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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
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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
propagation detection
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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Formal definition of cross-entr
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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Formal definition of cross-entr
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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Formal definition of cross-entr
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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Formal definition of cross-entr
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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Formal definition of cross-entr
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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Formal definition of cross-entr
( )FEntropy
(
),
(
)|
( ),C
GFEntropyJoint
GFEntropyCond. ECond.
GF
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Formal definition of cross-entr
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
propagation detection
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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IMI for rectangular matrice
HT
H
= square ma
Trecall pseudo-inverse f =( HH )
inversion operation associated with rank of
Tseigenvalue ( HH ) aluessingular v (
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IMI for rectangular matrice
object
channel
Hf
hardwarea
(me
field
propagation detection
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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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
Intro to Inverse Problems p-2
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
Intro to Inverse Problems p-3
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
Intro to Inverse Problems p-4
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
Intro to Inverse Problems p-5
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
Intro to Inverse Problems p-6
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
Intro to Inverse Problems p-7
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
Intro to Inverse Problems p-8
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
Intro to Inverse Problems p-9
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
Intro to Inverse Problems p-10
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
Intro to Inverse Problems p-11
Optical Inversion: incoher
( )yxI ,obj ( ) ( ) ( = xhyxIyxI ,, incohobjmeas
Linear problemLinear problem
object
intensityintensity measurement at the ou
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MIT 2.717
Intro to Inverse Problems p-12
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
Intro to Inverse Problems p-13
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
Intro to Inverse Problems p-14
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
Intro to Inverse Problems p-15
Cross-leaking power
A~
B~
ss
B
A
I
I
=
=
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MIT 2.717
Intro to Inverse Problems p-16
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
from www.aoc.nrao.edu
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VLA images
from www.aoc.nrao.edu
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
Apps of Stat Optics p-6
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from www.aoc.nrao.edu
MIT 2.717
Apps of Stat Optics p-7
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
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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
Apps of Stat Optics p-11www.fit
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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
Apps of Stat Optics p-12www.fit
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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
Apps of Stat Optics p-14
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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
Apps of Stat Optics p-16
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
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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
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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
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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
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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
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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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Coherent vs incoherent ima
field inoptical
system
Coherent fi
intensity in Incoherent inteoptical
system
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Coherent vs incoherent ima
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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Coherent vs incoherent ima
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
In homogeneous media,
light propagates in rectilinear path
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The concept of a polychromati
t=0(frozen)
energ
prett
all wav
propag
thwavefronts
In homogeneous media,
light propagates in rectilinear path
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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
these spherical
wavefrontsReview Lecture p-22
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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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Imaging condition: ray-tra
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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Imaging condition: ray-tra
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
7/25/2019 2717 spring 2002
130/132
7/25/2019 2717 spring 2002
131/132
7/25/2019 2717 spring 2002
132/132
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Electron
Nucleus
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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 .