Computer Graphics - BRDFs & Texturing - Hendrik Lensch - - PowerPoint PPT Presentation

Computer Graphics WS07/08 – BRDFs and Texturing

Computer Graphics

• BRDFs & Texturing -

Hendrik Lensch

Computer Graphics WS07/08 – BRDFs and Texturing

Overview

• Last time

– Radiance – Light sources – Rendering Equation & Formal Solutions

• Today

– Bidirectional Reflectance Distribution Function (BRDF) – Reflection models – Projection onto spherical basis functions – Shading

• Next lecture

– Varying (reflection) properties over object surface: texturing

Computer Graphics WS07/08 – BRDFs and Texturing

Reflection Equation - Reflectance

• Reflection equation
• BRDF

– Ratio of reflected radiance to incident irradiance

∫

+

Ω

=

i i i i

• i

r

• ω

d θ x L x f x L cos ) , ( ) , , ( ) , ( ω ω ω ω

) , ( ) , ( ) , , (

i i

• i
• r

x dE x dL x f ω ω ω ω =

Computer Graphics WS07/08 – BRDFs and Texturing

Bidirectional Reflectance Distribution Function

• BRDF describes surface reflection for light incident from direction

(θi,φi) observed from direction (θο,φο)

• Bidirectional

– Depends on two directions and position (6-D function)

• Distribution function

– Can be infinite

• Unit [1/sr]

i i i i

• i

i

• i
• r

d x dL x dL x dE x dL x f ω θ ω ω ω ω ω ω cos ) , ( ) , ( ) , ( ) , ( ) , , ( = =

Computer Graphics WS07/08 – BRDFs and Texturing

BRDF Properties

• Helmholtz reciprocity principle

– BRDF remains unchanged if incident and reflected directions are interchanged

• Smooth surface: isotropic BRDF

– reflectivity independent of rotation around surface normal – BRDF has only 3 instead of 4 directional degrees of freedom

) , ( ) , (

• i

r i

• r

f f ω ω ω ω = ) , , , (

i

• i

r x

f ϕ ϕ θ θ −

Computer Graphics WS07/08 – BRDFs and Texturing

BRDF Properties

• Characteristics

– BRDF units [sr--1]

• Not intuitive

– Range of values:

• From 0 (absorption) to ∞ (reflection, δ -function)

– Energy conservation law

• No self-emission
• Possible absorption

– Reflection only at the point of entry (xi = xo)

• No subsurface scattering

ϕ θ ω θ ω ω , 1 cos ) , , ( ∀ ≤

∫

Ω

• i
• r

d x f

Computer Graphics WS07/08 – BRDFs and Texturing

BRDF Measurement

• Gonio-Reflectometer
• BRDF measurement

– point light source position (θ,ϕ) – light detector position (θo ,ϕo)

• 4 directional degrees of freedom
• BRDF representation

– m incident direction samples (θ,ϕ) – n outgoing direction samples (θo ,ϕo) – m*n reflectance values (large!!!)

Stanford light gantry

Computer Graphics WS07/08 – BRDFs and Texturing

Reflectance

• Reflectance may vary with

– Illumination angle – Viewing angle – Wavelength – (Polarization, ...)

• Variations due to

– Absorption – Surface micro-geometry – Index of refraction / dielectric constant – Scattering

Magnesium; λ=0.5μm Aluminum; λ=0.5μm Aluminum; λ=2.0μm

Computer Graphics WS07/08 – BRDFs and Texturing

BRDF Modeling

• Phenomenological approach

– Description of visual surface appearance

• Ideal specular reflection

– Reflection law – Mirror

• Glossy reflection

– Directional diffuse – Shiny surfaces

• Ideal diffuse reflection

– Lambert’s law – Matte surfaces

Computer Graphics WS07/08 – BRDFs and Texturing

Reflection Geometry

• Direction vectors (normalize):

– N: surface normal – I: vector to the light source – V: viewpoint direction vector – H: halfway vector

H= (I + V) / |I + V|

– R(I): reflection vector

R(I)= I - 2(I•N)N

– Tangential surface: local plane

R(I) R(V) H V I N I I

• (

(I I•

• N

N) )N N

• (

(I I•

• N

N) )N N N R(I) I N R(I) V H R(V)

