diffraction opticsdial/ece425/notes9.pdf · diffraction optics physical basis • considers the...

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metrical optics

e wavefronts

ptics viewpoint,

response of of the exit pupil n)

DR. ROBERT A. SCHOWENGERDT [email protected]

DIFFRACTION OPTICS

Physical basis

• Considers the wave nature of light, unlike geo

• Optical system apertures limit the extent of th

• Even a “perfect” system, from a geometrical owill not form a point image of a point source

• Such a system is called diffraction-limited

Diffraction as a linear system

• Without proof here, we state that the impulsediffraction is the Fourier transform (squared) of the optical system (see Gaskill for derivatio

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nction (PSF)

rier transform of

ind and


• impulse response called the Point Spread Fu

• image of a point source

• The transfer function of diffraction is the Fouthe PSF

• called the Optical Transfer Function (OTF)

Diffraction-Limited PSF

• Incoherent light, circular aperture

where J1 is the Bessel function of the first k

the normalized radius r’ is given by,

PSF r'( ) 2J1 r'( )

r'--------------

2=

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1.22π


where

• The first zero occurs at , or

Compare to the course notes on 2-D Fourier transforms

r'πDλf-------r

πrλN--------= =

D aperture diameter=

f focal length=

N f-number=

λ wavelength of light=

r 1.22λfD------ 1.22λN== r' =

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8 10

)


0

0.2

0.4

0.6

0.8

1

0 2 4 6

PS

F

normalized radius r'

2-D view (contrast-enhanced“Airy pattern”

central bright region, to first-

zero ring, is called the “Airy disk”

radial profile

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• Example calculation of PSF size

• system specs:

D = 1cm

f = 50mm

N = f/D = 5

λ = 0.55µm (green)

• radius of PSF = 1.22λN = 3.36µm

• very small!

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ncy is given by,


Diffraction-Limited OTF

• Incoherent light, circular aperture

where the normalized radial spatial freque

and the cutoff frequency is given by,

where

OTF ρ'( ) 2π--- ρ'( )acos ρ' 1 ρ'

2––=

ρ' ρ ρc⁄=

ρcDλf------ 1

λN--------= =

D aperture diameter=

f focal length=

N f-number=

λ wavelength of light=

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” function

ter of spatial

1


In 2-D spatial frequency space, the OTF is nearly a “cone

OTF is a low-pass filfrequencies

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8

OT

F

normalized spatial frequency (ρ/ρc)

u

v

OTF

ρ = ρc

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363cycles/mm

utoff frequency of

al viewing


Example calculation of OTF cutoff frequency

• system specs:

D = 1cm

f = 50mm

N = f/D = 5

λ = 0.55µm (green)

• cutoff frequency = 1/λN = 0.363cycles/µm =

• very high! (the human vision system has a cabout 10cycles/mm (object scale) at normdistance)

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t and system

u,v

)

x

MTF

(

u,v

)

ut x( ) A B 2πuox( )sin+=

uox)


Modulation Transfer Function (MTF)

• Amplitude part of complex OTF

• signal modulation

• modulation is a measure of signal contrast

• for sinewave, , modulation = B/A

• Use MTF to predict image contrast, given objec

• output modulation(u,v) = input modulation(

• for sinewave input (object) to LSI system

the output (image) is

MTF u v,( ) OTF u v,( ) output signal modulationinput signal modulation------------------------------------------------------------= =

signal modulation max min–max min+--------------------------=

A B 2πuox( )sin+

inp

output x( ) A MTF uo( )B 2π(sin+=

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ulation MTF uo( ) BA---⋅=

u = 4u0

TFg system


and the image modulation at spatial frequency uo is image mod

MTF

u

1

input

output

u0 2u0 4u0

u = u0 u = 2u0

0.8

0.5

0.1

“typical” Mfor imagin

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LES

iffraction-


DIFFRACTION IMAGING EXAMP

Sine-wave imaged by circular aperture, dlimited

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cutoff-frequency, uc

atial frequency, u

convolution


opticssp

Input pattern

Output pattern

System response

(image)

