global illumination - vis center · · 2010-04-21unit solid angle). •measured in watts ......
TRANSCRIPT
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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.
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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
7
rl / • Solid angle
• Unit sphere has ?
2/ RA
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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
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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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