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Image Quality Metric

Image quality metrics

Mutual information (cross-entropy) m Intuitive definition Rigorous definition using entropy Example: two-point resolution pr Example: confocal microscopy

Square error metric Receiver Operator Characteristic (RO

Heterodyne detection

MIT 2.717Image quality metrics p-1

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Linear inversion model

object

channel

Hf

hardwarea

(me

field

propagation detection

inversion problem:determinef, given the measurement g =H

noise-to-signal ratio (NSR) =

powesignal(average

variance)(noise

normalizing signal power to 1


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Mutual information (cross-en

object

channel

Hf

hardwarea

(me

field


2

1ln=

n1

eigenvkC

+ of2

2

k=1


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The significance of eigenval

n

n-1

1

02

2

2

(aka

is worth)

=

=

n

k 12

1C

...

...

rank ofmeasurement

how manydimensions

the measurement l

n n1 2

eigenvalues of HMIT 2.717Image quality metrics p-4

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Precision of measuremen

2k

2

n11ln +

2 2 2t t 1

noise floor

C

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Formal definition of cross-entr

EntropyEntropy in thermodynamics (discrete systems): log2[how many are the possible states of the sy

E.g. two-state system: fair coin, outcome=heads

Entropy=log22=1

Unfair coin: seems more reasonable to weigh according to their frequencies of occurence (i.e.

)Entropy =

p( logstate p(state2states


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Fair coin: p(H)=1/2; p(T)=1/2

b1

1Entropy =

2

1 1 1

log2

2log2 =2

2

Unfair coin: p(H)=1/4; p(T)=3/4

1Entropy =

4

1 3log

4

4log2

3

= 81.02

4

Maximum entropyMaximum entropy Maximum uncMaximum unc


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Joint Entropy

Joint Entropylog2[how many are the possible states of a comb

obtained from the Cartesian product of two

EntropyJoint ( YX ) = yxp )log , ( , 2states states

Xx Yy

object EntroJointE.g.

hardware

channel

a(me

fieldpropagation detectionHf


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Conditional Entropy

Conditional Entropylog2[how many are the possible states of a comb

given the actual state of one of the two vari

EntropyCond. (Y |X ) = yxp )log( , 2states states

Xx Yy

object EntroCond.E.g.

hardware

channel

a(me

fieldpropagation detectionHf


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object

hardwarechannel

a

(me

field

propagation detectionHf

adds uncertainty to the measurement wrt tNoise adds uncertainty

eliminates informationeliminates information from the measurement


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uncertainty added due to noiserepresentation bySeth Lloyd, 2.100 EntropyCond. ( GF )|

Entropy( )F(

),C GFEn

information incontained c

in the object in the

EntropyCond. ( FG )

cr| (aka muinformation eliminated due to noise


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( )FEntropy

(

),

(

)|

( ),C

GFEntropyJoint

GFEntropyCond. ECond.

GF


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FF GGinformationinformation

sourcesource(object)(object)

infinf

rr(me(mea

Corruption source (Noise)Corruption source (Noise)

Physical ChannelPhysical Channel

(transform)(transform)

, |C( GF ) =Entropy(F ) EntropyCond. ( GF )

|=Entropy(G ) EntropyCond. ( FG )

=Entropy(F ) +Entropy(G ) EntJoint


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Entropy & Differential Entr

Discrete objects (can take values among a discrete set of

definition of entropy

( )log2 xpEntropy = xp ( )k kk

unit: 1 bit (=entropy value of a YES/NO question wi

uncertainty) Continuous objects (can take values from among a contin

definition of differential entropy

( )ln xp

EntropyDiff. =

xp ( )dx

( )X

unit: 1 nat (=diff. entropy value of a significant digitrepresentation of a random number, divided by ln10)


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Image Mutual Information (

object

channel

Hf

hardwarea

(me

field


Assumptions: (a) Fhas Gaussian statistics(b) white additive Gaussian noise (waGi.e. g=Hf+wwhere Wis a Gaussian random vector wcorrelation matrix

Then C GF,( )=n1 +

1ln

2

2k

k eige:=2 1k


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Mutual information &

degrees of freedom

n

n-1

1

0 2n

21n

22

...

...

rank ofmeasurement

mutual

=

+=

n

k

k

12

2

2

1C

H

2

MIT 2.717

1ln

informationAs noise increases one rank of is

overcomes a ne the remaining ran

Image quality metrics p-16

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Example: two-point resolut

Finite-NA imaging system, unit magnificatio

Two point-sources Two point-detectors

~AfA A

(object) (measurement)

x

~BfB B

Classica

noiseless

Ag Bg

intensitiesintensity

emitted @detector

plane


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Cross-leaking power

A~

B~

ss

B

A

g

g

=

=

s =


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IMI for two-point resolution p

=1 1s ( ) H = 2det 1H =

s=1 2s

1 s1 H

1

=

2

11

s

s

2(1 ) (1ln

1ln +1

2

1( GF ) s C + +=,2

2


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IMI vs source separation

(

)

