[PPT] - Annoucements Next Thursday: Chris software overview Reading: PowerPoint Presentation

SLIDE 1

Annoucements

Next Thursday: Chris software overview
Reading:

Henrik Wann Jensen, "Global Illumination using Photon Maps," In "Rendering Techniques '96". Eds. X. Pueyo and P. Schrder. Springer-Verlag, pp. 21-30, 1996 http://graphics.ucsd.edu/~henrik/papers/ewr7/ [web page] http://graphics.ucsd.edu/~henrik/papers/ewr7/egwr96.pdf [paper]

Photon Mapping

SIGGRAPH 2002 Course Notes

http://www.cs.cmu.edu/afs/cs.cmu.edu/academic/class/15864-

s04/www/slides/photonMappingCourse02.pdf

SLIDE 2

Local Illumination, Reflection, and BRDFs

Adapted from… Szymon Rusinkiewicz Princeton University C0S 526, Fall 2002

SLIDE 3

Overview

Radiometry and Photometry
Definition of BRDF
BRDF properties and common BRDFs
Rendering equation

SLIDE 4

Radiometric Units

Light is a form of energy – measured in Joules (J)
Power: energy per unit time

Measured in Joules/sec = Watts (W) Also called Radiant Flux (Φ)

SLIDE 5

Total radiant flux in Watts
How to define angular dependence?

Solid angle

Power per unit solid angle

Measured in Watts per steradian (W/sr)

Point Light Source

Area dA Solid angle dω = dA/r2

SLIDE 6

Light Falling on a Surface

Power per unit area – Irradiance (E)

Measured in W/m2

Move surface away from light

Inverse square law: E ~ 1/r2

Tilt surface away from light

Cosine law: E ~ n • l

SLIDE 7

Light Emitted from a Surface

Power per unit area per unit solid angle –

Radiance (L)

Measured in W/m2/sr Projected area – perpendicular to given direction dω dA

ω d dA d L Φ =

SLIDE 8

Total Light Emitted from a Surface

Radiance integrated over all directions
Called Radiosity (B)

Measured in W/m2

Ω

= ω θ φ θ d L B

cos

) , (

SLIDE 9

Radiometry vs. Photometry

These are all physical (radiometric) units
Don’t take perception into account
Eye sensitive to different colors

λ (nm)

400

(blue)

700

(red)

SLIDE 10

Photometric Units

Take human perception into account
Original unit: candle

Luminous intensity equal to a “standard candle”

Today: one of the basic SI units

One candela (cd) is the luminous intensity of a source producing 1/683 W at 555 nm (yellow-green).

SLIDE 11

Radiometric and Photometric Units

Luminosity Lux = lm / m2 Radiosity (B) W / m2 Luminance Nit = lm / m2 / sr Radiance (L) W / m2 / sr Illuminance Lux = lm / m2 Irradiance (E) W / m2 Luminous intensity Candela (cd) Radiant intensity (I) W / sr Luminous power Lumen (lm) = talbots / sec Radiant flux or power (Φ) Watt (W) = J / sec Luminous energy Talbot Radiant energy Joule (J)

SLIDE 12

Direct Illumination (i.e., Irradiance)

A E Φ =

A

SLIDE 13

Direct Illumination

A E Φ = ω I = Φ

Ι ω A

SLIDE 14

Direct Illumination

A E Φ = n ˆ l ˆ

r

ω I = Φ

2

) ˆ ˆ ( r A l n⋅ = ω

2

) ˆ ˆ ( r I E l n⋅ =

Ι

A

SLIDE 15

Imaging

Surface Lens Image Plane

(film, CCD)

SLIDE 16

Imaging

Area Asurf Radiance L Area Aaperture

surf

A L I =

2

surf aperture surf

d A A L = Φ

dsurf

SLIDE 17

Imaging

Area Asurf Radiance L Area Aaperture Area Aimg

surf

A L I =

dimg

img

A E Φ =

2

surf aperture surf

d A A L = Φ

dsurf

SLIDE 18

Punch line: cameras “see” radiance

Depends only

n camera

Imaging

surf

A L I =

2 surf aperture surf

d A A L = Φ

img

A E Φ =

img surf surf aperture

A d A A L E

2

=

2

=

img surf img surf

d d A A

2 img aperture

d A L E =

SLIDE 19

Surface Reflectance – BRDF

Bidirectional Reflectance Distribution Function
4-dimensional function: also written as

