Global Illumination

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Outline

• Light transport• Radiometric units

• Light received and reflected• Bidirectional reflectance distribution function

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Light Transport• Different energy transfer

processes:

• Conduction

• Convection

• Radiation

• Light transport is energy transfer.

• Through radiation.

• Through visible wavelength.

x

ωo

ωi

ωi

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Radiometric Units

• Radiant energy:• The amount of energy transferred through

radiation.

• Measured in Joules (J)

• Radiant flux or power (Φ):• Energy transferred within a unit time interval.

• Measured in Watts (W)

• Radiant intensity (I):• Power transferred along a given direction (per

unit solid angle).

• Measured in Watts per steradian (W/sr)

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Radiometric Units (Cont’d)

• Irradiance (E):• Incident radiant power on a unit projected area.

• Measured in Watts per square meter (W/m2).

• Radiosity or radiant exitance (B):• Exitant radiant power on a unit projected area.

• Also measured in W/m2.

• Radiance (L):• The power passes through a unit projected

surface area along a unit solid angle.

• Measured in W/m2/sr.

Solid Angle

• Area on the unit sphere that is spanned

• by a set of directions• Unit: steradian

• Directions usually denoted by

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Angle and Solid Angle

• Angle

• Unit circle has ? radian

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rl / • Solid angle

• Unit sphere has ?

2/ RA

Spherical Coordinate

• Unit direction vector in spherical coordinate

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Differential solid angle

• The differential area on the unit sphere around direction ω

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Differential solid angle

• Integrating the differential solid angle over the unit sphere yields the surface area of the sphere

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Differential solid angle

• Differential solid angle spanned by Differential area

at distance

and angle

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dA

r

2

cos

r

dAd

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Irradiance from a Single

Source• Assume the source is a

point light, which sends light uniformly to all directions.

• Less irradiance is received when surface is far away:

• E ~ 1/r2

• Maximum irradiance is received when surface is facing the light:

• E ~ n·l

r

nl

2

2

r

IE

r

AIA

Eln

ln

A

ω

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Energy Reflected• Bidirectional reflectance

distribution function:

• Depicts how a surface reflects light.

• Measures the ratio between the radiance reflected in direction ωo and that incident from direction ωi.

• BRDF is a 4D function

• 2 angles for input direction

• 2 for output direction.

iii

oooooiir

L

Lf

,

,,,,

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Properties of BRDF• Energy conservation:

• Total output energy is less than total input.

• Helmholtz reciprocity principle:• Stay the same if

travel direction is reversed.

• Isotropy vs. Anisotropic:• A BRDF is isotropy if

it stays the same when surface is rotated around the normal

• 3D vs. 4D function

1221 rr ff

1cos,,,

ooooiir df

2211

2211

,,,

,,,

r

r

f

f

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Idea Diffuse Surface• An idea diffuse surface

reflects equal amount of light to all directions:

• BRDF is a constant.

• The total energy reflected is a fraction (ρ) of total input.

• ρ is called albedo.

• kd is the diffuse parameter.

dooiir kf ,,,

L

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Relation between Diffuse

Parameter and Albedo

d

d

oood

ooood

ooooiir

k

k

dk

ddk

df

2

0

2

0

2

0

sincos2

sincos

cos,,,

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Idea Mirror• An idea mirror reflect

light from one direction to another.

• BRDF is 0 except where θo=θi & φo=φi+π.

• Can be defined using Dirac delta function (δ).

L

ioiom

ooiir

k

f ,,,

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Relation between Mirror

Parameter and Albedo

i

m

im

ooooioiom

ooooiir

k

k

ddk

df

cos

cos

sincos

cos,,,

2

0

2

0

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Rendering Equation for

Global Illumination

iiiioiroeoo dLfLL cos,,,,

xxxx

radiance Incoming:,

BRDF:,

radiance Emitted:,

radiance Outgoing:,

ii

oir

oe

oo

L

f

L

L

x

x

x

x

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Color Bleeding• Also called diffuse inter-

reflection.

• The light path:

• LD*E

Radiosity

• Based on heat transfer theory

• ssume diffuse reflections on all surfaces

• Solve for radiant exitance, or radiosity,(power per area) instead of radiance

• No directional variation, view independent

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Radiosity equation

• Radiosity:

• Diffuse BRDF

• Emitted Radiosity

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Finite Element Approach

• Express Solution as a weighted sum of basis functions

• Simplest Approach: constant basis function on a triangular mesh

• Solve for unknown coefficients

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Finite Element Approach

• Substitute into radiosityequation, we get

• Form factor

with area of triangles

• Form factor is fraction of light leaving triangle i incident on triangle j

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Intuition

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Intuition

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Intuition

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Intuition

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Intuition

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Matrix formulation

• Matrix form of

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Radiosity algorithm

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Form factors

• Area-to-area form factor

• Area to point

• Sample area-to-point form factors as an approximation to area-to-area factors

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Meshing

© Cohen and Wallace

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Meshing

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Meshing

• Increase mesh resolution

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Meshing

• Adaptive Meshing

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Gathering

• Rows of F times B

• Calculate one row of F and discard

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Examples

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Examples

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Examples

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Radosity summary

• Diffuse reflection only

• Finite element approach

• Yields linear system of equations

• Major challenges• Meshing

• Form factor computation

• Solution of linear system

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