SNR =

s0s1MIT 2.717Image quality metrics p-20

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IMI for rectangular matrice

H

= =

H

underdeterminedunderdetermined overdetermioverdetermi(more unknowns than (more measure

measurements) than unknow

eigenvalues cannot be computed, but insteadwe compute the singular valuessingular values of the

rectangular matrix


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HT

H

= square ma

Trecall pseudo-inverse f =( HH )

inversion operation associated with rank of

Tseigenvalue ( HH ) aluessingular v (


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object

channel

Hf

hardwarea

(me

field


under/over deter

n1+

k2

singulo1ln=

C

2

k=1


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Confocal microscope

Sm

Intensity

object

beam

splitter

pivirtual slice

detector

nhole

Dep

Lig

L

De

L

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Image quality metrics p-24

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Depth resolution vs. noi

point sources,

Object structure: mutually

incoherent

optical axis

sampling distance

Imaging method

correspondence intensity

measurements

CFM

object scanning

direction


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Depth resolution

vs. noise & pinhole size

units: Rayleigh distanceMIT 2.717Image quality metrics p-26

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IMI summary

It quantifies the number of possible states of the object thimaging system can successfully discern; this includes

the rank of the system, i.e. the number of object dimthe system can map

the precision available at each rank, i.e. how many s

digits can be reliably measured at each available dim An alternative interpretation of IMI is the game of 20 q

many questions about the object can be answered reliablimage information?

IMI is intricately linked to image exploitation for applicamedical diagnosis, target detection & identification, etc.

Unfortunately, it can be computed in closed form only foGaussian statistics of both object and image; other more models are usually intractable


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Other image quality metri

Mean Square Error (MSQ) between object and image Mean Square Error (MSQ)

2( f ) ofresult E f fk ==

k k

inversionobjectsamples

e.g. pseudoinverse minimizes MSQ in an overdeterm

obvious problem: most of the time, we dont know w

more when we deal with Wiener filters and regulariz

Receiver Operator ChaReceiver Operator Charracteacterriisstictic

measures the performance of a cognitive system (hum

computer program) in a detection or estimation task image data


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Receiver Operator Characte

Target detect

Example: med H0 (null hypno tumor H1 = tumor

TP = true posiidentification FP = false posalarm)


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MIT 2.717

Intro to Inverse Problems p-1

Introduction to Inverse Pro

What is an image? Attributes and Represen

Forward vs Inverse Optical Imaging as Inverse Problem

Incoherent and Coherent limits

Dimensional mismatch: continuous vs d

Singular vs ill-posed

Ill-posedness: a 22 example

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MIT 2.717


Basic premises

What you see or imprint on photographic film is a very

interpretation of the word image

Image is a representation of a physical object having cer

Examples of attributes

Optical image: absorption, emission, scatter, color w

Acoustic image: absorption, scatter wrt sound Thermal image: temperature (black-body radiation)

Magnetic resonance image: oscillation in response to

frequency EM field

Representation: a transformation upon a matrix of attribu

Digital image (e.g. on a computer file)

Analog image (e.g. on your retina)

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MIT 2.717


How are images formed

Hardware

elements that operate directly on the physical entity

e.g. lenses, gratings, prisms, etc. operate on the optic

e.g. coils, metal shields, etc. operate on the magnetic

Software

algorithms that transform representations

e.g. a radio telescope measures the Fourier transform

(representation #1); inverse Fourier transforming lea

representation in the native object coordinates (rep

#2); further processing such as iterative and nonlinea

lead to a cleaner representation (#3).

e.g. a stereo pair measures two aspects of a scene (re#1); a triangulation algorithm converts that to a bino

with depth information (representation #2).

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MIT 2.717


Who does what

In optics,

standard hardware elements (lenses, mirrors, prisms)

limited class of operations (albeit very useful ones);

operations are

linear in field amplitude for coherent systems

linear in intensity for incoherent systems a complicated mix for partially coherent systems

holograms and diffractive optical elements in genera

more general class of operations, but with the same l

constraints as above

nonlinear, iterative, etc. operations are best done wit

components (people have used hardware for these putends to be power inefficient, expensive, bulky, unre

these systems seldom make it to real life applications

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MIT 2.717


Imaging channels

PhysicsPhysics

AlgorithmsAlgorithms

Information generatorsInformation generators

Wave sourcesWave sources

Wave scatterersWave scatterers

ImagingImagingCommunicationCommunication

StorageStorage

Processing elementsProcessing elements

GOAL:GOAL:MaximizeMaximize informinform

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MIT 2.717


Generalized (cognitive) represent

Situation of

interest

encoded into

a scene

optical system produces a (geometrically

similar) image

Classical inverse problem viewClassical inverse problem view--pointpoint

Situation of

interestYY

/

encoded into

a scene

optical system produces an information-rich

light intensity patternan

NonNon--imaging or generalized sensor viewimaging or generalized sensor view--pointpoint

Advantages: - optimum resource allocation

- better reliability

- adaptive, attentive operation

if necessary (requires resou

e.g. is there a tanke.g. is there a tank

in the scene?in the scene?