(object)

(PSF)

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, diffraction-


Square-wave imaged by circular aperture

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cutoff-frequency, uc

spatial frequency, u

convolution


limited optics

Input pattern

Output pattern

System response

(image)

(object)

(PSF)

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sufficient energy

rce

se energy –>


Line and Edge Spread Functions

• PSF is often difficult to measure because of in

• Use a line source (slit) to increase energy –>

Line Spread Function (LSF)• Integrate PSF along direction of the line sou

• Use edge source (knife edge) to further increa

LSFx x( ) PSF x y,( ) yd

∞–

∞

∫=

LSFy y( ) PSF x y,( ) xd

∞–

∞

∫=

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ment

tem

f the PSF is not


Edge Spread Function (ESF)• Integrate LSF up to position of ESF measure

• Equivalent to step response in electronic sys

• ESF is a monotonic function of position

• LSF is the derivative of the ESF,

• Both LSF and ESF are orientation-dependent, iisotropic

ESFx x( ) LSFx α( ) αd

∞–

x

∫=

ESFy y( ) LSFy α( ) αd

∞–

y

∫=

LSFx x( )xd

dESFx x( )=

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tations to


• Must measure LSF and ESF at multiple orienreconstruct full 2-D PSF

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SF AND OTF

n are:

x(x)

Fx(x)

1-Dderivative


RELATIONSHIPS BETWEEN PSF, LSF, E

• For example, the relationships in the x-directio

PSF(x,y)

OTF(u,v) LSF

2-Done-sided

1-D

ES

1-DFourier

Transform

TransformFourier

OTF(u,0)profile

one-sided 1-D integration

integration

1-Dintegration

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MAGING


SECTION III – INTEGRATED ISYSTEM ANALYSIS

Scanning

Image Quality

System Simulation

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ctor(s)

s


SCANNING

Instantaneous image on focal plane dete

Scan the field-of-view

• Object-space scanning (mirror moves)

Show that the object-space scan angle itwice that of the mirror scan angle

object

θ

2θ

detector

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)


• Image-space scanning (detector moves)

• Translation scanning (camera or object moves

object

detector

object

detector

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te sensing

wavelengthdispersionnner

m scanner


Examples from airborne or satellite remosystems

What types of scanning are these examples?

1-D arraywhiskbroom scanner

GFOV

FOV cross-trackin-track

line scanner

2-D arraypushbroom sca

pushbroo

FOV: Field-Of-View (radians)

GFOV: Ground-projected FOV (km)

(aka “swath width”)

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ll area of the

mber of photons

terval equal to

ere W is the

ere is also image


Scanning is equivalent to convolution

• At each instant, the detector integrates a smaimage

• Integration time at each sample determines nucollected

• Generally, a scanned image is sampled at an indetector size

• undersampledWhy is one sample/detector element undersampled?

• PSF of scanning detector is rect(x/W,y/W), whdetector size

• If sample integration time is not negligible, thsmear

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is the amount of

gration time/ blurred)

r during pixel

tector width:

)---


• PSF of linear image smear is rect(x/S), where Ssmear during the integration time:

• Image smear sometimes used to increase intepixel, and therefore SNR (even though image is

• Total scanning PSF

• Combine scanning detector and image smeaintegration (normalize by volume PSF)

• if S = W, i.e. image smear is equal to one de

S image velocity x integration time=

vimgtint=

PSFscan x y,( ) rect x W⁄ y W⁄,( ) * rect x S⁄( )volumePSF

---------------------------------------------------------------------------=

rect x W⁄( ) * rect x S⁄( )[ ]rect y W⁄(volumePSF

------------------------------------------------------------------------------------------=

