Download TSBB09 Image Sensors The history of image formation Ancient

The history of image formation
• The idea of a camera is linked to how a
perceives the world with her eyes
y
human p
• But in the early days we had only vague or
incorrect ideas about
TSBB09 Image Sensors
– What is light
– How the eye maps objects in the 3D world to
the “image” that we perceive
2014-HT2
2014
HT2
Lecture A
I
Image
Formation
F
ti
• Prior to the camera: the artist/painter
TSBB09, Lecture A, Klas Nordberg, LiU
2
13th century Europe
Ancient Egypt
3
4
15th century
The Renaissance
• Early investigations in perspective started
alreadyy byy the ancient Greeks ((~500 BC))
and Arab scientists (~1000 AD)
• It was not until the 15th century that artists
began to use perspective as a basis for
their paintings
– Lines that are p
parallel in 3D should meet at a
single point in the image
– Brunelleschi (~1415)
( 1415)
TSBB09, Lecture A, Klas Nordberg, LiU
Christ handing the keys to Saint Peter, by Perugino 1481
5
Camera obscura
6
Camera obscura
• Since ancient times it has been known that
a brightly
g y illuminated scene can be
projected to an image
Full sized dedicated camera
obscura rooms were built in
mansions and castles in the
17th and 18th centuries
– In a dark room (Latin: camera obscura)
– Through a small hole (aperture)
– The image becomes rotated 180o
TSBB09, Lecture A, Klas Nordberg, LiU
7
From Diderot’s
Encyclopedia 1772
8
Camera obscura
Camera obscura
• Th
The fifirstt photograph
h t
h was taken
t k by
b a smallll
camera obscura in 1826 by Niépce
– 8h exposure time!
• Today
A camera obscura
at Melville Garden
in Massachusetts
around 1880
– Large sized: as tourist attractions
–S
Small
a ssized:
ed for
o hobby
obby p
photographs
o og ap s
TSBB09, Lecture A, Klas Nordberg, LiU
9
10
Laterna Magica
History of photography
• Devices that can project an image onto a screen
have been described since the 16th century
(in Europe, possibly earlier elsewhere?)
g
((magic
g lamp))
• Referred to as laterna magica
• We need only
–
–
–
–
An image painted on a transparent material (glass)
A strong light source
A lens
A suitable screen
From the Ars Lucis et Umbrae, 1671
11
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
1826: Niépce takes the first proper photograph. 8h exposure time!
1839: Daguerre develops the first practical method for photography
1839: Talbot invents the process for taking negative images that can be copied
1839: Herschel invents glass negatives
1861: Maxwell demonstrates color photographs
1878: Muybridge demonstrates moving images
1887: Celluloid film is introduced
1888: Kodak markets its first “easy-to-use” camera
1891: Edison patents his “kinetoscopic camera”
189 L
1895:
Lumiére
ié B
Bros. iinvent the
h ““cinématographe”
i é
h ”
1925: Leica introduces the 35mm film format for still images
1936: Kodachrome color film
1948: Land invents the Polaroid camera
1957 Fi
1957:
Firstt di
digitized
iti d iimage
1959: AGFA introduces the first automatic camera
1969: Boyle and Smith invent the first CCD chip for image capture (based on the ”bubble memory”)
1973: Fairchild Semiconductor markets the first CCD chip (100 × 100 pixels)
1975 B
1975:
Bayer att K
Kodak:
d k fifirstt single
i l chip
hi color
l CCD camera
1981: Sony markets the Mavica, the 1st consumer digital camera. Stores images on a floppy disc
1986: Kodak presents the first megapixel CCD camera
2005: Film based photography company AgfaPhoto files for insolvency
2006 D
2006:
Dalsa
l C
Corporation
ti presents
t a 111 M
Mpixel
i l CCD camera
2009: Kodak announces that it will discontinue production of Kodachrome film
12
Source: en.wikipedia.org
Basic physics
Frequency and wavelength
• Electromagnetic radiation consists of
g
waves
electromagnetic
• The relation between frequency and
g is
wavelength
– With energy
– That propagate through space
c=λν
• The waves consist of transversal electrical
and magnetic fields that alternate with a