Top view

Computer Graphics WS07/08 – BRDFs and Texturing

Ideal Specular Reflection

• Angle of reflectance equal to angle of incidence
• Reflected vector in a plane with incident ray and

surface normal vector

θ θo N R I θ = θo ϕ = ϕo + 180° ϕo ϕ

I

R+(-I) = 2 cosθ N = -2(I• N) N R(I) = I - 2(I• N) N

Computer Graphics WS07/08 – BRDFs and Texturing

Mirror BRDF

• Dirac Delta function δ(x)

– δ(x) : zero everywhere except at x=0 – Unit integral iff integration domain contains zero (zero otherwise)

• Specular reflectance ρs

– Ratio of reflected radiance in specular direction and incoming radiance – Dimensionless quantity between 0 and 1

) , ( ) ( cos ) , ( ) , , ( ) , ( ) ( cos ) cos (cos ) ( ) , , (

, ,

π ϕ θ θ ρ ω θ ϕ θ ω ω ω π ϕ ϕ δ θ θ θ δ θ ρ ω ω ± = = ± − ⋅ − ⋅ =

∫

+

Ω

• i

i s i i i i i i

• m

r

• i

i

• i

i s i

• m

r

L d L x f x L x f

L N R

θo θi

( ) ( ) ( )

i i

• i

s

θ Φ θ Φ = θ ρ

Computer Graphics WS07/08 – BRDFs and Texturing

Diffuse Reflection

• Light equally likely to be reflected in any output

direction (independent of input direction)

• Constant BRDF

– kd: diffuse coefficient, material property [1/sr]

E k d x L k d x L k x L k x f

d i i i i d i i i i d

• d

i

• d

r

= = = = =

∫ ∫

Ω Ω

ω θ ω ω θ ω ω ω ω cos ) , ( cos ) , ( ) , ( const ) , , (

,

N I Lo= const

Computer Graphics WS07/08 – BRDFs and Texturing

• Radiosity
• Diffuse Reflectance
• Lambert’s Cosine Law
• For each light source

– Lr,d = kd Li cosθi = kd Li (I•N)

Lambertian Diffuse Reflection

d d

k E B π ρ = =

θi N I

L Lr,d

r,d

• L

d L d x L B cos cos ) , ( π ω θ ω θ ω = = =

∫ ∫

Ω Ω

i i d d

E E B θ ρ ρ cos = =

Computer Graphics WS07/08 – BRDFs and Texturing

Lambertian Objects

Ω ⋅ ∝ Φ d L0

Self-Luminous spherical Lambertian Light Source dΩ

Ω ⋅ ⋅ ∝ Φ d L ϕ cos

1

Eye-light illuminated Spherical Lambertian Reflector dΩ ϕ

Computer Graphics WS07/08 – BRDFs and Texturing

Lambertian Objects II

⇒ Neither the Sun nor the Moon are Lambertian

• Absorption in photosphere
• Path length through photosphere

longer from the Sun’s rim

• Surface covered with fine dust
• Dust on TV visible best from

slanted viewing angle The Sun The Moon

Computer Graphics WS07/08 – BRDFs and Texturing

“Diffuse” Reflection

• Theoretical explanation

– Multiple scattering

• Experimental realization

– Pressed magnesium oxide powder – Almost never valid at high angles of incidence

Paint manufacturers attempt to create ideal diffuse paints

Computer Graphics WS07/08 – BRDFs and Texturing

Glossy Reflection

Computer Graphics WS07/08 – BRDFs and Texturing

Glossy Reflection

• Due to surface roughness
• Empirical models

– Phong – Blinn-Phong

• Physical models

– Blinn – Cook & Torrance

Computer Graphics WS07/08 – BRDFs and Texturing

Phong Reflection Model

• Cosine power lobe

– Lr,s = Li ks coske θRV

• Dot product & power
• Not energy conserving/reciprocal
• Plastic-like appearance

