Denoising via Wavele

8/8/2019 Denoising via Wavele

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DenoisingviaWaveletDenoisingviaWaveletTransformsTransforms

KatieMcCaffreyKatieMcCaffreyAdvisor:PatVanFleetAdvisor:PatVanFleet


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OverviewOverview ExplanationofDenoisingAlgorithmExplanationofDenoisingAlgorithm

WaveletShrinkageMethodsWaveletShrinkageMethods VisuShrinkVisuShrink

SureShrinkSureShrink MathematicaMathematica DemoDemo

OtherPossibleMethodsOtherPossibleMethods


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DenoisingAlgorithmDenoisingAlgorithmBeginwithanoisyvectorBeginwithanoisyvectoryy==vv+e,where+e,wherevv isthetrueisthetruesignalandsignalandeeisGaussianwhitenoise.isGaussianwhitenoise.

Step1:Step1: ComputeComputeii iterationsofthewavelettransformoniterationsofthewavelettransformonyy,,obtainingthenewvectorobtainingthenewvectorzz,madeoflowpassportion,madeoflowpassportionllandhighpassportionandhighpassportiondd..

Step2:Step2: Apply

threshold

rule

tohighpass

portion

Apply

threshold

rule

tohighpass

portion

ddof

ofzz,,

eithereithershrinkingshrinking thevalueorsettingitequaltozero.thevalueorsettingitequaltozero.

Step3:Step3: RejointhemodifiedhighpassportiontotheoriginalRejointhemodifiedhighpassportiontotheoriginallowpass,creatingthemodifiedvectorlowpass,creatingthemodifiedvector..

Step4:Step4: ComputeComputeii iterationsoftheinversewaveletiterationsoftheinversewavelettransformontransformon toobtaintoobtain..

TheNTheNvectorvector shouldbethedenoisedversionofy.shouldbethedenoisedversionofy.


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ThresholdRulesThresholdRules HardThresholdRuleHardThresholdRule

ChoosevalueforChoosevaluefor,thethresholdvalue.,thethresholdvalue.

If|xIf|xkk|,keepx,keepxkk..

SoftThresholdRuleSoftThresholdRule

If|xIf|xkk|,shrinkx,shrinkxkk byby..

For|xFor|xkk|>|>,thisisalinearfunctionf(,thisisalinearfunctionf()=|x)=|xkk||

LetLetstryitonthevector[1,1.2,3,6,2.4,5,1.1]withstryitonthevector[1,1.2,3,6,2.4,5,1.1]with =2.=2.

HardThreshold

Rule:

[0,

0,3,6,2.4,

5,0]Hard

Threshold

Rule:

[0,

0,3,6,2.4,

5,0]

SoftThresholdRule:[0,0,1,4,.4,3,0]SoftThresholdRule:[0,0,1,4,.4,3,0]


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AccurateShrinkageAccurateShrinkage

HowdowechooseanappropriatevalueHowdo

we

choose

an

appropriate

value

ofof foradataset?foradataset?

VisuShrinkVisuShrink

SureShrinkSureShrink Modifications?Modifications?


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VisuShrinkVisuShrink Remember,Remember,yy ==vv ++ee,where,whereee isGaussianwhiteisGaussianwhite

noisethat

isnormally

distributed

with

mean

0noisethatisnormallydistributedwithmean0

andvarianceandvariance22 andnoiselevelandnoiselevel.. WewanttominimizethemeansquarederrorWewanttominimizethemeansquarederrorfoundbyfoundbyDonohoDonoho andandJohnstoneJohnstone

UniversalthresholdUniversalthresholdunivuniv ==2ln(N),whereNis2ln(N),whereNisthenumberofvaluesinthenumberofvaluesinyy..

AsNAsN>> theMedianAbsoluteDeviationofthetheMedianAbsoluteDeviationofthehighpassvaluesconvergesto.6745highpassvaluesconvergesto.6745,sothe,sotheestimateforestimatefor isMAD(d)/.6745.isMAD(d)/.6745.

LetLetsseethisatworkinsseethisatworkinMathematicaMathematica


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ThresholdChangesThresholdChanges

QuadraticQuadratic

CubicCubic

FirmThresholdFirmThreshold

Using4data

sets

that

I

Using4data

sets

that

I

addedartificialnoiseto,Iaddedartificialnoiseto,Imeasuredtheerrorinmeasuredtheerrorineachofthecases,eachofthecases,resultinginthefollowingresultinginthefollowingvaluesvalues

ResultsResults

44332211

FirmFirm

CubicCubic

QuadQuad

LinearLinear

.0088.0088.0828.0828.0392.0392.163.163

.0097.0097.1079.1079.0393.0393.139.139

.0096.0096.1027.1027.0423.0423.158.158

.0082.0082.1051.1051.0420.0420.127.127

StillusingtheVisuShrinkmethod,ItestedotherStillusingtheVisuShrinkmethod,Itestedother

possiblethreshold

functions

(other

than

linear)possible

threshold

functions

(other

than

linear)