temporal
p
frequency
q
y ν ((Hertz)) and spatial
p
wavelength λ (meter)
TSBB09, Lecture A, Klas Nordberg, LiU
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Particles and energy
Energy
increases
with ν and
decreases
with λ
• E
Energy depends
d
d on th
the ffrequency ν
• Energy is preserved
c1
ν1
E=hν=hc/λ
h is Planck’s constant (≈ 6.623 · 10-34 Js)
TSBB09, Lecture A, Klas Nordberg, LiU
TSBB09, Lecture A, Klas Nordberg, LiU
14
Particles and energy
• Light can also be represented as particles,
p
photons
• The
Th energy off a photon
h t is
i
c is the speed of light and depends on the
medium c ≤ c0
medium,
• c0 = speed of light in vacuum ≈ 3·108 m/s
15
c2 and λ2 must
change with the
same factor relative
to c1 and λ1
c 2 < c1
λ1
ν2 = ν1
λ2 < λ1
• If the speed
p
of light
g changes
g from one medium to
another,
– the frequency ν is constant to make the energy constant
– the
th wavelength
l
th λ mustt change
h
TSBB09, Lecture A, Klas Nordberg, LiU
16
Spectrum
Spectrum
• In practice, light normally consists of
Less number
of photons
– photons with a range of energies, or
– waves with a range of frequencies
– This mix of frequencies/wavelengths/energies is
called the spectrum of the light
More number
of photons
E
E
• The spectrum gives the total amount of energy
for each frequency/wavelength/energy
• Monochromatic light consists of only one
frequency/wavelength
Same total energies
”Natural light”
TSBB09, Lecture A, Klas Nordberg, LiU
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Classification of light spectrum
λ
λ
– Can be produced by special light sources, e.g., lasers
Monochromatic light
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18
Polarization
• The
Th electromagnetic
l t
ti fifield
ld h
has a direction
di ti
– Perpendicular to the direction of motion
• The polarization of the light is defined as
the direction of the electric field
• Natural light is a mix waves with
polarization in all possible directions:
unpolarized light
• Special
S
i l lilight
ht sources or filt
filters can produce
d
polarized light of well-defined polarization
19
TSBB09, Lecture A, Klas Nordberg, LiU
20
Polarization
Polarization
• Circular/elliptical
Ci l / lli ti l polarization
l i ti
• Plane polarization
– The electric field vector rotates
– Can
C b
be constructed
t t d as th
the sum off two
t
plane
l
polarized
l i d
o
waves with 90 phase shift
– The electric field varies only
y in a single
g p
plane
Electric field
TSBB09, Lecture A, Klas Nordberg, LiU
21
Coherence
Radiometry
• The
Th phase
h
off th
the lilight
ht waves can either
ith b
be
– random: incoherent light (natural light)
– in a systematic relation: coherent light
• Light radiation has energy
– Each p
photon has a p
particular energy
gy related
to its frequency (E = h ν)
– The number of photons of a particular
frequency gives the amount of energy for this
frequency
– Described by the spectrum
– Unit:
U it Joule
J l (or
( Watt
W tt second)
d)
– Is usually not measured directly
• Coherent light is usually related to
monochromatic light sources
• Compare a red LED and a red laser
– Both produce light within a narrow range
– The
Th LED lilight
h iis iincoherent
h
– The laser light is coherent
TSBB09, Lecture A, Klas Nordberg, LiU
• Conversely: plane polarized light can be
decomposed as a sum of two circular polarized
22
waves that rotate in opposite directions
23
TSBB09, Lecture A, Klas Nordberg, LiU
24
Radiometry
Radiometry
• The power of the radiation, i.e., the energy
per unit time,, is the radiant flux
p
– Since the energy depends on the frequency,
so does the radiant flux
– Unit: Watt or Joule per second
– Is
I usually
ll nott measured
d di
directly
tl
TSBB09, Lecture A, Klas Nordberg, LiU
• The radiant flux per unit area is the flux density
– Since the flux depends on the frequency, so does the
flux density
– Unit: Watt per square meter
As the energy
through a specific
– Can be measured directly!