PSFscan x y,( ) tri x W⁄( )rect y W⁄( )volumePSF

----------------------------------------------------=

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systems

electronic

age quality

humanvision

subsystem

tina*

neuralnetwork

brain


IMAGE QUALITY

An imaging system consists of several sub

* points of signal transduction, optical <—>

Want to maintain the highest possible imthrough this process

lightsource

scene

imageacquisitionsubsystem

transmissionsubsystem

displaysubsystem*

optics detector* electronics

coder decoder

optics

re

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o closely spaced

mination

image

es when the first f the other pattern

easure


Spatial Resolution

• “Resolution” refers to the ability to detect twobjects in an image

• Many different ways to measure resolution

• Rayleigh Criterion

• quantitative; does not depend on visual exa

• appropriate for diffraction-limited optical

• two point sources —> two Airy patterns

• resolution is the separation of the two imagzero of one pattern coincides with the peak o

• does not incorporate noise into resolution m

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hey are just

λN


• Target-Specific Criteria

• separation of two particular objects when t“resolved”

• depends on visual examination

• incorporates noise into resolution measure

R Rayleigh = 1.22

Airypattern1

Airypattern2

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g imagery


targetimaging system PSF and resultin

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requency to

5 6

5 6


• Usually have fixed system PSF; vary target fevaluate resolution

target

imagingsystemPSF

image oftarget

1 2 3 4

1 2 3 4

profileof image

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e bar

et which is “just ually


• Define a line pair = one dark bar & one whit

• Define resolution to be measured by the targresolved” (correct number of bars can be visdistinguished along their entire length)

• Rbar = 1/d4 (line pairs/mm)

where d4 = mm/line pair for target 4

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5 6


• image noise lowers resolution

• Rbar = 1/d3 (line pairs/mm)

1 2 3 4

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ystem

it motion, in-


SENSOR SIMULATION

Landsat Thematic Mapper (TM) satellite s

Whiskbroom scanner

• side-to-side scanning, cross-track

• satellite orbtrack

whiskbroom scanner

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• Three major spatial response components


optics detector electronics

cross-track

in-trackcross-track

only

30m GIFOV

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al photography

to align orbit

scan tion - ixels


• Simulate input scene with high resolution aeri

rotatedwith TM

and direc2m p

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ample to 30m


• Apply each component PSF at 2m and downs

optics optics and detector

optics, detector and electronics

downsample2m —> 30m GSI

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ths later


Compare to real TM of same area, acquired 4 mon

simulated TM real TM (magnified)

(contrast-adjusted)

real TM

(magnified)

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f silver halide in

image

ia


PHOTOGRAPHIC GRANULARITY

• photographic density is aggregated “grains” odeveloped film

• causes random noise at every pixel in scanned

• Ex: aerial photograph of Dulles Airport, Virgin

enlargement

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versus DN


• Signal-dependent noise

• Standard deviation in Digital Numbers (DNs)

6

8

10

12

14

16

0 50 100 150 200

σ DN

µDN

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es

g


ELECTRO-OPTICAL SCANNER NOISE

• Due to

• line scan calibration (gain and offset) chang

• detector-to-detector calibration differences

• interference with other electronics

• saturation hysteresis

• Common problem

• Can be removed effectively by image processin

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erent noise

dom pixel noise


aerial video aperiodic scanline noise Landsat–4 MSS coh

Landsat–1 MSS bad scanline noise

Landsat–4 TM banding noise

AVIRIS scanline and ran

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Signal-to-Noise Ratio (SNR)

• Measure of image quality

• Several common definitions:

SN Rstd

σsignal

σnoise-----------------=

SN Rvar

σsignal2

σnoise2

-----------------=

SNRdB 10 SNR( )log=

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SNRstd 2=SNRvar 4=SNRdB 6=



• Generally correlated with visual quality



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noise


• Visual quality depends on noise type

noiseless

global random noise detector striping

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