Computer Vision February 2004 L1.1© 2004 by Davi Geiger

Image Formation

Light can change the image and appearances (images from D. Jacobs) What is the relation between pixel brightness and scene radiance?What is the relation between pixel brightness and scene reflectance ?

Computer Vision February 2004 L1.2© 2004 by Davi Geiger

http://www.acmi.net.au/AIC/CAMERA_OBSCURA.html (Russell Naughton)

Camera Obscura

"When images of illuminated objects ... penetrate through a small hole into a very dark room ... you will see [on the opposite wall] these objects in their proper form and color, reduced in size ... in a reversed position, owing to the intersection of the rays".

Da Vinci

Computer Vision February 2004 L1.3© 2004 by Davi Geiger

• Used to observe eclipses (eg., Bacon, 1214-1294)

• By artists (eg., Vermeer).

Computer Vision February 2004 L1.4© 2004 by Davi Geiger

http://brightbytes.com/cosite/collection2.html (Jack and Beverly Wilgus)

Jetty at Margate England, 1898.

Cameras

• First photograph due to Niepce

• First on record shown in the book - 1822

Computer Vision February 2004 L1.5© 2004 by Davi Geiger

Pinhole cameras

• Abstract camera model - box with a small hole in it

• Pinhole cameras work in practice

Computer Vision February 2004 L1.6© 2004 by Davi Geiger

LightSource emits photons

Photons travel in a straight line

When they hit an object they:

• bounce off in a new direction

• or are absorbed

• (exceptions later).

And then some reach the eye/camera.

Computer Vision February 2004 L1.7© 2004 by Davi Geiger

Irradiance, E

• Light power per unit area (watts per square meter) incident on a surface.

• If surface tilts away from light, same amount of light strikes bigger surface (less irradiance).

light

surface

Computer Vision February 2004 L1.8© 2004 by Davi Geiger

Radiance, L• Amount of light radiated from a surface into a given solid

angle per unit area (watts per square meter per steradian).

• Note: the area is the foreshortened area, as seen from the direction that the light is being emitted.

light

surface

Computer Vision February 2004 L1.9© 2004 by Davi Geiger

Image Formation

n̂

R

2

cos

R

A solid angle subtended by a small patch of area A.

A

L - radiance is the amount of light radiated from a surface per solid angle (power per unit area per unit solid angle emitted from a surface. )

E - irradiance is the amount of light falling in a surface (power per unit area incident in a surface. )

12 srmW

2mW

Computer Vision February 2004 L1.10© 2004 by Davi Geiger

n̂

A

I

zf

Surface Radiance and Image Irradiance

22 )cos/(

cos

)cos/(

cos

f

I

z

A

2

cos

cos

f

z

I

A

Same solid angle

Pinhole Camera Model

Computer Vision February 2004 L1.11© 2004 by Davi Geiger

n̂A

I

zf

d

Surface Radiance and Image Irradiance

3

2

2

2

cos4)cos/(

cos

4

z

d

z

d

Solid angle subtended by the lens, as seen by the patch A

coscos4

cos 32

z

dALALP

Power from patch A through the lens

4

2

32

cos4

coscos4

f

dL

z

d

I

AL

I

PE

Thus, we conclude

Computer Vision February 2004 L1.12© 2004 by Davi Geiger

Summary

4

2

cos4

f

dLE

•The irradiance at the image pixel is converted into the brightness of the pixel

•Image Irradiance is proportional to Scene Radiance

•Scene distance, z, does not affect/reduce image brightness (the model is too simplified, since in practice it does.)

•The angle of the scene patch with respect to the view ( reduces the brightness by the . In practice the effect is even stronger.

4cos

Computer Vision February 2004 L1.13© 2004 by Davi Geiger

The Bidirectional Reflectance Distribution Function (BRDF)

),(

),(),,,(

ii

eeeeii E

Lf

BRDF - How bright a surface appears when viewed from one direction while light falls on it from another.

),( ii

),( ee

n̂n̂

i

i

Usually f depends only on , true for matte surfaces and specularlyreflecting surfaces.

ie ei ,

Computer Vision February 2004 L1.14© 2004 by Davi Geiger

Extended Light Sources and BRDF

iiiiiii EE sin),(),( Light source radiance arriving through solid angle

iiiiiiiii EAEAP sincos),(),(cos Power arriving at patch A from

iii sin

i

i

A

Foreshortening

A

2,0i

,i

Computer Vision February 2004 L1.15© 2004 by Davi Geiger

Extended Light Sources and BRDF

thus the power arriving at patch A is

AEAEP iiiiiiE

0

2

0sincos),(

The radiance of a patch A at direction is thus, given by

2

0sincos),(),,,(),( iiiiiieeiiee EfL

),( ee

iiiiiiE AEP sincos),(

iii sin

i

i

A

Computer Vision February 2004 L1.16© 2004 by Davi Geiger

Special Cases of BRDF1. Lambertian Surfaces (matte)- appears equally bright from all viewing directions and reflects all incident light, absorbing none, i.e. the BRDF is constant and power arriving is the same as power leaving the surface or What is the constant f ?

2

0sincos eeee

Thus, the total “reflected power” from patch A becomes

we finally obtain1

f

2

0 0sincos),(),( EfEfL iiiiiiee

AEfALp eeeeeeL 0

2

0sincos),(

sinceForeshortening

.0EL

100

fAEAEfPp ELThus,

Computer Vision February 2004 L1.17© 2004 by Davi Geiger

Lambertian Surface Brightness

),( ii

),( ee

n̂

How bright will a Lambertian surface bewhen it is illuminated by a point sourceof radiance E? and by a “sky” of uniformradiance E?

For a point source the irradiance at the surface is and the radiance must then be

iee EEfL

cos1

),( 0

iEE cos0

Familiar cosine or “Lambert’s law” of reflection from matte surfaces(surfaces covered with finely powdered transparent materials such as barium sulfate or magnesium carbonate), and can approximate paper, snow and matte paint.

Finally, for a “sky” of uniform radiance E we obtain

!sincos1

),(2

0EEL iiiiee

Computer Vision February 2004 L1.18© 2004 by Davi Geiger

Special Cases: Lambertian Examples

Scene

(Oren and Nayar)

Lambertian sphere as the light moves.

(Steve Seitz)

Computer Vision February 2004 L1.19© 2004 by Davi Geiger

Special Cases of BRDF1. Specular Surfaces (mirrors) – reflects all light arriving from the direction into the direction . The BRDF is in this case proportional to the product of two impulses, and .What is the factor of proportionality ?

),( ii ),( ii)( ie

)( ie ),( iik

2

00 sincos),( iiiiiiEE

2

0sincos),( eeeeeeLL

eeeeee

iiiiiiieieiiee

Ek

EkL

sincos),(),(

sincos),()()(),(),(2

0

we finally obtainii

ieieeeiif

sincos

)()(),,,(

0ELand for specular surfacesii

iik

sincos

1),(

Computer Vision February 2004 L1.20© 2004 by Davi Geiger

Lambertian+Specular

(http://graphics.cs.ucdavis.edu/GraphicsNotes/Shading/Shading.html)