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)AppendixBshowsthattheestimatedvalueZ?kisgivenbyZ?k=E{Zk

2、y}T2?1=DσIS+

3、Cy(8.4)Ifwede?neavectorλ∈SasR2?1λ=σIS+CythenwecanexpresstheestimateasZ?=DTλkS=R(xa?xk)λaa=1N=R(xa?xk)Λaa=1whereΛ∈NRisavectorwhose?rstSelementsaregivenbyλ1,...,λSandwhoseotherelementsareallzero.Λrepresentsanimagethatisblankexceptattheobservationlocations.Theequationf

4、ortheestimatecanbeinterpretedas?lteringtheimageΛwiththe?lterh(x)=R(?x)andthenextractingthevalueatlocationxk.Weexpresstheestimateinthisformbecauseλ(andhenceΛ)doesnotdependonthelocationbeingestimatedandthereforepointestimatesforeverylocationaresimultaneouslygeneratedby

5、the?lteringofΛ.8.4Approximationtechniques8.4.1KrigingKrigingisacollectionofgeneralpurposeapproximationtechniquesforirregularlysampleddatapoints[13].Itsbasicform,knownasSimpleKriging,considersanestimatorKkfor8.4.APPROXIMATIONTECHNIQUES133therandomvariableZkthatisaline

6、arcombinationoftheobserveddatavalues:TKk=wYwherewisaS×1vectorcontainingthecoe?cientsofthelinearcombinationassociatedwithpositionxk.Thetechniqueisbasedontheassumptionthatthemeanandcovariancestructureofthedataisknown.Asbeforewesupposethatthedatahasbeenpreprocessedsotha

7、tthemeaniszero.ThecovarianceassumptionmeansthatweknowE{ZkYa}andE{YaYb}fora,b∈{1,...,S}.However,thisisallthatisassumedaboutthepriordistributionofthedata.Inparticular,thereisnoassumptionthatthedataisnecessarilydistributedaccordingtoamultivariateGaussian.Itisimpossiblet

8、ocalculatetheposteriordistributionbecausetheprecisepriordistri-butionisunknown.However,thereissu?cientinformationtocalculatetheestimatorwiththenicestpropertiesamongtherestictedchoiceofpurelylinearestimators.Moreprecisely,wischosentoachievetheminimumexpectedenergyofth

9、eerrorbetweentheestimateandthetruevalue.TheexpectedenergyoftheerrorFisgivenby:2F=E(Kk?Zk)SSS=EwaYawbYb?2EwaYaZk+E{ZkZk}a=1b=1