The Rendering Equation Direct ( local ) illumination Light directly - - PDF document

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CS348B Lecture 13 Pat Hanrahan, Spring 2009

The Rendering Equation

Direct (local) illumination

Light directly from light sources No shadows

Indirect (global) illumination

Hard and soft shadows Diffuse interreflections (radiosity) Glossy interreflections (caustics)

CS348B Lecture 13 Pat Hanrahan, Spring 2009

Radiosity

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CS348B Lecture 13 Pat Hanrahan, Spring 2009

Lighting Effects

Hard Shadows Soft Shadows Caustics Indirect Illumination

CS348B Lecture 13 Pat Hanrahan, Spring 2009

Challenge

To evaluate the reflection equation the incoming radiance must be known To evaluate the incoming radiance the reflected radiance must be known

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CS348B Lecture 13 Pat Hanrahan, Spring 2009

To The Rendering Equation Questions

• 1. How is light measured?
• 2. How is the spatial distribution of light energy

described?

• 3. How is reflection from a surface

characterized?

• 4. What are the conditions for equilibrium flow
• f light in an environment?

CS348B Lecture 13 Pat Hanrahan, Spring 2009

The Grand Scheme

Volume Rendering Equation Surface Rendering Equation Light and Radiometry Radiosity Equation Energy Balance

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Energy Balance

CS348B Lecture 13 Pat Hanrahan, Spring 2009

Balance Equation

Accountability [outgoing] - [incoming] = [emitted] - [absorbed]

Macro level

The total light energy put into the system must equal the energy leaving the system (usually, via heat).

Micro level

The energy flowing into a small region of phase space must

equal the energy flowing out.

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CS348B Lecture 13 Pat Hanrahan, Spring 2009

Surface Balance Equation

[outgoing] = [emitted] + [reflected]

CS348B Lecture 13 Pat Hanrahan, Spring 2009

Direction Conventions

BRDF Surface vs. Field Radiance

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CS348B Lecture 13 Pat Hanrahan, Spring 2009

Surface Balance Equation

[outgoing] = [emitted] + [reflected] + [transmitted]

BTDF

CS348B Lecture 13 Pat Hanrahan, Spring 2009

Two-Point Geometry

Ray Tracing

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CS348B Lecture 13 Pat Hanrahan, Spring 2009

Coupling Equations

Invariance of radiance

CS348B Lecture 13 Pat Hanrahan, Spring 2009

The Rendering Equation

Directional form Integrate over hemisphere of directions Transport operator i.e. ray tracing

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CS348B Lecture 13 Pat Hanrahan, Spring 2009

The Rendering Equation

Surface form Integrate over all surfaces Geometry term Visibility term

CS348B Lecture 13 Pat Hanrahan, Spring 2009

The Radiosity Equation

Assume diffuse reflection 1. 2.

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Integral Equations

CS348B Lecture 13 Pat Hanrahan, Spring 2009

Integral Equations

Integral equations of the 1st kind Integral equations of the 2nd kind

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CS348B Lecture 13 Pat Hanrahan, Spring 2009

Linear Operators

Linear operators act on functions like matrices act on vectors They are linear in that Types of linear operators

CS348B Lecture 13 Pat Hanrahan, Spring 2009

Solving the Rendering Equation

Rendering Equation Solution

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CS348B Lecture 13 Pat Hanrahan, Spring 2009

Formal Solution

Neumann series Verify

CS348B Lecture 13 Pat Hanrahan, Spring 2009

Successive Approximations

Successive approximations Converged

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CS348B Lecture 13 Pat Hanrahan, Spring 2009

Successive Approximation

Paths

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CS348B Lecture 13 Pat Hanrahan, Spring 2009

Light Path

CS348B Lecture 13 Pat Hanrahan, Spring 2009

Light Path

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CS348B Lecture 13 Pat Hanrahan, Spring 2009

Light Paths

CS348B Lecture 13 Pat Hanrahan, Spring 2009

Light Transport

Integrate over all paths of all lengths Question:

How to sample space of paths?

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CS348B Lecture 13 Pat Hanrahan, Spring 2009

Classic Ray Tracing

Forward (from eye): E S* (D|G) L

From Heckbert

CS348B Lecture 13 Pat Hanrahan, Spring 2009

Photon Paths

From Heckbert Caustics Radiosity

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CS348B Lecture 13 Pat Hanrahan, Spring 2009

How to Solve It?

Finite element methods

Classic radiosity

Mesh surfaces Piecewise constant basis functions Solve matrix equation

Not practical for rendering equation

Monte Carlo methods

Path tracing (distributed ray tracing) Bidirectional ray tracing Photon mapping