area during a
specific time interval
• Irradiance: flux density incident upon a surface
y emitted from
• Excitance or emittance: flux density
a surface
25
Radiometry
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26
Basic principle
• For point sources, or distant sources of
small extent,, the flux densityy can also be
measured per unit solid angle
• Based on preservation of energy
– A constant light
g source must p
produce the
same amount of energy through a solid angle
regardless
g
of distance to the source
• The radiant intensity is constant
• The radiant flux density decreases with the square
of the distance to the source
• The radiant intensity
intensit is the radiant flux
fl per
unit solid angle
– Unit: Watt per steradian
27
28
BREAK
The radiometric chain
Sensor
Light
source
Surface
29
The radiometric chain
30
The radiometric chain
Sensor
Light
source
TSBB09, Lecture A, Klas Nordberg, LiU
Surface 2
Sensor
Light
source 1
Surface 2
Light
source 2
Surface 1
TSBB09, Lecture A, Klas Nordberg, LiU
Surface 1
31
TSBB09, Lecture A, Klas Nordberg, LiU
32
Interaction between light and
matter
The radiometric chain
Sensor
Light
source 1
• Most types of light-matter interactions can
p
by
y
be represented
n = the material’s refractive index
α = the material’s
material s absorption coefficient
Surface 2
Light
source 2
Medium
• Both parameters depend on λ
• More complex interactions include
polarization effects or non-linear
non linear effects
Surface 1
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33
Light incident upon a surface
TSBB09, Lecture A, Klas Nordberg, LiU
34
Basic principle
• When
Wh light
li ht meets
t a surface
f
Based on preservation of energy:
E0 = E1 + E2 + E3
– Some part of it is transmitted through the new media
• Possibly
P
ibl with
ith another
th speed
d and
d di
direction
ti
– Some part of it is absorbed by the new media
E0 = incoming energy
• Usually: the light energy is transformed to heat
– Some part of it is reflected
E3 = absorbed energy
• All these effects are different for different
wavelengths!
E1 = transmitted energy
E2 = reflected energy
TSBB09, Lecture A, Klas Nordberg, LiU
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TSBB09, Lecture A, Klas Nordberg, LiU
36
Refraction
Absorption
• The light that is transmitted into the new
g in
medium is refracted due to the change
light speed
• Absorption implies attenuation of
g
transmitted or reflected light
• Materials get their colors as a result of
different amount of absorption for different
wavelengths
Snell’s law of refraction:
α1
sin α1
n1
c2
=
=
sin α2
n2
c1
– Ex: A green object attenuates wavelengths in
the green band less than in other bands.
α2
37
Absorption
38
Absorption spectrum
• Th
The absorption
b
ti off light
li ht iin matter
tt depends
d
d on th
the length
l
th
that the light travels through the material
• The spectrum of the reflected/transmitted
light
g is g
given by
y
a = e−αx
s2(ν)
( ) = s1(ν)
( ) a(ν)
( )
• a = attenuation of the light (0 ≤ a ≤ 1)
• α = the material’s absorption
p
coefficient
• x = length that the light travels in the material
TSBB09, Lecture A, Klas Nordberg, LiU
TSBB09, Lecture A, Klas Nordberg, LiU
s1 = incident spectrum
s2 = reflected/transmitted
fl t d/t
itt d spectrum
t
a = absorption
p
spectrum
p
((0 ≤ a(ν)
( ) ≤ 1))
39
TSBB09, Lecture A, Klas Nordberg, LiU
40
Reflection
Emission
• Highly dependent on the surface type
• Independent of its interaction with incident
g ((well,, almost…):
)
light
Light is reflected equally
much in all directions
independent of α
α
α
α
Mirror
Lambertian surface
– Any object, even one that is not considered a
light source
source, emits electromagnetic radiation
• Primarily in the IR-band, based on its
t
temperature
t
• More on this in the lecture on IR sensors
A real surface is often a mix between the two cases
TSBB09, Lecture A, Klas Nordberg, LiU
41
Scattering
TSBB09, Lecture A, Klas Nordberg, LiU
42
Scattering
• All mediums (other than vacuum) scatter light
– Examples: air, water, glass
• We can think of the medium as consisting of
small p
particles and with some p
probability
y they
y
reflect the light
–
–
–
–
In anyy p
possible direction
Different probability for different directions
Weak effect and roughly
g yp
proportional
p
to λ-4
In general, the probability depends also on the
distribution of particle sizes
TSBB09, Lecture A, Klas Nordberg, LiU
Medium
43
TSBB09, Lecture A, Klas Nordberg, LiU
44
Scattering
The plenoptic function
• Scattering is not an absorption
g ray
y does not travel
• It rather means that the light
along a straight line through the medium
• At a point x = (x1,x2,x3) in space we can
g energy
gy that
measure how much light
travels in the direction n = (n1,n2,n3),
knk = 1
– There is a p
probability
y that a certain p
photon exits the
medium in another direction than it entered.