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MIT 2.717


Forward problem

hardware

channel

at

(me

object

fieldpropagation detection

object me

The Forward Problem answers the following quest

Predict the measurement given the object attribu

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MIT 2.717


Inverse problem

hardware

channel

at

(me

object

fieldpropagation detection

object

representationme

The Inverse Problem answers the following questi

Form an object representation given the measurem

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MIT 2.717


Optical Inversion

amplitude object

(dark A on bright

background)

free space

(Fresnel)

propagation

free space

(Fresnel)

propagation

free space

(Fresnel)

propagation

lens lensarra

sen

amplitude

representationm

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MIT 2.717


Optical Inversion: cohere

( )yxf , ( ) ( ) ( coh ,,, = yxxhyxfyxI

Nonlinear problemNonlinear problem

object

amplitudeintensity measurement at the ou

Note: I could make the problem linear if I could m

amplitudes directly (e.g. at radio frequencies)

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MIT 2.717


Optical Inversion: incoher

( )yxI ,obj ( ) ( ) ( = xhyxIyxI ,, incohobjmeas

Linear problemLinear problem

object

intensityintensity measurement at the ou

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MIT 2.717


Dimensional mismatch

The object is a continuous function (amplitude or inten

assuming quantum mechanical effects are at sub-nanome

much smaller than the scales of interest (100nm or more

i.e. the object dimension is uncountably infinite

The measurement is discrete, therefore countable and

To be able to create a 1-1 object representation from thmeasurement, I would need to create a 1-1 map from a fi

integers to the set of real numbers. This is of course imp

the inverse problem is inherently ill-posed

We can resolve this difficulty by relaxing the 1-1 require

therefore, we declare ourselves satisfied if we sampl

with sufficient density (Nyquist theorem) implicitly, we have assumed that the object lives in a

dimensional space, although it looks like a continu

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MIT 2.717


Singularity and ill-posedn

Under the finite-dimensional object assumption, the linear in

is converted from an integral equation to a matrix eq

( ) ( ) ( )yyxxhyxfyxg d,,, = fg H=

If the matrix H is rectangular, the problem may be overco

underconstrained

If the matrix H is square and has det(H)=0, the problem is

can only be solved partially by giving up on some object di

(i.e. leaving them indeterminate)

If the matrix H is square and det(H) is non-zero but smallproblem may be ill-posed or unstable: it is extremely sensit

in the measurement f

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MIT 2.717


Resolution: a toy problem

x

Two point-sources

(object)

Two point-detectors

(measurement)

Finite-NA imaging system

A~

B~

A

B

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MIT 2.717


Cross-leaking power

A~

B~

ss

B

A

I

I

=

=

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MIT 2.717


Ill-posedness in two-point inv

=

1

1

s

sH

( )

2

1det s=

H

=

1

1

1

12

1

s

s

sH

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Applications of Statistical O

Radio Astronomy

Michelson Stellar Interferometry

Rotational Shear Interferometer (RS

Optical Coherence Tomography (OC

MIT 2.717

Apps of Stat Optics p-1

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Radio Telescope

(Very Large Array, VLA

27 Antennae (pa

diameter 25m, we

Y radius range

1km and 36km

wavelengths 90c

resolution 200-1smallest configura

arcsec in largest c

signals are multi

correlated at centr

obtain (x,y).

van Cittert-Zernwww.nrao.edu is used to invert th

and obtain the so

e.g. a constellation

MIT 2.717


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VLA images

These four images areimages of a large sola

17 June 1989. The r

images are optical im

superimposed contou

as seen with the VLA

GHz. The four image

times during the

progression toward m(bottom right). Thi

accompanied by a c

The two H alpha ribb

"footpoints" of an ar

which arch NE/SW

strongest toward the

sunspots appear dark event, the magnetic

emits radio waves

from www.aoc.nrao.edu magnetically conjuga

(b). The entire magneMIT 2.717 two footpointsApps of Stat Optics p-3

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VLA images

from www.aoc.nrao.edu

This is a radar image of Mars, made with the Goldstone-VLin 1988. Red areas are areas of high radar reflectivity. The

cap, at the bottom of the image, is the area of highest reflect

areas of high reflectivity are associated with the giant shield

Tharsis ridge. The dark area to the West of the Tharsis ridMIT 2.717 detectable radar echoes, and thus was dubbed the "SteaApps of Stat Optics p-4

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VLA images

The center of the Milky Way


MIT 2.717


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VLA images


The galaxy M81 is a spiral galaxy about 11 million light-years f

50,000 light-years across. This VLA image was made using datathe VLA's four standard configurations for a total of more than

time. The spiral structure is clearly shown in this image, whic

intensity of emission from neutral atomic hydrogen gas. In this

red indicates strong radio emission and blue weakerMIT 2.717


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MIT 2.717


This pair of images illustrates the need to study celes

wavelengths in order to get "the whole picture" of whaobjects. At left, you see a visible-light image of the M

This image largely shows light coming from stars in

radio image, made with the VLA, shows the hydrogen

of gas connecting the galaxies. From the radio image,

this is an interacting group of galaxies, not is

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Michelson Stellar Interferom