(the symbol ρ is also used sometimes)

) ( ) ( ) (

i i

i

r

dE dL f ω ω ω ω = → ) , ( ) , ( ) , , , (

i i i

i

i r

dE dL f ϕ θ ϕ θ ϕ θ ϕ θ =

SLIDE 20

Defining Surface Reflectance

Why is BRDF defined in this way?
Key point: BRDF is a differential quantity,

so limit must exist

Source Φsrc ωsrc Detector Φdet ωdet

SLIDE 21

Definition of BRDF

First attempt:

Source Φsrc ωsrc Detector Φdet ωdet

src det r

f Φ Φ =

SLIDE 22

Definition of BRDF

Should fr vary with ωsrc? No.

Source Φsrc ωsrc Detector Φdet ωdet

SLIDE 23

Definition of BRDF

Should fr vary with ωdet? Yes.

Source Φsrc ωsrc Detector Φdet ωdet

SLIDE 24

Definition of BRDF

Thus,

src det det r

f Φ Φ = ω

Source Φsrc ωsrc Detector Φdet ωdet

SLIDE 25

Definition of BRDF

What about surface area?

fr must be independent of surface area

Source Φsrc ωsrc Detector Φdet ωdet dA

SLIDE 26

Definition of BRDF

( )

E L dA dA f

src det det r

= Φ ⋅ Φ = ω

dA Source Φsrc ωsrc Detector Φdet ωdet

SLIDE 27

Energy conservation:
Helmholtz reciprocity:

(not always obeyed by “BRDFs” used in graphics)

Properties of the BRDF

1 cos ) , , , ( ≤

Ω
i

i r

d f ω θ ϕ θ ϕ θ ) ( ) (

i

r
i

r

f f ω ω ω ω → = →

SLIDE 28

Isotropy

A BRDF is isotropic if it stays the same when

surface is rotated around normal

Isotropic BRDFs are 3-dimensional functions:

) , , (

i
i

r

f ϕ ϕ θ θ −

SLIDE 29

Anisotropy

Anisotropic BRDFs do depend on surface rotation

SLIDE 30

Diffuse

The simplest BRDF is “ideal diffuse” or

Lambertian: just a constant

Note: does not include cos(θi )

Remember definition of irradiance

d

i

r

k f = → ) ( ω ω

SLIDE 31

Diffuse BRDF

Assume BRDF reflects a fraction ρ of light
The quantity ρ is called the albedo

π ρ ρ π ρ θ θ θ π ρ ϕ θ θ θ ρ ω θ ω ω

π π

θ π ϕ θ

= ∴ = = = = →

∈

∈ ∈ Ω Lambertian r d

d
d
i

Lambertian r

f k d k d d k d f

, ] .. [ ] 2 .. [ ] .. [ ,

2 2

cos sin 2 sin cos cos ) (

SLIDE 32

Ideal Mirror

All light incident from one direction is reflected

into another

BRDF is zero everywhere except where

π ϕ ϕ θ θ + = =

i

i

SLIDE 33

Ideal Mirror

To conserve energy,
So, BRDF is a delta function at direction of ideal

mirror reflection

1 cos ) (

,

= →

Ω
i

Mirror r

d f ω θ ω ω ) cos( ) ( ) (

, i

i
i

Mirror r

f θ ϕ ϕ δ θ θ δ − − =

SLIDE 34

Glossy Reflection

Non-ideal specular reflection
Most light reflected near ideal mirror direction

SLIDE 35

Phong BRDF

Phenomenological model for glossy reflection
Exponent n determines width of specular lobe
Constant ks determines size of lobe

n s Phong r

r l k f ) ˆ ˆ (

,

⋅ =

l is a vector to the light source r is the direction of mirror reflection

SLIDE 36

Torrance-Sparrow BRDF

Physically-based BRDF model

Originally used in the physics community Adapted by Cook & Torrance and Blinn for graphics

