problem:

• How do we aim those rays?
• What if all rays return the same color?

Introduction

Advanced Graphics – Light Transport 7

Anti-aliasing – Sampling Patterns

Adding anti-aliasing to a Whitted-style ray tracer: Send multiple primary rays through each pixel, and average their result. Problem:

• How do we aim those rays?
• What if all rays return the same color?

Introduction

Advanced Graphics – Light Transport 8

Anti-aliasing – Sampling Patterns

Introduction

Advanced Graphics – Light Transport 9

Anti-aliasing – Sampling Patterns

Introduction

Advanced Graphics – Light Transport 10

Anti-aliasing – Sampling Patterns

Adding anti-aliasing to a Whitted-style ray tracer: Send multiple primary rays through each pixel, and average their result. Problem:

• How do we aim those rays?
• What if all rays return the same color?

Introduction

Advanced Graphics – Light Transport 11

Whitted

Missing:

• Area lights
• Glossy reflections
• Caustics
• Diffuse interreflections
• Diffraction
• Polarization
• Phosphorescence
• Temporal effects
• Motion blur
• Depth of field

 Anti-aliasing

Introduction

Advanced Graphics – Light Transport 12

Distribution Ray Tracing*

*: Distributed Ray Tracing, Cook et al., 1984

Soft shadows

Introduction

Advanced Graphics – Light Transport 13

Distribution Ray Tracing*

*: Distributed Ray Tracing, Cook et al., 1984

Glossy reflections

Introduction

Advanced Graphics – Light Transport 14

Distribution Ray Tracing*

*: Distributed Ray Tracing, Cook et al., 1984

?

Introduction

Advanced Graphics – Light Transport 15

Distribution Ray Tracing*

*: Distributed Ray Tracing, Cook et al., 1984

Introduction

Advanced Graphics – Light Transport 16

Distribution Ray Tracing

Whitted-style ray tracing is a point sampling algorithm:

• We may miss small features
• We cannot sample areas

Area sampling:

• Anti-aliasing: one pixel
• Soft shadows: one area light source
• Glossy reflection: directions in a cone
• Diffuse reflection: directions on the hemisphere

Introduction

Advanced Graphics – Light Transport 17

Area Lights

Visibility of an area light source: 𝑊

𝐵 = න 𝐵

𝑊 𝑦, ꙍ𝑗 𝑒ꙍ𝑗 Analytical solution case 1: 𝑊

𝐵 = 𝐵𝑚𝑗𝑕ℎ𝑢 − 𝐵𝑚𝑗𝑕ℎ𝑢⋂𝑡𝑞ℎ𝑓𝑠𝑓

Analytical solution case 2: 𝑊

𝐵 = ?

Introduction

Advanced Graphics – Light Transport 18

Approximating Integrals

An integral can be approximated as a Riemann sum: 𝑊

𝐵 = න 𝐵 𝐶

𝑔(𝑦) 𝑒𝑦 ≈ ෍

𝑗=1 𝑂

𝑔 𝑢𝑗 𝛦𝑗 , where ෍

𝑗=1 𝑂

𝛦𝑗 = 𝐶 − 𝐵 Note that the intervals do not need to be uniform, as long as we sample the full interval. If the intervals are uniform, then ෍

𝑗=1 𝑂

𝑔 𝑢𝑗 𝛦𝑗 = 𝛦𝑗 ෍

𝑗=1 𝑂

𝑔 𝑢𝑗 = 𝐶 − 𝐵 𝑂 ෍

𝑗=1 𝑂

𝑔 𝑢𝑗 . Regardless of uniformity, restrictions apply to 𝑂 when sampling multi-dimensional functions (ideally, 𝑂 = 𝑁𝑒). Also note that aliasing may occur if the intervals are uniform.

Image from Wikipedia

A B

VA

Introduction

Advanced Graphics – Light Transport 19

Monte Carlo Integration

Alternatively, we can approximate an integral by taking random samples: 𝑊

𝐵 = න 𝐵 𝐶

𝑔(𝑦) 𝑒𝑦 ≈ 𝐶 − 𝐵 𝑂 ෍

𝑗=1 𝑂

𝑔 𝑌𝑗 Here, 𝑌1. . 𝑌𝑂 ∈ [𝐵, 𝐶]. As 𝑂 approaches infinity, 𝑊

𝐵 approaches the expected value of 𝑔.

Unlike in Riemann sums, we can use arbitrary 𝑂 for Monte Carlo integration, regardless of dimension.

