# Basic Principles of Surface Reflectance

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29 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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Basic Principles of Surface Reflectance

Thanks to Srinivasa Narasimhan, Ravi Ramamoorthi, Pat Hanrahan

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

(1) Solid Angle :

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 / m
2

)

Light Flux (power) incident per unit surface area.

Does not depend on where the light is coming from!

source

2

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

Lighting

Radiance is constant as it propagates along ray

Derived from conservation of flux

Fundamental in Light Transport.

Scene

Lens

Image

Camera

Electronics

Scene

Image

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?

f

z

surface patch

image plane

image patch

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

(1)

Solid angle subtended by lens:

(2)

f

z

surface patch

image plane

image patch

Flux received by lens from = Flux projected onto image

(3)

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

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

Measured

Pixel Values, I

(Grossberg and Nayar)

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.

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

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

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

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:

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 )

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)

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 = )

specular albedo

BRDFs of Glossy Surfaces

Delta Function too harsh a BRDF model

(valid only for polished mirrors and metals).

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

-
Shafer
-

Color of Source

(Specular reflection)

Color of Surface

(Diffuse/Body Reflection)

Does not specify any specific model for

Diffuse/specular reflection

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 = )

specular albedo

Delta Function too harsh a BRDF model

(valid only for highly polished mirrors and metals).

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