• Examples:
p
– The sky is blue because of scattering of the sun light
– A strong
g laser beam becomes visible in air
n
x
TSBB09, Lecture A, Klas Nordberg, LiU
45
The plenoptic function
46
A light camera
• A (light) camera is a device that samples
the plenoptic function in a particular way
• Different types of cameras sample in
different ways
• The plenoptic function is the
corresponding
p
g radiance intensity
y function
– p(x,n) (5-dim since x is 3-dim & n is 2-dim)
• Can also be a function of
– Pinhole-camera
– Orthographic camera
– Push-broom camera
– Light-field
Light field camera
–…
– Frequency ν
– Time t
– p(x,n,ν,t) (7-dim)
(7 dim)
– (Polarization)
TSBB09, Lecture A, Klas Nordberg, LiU
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47
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48
The pinhole camera
The pinhole camera model
Each point in the
image plane is
illuminated by a
single
i l ray passing
i
through the
aperture
• The most common camera model is the
pinhole camera
p
– Swedish: hålkamera
The aperture
through which all
light enters the
camera
• An ideal model of the camera obscura
The image plane
This is where we
measure the image
F an ideal
For
id l
pinhole camera
the aperture is a
single point
The camera
ffrontt
TSBB09, Lecture A, Klas Nordberg, LiU
49
The pinhole camera model
50
The pinhole camera model
The image
g p
plane and the camera
center define a camera-centered
coordinate system (x1,x2,x3):
• M
Mathematically
th
ti ll we need
d only
l kknow th
the
location of the image plane and the
aperture
– The rest is physics + practical implementation
– In fact, it suffices to know the aperture (why?)
x1,xx2 are parallel to the image
plane, x3 is perpendicular to the
plane and defines the viewing
direction of the camera
Principal or optical axis
P=(x1,x2,x3) is
a point in 3D
space
• In the literature
literature, the aperture point is also
called
– camera center
– camera focal point
TSBB09, Lecture A, Klas Nordberg, LiU
Q ( 1,y2) is
Q=(y
i
the projection
of P
51
f = focal distance, the
distance between the
image plane and the
52
camera center
The pinhole camera model
The pinhole camera model
• If we look at the camera coordinate system
along the x2 axis:
• R is the point where the optical axis
g p
plane
intersects the image
– The principal point or the image center
• The (x1,xx2) plane is the principal plane or
focal plane
• The green line is the projection line of
point P
– All points on the line are projected onto Q
– Alternatively:
Alt
ti l th
the projection
j ti liline off Q
Two similar
triangles give:
53
The pinhole camera model
!
Ã
y1
x1
= − xf
3 x2
y2
TSBB09, Lecture A, Klas Nordberg, LiU
or
y1 = − fxx1
3
54
The virtual image plane
• The projected image is rotated 180o relative to
how we “see” the 3D world
• Looking along the x1 axis gives a similar
expression
p
for y2
• This can be summarized as:
Ã
−y1
x1
=
f
x3
– Reflection in both y1 and y2 coordinates = rotation
• Must be de-rotated before we can view it
– In the film based camera, the image is manually
rotated
– In the digital camera this is taken care of by reading
out the pixels in the “rotated” order
!
• Mathematically this is equivalent to placing the
image plane in front of the focal point
55
TSBB09, Lecture A, Klas Nordberg, LiU
56
The virtual image plane
The virtual image plane
• Projection lines works as before: from P
g the focal p
point and intersect at Q
through
• This defines the virtual image plane
P=
A point in 3D
space
– Cannot
C
tb
be realized
li d iin practice
ti
– Produces the same image as the rotated
image from the real image plane
Q
O=
The camera
focal point
• Easier to draw?
TSBB09, Lecture A, Klas Nordberg, LiU
57
BREAK
The projection of P onto
the virtual image plane
Ã
!
Ã
y1
x1
= xf
3 x2
y2
!
58
Lenses vs
vs. infinitesimal aperture
• Th
The pinhole
i h l camera model
d ld
doesn’t
’t work
k iin
practice since
– If we make the aperture small, too little light
enters the camera
– If we make the aperture larger, the image
becomes blurred
• Solution: we replace the aperture with a
lens or a system of lenses
59
TSBB09, Lecture A, Klas Nordberg, LiU
60
Thin lenses
The object plane
• Th
The object
bj t plane
l
consist
i t off allll points
i t th
thatt appear
sharp when projected through the lens onto the
image plane
• The object plane is an ideal model of where the
“sharp
sharp points”
points are located
• The simplest model of a lens
• Focuses all points in an object plane onto
the image plane
object
plane
image
g
plane
a
b
61
Thin lenses
• The thin lens is characterized by a single
parameter: the focal length
p
g fL
1+1 = 1
a
b
fL
• To change a (distance to object plane), we
need to change
g b since f is constant
63
• a = ∞ for b = fL !