Introduction

Advanced Graphics – Light Transport 20

Monte Carlo Integration of Area Light Visibility

To estimate the visibility of an area light source, we take 𝑂 random point samples. In this case, 5 out of 6 samples are unoccluded: 𝑊 ≈ 1 6 1 + 1 + 1 + 0 + 1 + 1 = 5 6 In terms of Monte Carlo integration: 𝑊 = න

𝒯2𝑊(𝑞) 𝑒𝑞 ≈ 1

𝑂 ෍

𝑗=1 𝑂

𝑊 𝑄 With a small number of samples, the variance in the estimate shows up as noise in the image.

Introduction

Advanced Graphics – Light Transport 21

Monte Carlo Integration of Area Light Visibility

We can also use Monte Carlo to estimate the contribution of multiple lights:

• 1. Take the average of N samples from each light source;
• 2. Sum the averages.

𝐹 𝑦 ← = ෍

𝑗=1 2

𝑀𝑗 𝑊(𝑦 ↔ 𝑚𝑗) 𝑦

Introduction

Advanced Graphics – Light Transport 22

Monte Carlo Integration of Area Light Visibility

Alternatively, we can just take 𝑂 samples, and pick a random light source for each sample. 𝐹 𝑦 ← = 2 𝑂 ෍

𝑗=1 𝑂

𝑀𝑅 𝑊

𝑅 𝑄 ,

𝑅 ∈ {1,2} = 1 𝑂 ෍

𝑗=1 𝑂 𝑀𝑅 𝑊 𝑅 𝑄

0.5 𝑦 Probability of sampling light 𝑀𝑅

Introduction

Advanced Graphics – Light Transport 23

Monte Carlo Integration of Area Light Visibility

We obtain a better estimate with fewer samples if we do not treat each light equally. In the previous example, each light had a 50% probability of being sampled. We can use an arbitrary probability, by dividing the sample by this probability. 𝐹 𝑦 ← = 1 𝑂 ෍

𝑗=1 𝑂 𝑀𝑅 𝑊 𝑅 𝑄

𝜍𝑅 , ෍ 𝜍𝑅 = 1, 𝜍𝑅 > 0 𝑦

Introduction

Advanced Graphics – Light Transport 24

Distribution Ray Tracing

Key concept of distribution ray tracing: We estimate integrals using Monte Carlo integration. Integrals in rendering:

• Area of a pixel
• Lens area (aperture)
• Frame time
• Light source area
• Cones for glossy reflections
• Wavelengths
• …

Introduction

Advanced Graphics – Light Transport 25

Open Issues

Remaining issues:

• Energy distribution in the ray tree / efficiency
• Diffuse interreflections

Today’s Agenda:

• Introduction
• The Rendering Equation
• Light Transport

Rendering Equation

Advanced Graphics – Light Transport 27

Whitted, Cook & Beyond

Missing in Whitted:

• Area lights
• Glossy reflections
• Caustics
• Diffuse interreflections
• Diffraction
• Polarization
• Phosphorescence
• Temporal effects
• Motion blur
• Depth of field
• Anti-aliasing

Cook:  Area lights  Glossy reflections × Caustics × Diffuse interreflections × Diffraction × Polarization × Phosphorescence × Temporal effects  Motion blur  Depth of field  Anti-aliasing

Rendering Equation

Advanced Graphics – Light Transport 28

Whitted, Cook & Beyond

Cook’s solution to rendering: Sample the many-dimensional integral using Monte Carlo integration. න

𝐵𝑞𝑗𝑦𝑓𝑚

න

𝐵𝑚𝑓𝑜𝑡

න

𝑈𝑔𝑠𝑏𝑛𝑓

න

𝛻𝑕𝑚𝑝𝑡𝑡𝑧

න

𝐵𝑚𝑗𝑕ℎ𝑢

… Ray optics are still used for specular reflections and refractions: The ray tree is not eliminated.

(In fact: for each light, one or more shadow rays are produced)

Rendering Equation

Advanced Graphics – Light Transport 29

God’s Algorithm

1 room 1 bulb 100 watts 1020 photons per second Photon behavior:

• Travel in straight lines
• Get absorbed, or change direction:
• Bounce (random / deterministic)
• Get transmitted
• Leave into the void
• Get detected

Light Transport

Advanced Graphics – Light Transport 30

Rendering Equation

Advanced Graphics – Light Transport 31

God’s Algorithm - Mathematically

A photon may arrive at a sensor after travelling in a straight line from a light source to the sensor: 𝑀 𝑡 ← 𝑦 = 𝑀𝐹(𝑡 ← 𝑦) Or, it may be reflected by a surface towards the sensor: 𝑀 𝑡 ← 𝑦 = න

