2011 spring lecture(1)_digial imaging_fly

8/3/2019 2011 Spring Lecture(1)_Digial Imaging_FLY

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FlowMeasurement

Digitalimageanalysis

Workingprincipleofadigitalcamera

Digitalimageprocessing

LDV, LDA (continued)

FuLingYang

May,18,2011

Workingprincipleofadigitalcamera arrayofopticalsensorsthatconvertslightintensityintoelectricalsignals

2


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ScientificDigitalCamera

Digitalimagesensor Convertsanopticalimage(lightintensity)intoanelectricsignalby

CCDor

CMOS

sensors.

CCD:ChargeCoupledDevice

CMOS:ComplementaryMetalOxideSemiconductordetector

Digitalcamera Builtindigitalimageprocessingchiptocoverttherawdatafromthe

imagesensorintoacolorcorrectedimageinastandardimagefile

orma .

Commontypes:CMOScamera,CCDcamera,EMCCD(ElectronmultiplyingCCD)

camera;ICCD(ImageintensifiedCCD)camera.

3

CCDsensor

Chargecoupleddevicehasthreemajorcomponents:

(A)Asilicondiodephotosensor (aPixel)thatreceivesphotonsofvarious

n ens y.

e

nc en

p o ons

ave

su c en

energy

o

ag a e

an

electronmotionawayfromthesiliconlayer,whichgeneratesacharge.

(B)Thechargemovestoadownstream

chargestorageregion,generating

ananalogoussignal.

(A)

(C)Thequantityoftheaccumulated

charge,thesumof0or1(depending

on

the

incident

light

intensity),

is

then

amplifiedandtransmittedthrougha

clocksignal.

(B)

(C)

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CCDsensor (C),amplified

(A)

(B)

An image is projected by a lense on the diode photo-sensor, causing eachpixel to accumulate an electric charge proportional to the light intensity atthat location.

Once the array has been exposed to the image, a control circuit causes eachpixel to transfer its charges to its neighbor. The last capacitor in the storagesection sends its charge into a charge amplifier, which converts the chargeinto a voltage.

B re eatin this rocess scannin across the hoto-sensor , the controlsignal converts the charge information of the entire pixel array (in (A)) to asequence of voltages (output from (C)), which it samples, digitizes andstores in some memory format.

The algorithms for voltage conversion results in digital images/videos ofdifferent formats. These stored images can then be transferred to a printer,digital storage device or video display.

5

EMCCD

InaCCD,thereistypicallyonlyone

amplifieratthecorneroftheentire

array;thestoredchargeis

sequentiallytransferred

through

theparallelregisterstoalinear

serialregister,thentoanoutput

nodeadjacenttothereadout

amplifier.

uses m arstructuresto s, utpr orto e ngrea outatt e

outputnodethechargeisshiftedthroughanadditionalregister the

multiplicationregister inwhichthechargeisamplified.

Asignalcanthereforebeamplifiedabovethereadoutnoise oftheamplifier

andhenceanEMCCDcanhaveahighersensitivitythanaCCD.

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CMOSsensor

Eachcolumnofphoto

Arowofpixelscanbereadoutinparallelwiththerowselectedbyan

sensors

asan

amp er

associatedwithit(imagecan

betransmittedabovethe

noise).

columnmultiplexer(Xaddressing).

ACMOS

device

is

essentially

a

parallel

readout

device

and

therefore

can

achievehigherreadoutspeeds. However,compensatingforthevariations

inthecurrentstateoftheartCMOSdevicesisdifficult.

7

ICCDsensor

Thephotocathode issimilartothephotosensitiveregionofaPhotoMultiplierTubes

andspectrometers:

the

incident

photons

strikeoutelectronswiththeimpactenergy..

Theliberatedelectronsarethenaccelerated towardanelectron

multipliercomposedofaseriesofangledtubesknownastheMicro

ChannelPlate(MCP).

Undertheacceleratingpotentialofahighvoltage,theincident

electronsgainsufficientenergytoknockoffadditionalelectronsand

henceamplifiestheoriginalsignal.

Thissignalisthendetected/amplified/transmittedusingthe

techniquesdevelopedinotherCCDsensors. 8


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Remarksontheperformance

OneweaknessofaCCDstemsfromthefactthattheCCDisessentiallyaserialreadoutdevice andlownoiseperformanceisonlyachievedattheexpenseofslowreadoutspeeds.Thusashortexposuretime isthemajor

ott enec

or

t e

CCD

tec nique.

