2 basic imaging and radiometry
TRANSCRIPT
Basic Principles of Imaging and Lenses
Light
LightLightPhotonsPhotons
ElectromagneticElectromagneticRadiationRadiation
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 = hfwhere E is energy and h is Planck’s constant.
• But f = c/λ
• Putting these equations together, we see that
E = hc/λ
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.
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
The Multi-Wavelength Sun
X-Ray UV Visible
Infrared RadioComposite
EM Spectrum Relative Sizes
The Visible SpectrumThe Visible Spectrum
Light waves extend in wavelength from about 400 to 700 nanometers.
Transparent MaterialsTransparent Materials
Transparent - the term applied to materials through which light can pass in straight lines.
Opaque MaterialsOpaque 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 determineswhether we call it radio, infrared, visible, ultraviolet, X-ray or gamma-ray.
Camera Obscura, Gemma Frisius, 1558
1544A Brief History of Images
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
Lens Based Camera Obscura, 1568
15581568
A Brief History of Images
http://brightbytes.com/cosite/collection2.html (Jack and Beverly Wilgus)
Jetty at Margate England, 1898.
Slide credit: David Jacobs
Still Life, Louis Jaques Mande Daguerre, 1837
1558
1837
1568A Brief History of Images
Abraham Lincoln?
1558
1840?
1568A Brief History of Images
Silicon Image Detector, 1970
1558
1837
1568
1970
A Brief History of Images
1558
1837
1568
1970
1995
A Brief History of Images
Digital Cameras
1558
1837
1568
1970
1995
A Brief History of Images
Hasselblad HD2-39
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
(x,y)
screen scene
Is an image being formedon 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
Magnification
image plane
f’optical axis
y
x
zPinhole
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
dm
'
)()(
)'()'('22
22
=+
+==
δδδδ
),,(
),,(
zyyxxB
zyxA
δδ ++
)','',''('
)',','('
fyyxxB
fyxA
δδ ++
2mArea
Area
scene
image =
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
Note that B’ and C’ labels should be switched.
Size is inversely proportional to distance.
Parallel Lines Meet
Common to draw film planein front of the focal point.Moving the film plane merelyscales 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,z)→(fxz,fyz)
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
),(),,( yxszyx →
• s is constant for all points.
• Parallel lines no longer converge, they remain parallel.
Slide credit: David Jacobs
Model 2: Orthographic Projection
image plane
optical axis
y
x
z
),,( zyx=r
)',','(' fyx=r
zz∆
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?
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
Example: If f’ = 50mm,
= 600nm (red),
d = 0.36mm
λ'2 fd =
λ
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!
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
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
aperturediameter
aperture
foi
1
'
1
'
1 =+Gaussian Law:
Blur Circle, b
)'()()'(
)'( oofo
f
fo
fii −
−−=−
Blur Circle Diameter : )'('
iii
db −=
i
'i
o
'o
Problems with Lenses
Compound (Thick) Lens Vignetting
Chromatic Abberation Radial and Tangential Distortion
thickness
principal planes
nodal points
αα
1L2L3L B
A
more light from A than B !
RFBF GF
Lens has different refractive indicesfor different wavelengths.
image plane
ideal actual
ideal actual
Spherical Aberration
Spherical lenses are the only easy shape to manufacture, but are not correct for perfect focus.
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.
imageplane
lens 2 lens 1
object
intermediatevirtual image
1i
1o2i 2o2f 1f
finalimage
d
• Magnification: 1
1
2
2
o
i
o
im =
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
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
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