𝐵

𝑔

𝑠 𝑡 ← 𝑦 ← 𝑦′ 𝑀 𝑦 ← 𝑦′ 𝐻 𝑦 ↔ 𝑦′ 𝑒𝐵(𝑦′)

Those are the options. Adding direct and indirect illumination together: 𝑀 𝑡 ← 𝑦 = 𝑀𝐹 𝑡 ← 𝑦 + න

𝐵

𝑔

𝑠 𝑡 ← 𝑦 ← 𝑦′ 𝑀 𝑦 ← 𝑦′ 𝐻 𝑦 ↔ 𝑦′ 𝑒𝐵(𝑦′)

𝑀 𝑡 ← 𝑦 = 𝑀𝐹 𝑡 ← 𝑦 + න

𝐵

𝑔

𝑠 𝑡 ← 𝑦 ← 𝑦′ 𝑀 𝑦 ← 𝑦′ 𝐻 𝑦 ↔ 𝑦′ 𝑒𝐵(𝑦′)

Rendering Equation

Advanced Graphics – Light Transport 32

God’s Algorithm - Mathematically

Emission Hemisphere Reflection Indirect Geometry factor

Rendering Equation

Advanced Graphics – Light Transport 33

𝑀 𝑡 ← 𝑦 = 𝑀𝐹 𝑡 ← 𝑦 + න

𝐵

𝑔

𝑠 𝑡 ← 𝑦 ← 𝑦′ 𝑀 𝑦 ← 𝑦′ 𝐻 𝑦 ↔ 𝑦′ 𝑒𝐵(𝑦′)

The Rendering Equation:*

• Light transport from lights to sensor
• Physically based

The equation allows us to determine to which extend rendering algorithms approximate real-world light transport.

*: The Rendering Equation, Kajiya, 1986

Rendering Equation

Advanced Graphics – Light Transport 34

𝑀 𝑡 ← 𝑦 = 𝑀𝐹 𝑡 ← 𝑦 + න

𝐵

𝑔

𝑠 𝑡 ← 𝑦 ← 𝑦′ 𝑀 𝑦 ← 𝑦′ 𝐻 𝑦 ↔ 𝑦′ 𝑒𝐵(𝑦′)

Rasterization, according to the Rendering Equation:

• Visibility handled by z-buffer / Painter’s algorithm
• Visibility not supported for lights

𝑀 𝑡 ← 𝑦 = 𝑀𝐹 𝑡 ← 𝑦 + ෍

𝑗=1 𝑂𝑀

𝑔

𝑠 𝑡 ← 𝑦 ← 𝑀𝑗 𝑀𝑗 𝐻 𝑦 ↔ 𝑀𝑗

(note: this does not take into account approximations such as shadow maps and environment maps)

Rendering Equation

Advanced Graphics – Light Transport 35

𝑀 𝑡 ← 𝑦 = 𝑀𝐹 𝑡 ← 𝑦 + න

𝐵

𝑔

𝑠 𝑡 ← 𝑦 ← 𝑦′ 𝑀 𝑦 ← 𝑦′ 𝐻 𝑦 ↔ 𝑦′ 𝑒𝐵(𝑦′)

Whitted-style ray tracing, according to the Rendering Equation:

• Inter-surface reflections limited to specular surfaces
• Light sources limited to 𝑂 point lights
• Visibility is supported

𝑀 𝑡 ← 𝑦 = 𝑀𝐹 𝑡 ← 𝑦 + ෍

𝑗=1 𝑂𝑀

𝑔

𝑠 𝑡 ← 𝑦 ← 𝑀𝑗 𝑀𝑗 𝐻 𝑦 ↔ 𝑀𝑗

+ න

𝐵

𝑔

𝑠 𝜀 𝑡 ← 𝑦 ← 𝑦′ 𝑀 𝑦 ← 𝑦′ 𝐻 𝑦 ↔ 𝑦′ 𝑒𝐵(𝑦′)

(note: only specular recursive transport supported)

Today’s Agenda:

• Introduction
• The Rendering Equation
• Light Transport

Light Transport

Advanced Graphics – Light Transport 37

𝑀 𝑡 ← 𝑦 = 𝑀𝐹 𝑡 ← 𝑦 + න

𝐵

𝑔

𝑠 𝑡 ← 𝑦 ← 𝑦′ 𝑀 𝑦 ← 𝑦′ 𝐻 𝑦 ↔ 𝑦′ 𝑒𝐵(𝑦′)

Relation between real-world light transport and the RE:

• 1. Each sensor element registers an amount of photons arriving from the first surface visible

though that pixel.