R(I) R(V) H V I N I N R(I) V H

θ θRV

RV

θ θHN

HN

( )

e

k s i

• r

V I R k x f ⋅ = ) ( ) , , ( ω ω

Computer Graphics WS07/08 – BRDFs and Texturing

Phong Exponent ke

• Determines size of highlight

( )

e

k s i

• r

V I R k x f ⋅ = ) ( ) , , ( ω ω

Computer Graphics WS07/08 – BRDFs and Texturing

Blinn-Phong Reflection Model

• Blinn-Phong reflection model

– Lr,s = Li ks coske θHN – θRV ⇒ θHN – Light source, viewer far away – I, R constant: H constant

θHN less expensive to compute R(I) R(V) H V I N I N R(I) V H

θ θRV

RV

θ θHN

HN

( )

e

k s i

• r

N H k x f ⋅ = ) , , ( ω ω

Computer Graphics WS07/08 – BRDFs and Texturing

Phong Illumination Model

• Extended light sources: l point light sources
• Color of specular reflection equal to light source
• Heuristic model

– Contradicts physics – Purely local illumination

• Only direct light from the light sources
• No further reflection on other surfaces
• Constant ambient term
• Often: light sources & viewer assumed to be far away

(Blinn) ) ( ) ( L Phong) ( ) ) ( ( ) ( L

, r , r

∑ ∑ ∑ ∑

⋅ + ⋅ + = ⋅ + ⋅ + =

l k l l l s l l d a i a l k l l l s l l d a i a

e e

N H L k N I L k L k V I R L k N I L k L k

Computer Graphics WS07/08 – BRDFs and Texturing

Microfacet Model

• Isotropic microfacet collection
• Microfacets assumed as perfectly smooth reflectors
• BRDF

– Distribution of microfacets

• Often probabilistic distribution of orientation or V-groove assumption

– Planar reflection properties – Self-masking, shadowing

Computer Graphics WS07/08 – BRDFs and Texturing

Ward Reflection Model

• BRDF

฀ σ standard deviation (RMS) of surface slope – Simple expansion to anisotropic model (σx, σy) – Empirical, not physics-based – Inspired by notion of reflecting microfacets – Convincing results – Good match to measured data

2 2 2

4 ) / ) , ( tan exp( ) )( ( 1 πσ σ N H N V N I ρ π ρ f

s d r

∠ −

• +

=

N I V

viewer

H

θ

microfacet

surface

Computer Graphics WS07/08 – BRDFs and Texturing

Physics-inspired BRDFs

• Notion of reflecting microfacet
• Specular reflectivity of the form

– D : statistical microfacet distribution – G : geometric attenuation, self-shadowing – F : Fresnel term, wavelength, angle dependency of reflection along mirror direction – N•V : flaring effect at low angle of incidence

• Cook-Torrance model

– F : wavelength- and angle-dependent reflection – Metal surfaces

V N F G D f

i r

⋅ ⋅ ⋅ = π θ λ

λ

) , (

Computer Graphics WS07/08 – BRDFs and Texturing

Cook-Torrance Reflection Model

• Cook-Torrance reflectance model is based on the

microfacet model. The BRDF is defined as the sum of a diffuse and specular components: where ks and kd are the specular and diffuse coefficients.

• Derivation of the specular component ρs is based on a

physically derived theoretical reflectance model

1 ; ≤ + + =

s d s s d d r

k k k k f ρ ρ

Computer Graphics WS07/08 – BRDFs and Texturing

Cook-Torrance Specular Term

• D : Distribution function of microfacet orientations
• G : Geometrical attenuation factor

– represents self-masking and shadowing effects of microfacets

• Fλ : Fresnel term

– computed by Fresnel equation – relates incident light to reflected light for each planar microfacet

• N·V : Proportional to visible surface area
• N·I : Proportional to illuminated surface area

) )( ( I N V N DG F

s

⋅ ⋅ = π ρ

λ

N I V

viewer

H θ microfacet

surface

λ λ

)) ( 1 ( N V F ⋅ + ≈

Computer Graphics WS07/08 – BRDFs and Texturing

Microfacet Distribution Functions

• Isotropic Distributions

฀ α : angle to average normal of surface – Characterized by half-angle β

• Blinn
• Torrance-Sparrow
• Beckmann

– m : average slope of the microfacets – Used by Cook-Torrance

( )

[ ]2

/ tan 4 2 cos

4 1

m

e m D

α

α α

−

=

( )

