Basic Principles of Imaging and Lenses
Light
Electromagnetic
Radiation
Photons
Light
These three are the same…
• Light
* pure energy
• Electromagnetic Waves
* energy-carrying waves emitted by vibrating electrons
• Photons
* particles of light
EM Radiation Travels as a Wave
c = 3 x 108 m/s
EM Radiation Carries Energy
• Quantum mechanics tells us that for photons E = hf
where E is energy and h is Planck’s constant.
• But f = c/l
• Putting these equations together, we see that
E = hc/l
Electromagnetic Wave Velocity
• The speed of light is the same for all seven forms of light.
• It is 300,000,000 meters per second or 186,000 miles per
second.
The Electromagnetic Spectrum
•
•
•
•
•
•
•
Radio Waves - communication
Microwaves - used to cook
Infrared - “heat waves”
Visible Light - detected by your eyes
Ultraviolet - causes sunburns
X-rays - penetrates tissue
Gamma Rays - most energetic
The Multi-Wavelength Sun
X-Ray
UV
Composite
Infrared
Visible
Radio
EM Spectrum Relative Sizes
The Visible Spectrum
Light waves extend in wavelength from about 400 to 700 nanometers.
Transparent Materials
Transparent - the term applied to materials through which light can
pass in straight lines.
Opaque Materials
Opaque - the term applied to materials that absorb light.
• Are clouds transparent or opaque to visible
light?
– Answer: opaque
• Are clouds transparent or opaque to ultraviolet
light?
– Answer: almost transparent
Special Things About a Light Wave
• It does not need a medium through which to travel
• It travels with its highest velocity in a vacuum
• Its highest velocity is the speed of light, c,
equal to 300,000 km/sec
• The frequency (or wavelength) of the wave determines
whether we call it radio, infrared, visible, ultraviolet,
X-ray or gamma-ray.
A Brief History of Images
Camera Obscura, Gemma Frisius, 1558
1544
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
http://www.acmi.net.au/AIC/CAMERA_OBSCURA.html (Russell Naughton)
Slide credit: David Jacobs
A Brief History of Images
Lens Based Camera Obscura, 1568
1558
1568
Jetty at Margate England,
1898.
http://brightbytes.com/cosite/collection2.html (Jack and Beverly Wilgus)
Slide credit: David Jacobs
A Brief History of Images
1558
1568
1837
Still Life, Louis Jaques Mande Daguerre, 1837
A Brief History of Images
1558
1568
1840?
Abraham Lincoln?
A Brief History of Images
1558
1568
1837
Silicon Image Detector, 1970
1970
A Brief History of Images
1558
1568
1837
Digital Cameras
1970
1995
A Brief History of Images
1558
1568
1837
Hasselblad HD2-39
1970
1995
2006
Geometric Optics and Image Formation
TOPICS TO BE COVERED :
1) Pinhole and Perspective Projection
2) Image Formation using Lenses
3) Lens related issues
Pinhole Cameras
•
•
•
Pinhole camera - box with a small hole in it
Image is upside down, but not mirrored left-to-right
Question: Why does a mirror reverse left-to-right but not top-to-bottom?
Pinhole and the Perspective Projection
Is an image being formed
on the screen?
(x,y)
YES! But, not a “clear” one.
screen
scene
image plane
r  ( x, y , z )
y
optical
axis
effective focal length, f’
z
pinhole
x
r '  ( x' , y ' , f ' )
r' r

f' z
x' x

f' z
y' y

f' z
Magnification
y
f’
optical
axis
d’
image plane B’
A
z
x
planar scene
A' ( x' , y ' , f ' )
B' ( x'x' , y 'y ' , f ' )
y' y

f' z
x'x' x  x

f'
z
A( x, y, z )
B( x  x, y  y, z )
Pinhole
A’
From perspective projection:
x' x