– In practice: the object plane may be non-planar: e.g.
described by the surface of a sphere
– The shape of the object plane depends on the quality
of the lens (system)
– For thin lenses the object plane can often be
approximated as a plane
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62
Diffraction limited systems
• Due to the wave nature of light, even when
various lens effects are eliminated,, light
g
from a single 3D point cannot be focused
to an arbitrarily small point if it has passed
an aperture
• For coherent light:
– Huygens's
yg
p
principle:
p treat the incoming
g light
g
as a set of point light sources
– Gives diffraction pattern at the image plane
TSBB09, Lecture A, Klas Nordberg, LiU
64
Diffraction limited systems
Example: 1D
Diffraction limited systems
• E
Each
h point
i t along
l
th
the aperture,
t
att position
iti x’,
’
acts as a wave source
• In the image plane, at position x, each point
source contributes with a wave that has a
phase difference Δφ = 2π x’ sinθ / λ relative
the position at the centre of the aperture
• θ is the angle from point x to the aperture,
and assuming that x’ << x it follows that
sinθ ≈ x / f
• We g
get: Δφ
φ ≈ 2π x’x / ((λ f))
x’ = vertical position in the aperture
TSBB09, Lecture A, Klas Nordberg, LiU
65
Diffraction limited systems
66
Diffraction limited systems
• The principle of superposition means that
the resulting
g wave-function at the image
g
plane is a sum/integral of the contributions
from the different light sources:
Resulting wave-function
TSBB09, Lecture A, Klas Nordberg, LiU
• This phenomena generalizes to 2D:
– The resulting
g wave-function is the 2D FT of
the incoming spatial amplitude (function of x’)
• Example: a circular aperture of diameter D
Amplitude of incoming light
67
First order Bessel
function
68
The Airy disk from a single 3D point
The Airy disk
• The smallest resolvable distance in the
image
g p
plane,, Δx,, is given
g
byy
ψ’
Distance to first zero point in ψ(x)
ψ
lens focal length
lens diameter
camera front
focal plane
image plane
TSBB09, Lecture A, Klas Nordberg, LiU
The Airy disk,
the image of a
circular pattern
projected
j t d iinto
t
the image plane
light wavelength
69
The Airy disk
TSBB09, Lecture A, Klas Nordberg, LiU
70
The point spread function
Conclusions:
C
l i
• The image cannot have a better resolution
than ∆x
• No need to measure the image with higher
resolution than Δx !
• The
Th Airy
Ai disk
di k iis also
l called
ll d point
i t spread
d ffunction
ti
– or blur disk, circle of confusion
– Modulation
M d l ti transfer
t
f function
f
ti (MTF)
• In general the point spread function can be
related to several effects that make the image of
a point appear blurred
– Diffraction
– Lens imperfections
– Imperfections in the position of the image plane
• Be aware
a are of cameras with
ith high pi
pixel
el
resolution and high diffraction
– Image resolution is not defined by number of
pixels in the camera!