Assume surface consists of tiny “microfacets”

with mirror reflection off each

i

S T r

DGF f θ θ π cos cos

,

=

−

SLIDE 37

Torrance-Sparrow BRDF

D term is distribution of microfacets

(i.e., how many are pointing in each direction)

Beckmann distribution

β

β 4 2 ] / ) [(tan

cos 4

2

m e D

m −

=

β is angle between n and h h is halfway between l and v m is “roughness” parameter l n h v

SLIDE 38

Torrance-Sparrow BRDF

G term accounts for self-shadowing
⋅

⋅ ⋅ ⋅ ⋅ ⋅ = ) ( ) )( ( 2 , ) ( ) )( ( 2 , 1 min h v l n h n h v v n h n G

SLIDE 39

Torrance-Sparrow BRDF

F term is Fresnel term – reflection from an ideal

smooth surface (solution of Maxwell’s equations)

Consequence: most surfaces reflect (much) more

strongly near grazing angles

Metal

(note behavior at Brewster’s angle)

Dielectric

SLIDE 40

Aside: Brewster’s Angle

SLIDE 41

Other BRDF Features

BRDFs for dusty surfaces scatter light towards

grazing angles

SLIDE 42

Other BRDF Features

Retroreflection: strong reflection back towards

the light source

Can arise from bumpy diffuse surfaces
… or from corner reflectors

SLIDE 43

BRDF Representations

Physically-based vs. phenomenological models
Measured data
Desired characteristics:

Fast to evaluate Maintain reciprocity, energy conservation For global illumination: easy to importance sample

SLIDE 44

Beyond BRDFs

So far, have assumed 4D BRDF
Function of wavelength: 5D
Fluorescence (absorb at one wavelength,

emit at another): 6D

Phosphorescence (absorb now, emit later): 7D
Temporal dependence: 8D
Spatial dependence: 10D
Subsurface scattering: 12D
Polarization
Wave optics effects (diffraction, interference)
…

SLIDE 45

Rendering Equation

ω θ ω ω ω ω ω d x L x f x L x L

i i i

i

r

e
cos

) , ( ) , ( ) , ( ) , ( → + =

Ω

Outgoing radiance Emitted radiance BRDF Irradiance

SLIDE 46

Rendering Equation

Originally expressed by [Kajiya 1986] as

x x′ x′′

→

→ → + → = →

S r e

dA x x V x x I x x x f x x G x x I x x I ) ' , ( ) ' ( ) ' ' ' ( ) ' ' , ' ( ) ' ' ' ( ) ' ' ' (

SLIDE 47

Rendering Equation

Originally expressed by [Kajiya 1986] as
Integral is over all points in the scene
G(x,x’) is a geometry term:
→

→ → + → = →

S r e

dA x x V x x I x x x f x x G x x I x x I ) ' , ( ) ' ( ) ' ' ' ( ) ' ' , ' ( ) ' ' ' ( ) ' ' ' (

2

' cos cos ) ' , ( x x x x G

i

− ′ = θ θ

SLIDE 48

Rendering Equation

Originally expressed by [Kajiya 1986] as
Integral is over all points in the scene
V(x,x’) is a visibility term and is either 0 or 1
→

→ → + → = →

S r e

dA x x V x x I x x x f x x G x x I x x I ) ' , ( ) ' ( ) ' ' ' ( ) ' ' , ' ( ) ' ' ' ( ) ' ' ' (

SLIDE 49

Rendering Equation

Originally expressed by [Kajiya 1986] as
Integral is over all points in the scene
I(x→x’) is the two-point transport intensity:

(note: this is not the same I we’ve seen before…)

→

→ → + → = →

S r e

dA x x V x x I x x x f x x G x x I x x I ) ' , ( ) ' ( ) ' ' ' ( ) ' ' , ' ( ) ' ' ' ( ) ' ' ' ( ' ) ' , ( ) , ( ) ' ( dA dA x x G x L x x I ω = →

SLIDE 50

Rendering Equation

Next 3-4 weeks in the course: ways to solve the