06 learning and inference bobedits - penn state college …€¦ · · 2013-02-19if you want to...

Computervision:models,learningandinference

Chapter6LearningandInferenceinVision

Outline

22Computervision:models,learningandinference.©2011SimonJ.D.Prince

•  Computervisionmodels– Discriminativevsgenerative

•  Workedexample1:Regression

•  Workedexample2:Classification

•  Whichtypeshouldwechoose?

•  Applications

ComputerVisionInference

•  Observemeasureddata,x•  Drawinferencesfromitaboutstateof“world”,w

Examples:

–  Observeadjacentframesinvideosequence–  Infercameramotion

–  Observeimageofface–  Inferidentity

–  Observeimagesfromtwodisplacedcameras–  Infer3dstructureofscene


Regressionvs.Classification

•  Observemeasureddata,x

•  Drawinferencesfromitaboutworld,w

Whentheworldstatewiscontinuouswe’llcallthisregression


RegressionExamplePosefromImageSilhouette


inferjointangles(continuousquantities)

givenmeasuredsilhouetteandageometricbodymodel

RegressionExamplePosefromImage+Depth

6

infer3Djointpositions

givencolorimageandregistereddepthimage(andageometricbodymodel)

MicrosoftKinectSensor

RegressionExampleHeadposeestimation


inferrelativeorientationangleofthehead

givenmeasuredimagefeatures

Regressionvs.Classification

•  Observemeasureddata,x

•  Drawinferencesfromitaboutworld,w

Whentheworldstatewiscontinuouswe’llcallthisregression

Whentheworldstatewisdiscretewecallthisclassification


ClassificationExampleFaceDetection


yesornoresult:isthereafaceintheimage?

ifyouwanttoknowwherethefaceis,orcountmultiplefaces,runtheclassifieroveraslidingwindowofimagepatches

ClassificationExample:PedestrianDetection


sameideaasfacedetection.HoGfeaturedetectorbyDalaalandTriggsismostcommonimplementation.

ClassificationExampleFaceRecognition


amulti‐classdecision,butcanbeturnedintoasetofbinary“1‐vs‐All”classificationdecisions:

personAvseveryoneelsepersonBvseveryoneelsepersonCvseveryoneelse...

ClassificationExampleSemanticSegmentation


assignsemanticlabeltoeachpixelbycombiningseveral1‐vs‐Allbinaryclassifiers

Ambiguityofvisualworld

•  Unfortunatelyvisualmeasurementsmaybecompatiblewithmorethanoneworldstatew– Measurementprocessisnoisy–  Inherentambiguityinvisualdata

•  Conclusion:thebestwecandoiscomputeaprobabilitydistributionPr(w|x)overpossiblestatesofworld


Refinedgoalofcomputervision

•  Takeobservationsx•  ReturnprobabilitydistributionPr(w|x)overpossibleworldscompatiblewithdata

(notalwaystractable–mighthavetosettleforanapproximationtothisdistribution,samplesfromit,orthebest(MAP)solutionforw)


Componentsofsolution

Weneed

•  Amodelthatmathematicallyrelatesthevisualdataxtotheworldstatew.Modelspecifiesfamilyofrelationships,particularrelationshipdependsonparametersθ

•  Alearningalgorithm:fitsparametersθfrompairedtrainingexamplesxi,wi

•  Aninferencealgorithm:usesmodeltoreturnPr(w|x)givennewobserveddatax.


TypesofModel

Themodelmathematicallyrelatesthevisualdataxtotheworldstatew.Twomaincategoriesofmodel

1.  ModelcontingencyoftheworldonthedataPr(w|x) 2.  ModelcontingencyofdataonworldPr(x|w)


Generativevs.Discriminative

1.  ModelcontingencyoftheworldondataPr(w|x)(DISCRIMINATIVEMODEL)

2.ModelcontingencyofdataonworldPr(x|w)(GENERATIVEMODELS)


called“Generative”becausewhenwedrawsamplesfromthemodel,weGENERATEnewdata

Type1:ModelPr(w|x)‐Discriminative

HowtomodelPr(w|x)?

1.  ChooseanappropriateformforPr(w)2.  Makeparametersafunctionofx

3.  Functiontakesparametersθthatdefineitsshape

Learningalgorithm:learnparametersθfromtrainingdatax,w

Inferencealgorithm:justevaluatePr(w|x)


Type2:Pr(x|w)‐Generative

HowtomodelPr(x|w)?

1.  ChooseanappropriateformforPr(x)2.  Makeparametersafunctionofw



Inferencealgorithm:DefinepriorPr(w)andthencomputePr(w|x)usingBayes’rule


Summary

Twodifferenttypesofmodeldependonthequantityofinterest:1.  Pr(w|x)Discriminative2.  Pr(w|x)Generative

InferenceindiscriminativemodelseasyaswedirectlymodelposteriorPr(w|x).GenerativemodelsrequiremorecomplexinferenceprocessusingBayes’rule


Outline






•  Applications

Workedexample1:Regression

Considersimplecasewhere

•  wemakeaunivariatecontinuousmeasurementx•  usethistopredictaunivariatecontinuousstatew

(recall,whenworldstateiscontinuouswecallourinferencingprocedure“regression”)


SampleRegressionApplication

rownumber

halfhe

ightofp

ersoninpixels

Example:learningboundingboxheightdistributionasafunctionofimagerow

mean

stddev

GeandCollins,CVPR2009“MarkedPointProcessesforCrowdCounting”

X(data)

W(“world”)


HowtomodelPr(w|x)?







HowtomodelPr(w|x)?