• Often modeled as constant over the image
– Can be variable for poor optical systems
TSBB09, Lecture A, Klas Nordberg, LiU
71
TSBB09, Lecture A, Klas Nordberg, LiU
72
Depth of field
Depth of field
• We
W have
h
now placed
l
d a llens att th
the aperture
t
– Points that are off the object plane become
blurred proportional to the displacement from
the object plane
L)
d ≈ 2 ∆x a(a−f
Df
• Due to the point spread function, it makes
p blur in the order of Δx
sense to accept
L
– This blur will be there anyway due to diffraction
• Depth of field d is the displacement along
the optical axis from the object plane that
gives blur ≤ Δx
TSBB09, Lecture A, Klas Nordberg, LiU
• F
For a camera where
h
a < ∞, an
approximation (assuming d << a) for d is
73
a = distance
di t
ffrom lens
l
tto object
bj t plane
l
fL = lens focal length
D = lens diameter
Δx = required
q
image
g p
plane resolution
d = depth of field
Depth of field
74
The F-number
F number
• fL/D iis th
the F-number
F
b off the
th lens
l
or llens system
t
• For a lens where a = ∞, points that are
further awayy than dmin are blurred less
than Δx where
• Example
– A typical F number of a camera = 8
– Blue light = 420 nm wavelength
– Airy disk diameter Δx = 1.22 λ F ≈ 4 μm
fL D
dmin
i = 4 ∆x
• For a lens with fL = 15 mm we get
– d ≈ 0.6 m at a = 1.5 m
– dmin ≈ 1.8 m at a = ∞
TSBB09, Lecture A, Klas Nordberg, LiU
75
This means that the
depth of field is within
a manageable
bl range
TSBB09, Lecture A, Klas Nordberg, LiU
76
Lens distortion
Thin lenses and the pinhole camera
• b is
i th
the di
distance
t
ffrom th
the llens/aperture
/
t
to
t the
th image
i
plane
• A lens or a lens system can never map
g lines in the 3D scene exactly
y to
straight
straight lines in the image plane
• Depending on the lens type
type, a square
pattern will typically appear like a barrel or
a pincushion
– This is the focal length of the pinhole camera, had there not
been a lens. Same as the pin-hole camera focal length f
• a is the distance from the lens/aperture to the object
plane
• a and b are related by 1/a + 1/b ≈ 1/fL, where fL is the
focal length of the lens. Often b is variable
• All points within the field of depth will be projected with
maximum sharpness ∆x on the image plane
• The geometric effect of a lens in the aperture is that
– The camera center is placed at the center of the lens
– The effective focal distance of the pinhole camera becomes b
TSBB09, Lecture A, Klas Nordberg, LiU
77
Lens distortion
78
Radial lens distortion
• Thi
This effect
ff t is
i called
ll d lens
l
di
distortion
t ti (geometric
(
t i di
distortion)
t ti )
and can, in the simplest case, be modeled as a
radial distortion
Position according
Ob
Observed
d point
i t
to the pinhole
camera model
(y1, y2) = correct image
g coordinate
(y1, y2) = r (cos θ, sin θ)
(y’1, y’2) = real image coordinate
(y’1, y’2) = h(r) (cos θ, sin θ)
Barrel distortion
No distortion
TSBB09, Lecture A, Klas Nordberg, LiU
y2
y1
Pincushion distortion
• The observed positions of points in the image are
displaced in the radial direction relative the image center
as described by the pinhole camera model
model.
79
TSBB09, Lecture A, Klas Nordberg, LiU
80
Radial lens distortion
Lens distortion
• Whi
Which
h di
distortion
t ti ffunction
ti h is
i used
d
depends on the type of lens and other
practical considerations:
• h is approximately a linear function with
g
some non-linear deviation,, e.g.
The deviation from
f
a
linear function usually
grows with r
• Once modeled, we can compensate for
the distortion
81
– Number of parameters
– Invertibility
• More complicated distortion models
include angular dependent distortion
• Cheap lenses ⇒ significant distortion
• Almost no distortion ⇒ expensive lenses
TSBB09, Lecture A, Klas Nordberg, LiU
Vignetting
Vignetting
• Even if the light that enters the camera is
constant in all directions,, the image
g plane
p
will receive different amount of illumination
• Sometimes used as a photographic effect
• But is usually unwanted
• Can be compensated for in digital
cameras
82
• This effect is called vignetting
TSBB09, Lecture A, Klas Nordberg, LiU
83
Image from a digital camera
with a very light lens
84
Mechanical vignetting
B
The cos4 law
• W
We can see the
th aperture
t
as a light
li ht source iin th
the
form of a small area that illuminates the image
plane
α
Light from a larger solid angle
point A is focused
emitted from p
here
A
Light from a smaller solid angle
emitted from point B is focused
here
TSBB09, Lecture A, Klas Nordberg, LiU
85
The cos4 law
86
Chromatic aberration
• This effect exists also in lens-based
cameras
• This means that, in general, there is an
attenuation of the image towards the
edges of the image, approximately
according to cos4α
• Can be compensated for in a digital
camera
TSBB09, Lecture A, Klas Nordberg, LiU
– The flux density decreases with the square of the
distance to the light
g source: cos2 α
– The effective area of the detector relative to the
aperture varies as cos α
– The effective area of the aperture relative to the
detector varies as cos α
• The refraction index of matter (lenses) is
wavelength
g dependent
p
– Example: a prism can decompose the light
into its spectrum
87
– A ray of white light is decomposed into rays of
different colors that intersect the image
g p
plane
at different points
88
Chromatic aberration
Sometimes clearly visible if
you look close to the edges
through a pair of glasses
89