Optical version of the van Cittert-Zernicke theorem

Since multiplication cannot be performed directly, it is done throug

(Youngs interferometer) Extreme requirements on mechanical and thermal stability (better th

between the two arms)

Alternative: intensity interferometer (or Hanbury Brown Twiss

MIT 2.717 from www.physics.usydApps of Stat Optics p-8

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Hanbury Brown Twiss

interferometer

from www.physics.usyd.edu.au/astron/susi

MIT 2.717


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The Rotational Shear Interfer

folding mirror

folding mirror

beam splitter

dither

translation

stage

sensor

input aperture

rotating object

by David MIT 2.717

Apps of Stat Optics p-10www.fit

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Experimental RSI implementation (Univers

cooling fan

shutter

)

camera

platform linear bearings

mirr

long-travel platform (2

Princeton Instruments camera

Aerot

by David MIT 2.717


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Close-up view of the Interferometer Section o

shutter

input

aperture

magnetic

90

mirror

flexure

stage

coupling

shearing

by David MIT 2.717


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Mobile RSI

(University of Illinois a

Distant Focus Corporat

by David MIT 2.717


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EXPERIMENTAL RESULTEXPERIMENTAL RESULT

2-D spatial / 1-D spectral RSI reconstruct

Experimental Setup

Color Composite Image R

by David J. Brady, Duke University

www.fitzpatrick.duke.edu/disp/

Green (520-570 nm) BMIT 2.717


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The Rotational Shear Interfer

folding mirror

folding mirror

beam splitter

dither

translation

stage

sensor

input aperture

rotating object

by David MIT 2.717


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MIT 2.717


What does the RSI measur

Input field Folding prism

at Arm 1

Folding prism

at Arm 2 (=90o)

at

Input field

Arm 2

Arm 1To Camera

S

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Intensity on the RSI Sensor P

The field on arm 1 is:

E x y) =E x cos 2 +y sin 2, x sin 2 y cos 2)

The field on arm 2 is:

2 ( , o(

1 ( , o(

E x y) =E x cos 2 y sin 2,x sin 2 y cos 2)

2I x y) = E1 +E2

2

s( ,

2 *= + +E E2 +E E

*E1 E2 1 1 2

= +

I2

+

I1

=

2y sin 2 , y =

2xsin 2 ,x =

2x cos 2 +

x y y( , =o o

+*

by David MIT 2.717


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Coherence imaging using theCoherence imaging using the

, , ,) +

dc InterRe (x ,

yxy jk l i

),,,,( vyxyxS jilk

0),,,(

=

jilk yxyxJ

),,,( qyx

Re (

4-D Fourier transform

S

by David MIT 2.717

Apps of Stat Optics p-18www.fitz

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Mutual Intensity

Example RSI Images

2 point

sources

Experimental

by David MIT 2.717


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MIT 2.717J

wk1-b p-1

Welcome to ...

2.717J/M

Optical En

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This class is about

Statistical Optics

models of random optical fields, their propagation an

properties (i.e. coherence)

imaging methods based on statistical properties of lig

imaging, coherence tomography

Inverse Problems to what degree can a light source be determined by m

of the light fields that the source generates?

how much information is transmitted through an im

system? (related issues: what does _resolution_ reall

is the space-bandwidth product?)

MIT 2.717J

wk1-b p-2

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The van Cittert-Zernike theo

Very Large Array (VLA)radio

waves

+Fourier

Cross-Correlation

transform

Galaxy, ~100 million

light-years away

optical imageMIT 2.717J

wk1-b p-3

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Optical coherence tomogra

Coro

Image credits:

www.lightlabimaging.com

Intes

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wk1-b p-4

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Inverse Radon transform(aka Filtered Backprojection)

The hardware

The principle

Magnetic Resonance Imaging (MRI)Image credits:

www.cis.rit.edu/htbooks/mri/

www.ge.com

MIT 2.717J The image

wk1-b p-5

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You can take this class i

You took one of the following classes at MIT

2.996/2.997 during the academic years 97-98 and 99

2.717 during fall 00

2.710 during fall 01

OR

You have taken a class elsewhere that covered GeometriDiffraction, and Fourier Optics

Some background in probability & statistics is helpful bu

necessary

MIT 2.717J

wk1-b p-6

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Syllabus (summary)

Review of Fourier Optics, probability & statistics 4 week

Light statistics and theory of coherence 2 weeks

The van Cittert-Zernicke theorem and applications of sta

to imaging 3 weeks

Basic concepts of inverse problems (ill-posedness, regul

examples (Radon transform and its inversion) 2 weeks Information-theoretic characterization of imaging chann

Textbooks:

J. W. Goodman,Statistical Optics, Wiley.

M. Bertero and P. Boccacci,Introduction to Inverse Pro

Imaging, IoP publishing.

