Paper Summaries Any takers? Material Properties Assignments - - PDF document

1 Material Properties

Paper Summaries

• Any takers?

Assignments

• Checkpoint 2

– Due next Wednesday – Any questions? – Having trouble? Stick around after lecture.

Projects

• Project feedback
• Approx 22 projects
• Listing of projects now on Web
• Presentation schedule

– Presentations (15 min max) – Last 3 classes (week 10 + finals week) – Sign up

• Email me with 1st , 2nd , 3rd choices
• First come first served.

Plan for today

• Material Properties

– Bi-directional reflectance distribution functions (BRDFs) – Illumination Models – Using Empirical Data Computer Graphics as Virtual Photography

camera (captures light) synthetic image camera model (focuses simulated lighting)

processing

photo processing tone reproduction real scene 3D models Photography: Computer Graphics: Photographic print

2

Shading

• Computing the light that leaves a point
• Shading point - point under investigation
• Illumination model - function or algorithm used to

describe the reflective characteristics of a given surface.

• Shading model – algorithm for using an illumination

model to determine the color of a point on a surface.

• For efficiency’s sake, most illumination models are

approximations.

Reflections

• Ambient – light uniformly incident from the

environment

• Diffuse – light scattered equally in all directions
• Ambient and Diffuse – color of material plays a

part

• Specular – highlights connected with mirrorness
• Specular – mostly color of light

Bi-directional Reflectance Functions (BRDF)

BRDF Example

Sun behind observer Sun opposite observer

Note: Hidden shadows

BRDF Example

Sun behind observer Sun opposite observer

Note: Specular highlights

BRDF Example

Sun behind observer Sun opposite observer

Note: Specular highlights Note: Hidden shadows

3

BRDF

• Bi-directional Reflectance Function

) , , , (

r r i i r

f BRDF θ φ θ φ =

At a given point, gives relative reflected illumination in any direction with respect to incoming illumination coming from any direction; Note: The θ’s are elevation, ϕ’s are measured about the surface normal. The i’s refer to the incident ray; the r’s to the reflected ray.

BRDF Geometry BRDF

• Can return any positive value.
• Generally wavelength specific.

) , , , , ( λ θ φ θ φ

r r i i r

f BRDF =

BRDF

• Simplifying Assumptions wrt the BRDF

– Light enters and leaves from the same point.

• Not necessarily true
• Subsurface scattering
• Skin, marble

– Light of a given wavelength will only reflect back light of that same wavelength

• Not necessarily true
• Light Interference
• Oily patches, peacock feathers

Illumination Models

• Illumination model - function or algorithm

used to describe the reflective characteristics of a given surface.

• Revise to…

– function or algorithm used in approximating the BRDF.

Illumination Models

• Illumination Models and Viewing Direction

– Generally, BRDFs are independent of viewing direction – Most Illumination models take viewing direction into consideration

4

Illumination Models

• Geometry

N H S V R

reflection viewer normal Half-way source

Illumination Models

• Geometry

– N - normal vector – S - direction of incoming light – R - direction of perfect mirror reflection – H - halfway between light direction and viewing direction. – V - viewing direction.

Illumination Models

• Recall from Linear Algebra

θ u v

θ cos v u v u =

• Just one reason to normalize!

Illumination Models

• Lambertian
• Phong
• Cook-Torrance
• Ward
• Many others!

Illumination Models

• BRDF Viewer

– bv by Szymon Rusinkiewicz (Princeton) – http://graphics.stanford.edu/~smr/brdf/bv – SGI, Linux, and Java versions although not readily available for Java. I have it, if you want it, and you’ll need to load Java3D as well!

Lambertian Model

• Lambert Model

– Perfectly diffuse surface – reflection is constant in all directions (kd) – Independent of viewer direction

5

Lambertian Model

• Lambert Model

θ cos ) (

d Sk

L V L = ) ( ) ( S N k L V L

d S

• =

Lambertian Model

• Lambert Model

– why cos θ? – Surface has differential area dA – Intensity varies with projected area on surface – Projected area = cos θ

Lambertian Model

• BRDF Viewer

http://graphics.stanford.edu/~smr/brdf/bv

Phong Model

• Phong Model

– introduces specular (mirror-like) reflections – Viewer direction becomes more important – three components

• ambient - background light (ka)
• diffuse - Lambertian reflection (kd)
• specular – mirror-like reflection(ks)

Phong Model

specular diffuse ambient

V) R ( N) S ( ) (

∑ ∑

• +
• +

=

i k i i s i i i d a a

e

L k L k L k V L

Note: Ln are radiance terms, include both light and material info

Phong-Blinn Model

– Uses halfway angle rather than reflected

specular diffuse ambient

V) H ( N) S ( ) (

∑ ∑

• +
• +

=

i k i i s i i i d a a

e

L k L k L k V L

6

Phong-Blinn Model

• BRDF Viewer

http://graphics.stanford.edu/~smr/brdf/bv

Cook-Torrance Model

– based on physics of a surface

• Actually developed by Torrance & Sparrow,

physicists.

• Jim Blinn was the first to apply to CG
• Cook & Torrance’s was the first complete

implementation

– components

• microfacet model - describes geometry of surface
• Fresnel term - describes reflectance
• Roughness - describes microfacet distribution.

