2011 Spring Lecture(1)_Digial Imaging_FLY
Transcript of 2011 Spring Lecture(1)_Digial Imaging_FLY
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FlowMeasurement
Digitalimageanalysis
Workingprincipleofadigitalcamera
Digitalimageprocessing
LDV, LDA (continued)
FuLingYang
May,18,2011
Workingprincipleofadigitalcamera arrayofopticalsensorsthatconvertslightintensityintoelectricalsignals
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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.
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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.
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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.
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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.
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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.
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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.
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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
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Intensity/Truecolor(RGB)image
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Digitalimageformats
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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.
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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
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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]
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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 .
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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
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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.
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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).
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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
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Circle, used to locate a sphere center
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Postanalysis (1)
Afterweobtainthespherepositionintheconsecutiveimages,ostanal sis can be followed to obtain the var in velocit
x
y
andtheaccelerationcomponents
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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
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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.
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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
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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
????
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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.
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Example2.
PIVmeasurementofthemotionofaninjectedvortexring
C. Morales, Mexico Univ.
camera
L/D=2 L/D=4 53