3-D Computer Vision
CSc 83020
Image Formation and Optics
Image Formation & Optics
Image: 2D projection of a 3D scene.
We need to understand Geometric &
Radiometric relations between the scene
and its image.
Topics:
•
•
•
•
•
Pinhole & Perspective Projection.
Image Formation using Lenses.
Lens Related Issues.
Image Formation in the Eye.
Our Visual World.
Pinhole & the Perspective
Projection
(x,y)
SCENE
Is there an image being formed on the screen?
SCREEN
Pinhole Camera
• “Camera obscura” – known since antiquity
Image plane
Image
Pinhole
Object
Pinhole camera
Perspective Camera
From Trucco & Verri
r
(x,y,z)
r’
(X,Y,Z)
Center of
Projection
r =[x,y,z]T
r’=[X,Y,Z]T
r/f=r’/Z
f: effective focal length:
distance of image plane from O.
x=f * X/Z
y=f * Y/Z
z=f
Magnification
From Trucco & Verri
(x,y)
Center of
Projection
x/f=X/Z
y/f=Y/Z
(X,Y,Z)
d
(x+dx,y+dy)
(x+dx)/f=(X+dX)/Z
(y+dy)/z=(Y+dY)/Z
d’
(X+dX,Y+dY,Z)
=> dx/f=dX/Z
dy/f=dY/Z
Magnification
From Trucco & Verri
(x,y)
Center of
Projection
(X,Y,Z)
d
d’
(x+dx,y+dy)
(X+dX,Y+dY,Z)
Magnification: |m|=||d’||/||d||=|f/Z|
or m=f/Z
m is negative when image is inverted…
Magnification
• Area(image)/Area(scene)=?
• m can be assumed to be CONSTANT
if range of scene depth (ΔZ) is much smaller
than average scene depth (Z).
Implications For Perception*
Same size things get smaller, we hardly notice…
Parallel lines meet at a point…
* A Cartoon Epistemology: http://cns-alumni.bu.edu/~slehar/cartoonepist/cartoonepist.html
Vanishing Points
(from NALWA)
Consequences: Parallel lines meet
• There exist vanishing points
Marc Pollefeys
Vanishing points
H VPL
VPR
VP2
VP1
Different directions correspond
to different vanishing points
VP3
Marc Pollefeys
Question
• How many vanishing points are there in an
image?
1
2
3
6
100
∞
Approximations
• Linear approximation to perspective equations.
• Orthographic: (m=1 => x=X, y=Y).
Approximations
• Linear approximation to perspective equations.
• Weak-Perspective: m is CONSTANT.
x=f*X/Z  f*X/Zavg (Zavg average distance of points from
camera)
y=f*Y/Z  f*Y/Zavg
• Possible when Zavg is much smaller than ΔZ
(relative distance of points along the optical
axis).
Weak-Perspective Cont.
From Trucco & Verri
OBJECT
POINTS
Zavg
Zavg: average distance of points along the optical axis.
Weak
Perspective
Approximations
Para
Perspective
Ioannis Stamos – CSc 83020 Spring 2007
Pictorial Comparison
Weak perspective
Perspective
Marc Pollefeys
Problems with Pinholes
• Pinhole size (aperture) must be small.
• The smaller the size, the less light goes
through.
• If pinhole is comparable to wavelength λ of
light DIFFRACTION effects blur image.
• Pinhole diameter d=2*sqrt(f*λ) for sharp
images:
If f=50mm and λ=600nm (red light) then d=0.36mm.
Pinhole cameras
Lenses
Used to avoid problems associated with pinholes.
Ideal Lens: Same projection, but gathers more light!
From Trucco & Verri
f: point of convergence of rays that come from infinity.
Image plane
Thin Lens: Projection
optical axis
f
z
Spherical lense surface: Parallel rays are refracted to single point
Image plane
Thin Lens: Projection
optical axis
f
f
z
Spherical lense surface: Parallel rays are refracted to single point
Thin Lens: Properties
1. Any ray entering a thin lens parallel to
the optical axis must go through the
focus on other side
2. Any ray entering through the focus on
one side will be parallel to the optical
axis on the other side
Lenses
Used to avoid problems associated with pinholes.