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What you have to do

4 homeworks (1/week for the first 4 weeks)

3 Projects:

Project 1: a simple calculation of intensity statistics f

in Goodman (~2 weeks, 1-page report)

Project 2: study one out of several topics in the appli

coherence theory and the van Cittert-Zernicke theoreGoodman (~4 weeks, lecture-style presentation)

Project 3: a more elaborate calculation of informatio

imaging channels based on prior work by Barbastath

(~4 weeks, conference-style presentation)

Alternative projects ok

No quizzes or final exam

MIT 2.717J

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Administrative

Broadcast list will be setup soon Instructors coordinates

George Barbastathis

Please do not phone-call

Office hours TBA

Class meets

Mondays 1-3pm (main coverage of the material)

Wednesdays 2-3pm (examples and discussion)

presentations only: Wednesdays 7pm-??, pizza serve

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The 4F system

1f 1f 2f 2f

x y(

, )

G1

g1g1 yx f1,f1

object planeFourier plane I

MIT 2.717J

wk1-b p-10

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The 4F system

1f 1f 2f 2f

( )vuG ,1

x

y

x

v

u

sin

sin

=

=

x y(

, )

G1

g1g1 yx f1,f1

object planeFourier plane I

MIT 2.717J

wk1-b p-11

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The 4F system with FP aper

1f 1f 2f 2f

ryxcirc

( )G ,1

x

vu

( )

1G1

h

, g (

, )

f1 f1 Rg1 yx

object planeFourier plane: aperture-limited Imag

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The 4F system with FP aper

Transfer function: Impulse resp

circular aperture Airy functi

Rr

circ r

jinc

R f2

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wk1-b p-13

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Coherent vs incoherent ima

field inoptical

system

Coherent fi

intensity in Incoherent inteoptical

system

MIT 2.717J

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Coherent impulse response xh(field infield out)

(

,Coherent transfer function H ( vu ) =F(FT of field inFT of field out)

~Incoherent impulse response yxh ) =(

,(intensity inintensity out)

~Incoherent transfer function H ( vu ) =FT{,

(FT of intensity inFT of intensity out)=H (u

~H ( vu ) (MTFFunctionTransferModulation:,

~

H ( vu ) (OTF)FunctionTransferOptical:,MIT 2.717J

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1f 1f 2f 2f

2a

H( )uH

1 1

u

au 2uc cu =cf1

Coherent illumination Incoherent ill

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Aberrations: geometrica

P

Non-

ove

(G

im

Spheric

Origin of aberrations: nonlinearity of Snells law (n sin=const.,

relationship would have been n=const.)

Aberrations cause practical systems to perform worse than diffra

Aberrations are best dealt with using optical design software (Co

Zemax); optimized systems usually resolve ~3-5(~1.5-2.5m in

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Aberrations: wave

,Aberration-free impulse response h ndiffractio ( yx

limited

Aberrations introduce additional phase delay to the impu

, ,haberrated ( yx ) = h ndiffractio ( xlimited

c2u

( )~

1 (

uHunab

diffr

lim

Effect of aberrations

on the MTF

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Optics Overview

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What is light?

Light is a form of electromagnetic energy detected th

effects, e.g. heating of illuminated objects, conversion of

current, mechanical pressure (Maxwell force) etc.

Light energy is conveyed through particles: photons

ballistic behavior, e.g. shadows

Light energy is conveyed through waves

wave behavior, e.g. interference, diffraction

Quantum mechanics reconciles the two points of view, th

wave/particle duality assertion

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Particle properties of ligh

Photon=elementary light particle

Mass=0

Speed c=3108 m/sec

According to Special Relativity, a massAccording to Special Relativity, a mass--less particle tless particle tr

at light speed can still carry momentum!at light speed can still carry momentum!

relates the dual pEnergy E=h

nature of light;h=Plancks constant

is the temporal

=6.626210-34 J sec frequency of the MIT 2.71/2.710

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Wave properties of light

1/

angular frequency

: wavelength

(spatial period)

k=2/

wavenumber

: temporalfrequency

=2

E: electricfield

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Wave/particle duality for li

Photon=elementary light particle

Mass=0

Speed c=3108 m/sec

Energy E=h

h=Plancks constant

c=

Dispersion=6.626210-34 J sec

(holds in vacu=frequency (sec-1)

=wavelength (m)

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Light in matter

light in vacuum

light in matter

Speed c=3108 m/sec Speed c/n

n : refractive index(or index of refrac

Absorption coefficient 0 Absorption coeffic

energy decay coef

after distanceL : e

E.g. vacuum n=1, air n 1;

glass n1.5; glass fiber has 0.25dB/km=

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Materials classification

Dielectrics

typically electrical isolators (e.g. glass, plastics)

low absorption coefficient

arbitrary refractive index

Metals

conductivity

large absorption coefficient

Lots of exceptions and special cases (e.g. artificial diele

Absorption and refractive index are related through the K

Kronig relationship (imposed by causality)

absorption

refractive index

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Overview of light sourcenon-Laser

Thermal:polychromatic,

spatially incoherent

(e.g. light bulb)

Gas discharge: monochromatic,spatially incoherent

(e.g. Na lamp)

