Basic Principles of Surface Reflectance

Thanks to Srinivasa Narasimhan, Ravi Ramamoorthi, Pat Hanrahan

Radiometry and Image Formation

Image Intensities

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

Note: Image intensity understanding is an under-constrained problem!

source sensor

surfaceelement

normalNeed to considerlight propagation ina cone

Differential Solid Angle and Spherical Polar Coordinates

Radiometric Concepts

(1) Solid Angle :22

cos'

R

dA

R

dAd i ( steradian )

What is the solid angle subtended by a hemisphere?

d (solid angle subtended by )dA

R'dA

dA

(foreshortened area)

(surface area)

i

(2) Radiant Intensity of Source :

Light Flux (power) emitted per unit solid angle

dd

J

( watts / steradian )

(3) Surface Irradiance :dA

dE

( watts / m2 )

Light Flux (power) incident per unit surface area.

Does not depend on where the light is coming from!

source

2

(4) Surface Radiance (tricky) :

d

ddA

dL

r )cos(

(watts / m steradian )

dA

r

• Flux emitted per unit foreshortened area per unit solid angle.

• L depends on direction

• Surface can radiate into whole hemisphere.

• L depends on reflectance properties of surface.

r

The Fundamental Assumption in Vision

Surface Camera

No Change in

Radiance

Lighting

Radiance Properties

Radiance is constant as it propagates along ray– Derived from conservation of flux – Fundamental in Light Transport.

1 21 1 1 2 2 2d L d dA L d dA d

2 21 2 2 1d dA r d dA r

1 21 1 2 22

dAdAd dA d dA

r

1 2L L

SceneRadiance L

Lens ImageIrradiance E

CameraElectronics

Scene

ImageIrradiance E

Relationship between Scene and Image Brightness

Measured Pixel Values, I

Non-linear Mapping!

Linear Mapping!

• Before light hits the image plane:

• After light hits the image plane:

Can we go from measured pixel value, I, to scene radiance, L?

Relation Between Image Irradiance E and Scene Radiance L

f z

surface patchimage plane

id

sd

Ld

idA

sdA

image patch

si dd 22 )cos/(

cos

)cos/(

cos

z

dA

f

dA si 2

cos

cos

f

z

dA

dA

i

s

• Solid angles of the double cone (orange and green):

(1)

2

2

)cos/(

cos

4

z

dd L

• Solid angle subtended by lens:

(2)

Relation Between Image Irradiance E and Scene Radiance L

f z

surface patchimage plane

id

sd

Ld

idA

sdA

image patch

• Flux received by lens from = Flux projected onto image sdA idA

iLs dAEddAL )cos( (3)

• From (1), (2), and (3): 4

2

cos4

f

dLE

• Image irradiance is proportional to Scene Radiance!

• Small field of view Effects of 4th power of cosine are small.

Relation between Pixel Values I and Image Irradiance E

The camera response function relates image irradiance at the image plane to the measured pixel intensity values.

CameraElectronics

ImageIrradiance E

Measured Pixel Values, I

IEg :

(Grossberg and Nayar)

Radiometric Calibration

•Important preprocessing step for many vision and graphics algorithms such as photometric stereo, invariants, de-weathering, inverse rendering, image based rendering, etc.

EIg :1

•Use a color chart with precisely known reflectances.

Irradiance = const * ReflectanceP

ixel

Val

ues

3.1%9.0%19.8%36.2%59.1%90%

• Use more camera exposures to fill up the curve.• Method assumes constant lighting on all patches and works best when source is far away (example sunlight).

• Unique inverse exists because g is monotonic and smooth for all cameras.

0

255

0 1

g

?

?1g

The Problem of Dynamic Range

The Problem of Dynamic Range

• Dynamic Range: Range of brightness values measurable with a camera

(Hood 1986)

High Exposure Image Low Exposure Image

• We need 5-10 million values to store all brightnesses around us.• But, typical 8-bit cameras provide only 256 values!!

• Today’s Cameras: Limited Dynamic Range

Images taken with a fish-eye lens of the sky show the wide range of brightnesses.

High Dynamic Range Imaging

• Capture a lot of images with different exposure settings.

• Apply radiometric calibration to each camera.

• Combine the calibrated images (for example, using averaging weighted by exposures).

(Debevec)

(Mitsunaga)

Computer Vision: Building Machines that See

Lighting

Scene

Camera

Computer

Physical Models

Scene Interpretation

We need to understand the Geometric and Radiometric relationsbetween the scene and its image.

Computer Graphics: Rendering things that Look Real

Lighting

Scene

Camera

Computer

Physical Models

Scene Generation

We need to understand the Geometric and Radiometric relationsbetween the scene and its image.

Basic Principles of Surface Reflection

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 (Isotropy):

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

),,( ririf

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

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

Differential Solid Angle and Spherical Polar Coordinates

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:

Mechanisms of Surface 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

BRDFs of Glossy Surfaces

• Delta Function too harsh a BRDF model (valid only for polished mirrors and metals).

• Many glossy surfaces show broader highlights in addition to specular reflection.

• Example Models : Phong Model (no physical basis, but sort of works (empirical))

Torrance Sparrow model (physically based)

Phong Model: An Empirical Approximation

• An illustration of the angular falloff of highlights:

• Very commonly used in Computer Graphics

shinynsIL )(cos

Phong Examples

• These spheres illustrate the Phong model as lighting

• direction and nshiny are varied:

Components of Surface Reflection

A Simple Reflection Model - Dichromatic ReflectionObserved Image Color = a x Body Color + b x Specular Reflection Color

R

G

B

Klinker-Shafer-Kanade 1988

Color of Source(Specular reflection)

Color of Surface(Diffuse/Body Reflection)

Does not specify any specific model forDiffuse/specular reflection

Dror, Adelson, Wilsky

Specular Reflection and Mirror BRDF - RECALL

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

• Delta Function too harsh a BRDF model (valid only for highly polished mirrors and metals).

• Many glossy surfaces show broader highlights in addition to mirror reflection.

• Surfaces are not perfectly smooth – they show micro-surface geometry (roughness).

• Example Models : Phong model

Torrance Sparrow model

Glossy Surfaces

Blurred Highlights and Surface Roughness

Roughness

Phong Model: An Empirical Approximation

• How to model the angular falloff of highlights:

Phong Model Blinn-Phong Model

• Sort of works, easy to compute

• But not physically based (no energy conservation and reciprocity).

• Very commonly used in computer graphics.

shinyns ERIL ).(

-SR

E

HNN

shinyns HNIL ).(

NSNSR ).(2 2/)( SEH

Phong Examples

• These spheres illustrate the Phong model as lighting direction and nshiny are varied:

Those Were the Days

“In trying to improve the quality of the synthetic images, we do not expect to be able to display the object exactly as it would appear in reality, with texture, overcast shadows, etc. We hope only to display an image that approximates the real object closely enough to provide a certain degree of realism.”

– Bui Tuong Phong, 1975