Ideal Lens: Same projection, but gathers more light!
Ray of light
From Trucco & Verri
Optical Axis
Ioannis Stamos – CSc 83020 Spring 2007
Lenses
Gaussian Lens Formula for thin lenses: 1/Ž + 1/ž = 1/f
Ž=Z+f, ž=z+f
f: focal length of lens: ability to bend light
Ray of light
Optical Axis
From Trucco & Verri
Example: if f=50mm, Ž=300mm, then image distance
ž =60mm.
Blur Circle (Defocus)
IMAGE PLANE
APERTURE
Blur Circle w/
diameter b
P
p
d
(aperture)
OPTICAL
AXIS
Ž
ž
ž’
Ž’
Blur Circle (Defocus)
IMAGE PLANE
APERTURE
Blur Circle w/
diameter b
P
p
d
OPTICAL
AXIS
3D SCENE
ž’
Ž’
Blur Circle (Defocus)
IMAGE PLANE
APERTURE
Blur Circle w/
diameter b
OPTICAL
AXIS
3D SCENE
Ž
ž
ž’
Ž’
Blur Circle (Defocus)
IMAGE PLANE
APERTURE
Blur Circle w/
diameter b
P
p
d
(aperture)
OPTICAL
AXIS
Ž
ž
ž’
Ž’
b=?
1/ ž+1/ Ž=1/f
1/ ž’+1/ Ž’=1/f
ž=Ž*f/(Ž-f)
ž’=Ž’*f/(Ž’-f)
(ž’- ž)=[f/(Ž’-f)]*[f/(Ž-f)]*(Ž- Ž’)
Blur Circle Diameter b= | ž’-ž | d / ž’
(from similar triangles)
--------------------------------------------------------------------------Depth of Field
Range of object distances (Ž- Ž’) over which
image is “sufficiently well” focused.
i.e. b is less than resolution of imaging sensor.
Note that b is proportional to d (aperture).
Aperture & DOF
d=
(From KODAK)
Ioannis Stamos – CSc 83020 Spring 2007
Blur Circle (Defocus)
IMAGE PLANE
APERTURE
Blur Circle w/
diameter b
P
p
d
(aperture)
OPTICAL
AXIS
Ž
ž
ž’
Ž’
Ioannis Stamos – CSc 83020 Spring 2007
Focusing
• Defocused image can be made focused
by:
–Moving image plane.
–Moving the lens.
–Both as a single unit.
Ioannis Stamos – CSc 83020 Spring 2007
Two Lens System
From Shree Nayar’s notes
Ž2
Ž1
ž1
ž2
Magnification: m=x’’/x=(i2/o2)*(i1/o1)
Zooming: Varying magnification without moving object or
image plane.
Example: Move LENS2 to change m and then move LENS1
and LENS2 together to re-focus.
ZOOMING=CHANGING EFFECTIVE FOCAL LENGTH
Thick Lens
From Horn
Ioannis Stamos – CSc 83020 Spring 2007
Vignetting
From Horn
Vignetting
Effect: Darkens pixels near the image boundary
From Shree Nayar’s notes
Ioannis Stamos – CSc 83020 Spring 2007
Distortion
magnification/focal length different
for different angles of inclination
pincushion
(tele-photo)
barrel
(wide-angle)
Can be corrected! (if parameters are know)
Marc Pollefeys
Chromatic Aberration
rays of different wavelengths focused
in different planes
cannot be removed completely
Marc Pollefeys
Image Formation in the Eye
• Optics in the Eye: Iris, Lens, Retina…
• Defects in the Eye’s Lens:
– Myopia (Near-Sighted)
– Hyperopia (Far-Sighted)
• Accomodation (Focusing)
Ioannis Stamos – CSc 83020 Spring 2007
THE HUMAN EYE!
Ioannis Stamos – CSc 83020 Spring 2007
Ioannis Stamos – CSc 83020 Spring 2007
Ioannis Stamos – CSc 83020 Spring 2007
Our Visual World
• Image Formation: 3D => 2D
– Can we recover 3D Scene from 2D Image?
• We live in a special world!
– Medium (Air): Transparent & Homogeneous.
– Objects: Opaque & Reflective.
• We need to recover surfaces, not
volumes.
• Is one image enough?