The Light Field Light field radiance function on

The Light Field Light field = radiance function on rays Conservation of radiance Measurement equation Throughput and counting rays Conservation of throughput Area sources and irradiance Form factors and radiosity From London and Upton CS 348 B Lecture 5 Pat Hanrahan, 2005

Light Field = Radiance(Ray)

Field Radiance Definition: The field radiance (luminance) at a point in space in a given direction is the power per unit solid angle per unit area perpendicular to the direction Radiance is the quantity associated with a ray CS 348 B Lecture 5 Pat Hanrahan, 2005

Properties of Radiance

Properties of Radiance 1. Fundamental field quantity that characterizes the distribution of light in an environment. 1. Radiance is a function on rays 2. All other field quantities are derived from it 2. Radiance invariant along a ray. 1. 5 D ray space reduces to 4 D 3. Response of a sensor proportional to radiance. CS 348 B Lecture 5 Pat Hanrahan, 2005

1 st Law: Conversation of Radiance The radiance in the direction of a light ray remains constant as the ray propagates CS 348 B Lecture 5 Pat Hanrahan, 2005

1 st Law: Conversation of Radiance The radiance in the direction of a light ray remains constant as the ray propagates CS 348 B Lecture 5 Pat Hanrahan, 2005

Spherical Gantry 4 D Light Field Capture all the light leaving an object - like a hologram CS 348 B Lecture 5 Pat Hanrahan, 2005

Two-Plane Light Field 2 D Array of Cameras CS 348 B Lecture 5 2 D Array of Images Pat Hanrahan, 2005

Multi-Camera Array Light Field CS 348 B Lecture 5 Pat Hanrahan, 2005

Throughput Counts Rays Define an infinitesimal beam as the set of rays intersecting two infinitesimal surface elements T measures/count the number of rays in the beam CS 348 B Lecture 5 Pat Hanrahan, 2005

Conservation of Throughput n Throughput conserved during propagation n Number of rays conserved n Assuming no attenuation or scattering n n 2 (index of refraction) times throughput invariant under the laws of geometric optics n Reflection at an interface n Refraction at an interface n n Causes rays to bend (kink) Continuously varying index of refraction n Causes rays to curve; mirages CS 348 B Lecture 5 Pat Hanrahan, 2005

Conservation of Radiance is the ratio of two quantities: 1. Power 2. Throughput Since power and throughput are conserved, Radiance conserved CS 348 B Lecture 5 Pat Hanrahan, 2005

Quiz Does radiance increase under a magnifying glass? CS 348 B Lecture 5 Pat Hanrahan, 2005

Quiz Does radiance increase under a magnifying glass? No!! CS 348 B Lecture 5 Pat Hanrahan, 2005

Radiance: 2 nd Law The response of a sensor is proportional to the radiance of the surface visible to the sensor. Aperture Sensor L is what should be computed and displayed. T quantifies the gathering power of the device; the higher the throughput the greater the amount of light gathered CS 348 B Lecture 5 Pat Hanrahan, 2005

Quiz Does the brightness that a wall appears to the sensor depend on the distance? CS 348 B Lecture 5 Pat Hanrahan, 2005

Measuring Rays = Throughput

Throughput Counts Rays Define an infinitesimal beam as the set of rays intersecting two infinitesimal surface elements Measure/count the number of rays in the beam CS 348 B Lecture 5 Pat Hanrahan, 2005

Parameterizing Rays Parameterize rays wrt to receiver CS 348 B Lecture 5 Pat Hanrahan, 2005

Parameterizing Rays Parameterize rays wrt to source CS 348 B Lecture 5 Pat Hanrahan, 2005

Parameterizing Rays Tilting the surfaces reparameterizes the rays All these throughputs must be equal. CS 348 B Lecture 5 Pat Hanrahan, 2005

Projected Solid Angle CS 348 B Lecture 5 Pat Hanrahan, 2005

Parameterizing Rays: S 2 × R 2 Parameterize rays by Projected area Measuring the number or rays that hit a shape Sphere: CS 348 B Lecture 5 Pat Hanrahan, 2005

Parameterizing Rays: M 2 × S 2 Parameterize rays by Sphere: Crofton’s Theorem: CS 348 B Lecture 5 Pat Hanrahan, 2005

Incident Surface Radiance Definition: The incoming surface radiance (luminance) is the power per unit solid angle per unit projected area arriving at a receiving surface CS 348 B Lecture 5 Pat Hanrahan, 2005

Exitant Surface Radiance Definition: The outgoing surface radiance (luminance) is the power per unit solid angle per unit projected area leaving at surface Alternatively: the intensity per unit projected area leaving a surface CS 348 B Lecture 5 Pat Hanrahan, 2005

Irradiance from a Uniform Area Source

Irradiance from the Environment CS 348 B Lecture 5 Pat Hanrahan, 2005

Uniform Area Source CS 348 B Lecture 5 Pat Hanrahan, 2005

Uniform Disk Source Geometric Derivation CS 348 B Lecture 5 Algebraic Derivation Pat Hanrahan, 2005

Spherical Source Geometric Derivation CS 348 B Lecture 5 Algebraic Derivation Pat Hanrahan, 2005

The Sun Solar constant (normal incidence at zenith) Irradiance 1353 W/m 2 Illuminance 127, 500 lm/m 2 = 127. 5 kilolux Solar angle =. 25 degrees =. 004 radians (half angle) = 6 x 10 -5 steradians Solar radiance CS 348 B Lecture 5 Pat Hanrahan, 2005

Polygonal Source CS 348 B Lecture 5 Pat Hanrahan, 2005

Polygonal Source CS 348 B Lecture 5 Pat Hanrahan, 2005

Polygonal Source CS 348 B Lecture 5 Pat Hanrahan, 2005

Form Factors

Types of Throughput 1. Infinitesimal beam of rays (radiance) 2. Infinitesimal-finite beam (irradiance calc. ) 3. Finite-finite beam (radiosity calc. ) CS 348 B Lecture 5 Pat Hanrahan, 2005

Probability of Ray Intersection Probability of a ray hitting A’ given that it hits A CS 348 B Lecture 5 Pat Hanrahan, 2005

Another Formulation CS 348 B Lecture 5 Pat Hanrahan, 2005

Form Factor Probability of a ray hitting A’ given it hits A Form factor definition Form factor reciprocity CS 348 B Lecture 5 Pat Hanrahan, 2005

Radiosity Power transfer from a constant radiance source Set up system of equations representing power transfers between objects CS 348 B Lecture 5 Pat Hanrahan, 2005

Form Factors and Throughput Form factors represent the probability of ray leaving a surface intersecting another surface n Only a function of surface geometry Differential form factor n Irradiance calculations Form factors n Radiosity calculations (energy balance) CS 348 B Lecture 5 Pat Hanrahan, 2005