Basic Principles of Surface Reflectance

baconossifiedMechanics

Oct 29, 2013 (3 years and 9 months ago)

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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

surface

element

normal

Need to consider

light propagation in

a cone

Differential Solid Angle and Spherical Polar Coordinates

Radiometric Concepts

(1) Solid Angle :

( steradian )

What is the solid angle subtended by a hemisphere?

(solid angle subtended by )

(foreshortened area)

(surface area)

(2) Radiant Intensity of Source :

Light Flux (power) emitted per unit solid angle

( watts / steradian )

(3) Surface Irradiance :

( watts / m
2

)

Light Flux (power) incident per unit surface area.



Does not depend on where the light is coming from!

source

2

(4) Surface Radiance (tricky) :

(watts / m steradian )



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.


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.





Scene

Radiance L

Lens


Image

Irradiance E


Camera

Electronics


Scene


Image

Irradiance 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 patch

image plane

image patch



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

(1)



Solid angle subtended by lens:

(2)

Relation Between Image Irradiance E and Scene Radiance L

f

z

surface patch

image plane

image patch



Flux received by lens from = Flux projected onto image

(3)



From (1), (2), and (3):



Image irradiance is proportional to Scene Radiance!




Small field of view


Effects of 4
th

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.


Camera

Electronics


Image

Irradiance E


Measured

Pixel Values, I

(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.


Use a color chart with precisely known reflectances.

Irradiance = const * Reflectance

Pixel Values

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

?

?

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

relations

between 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

relations

between 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

surface

element

normal

BRDF: Bidirectional Reflectance Distribution Function

x

y

z

source

viewing

direction

surface

element

normal

incident

direction

Irradiance at Surface in direction

Radiance of Surface in direction

BRDF :

Important Properties of BRDFs

x

y

z

source

viewing

direction

surface

element

normal

incident

direction

BRDF is only a function of 3 variables :



Rotational Symmetry (Isotropy):




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



Helmholtz Reciprocity: (follows from 2
nd

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!

From the definition of BRDF:

Write Surface Irradiance in terms of Source Radiance:

Integrate over entire hemisphere of possible source directions:

Convert from solid angle to theta
-
phi representation:

Mechanisms of Surface Reflection

source

surface

reflection

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

Mechanisms of Surface Reflection

Body Reflection:



Diffuse Reflection


Matte Appearance


Non
-
Homogeneous Medium


Clay, paper, etc

Surface Reflection:



Specular Reflection


Glossy Appearance


Highlights


Dominant for Metals

Many materials exhibit both Reflections:

Diffuse Reflection and Lambertian BRDF

viewing

direction

surface

element

normal

incident

direction



Lambertian BRDF is simply a constant :

albedo



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



Surface Radiance :



Commonly used in Vision and Graphics!

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)

Specular Reflection and Mirror BRDF

source intensity
I

viewing

direction

surface

element

normal

incident

direction

specular/mirror


direction



Mirror BRDF is simply a double
-
delta function :



Very smooth surface.




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



Surface Radiance :

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

Phong Examples


These spheres illustrate the Phong model as
lighting


direction

and
n
shiny

are varied:

Components of Surface Reflection

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

Dror, Adelson, Wilsky

Specular Reflection and Mirror BRDF
-

RECALL

source intensity
I

viewing

direction

surface

element

normal

incident

direction

specular/mirror


direction



Mirror BRDF is simply a double
-
delta function :



Very smooth surface.




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



Surface Radiance :

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.

-
S

R

E

H

N

N

Phong Examples


These spheres illustrate the Phong model as
lighting direction

and
n
shiny

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