This document provides an overview of basic principles of imaging and lenses. It discusses how light behaves as electromagnetic waves and photons. It explains image formation using pinholes, lenses, and the human eye. Key points covered include how lenses form images by gathering more light than pinholes, issues with lenses like chromatic aberration, and comparisons between the human eye and camera imaging systems.

1. Basic Principles of Imaging and Lenses

2. Light

3. LightLightPhotonsPhotons
ElectromagneticElectromagnetic
RadiationRadiation

4. These three are the same…
• Light
* pure energy
• Electromagnetic Waves
* energy-carrying waves emitted by vibrating electrons
• Photons
* particles of light

6. EM Radiation Travels as a Wave
c = 3 x 108
m/s

8. 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/λ
• Putting these equations together, we see that
E = hc/λ

9. Electromagnetic Wave VelocityElectromagnetic 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.

10. The Electromagnetic SpectrumThe 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

12. The Multi-Wavelength Sun
X-Ray UV Visible
Infrared Radio
Composite

13. EM Spectrum Relative Sizes

14. The Visible SpectrumThe Visible Spectrum
Light waves extend in wavelength from about 400 to 700 nanometers.

15. Transparent MaterialsTransparent Materials
Transparent - the term applied to materials through which light can
pass in straight lines.

16. Opaque MaterialsOpaque Materials
Opaque - the term applied to materials that absorb light.

17. • Are clouds transparent or opaque to visible
light?
– Answer: opaque
• Are clouds transparent or opaque to ultraviolet
light?
– Answer: almost transparent

18. 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.

19. Camera Obscura, Gemma Frisius, 1558
1544A Brief History of Images

20. 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
Slide credit: David Jacobs

21. Lens Based Camera Obscura, 1568
1558
1568
A Brief History of Images

22. http://brightbytes.com/cosite/collection2.html (Jack and Beverly Wilgus)
Jetty at Margate England,
1898.
Slide credit: David Jacobs

23. Still Life, Louis Jaques Mande Daguerre, 1837
1558
1837
1568
A Brief History of Images

24. Abraham Lincoln?
1558
1840?
1568
A Brief History of Images

25. Silicon Image Detector, 1970
1558
1837
1568
1970
A Brief History of Images

26. 1558
1837
1568
1970
1995
A Brief History of Images
Digital Cameras

27. 1558
1837
1568
1970
1995
A Brief History of Images
Hasselblad HD2-39
2006

28. Geometric Optics and Image Formation
TOPICS TO BE COVERED :
1) Pinhole and Perspective Projection
2) Image Formation using Lenses
3) Lens related issues

29. 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?

30. Pinhole and the Perspective Projection
(x,y)
screen scene
Is an image being formed
on the screen?
YES! But, not a “clear” one.
image plane
effective focal length, f’optical
axis
y
x
z
pinhole
),,( zyx=r
z
y
f
y
z
x
f
x
==
'
'
'
'
zf
rr
=
'
'
)',','(' fyx=r

31. Magnification
image plane
f’optical
axis
y
x
z
Pinhole
planar scene
A
B
A’
B’
d
d’
z
yy
f
yy
z
xx
f
xx
z
y
f
y
z
x
f
x
δδδδ +
=
++
=
+
==
'
''
'
''
'
'
'
'
From perspective projection: Magnification:
z
f
yx
yx
d
d
m
'
)()(
)'()'('
22
22
=
+
+
==
δδ
δδ
),,(
),,(
zyyxxB
zyxA
δδ ++
)','',''('
)',','('
fyyxxB
fyxA
δδ ++
2
m
Area
Area
scene
image
=

32. 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

33. Distant Objects are Smaller
Note that B’ and C’ labels should be switched.
Size is inversely proportional to distance.

34. Parallel Lines Meet
Common to draw film plane
in front of the focal point.
Moving the film plane merely
scales the image.

35. 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.

37. Model 0: Pinhole Projection

38. 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,z)→(f
x
z
,f
y
z
)
3D object point  2D image point

39. 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

40. The Equation of Weak Perspective
),(),,( yxszyx →
• s is constant for all points.
• Parallel lines no longer converge, they remain parallel.
Slide credit: David Jacobs

41. Model 2: Orthographic Projection
image plane
optical
axis
y
x
z
),,( zyx=r
)',','(' fyx=r
z
z∆
xmx =' ymy ='Magnification:
When m = 1, we have orthographic projection
This is possible only when zz ∆>>
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?

42. 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

43. 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
Example: If f’ = 50mm,
= 600nm (red),
d = 0.36mm
λ'2 fd =
λ

44. The Reason for Lenses

45. Image Formation using (Thin) Lenses
• Lenses are used to avoid problems with pinholes.
• Ideal Lens: Same projection as pinhole but gathers more light!
i o
foi
111
=+Gaussian Lens Formula:
• 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!
P
P’
f

46. Focus and Defocus
foi
111
=+
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.
d
aperture
diameter
aperture
foi
1
'
1
'
1
=+Gaussian Law:
Blur Circle, b
)'(
)()'(
)'( oo
fo
f
fo
f
ii −
−−
=−
Blur Circle Diameter : )'(
'
ii
i
d
b −=
i
'i
o
'o

47. Problems with Lenses
Compound (Thick) Lens Vignetting
Chromatic Abberation Radial and Tangential Distortion
thickness
principal planes
nodal points
α
α
1L2L3L B
A
more light from A than B !
RF
BF GF
Lens has different refractive indices
for different wavelengths.
image plane
ideal actual
ideal
actual

48. Spherical Aberration
Spherical lenses are the only easy shape to manufacture, but are not correct for perfect focus.

49. Two Lens System
• 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.
image
plane
lens 2 lens 1
object
intermediate
virtual image
1i
1o
2i 2o2f 1f
final
image
d
• Magnification:
1
1
2
2
o
i
o
i
m =
Exercises: What is the combined focal length of the system?
What is the combined focal length if d = 0?

50. Lens systems
• A good camera lens may
contain 15 elements and cost
a many thousand dollars
• The best modern lenses may
contain aspherical elements

51. Insect Eye
We make cameras that act “similar” to the human eye
Fly
Mosquito

52. http://www.cas.vanderbilt.edu/bsci111b/eye/human-eye.jpg
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
Slide credit: David Jacobs

53. 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

54. Human Eye
• Cones
– Three types perceive different portions of the visible
light spectrum

55. 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

56. Human Eye vs. the Camera
• We make cameras that act “similar” to the human eye

57. CCD Cameras
http://huizen.ddsw.nl/bewoners/maan/imaging/camera/ccd1.gif
Slide credit: David Jacobs