2

2 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ −

=

α β

α e D

( )

α α

β cos ln 2 ln

cos = D

( )

2 1 = β D

( ) ( )

H N⋅ = ⇒ α α ω D D

N I V

viewer

H θ microfacet

surface

Computer Graphics WS07/08 – BRDFs and Texturing

Beckman Microfacet Distribution Function

m=0.2 m=0.6

Computer Graphics WS07/08 – BRDFs and Texturing

Geometric Attenuation Factor

• V-shaped grooves
• Fully illuminated and visible
• Partial masking of reflected light
• Partial shadowing of incident light

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ = ) ( ) )( ( 2 , ) ( ) )( ( 2 , 1 min H V I N H N H V V N H N G 1 = G ) ( ) )( ( 2 H V V N H N G ⋅ ⋅ ⋅ = ) ( ) )( ( 2 H V I N H N G ⋅ ⋅ ⋅ =

Computer Graphics WS07/08 – BRDFs and Texturing

Comparison Phong vs. Torrance

Computer Graphics WS07/08 – BRDFs and Texturing

Texturing

Computer Graphics WS07/08 – BRDFs and Texturing

Simple Illumination

• No illumination
• Constant colors
• Parallel light
• Diffuse reflection

Computer Graphics WS07/08 – BRDFs and Texturing

Standard Illumination

• Parallel light
• Specular reflection

Object properties constant over surface

• Multiple local light sources
• Different BRDFs

Computer Graphics WS07/08 – BRDFs and Texturing

Texturing

Locally varying

• bject characteristics
• 2D Image Textures
• Shadows
• Bump-Mapping
• Reflection textures

Computer Graphics WS07/08 – BRDFs and Texturing

Texture-modulated Quantities

• Modulation of object surface properties
• Reflectance

– Color (RGB), diffuse reflection coefficient kd – Specular reflection coefficient ks

• Opacity (α)
• Normal vector

– N(P)= N(P+ t N) or N= N+dN – „Bump mapping“ or „Normal mapping“

• Geometry

– P= P + dP – „Displacement mapping“

• Distant illumination

– “Environment mapping“, “Reflection mapping“

Computer Graphics WS07/08 – BRDFs and Texturing

Texture Mapping Transformations

Texture Space (2-D) Object Space (3-D) Image Space (2-D)

Texture-Surface Transformation Viewing/Projection Transformation The texture is mapped onto a surface in 3-D object space, which is then mapped to the screen by the viewing projection. These two mappings are composed to find the overall 2-D texture space to 2-D image space mapping, and the intermediate 3-D space is often

• forgotten. This simplification suggests texture mapping’s close ties with

image warping and geometric distortion. Texture space (u,v) Object space (xo,yo,zo) Screen space (x,y)

Computer Graphics WS07/08 – BRDFs and Texturing

2D Texturing

• 2D texture mapped onto object
• Object projected onto 2D screen
• 2D→2D: warping operation
• Uniform sampling ?
• Hole-filling/blending ?

Computer Graphics WS07/08 – BRDFs and Texturing

Texture Mapping in a Ray Tracer

• approximation:

– ray hits surface – surface location corresponds to coordinate inside a texture

Computer Graphics WS07/08 – BRDFs and Texturing

Texture Mapping in a Ray Tracer

• approximation:

– ray hits surface – surface location corresponds to coordinate inside a texture

Computer Graphics WS07/08 – BRDFs and Texturing

Interpolation 1D

• How to interpolate the color of the pixel?

c3 c2 c0 c1

Computer Graphics WS07/08 – BRDFs and Texturing

Interpolation 1D

• How to interpolate the color of the pixel?

s 1-s

c3 c2 c0 c1

Computer Graphics WS07/08 – BRDFs and Texturing

Interpolation 2D

• How to interpolate the color of the pixel?

s 1-s

c3 c2 c0 c1

Computer Graphics WS07/08 – BRDFs and Texturing

Interpolation 2D

• How to interpolate the color of the pixel?
• 1D: i0 = (1-t)c0 + tc1

i1 = (1-t)c3 + tc2

• 2D: c = (1-s) i0 + s i1

t t 1-t s 1-s

c3 c2 c0 c1