Light emitting diodes (LEDs):

monochromatic, spatially

incoherent

La

Continuous wa

strictly monoch

spatially cohere

(e.g. HeNe, Ar+

Pulsed: quasi-m

spatially cohere

(e.g. Q-switche

~nsec

pulse

mono/poly-chromatic = single/multi co

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Monochromatic, spatially coh

1/ , we

descript

light nice, r

stabiliz

good ap

most o

rough ap pulsed

laser sou

more co

Incoherent: random, irregular waveform

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The concept of a monochrom

ray

direc

energy p

lig

t=0(frozen)

wavefronts

In homogeneous media,

light propagates in rectilinear path

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The concept of a monochrom

ray

direc

energy p

lig

t=t(advanced)

wavefronts



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The concept of a polychromati

t=0(frozen)

energ

prett

all wav

propag

thwavefronts



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Fermat principle

light

ray

P

P

is chosen to minimi

zyxn ) dl path integral, comp( , ,alternative path

(aka minimum pathprinciple)

Consequences: law of reflection, law of refr

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The law of reflection

P

P

O

O

a)

instead of P

b) Alternative p

longer than POP

c) Therefore, lig

symmetric path P

P

mirror

Consider virt

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The law of refraction

n n

reflected

refrac

incident

=

sinsin nn Snells Law of

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Optical waveguide

TIR

Tn

n

nn

=1.51

=1.5105

=1.511.00

Planar version: integrated optics

Cylindrically symmetric version:fiber optics

Permit the creation of light chips and light cables, resp

light is guided around with few restrictions

Materials research has yielded glasses with very low losse Basis for optical telecommunications and some imaging (e

and sensing (e.g. pressure) systems

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Refraction at a spherical sur

point

source

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Imaging a point source

point

source

Lens

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Model for a thin lens

1st FP

at 1st FP

f

point object

focal length

plane wave (or parallel ra

image at infinity

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Model for a thin lens

at 2nd F

f

point im

focal length

plane wave (or parallel ray bundle);

object at infinity

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Huygens principle

optical

MIT 2.71/2.710

Each point on thacts as a second

emitting a sphe

The wavefront

propagation dis

result of superim

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Why imaging systems are ne

Each point in an object scatters the incident illumination into a

according to the Huygens principle.

A few microns away from the object surface, the rays emanatin

object points become entangled, delocalizing object details.

To relocalize object details, a method must be found to reassig

the rays that emanated from a single point object into another p

(the image.)

The latter function is the topic of the discipline of Optical Imag

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Imaging condition: ray-tra

2nd FP

1st FP

object

chiefray

thin lens (+)

Image point is located at the common intersection of all

emanate from the corresponding object point

The two rays passing through the two focal points and thcan be ray-traced directly

The real image is inverted and can be magnified or dem

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2nd FP

1st FP

object

chiefray

thin lens (+)

s s

ox

io

Lateral Angular Lens Law

magnification magnification

1 1 1M

s

s

x

x

oi=

o i

M

s

s

i=

o

+ = =fsso

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i

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2nd FP

1st FP

object

(virtual)

chiefray

thin lens (+)

image

The ray bundle emanating from the system is divergent;

image is located at the intersection of the backwards-exten

The virtual image is erect and is magnified When using a negative lens, the image is always virtual,

demagnified

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Tilted object:

the Scheimpflug conditio

The object plane and the image plane

intersect at the plane of the thin lens.

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Lens-based imaging

Human eye

Photographic camera

Magnifier

Microscope

Telescope

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The human eye

Remote object (unaccommodated eye)

Near object (accommodated eye)

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The photographic camer

F

)

detector array digital imaging

meniscus

lens

or (nowadays

zoom lens

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The pinhole camera

object

imageopaquescreen

pin

hole

The pinhole camera blocks all but one ray per object point from

image spacean image is formed (i.e., each point in image sp

a single point from the object space). Unfortunately, most of the light is wasted in this instrument.

Besides, light diffracts if it has to go through small pinholes as

diffraction introduces undesirable artifacts in the image.

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Field of View (FoV)

FoV=angle that the chief ray from an object ca

towards the imaging system

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Numerical Aperture

n

medium of

refr. index

Numerical Ape

: half-angle subtended by (NA) = n sin

the imaging system froman axial object Speed (f/#)=1/2

pronounced f-number

f/8 means (f/#)=8.

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Resolution

?

x

How far can two distinct point objects

before their images cease to be distinguis

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Factors limiting resolution i

imaging system

Intricately related; assessmen

quality depends on the degree th

Diffraction

Aberrations problem is solvable (i.e. its2.717 sp02 for detai Noise

electronic noise (thermal, Poisson) in camer

multiplicative noise in photographic film

stray light

speckle noise (coherent imaging systems on

Sampling at the image plane

camera pixel size

photographic film grain size

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Point-Spread Function

Light distribution

near the Gaussian = PS(geometric) focus

Point source

(ideal)

2z ~

NA

2

The finite extent of the PSF causes blur in t

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Diffraction limited resolut

object

spacing

)

)

x

lateral coordinate at image plane (arbitrary units

lightintensity

(arbitraryu

nits

Point objects justx

22.1 Rayleig

resolvable when (NA) cri

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Wave nature of light

Diffraction

broadening of

point images

diffracti

Inteference

??