f' z
d
B
Magnification:
d'
m 
d
y 'y ' y  y

f'
z
(x' ) 2  (y ' ) 2
(x) 2  (y ) 2
Areaimage
Areascene
 m2

f'
z
Properties of Projection
•
•
•
•
•
Points project to points
Lines project to lines
Planes project to the whole or half image
Angles are not preserved
Degenerate cases
– Line through focal point projects to a point.
– Plane through focal point projects to line
Distant Objects are Smaller
Size is inversely proportional to distance.
Note that B’ and C’ labels should be switched.
Parallel Lines Meet
Common to draw film plane
in front of the focal point.
Moving the film plane merely
scales the image.
Vanishing Points
•
Each set of parallel lines meets at a
different point
–
•
The vanishing point for this direction
Sets of parallel lines on the same
plane lead to collinear vanishing
points.
–
The line is called the horizon for that
plane
•
Good ways to spot faked images
–
–
–
scale and perspective don’t work
vanishing points behave badly
supermarket tabloids are a great
source.
Model 0: Pinhole Projection
The Equation of Pinhole Projection
• Cartesian coordinates:
– We have, by similar triangles, that
(x, y, z) -> (f x/z, f y/z, f)
[multiply by f/z]
– Ignore the third coordinate, and get
x y
(x, y, z)  ( f , f )
z z
3D object point  2D image point
Model 1: Weak Perspective Projection
•
Issue
– Perspective effects, but not over
the scale of individual objects
– Collect points into a group at
about the same depth, then divide
each point by the depth of its
group
– Advantage: EASY
– Disadvantage: WRONG
The Equation of Weak Perspective
( x, y , z )  s ( x, y )
• s is constant for all points.
• Parallel lines no longer converge, they remain parallel.
Slide credit: David Jacobs
Model 2: Orthographic Projection
Magnification:
x'  m x
y'  m y
When m = 1, we have orthographic projection
r  ( x, y , z )
r '  ( x' , y ' , f ' )
optical
axis
y
z
x
z
z
image plane
This is possible only when z  z
In other words, the range of scene depths is assumed to be much
smaller than the average scene depth.
But, how do we produce non-inverted images?
Pros and Cons of These Models
•
Weak perspective has simpler math.
– Accurate when object is small and distant.
– Most useful for recognition.
•
Pinhole perspective much more accurate for scenes.
– Used in structure from motion.
•
When accuracy really matters, we must model the real camera
– Use perspective projection with other calibration parameters (e.g., radial lens
distortion)
Slide credit: David Jacobs
Problems with Pinholes
•
Pinhole size (aperture) must be “very small” to obtain a clear image.
•
However, as pinhole size is made smaller, less light is received by image plane.
•
If pinhole is comparable to wavelength of incoming light, DIFFRACTION
effects blur the image!
•
Sharpest image is obtained when:
pinhole diameter d  2
f 'l
Example: If f’ = 50mm,
l
= 600nm (red),
d = 0.36mm
The Reason for Lenses
Image Formation using (Thin) Lenses
•
Lenses are used to avoid problems with pinholes.
•
Ideal Lens: Same projection as pinhole but gathers more light!
o
i
P
P’
f
Gaussian Lens Formula:
1 1 1
 
i o f
• f is the focal length of the lens – determines the lens’s ability to bend (refract) light
• f different from the effective focal length f’ discussed before!
Focus and Defocus
aperture
Blur Circle,
aperture
diameter
b
d
o
i
i'
Gaussian Law:
1 1 1
 
i o f
o'
1 1 1
 
i ' o' f
Blur Circle Diameter :
(i 'i) 
b
f
f
(o  o' )
(o' f ) (o  f )
d
(i '  i )
i'
Depth of Field: Range of object distances over which image is sufficiently well focused,
i.e., range for which blur circle is less than the resolution of the imaging sensor.
Problems with Lenses
Compound (Thick) Lens
Vignetting
B
L3 L2 L1
principal planes


A
nodal points
thickness
Chromatic Abberation
more light from A than B !
Radial and Tangential Distortion
ideal
FB FG
FR
actual
ideal
actual
image plane
Lens has different refractive indices
for different wavelengths.
Spherical Aberration
Spherical lenses are the only easy shape to manufacture, but are not correct for perfect focus.
Two Lens System
d
object
final
image
f2
i2
o2
i1
f1
o1
image
plane
intermediate
virtual image
lens 2
lens 1
• Rule : Image formed by first lens is the object for the second lens.
• Main Rays : Ray passing through focus emerges parallel to optical axis.
Ray through optical center passes un-deviated.
• Magnification:
m
i2 i1
o2 o1
Exercises: What is the combined focal length of the system?
What is the combined focal length if d = 0?
Lens systems
• A good camera lens may
contain 15 elements and cost
a many thousand dollars
• The best modern lenses may
contain aspherical elements
Insect Eye
We make cameras that act “similar” to the human eye
Fly
Mosquito
Human Eye
•
The eye has an iris like a
camera
•
Focusing is done by
changing shape of lens
•
Retina contains cones
(mostly used) and rods (for
low light)
•
The fovea is small region
of high resolution
containing mostly cones
•
Optic nerve: 1 million
flexible fibers
http://www.cas.vanderbilt.edu/bsci111b/eye/human-eye.jpg
Slide credit: David Jacobs
Human Eye
• Rods
– Intensity only
– Essentially night vision and peripheral vision only
– Since we are trying to fool the center of field of view of
human eye (under well lit conditions) we ignore rods
Human Eye
• Cones
– Three types perceive different portions of the visible
light spectrum
Human Eye
• Because there are only 3 types of cones in
human eyes, we only need 3 stimulus values to
fool the human eye
– Note: Chickens have 4 types of cones
Human Eye vs. the Camera
• We make cameras that act “similar” to the human eye
CCD Cameras
http://huizen.ddsw.nl/bewoners/maan/imaging/camera/ccd1.gif
Slide credit: David Jacobs