• 2. This surface may be emissive, in which case it produced the sensed photons.
• 3. This surface may also reflect photons, arriving from other surfaces in the scene.
• 4. For the other surfaces: goto 2.

Light Transport

Advanced Graphics – Light Transport 38

Light Transport Quantities

Radiant flux - 𝛸 : “Radiant energy emitted, reflected, transmitted or received, per unit time.” Units: watts = joules per second 𝑋 = 𝐾 𝑡−1 . Simplified particle analogy: number of photons.

Note: photon energy depends on electromagnetic wavelength: E = hc

λ , where h is Planck’s constant, c is the speed of light,

and λ is wavelength. At λ = 550nm (yellow), a single photon carries 3.6 ∗ 10−19 joules.

Light Transport

Advanced Graphics – Light Transport 39

Light Transport Quantities

In a vacuum, radiant flux emitted by a point light source remains constant over distance: A point light emitting 100W delivers 100W to the surface of a sphere of radius r around the light. This sphere has an area of 𝜌𝑠2; energy per surface area thus decreases by 1/𝑠2. In terms of photons: the density of the photon distribution decreases by 1/𝑠2.

Light Transport

Advanced Graphics – Light Transport 40

Light Transport Quantities

A surface receives an amount of light energy proportional to its solid angle: the two-dimensional space that an object subtends at a point. Solid angle units: steradians (sr). Corresponding concept in 2D: radians; the length of the arc on the unit sphere subtended by an angle.

Light Transport

Advanced Graphics – Light Transport 41

Light Transport Quantities

Radiance - 𝑀 : “The power of electromagnetic radiation emitted, reflected, transmitted or received per unit projected area per unit solid angle.” Units: 𝑋𝑡𝑠−1𝑛−2 Simplified particle analogy: Amount of particles passing through a pipe with unit diameter, per unit time. Note: radiance is a continuous value: while flux at a point is 0 (since both area and solid angle are 0), we can still define flux per area per solid angle for that point. 𝑀

Light Transport

Advanced Graphics – Light Transport 42

Light Transport Quantities

Irradiance - 𝐹 : “The power of electromagnetic radiation per unit area incident on a surface.” Units: Watts per 𝑛2 = joules per second per 𝑛2 𝑋𝑛−2 = 𝐾𝑛−2𝑡−1 . Simplified particle analogy: number of photons arriving per unit area per unit time, from all directions. 𝑂

Light Transport

Advanced Graphics – Light Transport 43

Light Transport Quantities

Converting radiance to irradiance: 𝐹 = 𝑀 cos 𝜄 𝑀 𝑀 𝑂 𝜄

Light Transport

Advanced Graphics – Light Transport 44

Pinhole Camera

A camera should not accept light from all directions for a particular pixel on the film. A pinhole ensures that only a single direction is sampled. In the real world, an aperture with a lens is used to limit directions to a small range, but only on the focal plane.

Light Transport

Advanced Graphics – Light Transport 45

Light Transport

𝑀 𝑡 ← 𝑦 = 𝑀𝐹 𝑡 ← 𝑦 + න

𝐵

𝑔

𝑠 𝑡 ← 𝑦 ← 𝑦′ 𝑀 𝑦 ← 𝑦′ 𝐻 𝑦 ↔ 𝑦′ 𝑒𝐵(𝑦′)

𝑀𝑝 𝑦, 𝜕𝑝 = 𝑀𝐹 𝑦, 𝜕𝑝 + න

𝛻

𝑔

𝑠 𝑦, 𝜕𝑝, 𝜕𝑗 𝑀𝑗 𝑦, 𝜕𝑗 cos 𝜄𝑗 𝑒𝜕𝑗

Radiance Radiance Radiance Irradiance BRDF

Light Transport

Advanced Graphics – Light Transport 46

Bidirectional Reflectance Distribution Function

BRDF: function describing the relation between radiance emitted in direction 𝜕𝑝 and irradiance arriving from direction 𝜕𝑗: 𝑔

𝑠 𝜕𝑝, 𝜕𝑗 = d𝑀𝑝(𝜕𝑝)

d𝐹𝑗(𝜕𝑗) = d𝑀𝑝(𝜕𝑝) 𝑀𝑗 𝜕𝑗 cos θ𝑗 𝑒𝜕𝑗 = 𝑝𝑣𝑢𝑕𝑝𝑗𝑜𝑕 𝑠𝑏𝑒𝑗𝑏𝑜𝑑𝑓 𝑗𝑜𝑑𝑝𝑛𝑗𝑜𝑕 𝑗𝑠𝑠𝑏𝑒𝑗𝑏𝑜𝑑𝑓 Or, if spatially variant: 𝑔

