Basic Principles of Surface Reflectance

Lecture #3

Thanks to Shree Nayar, Ravi Ramamoorthi, Pat Hanrahan

Methods Relying on Surface Reflectance

Shape from Shading

Photometric Stereo Reflection Separation

Texture Modeling

Computer Vision: Building Machines that See

Lighting

Scene

Camera

Computer

Physical Models

We need to understand the relation between the lighting, surface reflectance and medium and the image of the scene.

Surface Appearance

Image intensities = f ( normal, surface reflectance, illumination )

Surface Reflection depends on both the viewing and illumination direction.

source sensor

surfaceelement

normal

BRDF: Bidirectional Reflectance Distribution Function

x

y

z

source

viewingdirection

surfaceelement

normal

incidentdirection

),( ii ),( rr

),( iisurfaceE

),( rrsurfaceL

Irradiance at Surface in direction ),( ii Radiance of Surface in direction ),( rr

BRDF :),(

),(),;,(

iisurface

rrsurface

rrii E

Lf

Important Properties of BRDFs

x

y

z

source

viewingdirection

surfaceelement

normal

incidentdirection

),( ii ),( rr

BRDF is only a function of 3 variables :

),;,(),;,( iirrrrii ff

• Rotational Symmetry:

BRDF does not change when surface is rotated about the normal.

),,( ririf

• Helmholtz Reciprocity: (follows from 2nd Law of Thermodynamics)

BRDF does not change when source and viewing directions are swapped.

Derivation of the Scene Radiance Equation

),( iisrcL

),( rrsurfaceL

),;,(),(),( rriiiisurface

rrsurface fEL

From the definition of BRDF:

Derivation of the Scene Radiance Equation – Important!

),;,(),(),( rriiiisurface

rrsurface fEL

iirriiiisrc

rrsurface dfLL cos),;,(),(),(

From the definition of BRDF:

Write Surface Irradiance in terms of Source Radiance:

Integrate over entire hemisphere of possible source directions:

2

cos),;,(),(),( iirriiiisrc

rrsurface dfLL

2/

0

sincos),;,(),(),( iiiirriiiisrc

rrsurface ddfLL

Convert from solid angle to theta-phi representation:

Differential Solid Angle and Spherical Polar Coordinates

Mechanisms of Reflection

source

surfacereflection

surface

incidentdirection

bodyreflection

Body Reflection:

Diffuse ReflectionMatte AppearanceNon-Homogeneous MediumClay, paper, etc

Surface Reflection:

Specular ReflectionGlossy AppearanceHighlightsDominant for Metals

Image Intensity = Body Reflection + Surface Reflection

Mechanisms of Surface Reflection

Body Reflection:

Diffuse ReflectionMatte AppearanceNon-Homogeneous MediumClay, paper, etc

Surface Reflection:

Specular ReflectionGlossy AppearanceHighlightsDominant for Metals

Many materials exhibit both Reflections:

Diffuse Reflection and Lambertian BRDF

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

White-out Conditions from an Overcast Sky

CAN’T perceive the shape of the snow covered terrain!

CAN perceive shape in regions lit by the street lamp!!

WHY?

Diffuse Reflection from Uniform Sky

• Assume Lambertian Surface with Albedo = 1 (no absorption)

• Assume Sky radiance is constant

• Substituting in above Equation:

Radiance of any patch is the same as Sky radiance !! (white-out condition)

2/

0

sincos),;,(),(),( iiiirriiiisrc

rrsurface ddfLL

skyii

src LL ),(

1

),;,( rriif

skyrr

surface LL ),(

Specular Reflection and Mirror BRDF

source intensity I

viewingdirectionsurface

element

normal

incidentdirection n

v

s

rspecular/mirror direction

),( ii ),( vv

),( rr

• Mirror BRDF is simply a double-delta function :

• Very smooth surface.

• All incident light energy reflected in a SINGLE direction. (only when = )

• Surface Radiance : )()( vivisIL

v r

)()(),;,( vivisvviif specular albedo

Specular Reflections in Nature

Compare sizes of objects and their reflections!

The reflections when seen from a lower viewpoint are always longer than when viewedfrom a higher view point.

It's surprising how long the reflections are when viewed sitting on the river bank.

Specular Reflections in Nature

The reflections of bright objects have better perceived contrast.

Intensity of reflected light is a fraction of the direct light – [Fresnel term (derivation in a later class)]

Papers to Read

http://grail.cs.washington.edu/projects/sam/

Shape and Materials by Example: A Photometric Stereo Approach

http://www.eecs.harvard.edu/~zickler/helmholtz.html

Helmholtz Stereopsis

http://www.eecs.harvard.edu/~zickler/dichromaticediting.html

http://www.eecs.harvard.edu/~zickler/projects/colorsubspaces.html

Color Subspaces as Photometric Invariants

Specularity Removal and Dichromatic Editing