Torrance Sparrow Model of Reflectance + Oren Nayar Model of Reflectance

Torrance Sparrow Model of Reflectance

+

Oren Nayar Model of Reflectance

Torrance-Sparrow Model – Main Points

•Physically Based Model for Surface Reflection.

•Based on Geometric Optics.

•Explains off-specular lobe (wider highlights).

•Works for only rough surfaces.

•For very smooth surfaces, electromagnetic nature of light must be used

Beckmann-Spizzichinno model.

Beyond the scope of this course.

Modeling Rough Surfaces - Microfacets

• Roughness simulated by Symmetric V-groves at Microscopic level.

• Distribution on the slopes of the V-grove faces are modeled.

• Each microfacet assumed to behave like a perfect mirror.

Coordinate System needed to derive T-S model

Torrance-Sparrow or Cook-Torrance BRDF

Physically based model of a reflecting surface. Assumes a surface is a collection of planar microscopic facets, microfacets. Each microfacet is a perfectly smooth reflector.

• D describes the distribution of microfacet orientations.

• G describes the masking and shadowing effects between the microfacets.

• F term is a Fresnel reflection term related to material’s index of refraction.


Microfacet Distribution Function

• Statistical model of the microfacet variation in the halfway-vector H direction

• Based on a Beckman distribution function

• Consistent with the surface variations of rough surfaces

• β - the angle between N and H

• m - the root-mean-square slope of the microfacets

Beckman’s Distribution:



Geometric Attenuation Factor:

The geometric attenuation factor G accounts for microfacet shadowing. The factor G is in the range from 0 (total shadowing) to 1 (no shadowing). There are many different ways that an incoming beam of light can interact with the surface locally. The entire beam can simply reflect, shown here.



A portion of the outgoing beam can be blocked. This is called masking.



A portion of the incoming beam can be blocked. This is called shadowing.



In each case, the geometric configurations can be analyzed to compute the percentage of light that actually escapes from the surface.

Geometric Attenuation Factor


Fresnel Factor:

The Fresnel effect is wavelength dependent. It behavior is determined by the index-of-refraction of the material (taken as a complex value to allow for attenuation). This effect explains the variation in colors seen in specular regions particular on metals (conductors). It also explains why most surfaces approximate mirror reflectors when the light strikes them at a grazing angle.

Coordinate System needed to derive T-S model

Components of Surface Reflection – Moving Light Source

Components of Surface Reflection – Moving Camera

Split off-specular Reflections in Woven Surfaces

Next Class – Rough Diffuse Surfaces

Same Analysis of Roughness for Diffuse Objects – Oren Nayar Model

Dror, Adelson, Wilsky

Diffuse Reflections from Rough Surfaces

Diffuse Reflection and Lambertian BRDF - Recap

viewingdirection

surfaceelement

normalincidentdirection

in

v

s

d

rriif ),;,(• Lambertian BRDF is simply a constant :

albedo

• Surface appears equally bright from ALL directions! (independent of )

• Surface Radiance :

v

• Commonly used in Vision and Graphics!

snIIL di

d .cos

source intensity

source intensity I

Diffuse Reflection and Lambertian BRDF - Recap

Radiance decreases with increase in angle between surface normal and source

Rendered Sphere with Lambertian BRDF

• Edges are dark (N.S = 0) when lit head-on

• See shading effects clearly.

Why does the Full Moon have a flat appearance?

• The moon appears matte (or diffuse)

• But still, edges of the moon look bright(not close to zero).

Why does the Full Moon have a flat appearance?

Lambertian Spheres and Moon Photos illuminated similarly

Surface Roughness Causes Flat Appearance

Actual Vase Lambertian Vase

Surface Roughness Causes Flat Appearance – More Examples

Surface Roughness Causes Flat Appearance

Increasing surface roughness

Lambertian model

Valid for only SMOOTH MATTE surfaces.

Bad for ROUGH MATTE surfaces.

Blurred Highlights and Surface Roughness - RECAP

Roughness

Oren-Nayar Model – Main Points

•Physically Based Model for Diffuse Reflection.

•Based on Geometric Optics.

•Explains view dependent appearance in Matte Surfaces

•Take into account partial interreflections.

•Roughness represented like in Torrance-Sparrow Model

•Lambertian model is simply an extreme case withroughness equal to zero.

Modeling Rough Surfaces - Microfacets

•Roughness simulated by Symmetric V-groves at Microscopic level.

•Distribution on the slopes of the V-grove faces are modeled.

•Each microfacet assumed to behave like a perfect Lambertian surface.

View Dependence of Matte Surfaces - Key Observation

• Overall brightness increases as the angle between the source and viewing direction decreases. WHY?

• Pixels have finite areas. As the viewing direction changes, different mixes between dark and bright are added up to give pixel brightness.

Torrance-Sparrow BRDF – Different Factors (RECAP)

( ) ( , ) ( )

4cos( )cos( )

i i r h

i r

F G Df

Fresnel term:allows for

wavelength dependency

Geometric Attenuation:reduces the output based on the amount of shadowing or masking

that occurs.

Distribution:distribution

function determines what

percentage of microfacets are

oriented to reflect in the viewer

direction.

How much of the macroscopic

surface is visible to the light source

How much of the macroscopic surface is visible

to the viewer

Slope Distribution Model

• Model the distribution of slopes as Gaussian.

• Mean is Zero, Variance represents ROUGHNESS.

Geometric Attenuation Factor

• No interreflections taken into account in above function.• Derivation found in 1967 JOSA paper (read if interested).

Torrance-Sparrow BRDF – Different Factors (RECAP)

( ) ( , ) ( )

4cos( )cos( )

i i r h

i r

F G Df




that occurs.





direction.




to the viewer

Oren-Nayar Model – Different Factors

( ) ( , ) ( )

4cos( )cos( )

i i r h

i r

F G Df




that occurs.





direction.




to the viewer

Oren-Nayar Model – Different Factors

( ) ( , ) ( )

4cos( )cos( )

i i r h

i r

F G Df




that occurs.



fraction of the surface area do the facets of the same orientation

cover?




to the viewer

Oren-Nayar Model – Different Factors (contd.)

• Take into account two light bounces (reflections).

• Hard to solve analytically, so they find a functional approximation.

Oren-Nayar Model – Final Expression

Lambertian model is simply an extreme case with roughness equal to zero.

Comparison to Ground Truth

Comparison to Ground Truth

Real Objects Renderings

Summary of Surfaces and BRDFs

Smooth Rough

Diffuse

SpecularMirror BRDF Torrance-Sparrow BRDF

Lambertian BRDF Oren-Nayar BRDF

No view dependence Models view dependence

Delta FunctionSpeck of reflection

Broader HighlightsOff-specular lobe

Many surfaces may be rough and show both diffuse and surface reflection.

Summary of Surfaces and BRDFs

Smooth Rough

Diffuse

SpecularMirror BRDF Torrance-Sparrow BRDF

Lambertian BRDF Oren-Nayar BRDF

No view dependence Models view dependence

Delta FunctionSpeck of reflection

Broader HighlightsOff-specular lobe

Many surfaces may be rough and show both diffuse and surface reflection.

Documents

Torrance Sparrow Model of Reflectance + Oren Nayar Model of Reflectance