Cook-Torrance Model

• Microfacets

– surface is composed of V shaped grooves (microfacets) – Light interactions with microfacets

• Reflect - causes specular reflections
• Scatter - causes diffuse reflections

Cook-Torrance Model

• Microfacets

Cook-Torrance Model

• Microfacets – GeometryTerm

– Some microfacets may shadow others

⎭ ⎬ ⎫ ⎩ ⎨ ⎧

• =

H) (V S) H)(N N ( 2 , H) (V V) H)(N N ( 2 , 1 min G

Note: S from before is the L in these diagrams

Cook-Torrance Model

• Fresnel Equation for polarized light

– Describes reflectance as a function of:

• Wavelength of incident light (λ)
• Index of refraction (η(λ))
• Extinction coefficient (ease at which wave can

penetrate a surface) (κ(λ))

• Angle of incidence (θ)

7

Cook-Torrance Model

• Fresnel equations for polarized light

θ θ θ θ

2 2 2 2 2 2

cos cos 2 cos cos 2 + + + + − + = a b a a b a Fs θ θ θ θ θ θ θ θ

2 2 2 2 2 2 2 2

tan sin tan sin 2 tan sin tan sin 2 + + + + − + = a b a a b a F F

s p

a, b are functions

• f η, κ, and θ

η, κ are functions

• f λ

p s

F F F 2 1 2 1 + =

F is total reflectance

Perpendicular component Parallel component

Cook-Torrance Model

• Fresnel

– If all quantities known, use Fresnel equations – If not, approximate using reflectance off normal

• See [Glassner] or [Cook/Torrance81] for details

Cook-Torrance Model

• Roughness

– Characterizes the distribution of the slopes of the microfacets – Roughness parameter, m

• m between 0 -1
• small m - smooth surface, specular reflectance
• large m - rough surface, diffuse reflectance

– Statistical models

Cook-Torrance Model

• Roughness

2

) / ( m

ce D

α −

=

α

α

4 2 ) / ) ((tan

cos

2

m e D

m −

=

Gaussian Model c is arbitrary constant Beekman Model

Cook-Torrance Model

• Roughness

Cook-Torrance Model

• Putting it all together

π 1 =

d

f

V) S)(N N ( 1

• ×

× = G D F fs π

diffuse specular d s r

df sf f + =

total reflectance

Where D is the roughness function, F is the Fresnel function, and G is the geometrical attenuation factor from previous pages

8

Cook-Torrance Model

• Complete Cook-Torrance Model

∑

• +

=

i i r i a a r

d f L R L L ϖ ) Si)( N (

• Parameters for fr:

m – roughness value Type of material (determines terms for Fresnel eqn) Wavelength of incident light (determines terms for Fresnel eqn) Diffuse / specular contribution constants La Ra is the ambient radiance reflected by Ra Li is the light’s radiance

Cook-Torrance Models

• examples

Cook-Torrance Models

• BRDF Viewer

http://graphics.stanford.edu/~smr/brdf/bv

Cook-Torrance Model

• Summary

– Complicated model based on physics – Components

• Microfacets
• Fresnel equation
• Roughness

– Want accuracy? Go to the source!

• Break.

Anisotropic Models

• Anisotropy

– Isotropic - surfaces reflect equally from any direction of view – Anisotropic - reflection varies not only with angle of incidence, but also with the angle of the incident light w.r.t some viewing angle.

• Surfaces considered to possess an intrinsic grain
• Examples: satin, velvet, hair, brushed aluminum

Anisotropic Models

• Anisotropic reflection -- example

9

Anisotropic Models

• Ward Model [Ward92]

– Designed for both accuracy and ease of use – Includes model for anisotropic reflection

Anisotropic Models

• Ward Model - Isotropic

specular 2 / ) (tan diffuse

) 4 cos cos 1 (

2 2

πα δ θ ρ π ρ ρ

α γ −

• +

= e

s d

Anisotropic Model

• Ward Model

– ρd - Diffuse reflectance coefficient (can vary with wavelength) – ρs - Specular reflectance coefficient (can vary with wavelength) – α - Standard deviation of surface slope

Anisotropic Models

• Ward Model -- anisotropic

specular )) / sin / (cos (tan diffuse

) 4 cos cos 1 (

2 2 2 2 2

y x s d

y x

e α πα δ θ ρ π ρ ρ

α φ α φ γ + −

• +

=

Anisotropic Models

• Ward Model w/ ansiotropy

– αx - Standard deviation of surface slope in x- direction – αy - Standard deviation of surface slope in y- direction

Anisotropic Models

• Ward Model - example

Photo Isotropic Anisotropic

10

Anisotropic Models

• Other anisotropic models (all based on

physics)

– [Kajia85] – [Poulin90] – [He91]

Illumination Models

• There are many other illumination models -

both empirical approximations and rigorous physically based solutions.

Measuring BRDFs

• Can use empirical data
• BRDFs measured using a goniometer
• See [Ward92]

Measuring BRDFs

Light Receptor

Measuring BRDFs

• Storage using spherical sampling

Measuring BRDFs

• BRDF Databases

– Cornell

• http://www.graphics.cornell.edu/online/measurements

– Columbia-Utrecht

• http://www.cs.columbia.edu/CAVE/curet

11

Measuring BRDFs

• Problems with measured BRDFs

– Large – Difficult to control measuring device – Can be noisy, due to measurement device

Summary

• BRDFs - defines reflection off surface in

each direction as result from light arriving at each direction.

• Illumination models - approximations to

BRDF

• Can use measured BRDFs

Summary

• BRDF
• Approximations

– Lambert Model – Phong Model – Cook-Torrance (heavy on the physics) – Anisotropic Reflection

• Empirical Data

Further Reading

• Glassner, Principles of Digital Image

Synthesis, Chapter 15.

• See paper list (on Web) for papers on

individual models

– [Cook81] – [Ward92] [Kajiya85] [Poulin90][He91] – [Strauss90]