?Fabry-Perot interferometer

Michelson interferometer

(o

Polarization: polaroids, dichroics, liquid crysta

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Diffraction grating

incident Grating spatial frequenplane

Angular separation between diffractwave

m

=1

m=3

m=2

m=1

m=2

m=3

m=0 straight-through or

Condition for cons

=

m22

sin =m

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Grating dispersion

An

(or

dis

polychromatic

(white)

lightGlass prism:

normal dispersion

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Fresnel diffraction formul

xy

z

xy

outg(

)g ,in yx

2

(

x

x) +

1

x

y

z

x

y

outG(

)G ,in vu

z (

, )gout (

zyx , ;

)

2

i=

exp i

g yx expinzi z

Gout , ;( zvu )

=exp i

Gin

{ (-exp i uzz ( vu ) 22 +v,MIT 2.71/2.710

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1

Fresnel diffractionas a linear, shift-invariant system

2x +

y2

z

Thin transparencyyxh =

1

)( , i

exp i

z

2) exp( yxt , zi

( )

(

)

),(,

),(

1

2

g

g

=

=

yxtyx

yx

(2g

g

=

impulse response

convolution

g yx,

Fourier

transform

(

) (

)G ,2

2G

G

=multiplication

plane wave

spectrum vu

transfer function

exp{ (u ) z( vu ) 2 + 2H =exp i2 i v z,MIT 2.71/2.710

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2 . b ) D e r i v e a n d p l o t t h e i n t e n s i t y d i s t r i b u t i o n a t t h e o u t p u t p l a n e u s i n g t h e

a b o v e a s s u m p t i o n .

t(x)

1

...X/2 X/2

...

x

0

F i g u r e 2 A

input nonlinear output

ff f f

transparencyplane plane

illumination

F i g u r e 2 B

Iout

IinIthr

Isat

F i g u r e 2 C

2

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3 . A c o m m o n l y c i t e d q u a n t i t y d e t e r m i n i n g t h e s e r i o u s n e s s o f a b e r r a t i o n s o f a n o p -

t i c a l s y s t e m i s t h e S t r e h l n u m b e r D , w h i c h i s d e n e d a s t h e r a t i o o f t h e l i g h t

i n t e n s i t y a t t h e m a x i m u m o f t h e p o i n t - s p r e a d f u n c t i o n o f t h e s y s t e m w i t h a b e r r a -

t i o n s t o t h a t s a m e m a x i m u m f o r t h a t s y s t e m i n t h e a b s e n c e o f a b e r r a t i o n s ( i . e . ,

t h e d i r a c t i o n - l i m i t e d c a s e b o t h m a x i m a a r e a s s u m e d t o e x i s t o n t h e o p t i c a l

a x i s ) .

3 . a ) P r o v e t h a t D i s e q u a l t o t h e n o r m a l i z e d v o l u m e u n d e r t h e o p t i c a l t r a n s f e r

f u n c t i o n o f t h e a b e r r a t e d i m a g i n g s y s t e m t h a t i s , p r o v e

R R

H

+ 1

a b e r r a t e d

( u v ) d u d v

D =

R R

1

H

+ 1

d i r { l i m

( u v ) d u d v

1

3 . b ) A r g u e t h a t D i s a r e a l n u m b e r a n d t h a t D 1 a l w a y s .

4 . S k e t c h t h e u a n d v c r o s s - s e c t i o n s o f t h e o p t i c a l t r a n s f e r f u n c t i o n o f a n i n c o h e r e n t

i m a g i n g s y s t e m h a v i n g a s a p u p i l f u n c t i o n t h e t w o - p i n h o l e c o m b i n a t i o n s h o w n

i n F i g u r e 4 . A s s u m e w < d . D o n o t u s e M a t l a b f o r t h i s c a l c u l a t i o n . E x p l a i n

b r i e y t h e a p p e a r a n c e o f y o u r s k e t c h e s , a n d b e s u r e t o l a b e l t h e v a r i o u s c u t o

f r e q u e n c i e s a n d c e n t e r f r e q u e n c i e s .

x

y

d2

2w

2w

F i g u r e 4

3

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5 . A n o j b e c t w i t h s q u a r e - w a v e a m p l i t u d e t r a n s m i t t a n c e i d e n t i c a l t o t h e g r a t i n g o f

F i g u r e 2 A i s i m a g e d b y a l e n s w i t h a c i r c u l a r p u p i l f u n c t i o n . T h e f o c a l l e n g t h o f

t h e l e n s i s 1 0 c m , t h e f u n d a m e n t a l f r e q u e n c y o f t h e s q u a r e w a v e i s 1 = X = 1 0 0 c y -

c l e s / m m , t h e o b j e c t d i s t a n c e i s 2 0 c m , a n d t h e w a v e l e n g t h i s 1 m . W h a t i s

t h e m i n i m u m l e n s d i a m e t e r t h a t w i l l y i e l d a n y v a r i a t i o n s o f i n t e n s i t y a c r o s s t h e

i m a g e p l a n e f o r t h e c a s e s o f

5 . a ) C o h e r e n t o b j e c t i l l u m i n a t i o n ?

