Complex Wavelet Based image Analysis and Synthesis.0016

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1、132CHAPTER8.INTERPOLATIONANDAPPROXIMATIONA-PDFSplitDEMO:Purchasefromwww.A-PDF.comtoremovethewatermarkcovariancesbetweentheobservationlocationsandlocationxk.Da=EZs(a)Zk=Bs(a),k=R(xs(a)−xk)=R(xa−xk)AppendixBshowsthattheestimatedvalueZˆkisgivenbyZˆk=E{Zk

2、y}

3、T2−1=DσIS+Cy(8.4)Ifwedefineavectorλ∈SasR2−1λ=σIS+CythenwecanexpresstheestimateasZˆ=DTλkS=R(xa−xk)λaa=1N=R(xa−xk)Λaa=1whereΛ∈NRisavectorwhosefirstSelementsaregivenbyλ1,...,λSandwhoseotherelementsareallzero.Λrepresentsanimagethatisblankexceptattheobservation

4、locations.TheequationfortheestimatecanbeinterpretedasfilteringtheimageΛwiththefilterh(x)=R(−x)andthenextractingthevalueatlocationxk.Weexpresstheestimateinthisformbecauseλ(andhenceΛ)doesnotdependonthelocationbeingestimatedandthereforepointestimatesforeveryloc

5、ationaresimultaneouslygeneratedbythefilteringofΛ.8.4Approximationtechniques8.4.1KrigingKrigingisacollectionofgeneralpurposeapproximationtechniquesforirregularlysampleddatapoints[13].Itsbasicform,knownasSimpleKriging,considersanestimatorKkfor8.4.APPROXIMATIO

6、NTECHNIQUES133therandomvariableZkthatisalinearcombinationoftheobserveddatavalues:TKk=wYwherewisaS×1vectorcontainingthecoefficientsofthelinearcombinationassociatedwithpositionxk.Thetechniqueisbasedontheassumptionthatthemeanandcovariancestructureofthedataiskno

7、wn.Asbeforewesupposethatthedatahasbeenpreprocessedsothatthemeaniszero.ThecovarianceassumptionmeansthatweknowE{ZkYa}andE{YaYb}fora,b∈{1,...,S}.However,thisisallthatisassumedaboutthepriordistributionofthedata.Inparticular,thereisnoassumptionthatthedataisnece

8、ssarilydistributedaccordingtoamultivariateGaussian.Itisimpossibletocalculatetheposteriordistributionbecausetheprecisepriordistri-butionisunknown.However,thereissufficientinformationtocalculatetheestimatorwiththenicestpropertiesamongtherestictedchoiceofpurely

9、linearestimators.Moreprecisely,wischosentoachievetheminimumexpectedenergyoftheerrorbetweentheestimateandthetruevalue.TheexpectedenergyoftheerrorFisgivenby:2F=E(Kk−Zk)SSS=EwaYawbYb−2EwaYaZk+E{ZkZk}a=1b=1