CMOScamerascanachievehighframerateswithmoderatesensitivity.The precisions,however,iscomparativelylowduetothenonlinearmodulationofsignalattransmission.

Duetotheaccelerationsection,IntensifiedCCDCamerascanachieveultrashortexposuretimes.ButtheMCPsmaketheunitexpensive,heavy,and

.

Sincetheelectrontransmissionissensitivetotemperature,overheatingmay

degrade

the

performance

of

a

high

speed

CCD

image

acquisition

facility.Extraexpensesispaidforintegratedcoolingsystem.

9

Colorimagesensor

Differedbythemeansofthecolorseparationmechanisms:

, , pixels.

The

image

is

reconstructed

using

a

specific

algorithm.

FoveonX3sensor: layeredsensoratonepixelthatrespondstoR,G,Blightdifferently.

3CCD: thelightisfirstseparatedbyadichroicprism,whichgenerates, , .

quality,butmostexpensive)

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Bayerfilter(GRGB)

Acolorfilterarrayatopofasquaregrid

ofCCDsensorsthatredistributeR,G,B

lightsinaspecificarrangements:

EachpixeloftheBayerpatternimageonlycarriesinformationofONE

Tomimichumaneyesthataremoresensitivetogreenlights,thefilteredimage

is50%green,25%red,and25%blue,

resultingaBayerpatternimage.

co or. us,someco or a a sm ss ngan as o erecons ruc e withsomemathematicalalgorithms(artificialcompensation).

Tointerpolate

a

complete

image

from

a

partial

raw

data,

Demosaicing

(incontrarytoaMosaicpattern)algorithmcanbethemeanofnearestneighborhood,linearinterpolation,,etc.

11

Foveonverticalcolorfilter

Atopeachpixel,thereare3stackedactivesensorswithdifferent

siliconlayerdeposition.TheR,G,Bcomponentsarefilteredin

se uenceandthefilteredcom ositionof R G B.

Thetransmissionrateofeachcolorelementisbasedonthe

wavelengthdependentabsorptionoflightofthesilicon.

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Trichroicprismassembly(TPA)

Afteralightrayentersthefirstprism(A),theblue

componentisreflectedbyacoating(F1)attheAB

interface(longwavelengthbandwidth,high

TheremaininggreencomponentofthebeamtravelsthroughprismC.

frequency).

Thetransmittedlightray(oflowerfrequency)enters

asecondprism(B)andissplitagainattheBCinterface,

withacoating(F2)designatedfortheredcomponent.

EachofR,G,Braysundergoesatotalinternalreflection Thethreecolorcomponentsarethen

Note:TPAcanbeusedinreversetocombinered,greenand

bluebeamsintoacolorimageandisusedinmanyprojector

devices.

13

ColorimageformatsInadditiontoRGBdecomposition,HSL(Hue,Saturation,Lightness)andHSV(

,,value)presentationsarealsopopular.

Atruecolorcanalsobecomposedbythecombinationof

hue

()

thesaturation()

thelightness/value()

NotethatHSL/Vformatsarenottherawdatafromadigitalcamera. 14


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StructureofIndexedimage(8/16 bit)

matrix

ColorMap

15

Intensity/Truecolor(RGB)image

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Digitalimageformats

17

Digitalimageformats

Twomaincategories:

as er

mage

,

,

,

,

, 2dimensional

Vectorimage(SVGscalableVectorGraphics)

Alsoknownasgeometricmodeling,objectorientedgraphicsthatutilizesgeometricalprimitives(point,line,triangle,curve,polygon)

torepresentanimageincomputergraphics.

,theconstructingelementstogenerateanimagewrt.aspecific

screen/imageresolution.

Theresulting

image

in

general

preserves

good

quality

than

a

2D

flatrasterimagewhentheimagesareenlarged.

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Imagecompression

Losslesscompression:

LossycompressionDiscardinformationthatcannotberecognizedbyhuman

eyes;differentlossycompressionalgorithmsutilizevariouslevelsofqualitycompressiontoreducetheimagefilesize

Maycausedeterioration(compressionartifacts)

er mageApplyaninternationallyacceptednoisetothedatato

reducethe

artificial

error

induced

by

quantization.