1.  Choosenormaldistributionoverw2.  Makemeanµalinearfunctionofx

Letvariancebeaconstant

3.  Modelparametersareφ0,φ1,σ2.

note:Thisisalinearregressionmodel.

Parameters arey‐offset,slopeandvariance


Learningalgorithm:learnθfromtrainingpairs(xi,wi).E.g.MAP


Inferencealgorithm:justevaluatePr(w|x)fornewdatax


note:thisgivesusawholenormaldistributionoverw,butwecouldthenusethemeantomakeaprediction..


HowtomodelPr(x|w)?

1.  ChooseanappropriateformforPr(x)

2.  Makeparametersafunctionofw3.  Functiontakesparametersθthatdefineitsshape


Inferencealgorithm:DefinepriorPr(w)andcomputePr(w|x)usingBayes’rule



HowtomodelPr(x|w)?



1.  Choosenormaldistributionoverx2.  Makemeanµlinearfunctionofw

(varianceconstant)

3.  Parameterareφ0,φ1,σ2.


Learningalgorithm:learnθfromtrainingpairs(xi,wi).


Pr(x|w)xPr(w)=Pr(x,w)jointprobability


canalsolearnpriordistributionfromthewicomponentsofthetrainingdatapairs,orspecifyinsomeotherway

Inferencealgorithm:computePr(w|x)usingBayesrule


InterestingObservation


Inthisexample,bothgenerativeanddiscriminativemodelsleadtothesameposteriornormaldistributionp(w|x)ifMLEisusedtoestimatemodelparameters.

Thisisdueto:*bothxandwbeingcontinuous*theyarerelatedbyalinearmodel*normaldistributionsweretorepresentalluncertainties.

MAPestimationwouldhaveledtodifferencesinthegenerativeanddiscriminativeresultsbecausewewouldplacepriorsontheparameters(e.g.φ0,φ1,σ2)andtheparametershavedifferentmeaningsinthetwomodels.

Outline






•  Applications

Workedexample2:Classification

Considersimplecasewhere

•  wemakeaunivariatecontinuousmeasurementx•  usethistopredictadiscretebinaryworldw

(recall:wearecallinginference“classification”whentheworldstateisdiscrete)


ClassificationExample:SkinDetection


redchannelhistogram

skinnonskininputRGBimage

classifyeachpixelasskinvsnotskinofflinelearningfromlabeled

trainingdata

+


HowtomodelPr(w|x)?

–  ChooseanappropriateformforPr(w)

–  Makeparametersafunctionofx–  Functiontakesparametersθthatdefineitsshape




Type1:ModelPr(w|x)‐DiscriminativeHowtomodelPr(w|x)?

1.  ChooseanappropriateformforPr(w)

2.  Makeparametersafunctionofx3.  Functiontakesparametersθthatdefineitsshape

39

Computervision:models,learningandinference.©2011SimonJ.D.Prince

1.  ChooseBernoullidist.forPr(w)2.  Makeparameterλafunctionofx

3.  Functiontakesparametersφ0andφ1

note:Thismodeliscalledlogisticregression(eventhoughwearedoingclassificationherenotregression)

Background:Logisticor“sig”Function

40

t

P(t)

Mapsrealnumberlineintorange[0,1]

Learnthetwomodelparametersfromtrainingpairs(xi,wi)bystandardmethods(ML,MAP,Bayesian)

Inference:JustevaluatePr(w|x)

41

Computervision:models,learningandinference.©2011SimonJ.D.Prince


HowtomodelPr(x|w)?







HowtomodelPr(x|w)?



1.  ChooseaGaussiandistributionforPr(x)

2.  Makeparametersafunctionofdiscretebinaryw

3. Functiontakesparametersµ0, µ1, σ2

0, σ21thatdefineitsshape



Learnmodelparametersµ0, µ1, σ20, σ2

1fromtrainingdata

thisisreallytwodifferentnormaldistributions,oneforP(x|w=0)andoneforP(x|w=1).



likelihood

*

prior

posterior


ComparingPosteriorsdiscriminative generative

Thistimetheposteriorsarenotequivalent.Thisispartlyduetothe“asymmetry”betweenworldstate(discretedata)andmeasurements(continuousdata).Also,theshapesofdiscriminativeposteriorstendbydefinitiontobesimplerthangenerativeones.

Outline





•  Whichtypeofmodelshouldwechoose?

•  Applications

Whichtypeofmodeltouse?1.  Generativemethodsmodeldata–costlyand

manyaspectsofdatamayhavenoinfluenceonworldstate


BayesRule

mightbesimplertodirectlymodeltheposterior

needlotsofdatatoaccuratelymodelcomplicatedmultimodallikelihoodfunctions

Whichtypeofmodeltouse?

2.  Inferencetendstobesimplerandfasterwhenusingdiscriminativemodels

3.  Datareallyisgeneratedfromtheworld–generativemodelingmatchesthis

4.  Ifmissingdata,thengenerativepreferred

5.  Generativeallowsimpositionofpriorknowledgespecifiedbyuser


Conclusion


•  Todocomputervisionwebuildamodelrelatingtheimagedataxtotheworldstatethatwewishtoestimatew

•  Twotypesofmodel•  ModelPr(w|x)‐‐discriminative•  ModelPr(w|x)–generative

FuturePlan

•  Seentwotypesofmodel

–  Probabilitydensityfunction–  Linearregression–  Logisticregression

•  Nextthreechaptersconcernthesemodels


06 learning and inference bobedits - penn state college …€¦ · · 2013-02-19if you want to...

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