3-principles of surface reflection

8/13/2019 3-Principles of Surface Reflection

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Basic Principles of Surface Reflectance

Thanks to Srinivasa Narasimhan, Ravi Ramamoorthi, Pat Hanrahan


2/46

Radiometry and Image Formation


3/46

Image Intensities

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

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

source sensor

surface

element

normal Need to considerlight propagation in

a cone


4/46


5/46

Differential Solid Angle and Spherical Polar Coordinates


6/46


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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

d

dJ

( 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 areaper unit solid angle.

Ldepends on direction

Surface can radiate into whole hemisphere.

Ldepends on reflectance properties of surface.

r


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The Fundamental Assumption in Vision

Surface Camera

No Change in

Radiance

Lighting


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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 2

1 2 2 1d dA r d dA r

1 2

1 1 2 22dA dAd dA d dA

r

1 2L L


10/46

Scene

Radiance LLens

Image

Irradiance E

CameraElectronics

Scene

ImageIrradiance E

Relationship between Scene and Image Brightness

MeasuredPixel 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?


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Relation Between Image Irradiance E and Scene Radiance L

f z

surface patchimage plane

id

sd

Ld

idA

s

dA

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)


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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 images

dAi

dA

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 4thpower of cosine are small.


13/46

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.

Camera

Electronics

Image

Irradiance E

Measured

Pixel Values, I

IEg :

(Grossberg and Nayar)


14/46

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 * Reflectance

PixelValues

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


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The Problem of Dynamic Range


16/46

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!!

Todays Cameras: Limited Dynamic Range


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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)


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Computer Vision: Building Machines that See

Lighting

Scene

Camera

Computer

Physical Models

Scene Interpretation

We need to understand the Geometric and Radiometric relations

between the scene and its image.


19/46

Computer Graphics: Rendering things that Look Real

Lighting

Scene

Camera

Computer

Physical Models

Scene Generation

We need to understand the Geometric and Radiometric relations

between the scene and its image.


20/46

Basic Principles of Surface Reflection


21/46

Surface Appearance

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

Surface Reflection depends on both the viewing and illumination direction.

source sensor

surface

element

normal


22/46

BRDF: Bidirectional Reflectance Distribution Function

x

y

z

source

viewing

direction

surfaceelement

normal

incidentdirection

),( ii ),( rr

),( iisurfaceE

),(rr

surfaceL

Irradiance at Surface in direction ),( ii

Radiance of Surface in direction ),( rr

BRDF :

),(

),(),;,(

ii

surface

rr

surface

rriiE

Lf


23/46

Important Properties of BRDFs

x

y

z

source

viewing

direction

surface

element

normal

incident

direction

),( 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 2ndLaw of Thermodynamics)

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


24/46


25/46

Differential Solid Angle and Spherical Polar Coordinates


26/46


27/46

Derivation of the Scene Radiance EquationImportant!

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

rr

surface

fEL

iirriiii

src

rr

surface 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

rr

surface dfLL

2/

0

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

rr

surface ddfLL

Convert from solid angle to theta-phi representation:

M h i f S f R fl ti


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Mechanisms of Surface Reflection

source

surfacereflection

surface

incident

direction

body

reflection

Body Reflection:

Diffuse Reflection

Matte Appearance

Non-Homogeneous Medium

Clay, paper, etc

Surface Reflection:

Specular Reflection

Glossy Appearance

Highlights

Dominant for Metals

Image Intensity = Body Reflection + Surface Reflection


29/46

Mechanisms of Surface Reflection

Body Reflection:

Diffuse ReflectionMatte Appearance

Non-Homogeneous Medium

Clay, paper, etc

Surface Reflection:

Specular ReflectionGlossy Appearance

Highlights

Dominant for Metals

Many materials exhibit both Reflections:

Diff R fl ti d L b ti BRDF


30/46

Diffuse Reflection and Lambertian BRDF

viewing

direction

surface

element

normalincident

direction

i

n

v

s

drriif ),;,( 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 did .cos

source intensity

source intensity I


31/46

Diffuse Reflection and Lambertian BRDF


32/46

White-out Conditions from an Overcast Sky

CANT perceive the shape of the snow covered terrain!

CAN perceive shape in regions

lit by the street lamp!!

WHY?


33/46

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

rr

surface ddfLL

sky

ii

src LL ),(

1

),;,( rriif

sky

rr

surface LL ),(

Specular Reflection and Mirror BRDF


34/46

Specular Reflection and Mirror BRDF

source intensity I

viewing

directionsurface

element

normal

incident

direction 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


35/46

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)


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Phong Model: An Empirical Approximation

An illustration of the angular falloff of highlights:

Very commonly used in Computer Graphics

shinyn

sIL )(cos


37/46

Phong Examples

These spheres illustrate the Phong model as lighting

directionand nshinyare varied:


38/46

Components of Surface Reflection

A Si l R fl ti M d l Di h ti R fl ti


39/46

A Simple Reflection Model - Dichromatic Reflection

Observed 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 for

Diffuse/specular reflection


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Dror, Adelson, Wilsky

Specular Reflection and Mirror BRDF - RECALL


41/46

Specular Reflection and Mirror BRDF - RECALL

source intensity I

viewing

directionsurface

element

normal

incident

direction 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 : )()(vivis

IL

v r

)()(),;,( vivisvviif

specular albedo

G S f


42/46

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 smooththey show micro-surface geometry (roughness).

Example Models : Phong model

Torrance Sparrow model

Glossy Surfaces


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Blurred Highlights and Surface Roughness

Roughness


44/46

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.

shinyn

s ERIL ).(

-SR

E

HNN

shinyn

s HNIL ).(

NSNSR ).(2 2/)( SEH


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Phong Examples

These spheres illustrate the Phong model as lighting directionand

nshinyare varied:


46/46

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

3-principles of surface reflection

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