19

Rasterimagetypes

Eachimageisstoredasambynmatrix.Eachentry

relationtolightintensity)withoneofthefollowing

formats:

Binary:0/1(black/white)image

8bit/16bit: 1256/1 162

Grayscale: 01

Color: OnepopularformatisRGBcolor formats.Eachpixelhas

threedigits

(8

/16

bits)

that

store

the

intensity

of

each

color

(R,G,B)

element. OtherformatincludesHSL,HSV.

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Rasterimageformat(1)

JPEG(JointPhotographicExpertsGroups) Lossy format(8bitpalette;256colorsintotal);degradationafterrepetitive

editing/saving

BMP(Windowsbitmap) Losslesscompression(developedforMSsystem,thuswidelyaccepted)

GIF(GraphicsInterchangeFormat) Losslesscompression;8bitpalette

Suitabletostoreimagesofsimplecolorcombination;

Raw Losslesscompressionbutdifferamongdevices,thusnotaccessibletonormalsoftware

21

Rasterimageformat(2)

TIFF(TaggedImageFileFormat) LossyorLosslessformat(8/16 bitspercolor)

Recorddevicespecifiedcolorspaces,notsupportedbynormalsoftware

PNG(PortableNetworkGraphics) Losslessformatisexcellentforediting

Lossyformatusedwhendistributingimages(aspopularasJPEG)

Supportstruecolor(16millioncolors)images;whilethesimilarGIFonlyaccepts256colors.

Otherformats:

EPS(EncapsulatedPostScript);PDF;WindowsMetafile

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LZWfundamentals(19771979)

Thefirst256codes(8bitcharacters)arebydefaultassignedtothestandardcharacterset.Theremainingcodes(dependingonthetypeof

codes,eg:

12bit,

16

bit)

are

assigned

to

strings

of

new

characters

as

the

algorithmproceeds.

Fora12bitcodes,themain0255codesrefertoindividualcharactersof

usualapplications,whilecodes2564095refertosubstringsgeneratedfor

eachdataset.

Compression:outputanewcodeonlywhenanewcharacterisdetected.Theresultingnewstringwillbeaddedtothestringtable(04095).

[OPTIONAL]

23

LZWdecompression[OPTIONAL]

Input Codes: / W E D 256 E 260 261 257 B 260 TInput/

NEW_CODE OLD_CODESTRING/Output

CHARACTER New table entry

/ / /W / W W 256 (= /W)E W E E 257 (= WE)D E D D 258 (= ED)

256 D /W / 259 (= D/)E 256 E E 260 (= /WE)

260 E /WE / 261 (= E/)261 260 E/ E 262 (= /WEE

ReadOLD_CODE

outputOLD_CODE

WHILEtherearestillinputcharactersDO

ReadNEW_CODE

STRING=gettranslationofNEW_CODE

257 261 WE W 263 (= E/W)B 257 B B 264 (= WEB)

260 B /WE / 265 (= B/)T 260 T T 266 (= /WET)

_

outputSTRING

CHARACTER=firstcharacterinSTRING

addOLD_CODE+CHARACTERtothetranslationtable

OLD_CODE=NEW_CODE

ENDofWHILE

Justlikethecompressionalgorithm,itaddsanewstringtothestringtableeachtimeitreadsinanewcode.Allitneedstodoinadditiontothatistranslateeachincomingcodeintoastringandsendittotheoutput.

Outputcodevalue

Astringofoutputcodevalueisgenerated,whichcanbeconvertedintotheinputstringusingthestringtablegeneratedduringcompression..Theoutputstringisidenticaltotheinputstringfromthecompressionalgorithm.

OnereasonfortheefficiencyoftheLZWalgorithmisthatitdoesnotneedtopassthestringtabletothe

decompressioncode.Thetablecanbebuiltexactlyasitwasduringcompression,usingtheinputstreamasdata.

ThisispossiblebecausethecompressionalgorithmalwaysoutputstheSTRINGandCHARACTERcomponentsofa

codebeforeitusesitintheoutputstream.Thismeansthatthecompresseddataisnotburdenedwithcarryinga

largestringtranslationtable.