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SureShrinkSureShrink SteinSteinsUnbiasedRiskEstimatorremovesthesUnbiasedRiskEstimatorremovestheerrorobtainedbyerrorobtainedbyunivuniv whichdependsonthewhichdependsonthesizeofthedatasetsizeofthedataset

MinimizethemeansquarederrorbyminimizingMinimizethemeansquarederrorbyminimizingthefunctionthefunction

f(f()=N+||g(x)||)=N+||g(x)||22 +2+2d/dxd/dxkk(g(gkk(x(x))))wherewhereggkk(x(x)isthethresholdfunctionminusthe)isthethresholdfunctionminusthevalueforeachvalueofxvalueforeachvalueofxkk,k=1,2,,k=1,2, NN

Oncethis

value

for

Oncethisvalueforsuresure

hasbeen

chosen,

use

ithasbeenchosen,useit

intheoriginalthresholdfunctiontoshrinktheintheoriginalthresholdfunctiontoshrinkthehighpassportionandcontinuetheprocessofhighpassportionandcontinuetheprocessofdenoisingdenoising


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SureShrinkSureShrink Withthelinearsoftthresholdfunction,thefunctionWiththelinearsoftthresholdfunction,thefunctionsimplifiesnicelyassimplifiesnicelyas

f(f()=N)=N 22 #{k:|x#{k:|xkk|| }+}+ min(xmin(xkk22,,22)) ThisgivesdifferentfunctionsofThisgivesdifferentfunctionsof betweenxbetweenxkkandxandxk+1k+1

withaminimumatxwithaminimumatxkk eachtimeeachtime Example:Example: xx=[1,1.1,2,2,2.4]=[1,1.1,2,2,2.4]TT

For1


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AlternativeThresholdsAlternativeThresholds ButwhataboutdifferentthresholdButwhataboutdifferentthreshold

functions?functions? Quadratic;f(Quadratic;f())

Firmthreshold;h(Firmthreshold;h(11,,22))

Goal:FindandminimizefandhGoal:Findandminimizefandh


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FindingEquationsFindingEquations f(f()=N+||g(x)||)=N+||g(x)||22 +2+2d/dxd/dxkk(g(gkk(x(x))))

Quadratic:s(xQuadratic:s(xkk)=(|x)=(|xkk|| ))22,|x,|xkk|>|>;;0,|x0,|xkk|


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CurrentlyCurrently TheexampleIusedwiththelinearthresholdTheexampleIusedwiththelinearthresholdfunctionhad5cases,oneforeachvalueofxfunctionhad5cases,oneforeachvalueofx

kkinin

placeforplacefor.. h(h(11,,22)hasmanymoreoptionsbecauseofthe)hasmanymoreoptionsbecauseofthepossiblecombinationsofpossiblecombinationsofxxkkss..

Eachcombination

provides

an

equation

to

Eachcombination

provides

an

equation

to

minimizeatthevaluesofxminimizeatthevaluesofxkk Currently,IhavefoundthecasesforseveralCurrently,Ihavefoundthecasesforseveralvectorsandamintheprocessoffindingthevectorsandamintheprocessoffindingthe

minimumvaluesforminimumvaluesfor11andand22 Afterallthework,itboilsdowntousingAfterallthework,itboilsdowntousingMultivariableCalculus!MultivariableCalculus!


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CurrentPlaceCurrentPlace Vectors:Vectors:

[1,2,3,4,5]21

cases[1,

2,3,4,5]21

cases

[1,2,3]10cases[1,2,3]10cases

[1,1,2,3],[1,2,2,3],[1,1,2,3],[1,2,2,3],

[1,2,3,3]10cases[1,2,3,3]10cases

CurrentQuestions:CurrentQuestions: Inthisgraph,whichcasesareonthediagonal?Inthisgraph,whichcasesareonthediagonal?

HowdorepeatedvectorvalueseffecttheresultantHowdorepeatedvectorvalueseffecttheresultant11

andand22?? HowdoHowdo11andand22 fromsmallerrepeatedvaluesdifferfromsmallerrepeatedvaluesdifferfromfrom11andand22 fromlargerones?fromlargerones?


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OtherApplicationsOtherApplications SomeotherapplicationsofdenoisingviawaveletSomeotherapplicationsofdenoisingviawavelettransformationstransformations DeDenoiseaudiofilesnoiseaudiofiles

DetectedgesDetectedges IncreasequalityofdigitalimagesIncreasequalityofdigitalimages

MyapplicationsinGeologyMyapplicationsinGeology CrossCrosssectionofasedimentarydepositfromSt.AnthonyFallssectionofasedimentarydepositfromSt.AnthonyFallsLabLabJurassicTankJurassicTank

Howwerethesedimentsdeposited?Howwerethesedimentsdeposited?

PatternsincrossPatternsincrosssection,channelbeds,etc.section,channelbeds,etc.


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Questions?Questions?

Denoising via Wavele

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