© 2003 by davi geigercomputer vision september 2003 l1.1 image formation light can change the image...
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Computer Vision September 2003 L1.1© 2003 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 September 2003 L1.2© 2003 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 September 2003 L1.3© 2003 by Davi Geiger
• Used to observe eclipses (eg., Bacon, 1214-1294)
• By artists (eg., Vermeer).
Computer Vision September 2003 L1.4© 2003 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 September 2003 L1.5© 2003 by Davi Geiger
Pinhole cameras
• Abstract camera model - box with a small hole in it
• Pinhole cameras work in practice
Computer Vision September 2003 L1.6© 2003 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 September 2003 L1.7© 2003 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 September 2003 L1.8© 2003 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 September 2003 L1.9© 2003 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 September 2003 L1.10© 2003 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 September 2003 L1.11© 2003 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 September 2003 L1.12© 2003 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 September 2003 L1.13© 2003 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 September 2003 L1.14© 2003 by Davi Geiger
Extended Light Sources and BRDF
iii sin
i
i
Aiiiiiii EE sin),(),(
Light source radiance arriving through solid angle
iiiiiiiii EAEAP sincos),(),(cos Power arriving at patch A from
thus the irradiance arriving at patch A is
2
00 sincos),( iiiiiiEA
PE
The radiance of a patch A at direction is thus, given by
2
0sincos),(),,,(),( iiiiiieeiiee EfL
Foreshortening
),( ee
Computer Vision September 2003 L1.15© 2003 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 . What constant f ?
2
0sincos eeee
Thus, the total “reflected power” from patch A becomes
Using that
we finally obtain1
f
2
0 0sincos),(),( EfEfL iiiiiiee
AEfALp eeeeee 0
2
0sincos),(
sinceForeshortening
.0EL
,0EL
0EfA
pL and for Lambertian surfaces
Computer Vision September 2003 L1.16© 2003 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 September 2003 L1.17© 2003 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 September 2003 L1.18© 2003 by Davi Geiger
Special Cases: Lambertian Examples
Scene
(Oren and Nayar)
Lambertian sphere as the light moves.
(Steve Seitz)
Computer Vision September 2003 L1.19© 2003 by Davi Geiger
Lambertian+Specular
(http://graphics.cs.ucdavis.edu/GraphicsNotes/Shading/Shading.html)