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%Compressioncode

Initiation:STRING=getinputcharacter

WHILEtherearestillinputcharactersDO

CHARACTER=getinputcharacter

IFSTRING+CHARACTERisinthestringtablethen

LZWcompression[OPTIONAL]

InputString=/WED/WE/WEE/WEB/WET

Character

Input

Code

Output

Newcode

value

New

String

/W / 256 /W

E W 257 WE

D E 258 ED

/ D 259 D/

WE 256 260 /WE

= c arac er

ELSE

outputthecodeforSTRING

addSTRING+CHARACTERtothe

stringtable

STRING=CHARACTER

ENDofIF

ENDofWHILE

outputthecodeforSTRING

TheinputstringisashortlistofEnglishwordsseparatedbythe'/'character.

WEE 260 262 /WEE

/W 261 263 E/W

EB 257 264 WEB

/ B 265 B/

WET 260 266 /WET

EOF T

1. Readin/andWasthefirststringandcharacter.Thestring/Wisnotinthedefaultstringtable,generatea

newcodevalue(256)forandaddittothestringtable.ThereferencestringisnowW.

2. KeepreadinginEasthenewcharacter. Wisoutputasanexistingstring.ThenewstringWEisnewtothe

stringtable,

thus

added

in

and

assigned

to

a

new

code

value

(257).

.The

reference

string

is

now

E.

3. WhenwereadW at asthenewcharacter.Itcombineswiththereferencestring/intoastring/Won

thestringtableandacodevalue256isoutput.TheprogramreadsinthenextcharacterEandcombinesit

withthereferencestring/Wtoformanewstring/WEwithanewcodevalue(260).

4. Theprocesscontinuesuntilthestringisexhaustedandallofthecodeshavebeenoutputtoastringtable. 25

Digitalvideoformat

Adigitalvideoisgeneratedbythreadingaseriesofdigitalimagesintime.

Differedbythealgorithmsforaudioandvideodatacompression:

MPEG(MovieProfessionalExpertGroup)

MPEG1,MPEG2(forbroadcastqualitytelevision),MPEG4

AVI(AudioVideoInterleave)

especiallydesignedforMSenvironment,developedupontheResourceInterchangeFileFormat(RIFF).

,coded/decodedbyaspecificCODEC.

QuickTime

MultimediaframeworkdevelopedbyApple.

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Analysisofadigitalvideo

ConverttheAVI,MPEG,..etcvideosintoasequenceofdigitalimages(grayscaleortruecolor).Analyzetheconsecutiveimagestoobtainthe

. Becarefulabouttheconversionrate:

Recrededat20fps

1sec

Convertedat

10

fps

for

1sec

Convertedat3fpsfor1sec

t=1/20sec

t=1/10sec

t=1/3sec 27

Fundamentalsofimageanalysis

Conversionbetweenimageformats

Definethetargetedimagecomponents

consecutiveimage

frames.

Theliquidmotioncanbeextrapolatedbythedisplacementoftheseedingparticles betweentwoframes

Thevolumefractionofasolidliquidmixturescanbeestimatedbythetotalareaofthesolidobjects

Developaneffectivealgorithmforthetaskbymanipulatingtheimages.

Software:MATLAB,ImageJ,

Byanalyzingthesequentialimages,thetransientphenomenacanbe

extrapolated.Thus,

the

knowledge

of

time

duration

between

consecutive

imagesisessentialforarealtransientphenomenon.

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Digitalimageacquisition

Employadigitalcameratorecordatransientoradynamicphenomena.

Theobtaineddigitalvideoistransformedintoasequenceofdigitalimages

witha

designated

rate.

Boththefilmingandexportframeratecombinetodeterminetheactual

timedurationbetweenthetwoconsecutiveimages,whichinformationis

importantinextrapolatingthetransientinformation.

Thecamerafilmingratedetermineshowfasttheinformationoflight

intensityisexportedfromaCCDsensor.Thus,astronglightsourceis

requ re a g ramera e srequ re .

29

Digitalimageprocessingandanalysis

Generallycontainsthreemajorcategories:

(1)Image

conversion

changeofformats,compression(toreducetheamountmemoryneeded

tostoreadigitalimage);

(2)Imageenhancementandrestoration

imagedefectscausedbydigitizationprocessorbythesystemsetupmay

becorrectedorenhancedinthissteptoobtainanimageappropriatefor

atermeasurements

(3)Measurementextraction

furthermanipulation

may

be

needed

to

achieve

the

desired

measurement

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Imageprocessing

Adigitalimage isacommonlystoredasaMxNx3(orx1)matrixof

numbers,whichvaluecorrespondtothelightintensity(EMwave)ofthe

object.