5 . b ) I n c o h e r e n t o b j e c t i l l u m i n a t i o n ?

H i n t T h e F o u r i e r s e r i e s e x p a n s i o n o f t h e s q u a r e w a v e o f F i g u r e 2 A i s

o

X

1

n = + 1

n

n

n x

t ( x ) = s i n c e x p i 2

2 2 X

n = 1

w h e r e s i n c ( ) = s i n ( ) = ( ) .

4

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4 . a ) S h o w t h a t t h e P E i s m i n i m i z e d i f w e s e l e c t

V

V

1

+ V

2

0

= :

2

4 . b ) U s i n g t h e o p t i m u m t h r e s h o l d , c a l c u l a t e t h e P E i n t e r m s o f t h e \ e r r o r f u n c -

t i o n "

Z

2

z

e r f ( z ) =

p

e

t

2

d t :

0

N o t e s : ( 1 ) T h e a b o v e - d e s c r i b e d p r o c e s s o f s e l e c t i n g a d e t e c t i o n t h r e s h o l d i s k n o w n

a s \ B a y e s d e c i s i o n . " ( 2 ) T h e e r f d e n i t i o n a b o v e i s a f t e r A b r a m o w i t z & S t e g u n ,

H a n d b o o k o f M a t h e m a t i c a l F u n c t i o n s , D o v e r 1 9 7 2 ( p . 2 9 7 ) . T h e c o n s t a n t s a n d

i n t e g r a l l i m i t s a r e s o m e t i m e s d e n e d d i e r e n t l y i n t h e l i t e r a t u r e .

5 . N o r m a l i z a t i o n . L e t f X

k

g b e a s e q u e n c e o f m u t u a l l y i n d e p e n d e n t r a n d o m v a r i -

a b l e s w i t h a c o m m o n d i s t r i b u t i o n . S u p p o s e t h a t t h e X

k

a s s u m e o n l y p o s i t i v e

X

1

v a l u e s a n d t h a t E V f X

k

g = x

k

= a a n d E V = b e x i s t . L e t

k

S

n

= X

1

+ : : : + X

n

:

P r o v e t h a t E V f S

1

i s n i t e a n d t h a t g

n

X

k

1

E V = f o r k = 1 : : : n :

S

n

n

6 . U n b i a s e d e s t i m a t o r . L e t X

1

: : : X

n

b e m u t u a l l y i n d e p e n d e n t r a n d o m v a r i -

a b l e s w i t h a c o m m o n d i s t r i b u t i o n l e t i t s m e a n b e , i t s v a r i a n c e

2

. L e t

X

1

+ : : : + X

n

X = :

n

P r o v e t h a t

n

X

1

2

=

n 1

E V

(

X

k

X

2

)

:

k = 1

( N o t e : I n s t a t i s t i c s , X i s c a l l e d a n u n b i a s e d e s t i m a t o r o f x = E V f X g , a n d

P

2

X

k

X = ( n 1 ) i s a n u n b i a s e d e s t i m a t o r o f

2

.

2

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E in

EoutM

m

Electron

Nucleus

b

k

5 . W h y i s t h e s k y b l u e ? I n 1 8 9 9 , L o r d R a y l e i g h o b s e r v e d t h a t w h e n w e l o o k a t t h e

s k y , w e s e e l i g h t s c a t t e r e d f r o m p a r t i c l e s i n t h e a t m o s p h e r e , p r i m a r i l y n i t r o g e n .

H e t h e n p r o p o s e d t h e m o d e l s h o w n a b o v e i n o r d e r t o q u a n t i f y t h e s c a t t e r i n g

p r o c e s s . T h e g u r e s h o w s a n e l e c t r o n w i t h m a s s m b o u n d t o t h e n u c l e u s w i t h

a s p r i n g w i t h s p r i n g c o n s t a n t k a n d f r i c t i o n c o e c i e n t b . T h e n u c l e u s h a s m a s s

M m . A f o r c e i s a p p l i e d t o t h e e l e c t r o n d u e t o t h e e l e c t r i c e l d o f t h e

i n c i d e n t s u n l i g h t . T h e s c a t t e r e d e l d i s t h e n p r o p o r t i o n a l t o t h e a c c e l e r a t i o n o f

t h e e l e c t r o n .

5 . a ) F o r m u l a t e a o n e - d i m e n s i o n a l m o d e l f o r t h e s c a t t e r i n g p r o c e s s d e s c r i b e d

a b o v e . ( H i n t : m o d e l t h e n u c l e u s a s i m m o b i l e . )

5 . b ) A s s u m i n g t h a t t h e p o w e r s p e c t r a l d e n s i t y o f s u n l i g h t i s p r e t t y m u c h c o n -

s t a n t o v e r t h e e n t i r e e l e c t r o m a g n e t i c s p e c t r u m , d e r i v e a n e x p r e s s i o n f o r t h e

p o w e r s p e c t r u m o f t h e s c a t t e r e d l i g h t .

2717 spring 2002

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