𝑠 𝑦, 𝜕𝑝, 𝜕𝑗 = d𝑀𝑝(𝑦, 𝜕𝑝)

d𝐹𝑗(𝑦, 𝜕𝑗) = d𝑀𝑝(𝑦, 𝜕𝑝) 𝑀𝑗 𝑦, 𝜕𝑗 cos θ𝑗 𝑒𝜕𝑗 Properties:

• Should be positive: 𝑔

𝑠 𝜕𝑝, 𝜕𝑗 ≥ 0

• Helmholtz reciprocity should be obeyed: 𝑔

𝑠 𝜕𝑝, 𝜕𝑗 = 𝑔 𝑠 𝜕𝑗, 𝜕𝑝

• Energy should be conserved: ׬

𝛻 𝑔 𝑠 𝜕𝑝, 𝜕𝑗 cos 𝜄𝑝 𝑒𝜕𝑝 ≤ 1

Light Transport

Advanced Graphics – Light Transport 47

Bidirectional Reflectance Distribution Function

Diffuse / Lambert*:

• 1. Reflects irradiance equally in all directions.
• 2. Reflects irradiance with a cosine angular distribution.

Q: Which one is correct? A: Both:

• Outgoing light is distributed proportional to 𝑜 ∙ 𝜕𝑝 .
• But 𝑀 = 𝐹/(𝑜 ∙ 𝜕𝑝), and 𝑀 is therefore independent of 𝜕𝑝 .

Or, stated differently: At an angle, the solid angle becomes smaller, and therefore the radiance per unit solid angle increases proportional to the cosine of the angle.

*: See: http://www.oceanopticsbook.info/view/surfaces/lambertian_brdfs

𝒐 𝝏𝒋

Light Transport

Advanced Graphics – Light Transport 48

Bidirectional Reflectance Distribution Function

The diffuse BRDF is: 𝑔

𝑠 𝜕𝑝, 𝜕𝑗 = 𝑏𝑚𝑐𝑓𝑒𝑝

π So, for a total irradiance 𝐹 at surface point 𝑦, the

• utgoing radiance 𝑀𝑝 = 𝐹𝑗

𝑏𝑚𝑐𝑓𝑒𝑝 𝜌

. Why the 𝜌? Energy conservation: 𝐹𝑝 ≤ 𝐹𝑗 Suppose we have a directional light parallel to 𝑜, with intensity 1. Then: 𝐹𝑗 = 𝑀𝑗 = 1. Suppose our BRDF =

𝑏𝑚𝑐𝑓𝑒𝑝 1

. Then: 𝐹𝑝 = ׬

𝛻 𝐹𝑗 𝑔 𝑠(𝜕𝑝 , 𝜕𝑗) cos 𝜕𝑝 𝑒𝜕𝑝 = ׬ 𝛻 cos 𝜕𝑝 𝑒𝜕𝑝

Now: ׬

𝛻 cos 𝜕𝑝 𝑒𝜕𝑝 = ׬ ϕ=0 2𝜌 ׬ θ=0 𝜌/2 cos θ sin θ 𝑒θ𝑒ϕ = 𝜌  𝐹𝑝 = 𝜌 𝐹𝑗.

𝒐 𝝏𝒋

Light Transport

Advanced Graphics – Light Transport 49

Bidirectional Reflectance Distribution Function

Mirror / Perfect specular: Reflects light in a fixed direction. For a given incoming direction 𝜕𝑗, all light is emitted in a single infinitesimal set of directions. The specular BRDF is thus 𝑔

𝑠 𝑦, 𝜕𝑝, 𝜕𝑗 = ቊ∞, along reflected vector

0, otherwise. This is not practical, and therefore we will handle the pure specular case (reflection and refraction) separately.

𝒐 𝝏𝒑 𝝏𝒋

Light Transport

Advanced Graphics – Light Transport 50

Flux / Particle Count / Probability

So far, we measured light transport as energy: Radiance = Joules per second per square meter; Irradiance = Radiance projected to the surface. Alternatively, we can imagine this as a large number

• f particles (say, 1020). How does this affect the

concept of radiance, irradiance and BRDF? Alternatively, we can think of irradiance and the BRDF as a probability distribution. How does this affect the concept of irradiance and the BRDF?

Today’s Agenda:

• Introduction
• The Rendering Equation
• Light Transport

INFOMAGR – Advanced Graphics

Jacco Bikker - November 2017 - February 2018

END of “Light Transport”

next lecture: “Path Tracing”