Imageprocessing/editingisperformingmathematicaloperations on

thesenumbersinordertoachievedesiredformatortoextrapolate

informationfromthearraysofnumbers:

Forexample,thefollowingalgorithmfollowsNTSCstandardto

calculatetheluminanceofaRGBcolorimagestoredasP(M,N,3):

I(m,n)=0.2989P(m,n,1)+0.587 P(m,n,2)+0.114P(m,n,3)

R G B

31

Manipulation/Conversion

Histogram:agraphicalrepresentationoftonal(grayscaleor

luminosity

luminosity)distribution

of

a

digital

image

Bychangingthehistogram,image

enhancementandmanipulationcan

beachieved.

R,G,B

Bri htness mean luminosity of the image

Threshold change a grayscale image into a BW (binary) image:

For each pixelI >threshold, make it 1 (Black pixel)

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Manipulation/dataextraction

Edgedetection:onagrayscaleimage(imageluminositystoredinanarrayofnumbersbetween0and1),theimagelightgradientcanbe

calculate(use

the

difference

between

the

neighboring

pixels)

.Use

the

localmaximumofgradienttopresenttheedge.

Different algorithms can be developed

to outline the image.

This class of approach requires

(1) Gradient calculation (mathematical

description, difference equation)

(2) Criterion for the characteristic

gradient of an edge33

Sharpen/Bluralgorithmsusingthegradientfieldoftheoriginalgrayscaleimage

Manipulation/dataextraction

Watershed algorithm(Segmentationmethod)tosplitthewholeimageintosmallsegments

Fill from the minimum (local or global) upto a specific level to reveal the boarderlines (which can be used to regeneratedthe watershed).

Due to the shape homogeneity, equaldistance between the boardershapes can be applied as one extraalgorithm to separate the objects

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example

Enhancecontrast:bysubtractingan

averagedbackground(2),whichisgenerated

bydilating/smearingouttheoriginalimage(1) (2) FFT of (1)

(1) (2) (3) Remove peaks (4) inverse FFT

(3) (4)

Removal the regular unwanted

pattern with the help of FFT (fastFourier Transformation) thatreveals the underlying relation(distribution) of the number array.

35

Example1.

Wedliketoanalyzethemotionofthetwospheresfromtherecordedvideo.

Objective:Locatethespherepositionineachframe.

Goal:By

the

displacement

between

two

consecutive

images,

the

sphere

velocitycanbeestimatedwithagiven elapsedtime.

Conert an AVI video into an image sequence at a specific frame rate.36


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Preparation:generateimagesequence,obtaint

t

Crop the image of interest. To savememory of your pc. 37

Imageprocessing:

Converttheimagetype:RGBtoGrayscale(intensity)tomanipulatetheimage

x

y

+ brightness + contrast

into BW image:

For each pixelI >threshold, make it 1 (Black pixel)

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ImageProcessing (3)

Method1:

Usethepixelswith011or110neighborstolocatetheedge ofthe

blackpixels.

Thenusethecircumferentialpixelstolocatethespherecenter.

(x1 , y1)

(x , y)=??

x

(x2 , y2)

(x3 , y3)

Large deviation due to the thin line that defines the edge of the sphere39

ImageProcessing (4)

Method2: Fillholes:nowalltheblackpixelspresentthesolidspherewhilewhite

y

.

Averageoutthecoordinates(xi,yi)ofalltheblackpixelswouldgiveanaveragedpositionforthesphere.

(x , y)=??

x(xi , yi)

Due to the greater number of solid pixels, the estimated sphere center isless sensitive to the BW image manipulations.

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Smalldefectduetounevennessofthelightintensity

ImageProcessing (5)

Dilatethenerodetheimages(Close)

Erodethendilatetheimage(Open)

Dilate 3 times (radially) to diminish the influences of thedefect on the averaged coordinate of the sphere (moreblack/less white pixels).

