The rendering equation
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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). Early Radiosity. Lighting Effects. Hard Shadows.

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The Rendering Equation

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The rendering equation

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)

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Early radiosity

Early Radiosity

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Lighting effects

Lighting Effects

Hard Shadows

Soft Shadows

Caustics

Indirect Illumination

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Challenge

Challenge

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

  • To evaluate the incoming radiance the reflected radiance must be known

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


To the rendering equation

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 of light in an environment?

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


The grand scheme

The Grand Scheme

Light and Radiometry

Energy Balance

Surface

Rendering Equation

Volume

Rendering Equation

Radiosity Equation

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Balance equation

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.

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Surface balance equation

Surface Balance Equation

  • [outgoing] = [emitted] + [reflected]

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Direction conventions

Direction Conventions

Surface vs. Field Radiance

BRDF

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Surface balance equation1

Surface Balance Equation

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

BTDF

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Two point geometry

Two-Point Geometry

Ray Tracing

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Coupling equations

Coupling Equations

Invariance of radiance

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


The rendering equation1

The Rendering Equation

  • Directional form

Transport operator

i.e. ray tracing

Integrate over

hemisphere of

directions

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


The rendering equation2

The Rendering Equation

  • Surface form

Geometry term

Integrate over

all surfaces

Visibility term

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


The radiosity equation

The Radiosity Equation

  • Assume diffuse reflection

    1.

    2.

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Integral equations

Integral Equations

  • Integral equations of the 1st kind

  • Integral equations of the 2nd kind

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Linear operators

Linear Operators

  • Linear operators act on functions like matrices act on vectors

  • They are linear in that

  • Types of linear operators

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Rendering operators

Rendering Operators

  • Scattering operator

  • Transport operator

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Solving the rendering equation

Solving the Rendering Equation

  • Rendering Equation

  • Solution

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Formal solution

Formal Solution

  • Neumann series

  • Verify

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Successive approximations

Successive Approximations

  • Successive approximations

  • Converged

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Successive approximation

Successive Approximation

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Light path

Light Path

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Light path1

Light Path

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Light paths

Light Paths

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Light transport

Light Transport

  • Integrate over all paths of all lengths

  • Question:

    • How to sample space of paths?

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Classic ray tracing

Classic Ray Tracing

  • Forward (from eye): E S* (D|G) L

From Heckbert

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


Photon paths

Photon Paths

Radiosity

Caustics

From Heckbert

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


How to solve it

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)

      • Randomly trace ray from the eye

    • Bidirectional ray tracing

    • Photon mapping

University of Texas at Austin CS395T - Advanced Image Synthesis Spring 2007 Don Fussell


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