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