41

Concept:shapedetectioninparameterspace

Line:

ImageProcessing (6):Houghtransformation(2nd method)

0 1 2 3 4 54

6

8

10

12

14

x

y

(1,6)

(2,8)

(3,10)

(4,12)

8

10

12

14

y

y=2x+4

0),( baxyyxf

420 0.5 1 1.5 2 2.5 3

0

2

4

6

8

10

12

a

b

(2,4)

0),( yaxbbaf0 1 2 3 4 5

4

6

x


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0)()(),( 222 rbyaxyxf

Circle

0)()(),( 222 rybxabaf

1)2()2(22 yx

43

Circle, used to locate a sphere center

44


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Postanalysis (1)

Afterweobtainthespherepositionintheconsecutiveimages,ostanal sis can be followed to obtain the var in velocit

x

y

andtheaccelerationcomponents

45

Velocity: differentiation of the digitized position-time data

To express : use

PostAnalysis(2)

df dx

x

y

' x h x

212!

0 0

(3) 21 12! 3!

0

( ) ( ) 1lim lim ( ) '( ) ''( ) ( )

'( ) lim ''( ) ( )

h h

h

df f x h f xf x f x h f x h f x

dx h h

f x f x h f x h

appx

hus

' ( ) ( )Error ( ) '( ) '( ) ( ) . . .app

f x h f xf x f x f x o h H O T

h

Forward-difference scheme

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Postanalysis(3): Finitedifferenceequation

( ) ( )df f x h f xo h

To express first order differential equation

( ) ( )

( )

dx hdf f x f x h

o hdx h

Central difference

Backward difference

More accurate21 ( )

2 F B

df df df o h

dx dx dx

0

0

212!

2102!

1 1 ( ) ( ) ( ) ( )lim

2 2

1lim ( ) ( )

2

( ) '( ) ''( )1lim

2 ( ) '( ) ''( )

hC F B

h

h

df df df f x h f x f x f x h

dx dx dx h h

f x h f x hh

f x f x h f x h

h f x f x h f x h

47

Post-analysis (4)

x

y

Velocity, if taking forward difference scheme:

Toobtaintheactualvelocity,needconversionbetweenthedigitalandtheactualsizesandtimeduration:

(1) Length:measurehowmanypixelscorrespondtoaspherediameter(flowcharacteristiclen thscalethatiseas tomeasure

(pixel/frame)( )i N ii

xuN

N: Number of elapsed frames

(2) Time:byknowtheactualtimedurationbetweentwoprocessedimages(Nofelapsedframes/recordingframerate)

1 5

2

(

7

) ?i ii

mm ms

pixels n frame

x x

t su

(mm/s)

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Errorestimation

Sincethetimedifferencebetweeneachframeispreciselyset(viadigitalcameraandtheconversionsoftware),themain

sourceo

error

n

e erm n ng

e

sp ere

ve oc y

comes

rom

theimageanalysis! Thisiswhyweneedtobecarefulaboutimagemanipulations.

Repeatseveralmeasurements,thenanaveragedvelocityprofilewillbeobtained.Thestandarddeviationoftheresiduesofeachmeasurementfromthemeanvaluecanbe

.

Ifa

line

is

fitted

to

the

empirical

data,

then

a

fitting

deviation

canbecalculatedlikewise.

49

ErrorEstimation

Assume we want to measure a quantity that requires two actualmeasurements on x, y, andz from the experiment.

( , , )F x y z

F x z , , .A general error estimation can be obtained as the following.

Denote the uncertainty in the measuredx, y, andz by .

The overall uncertainty in F is thus:

, , andx y z

22 2

2

, ,,y z x yx z

F F FdF x y z

x y z

Normalizing the uncertainty with the measured value ofFgives thegeneralized error estimation

2

2 dFE

F

50


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ErrorEstimation example

2 1U U

Assume that we want to calculate the coefficient of restitution for a

collision between two solid spheres, defined as

1 2 1 2, , , andV V U U

2 1V V

2 2 2 2

where are the velocities of each sphere before and after

the collision measured in the experiment.

From the previous formula, the error would be

2

2 deE

e

2

2 1 2 1

2 1 2 1

e e e ede U U V V

U U V V

2 2 2 222 1 2 1

2 22

2 1 2 1

+ +

U U V V deE

e U U V V

????

51

ErrorEstimation example

For example, we can track the sphere motion in time, which gives the

following plot:

We can fit a line through 4 position data points and use the slope for the

sphere velocity.

1U 2U1V 2V

2

2 _predicted 2 _measured12

( )N

i iiy y t

UN

the standard deviation of line-fitting.

52


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Example2.

PIVmeasurementofthemotionofaninjectedvortexring

C. Morales, Mexico Univ.

camera

L/D=2 L/D=4 53

2011 spring lecture(1)_digial imaging_fly

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