Complex Wavelet Based image Analysis and Synthesis.0016

《Complex Wavelet Based image Analysis and Synthesis.0016》由会员上传分享，免费在线阅读，更多相关内容在行业资料-天天文库。

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}T2−1=DσIS+

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

4、ortheestimatecanbeinterpretedasfilteringtheimageΛwiththefilterh(x)=R(−x)andthenextractingthevalueatlocationxk.Weexpresstheestimateinthisformbecauseλ(andhenceΛ)doesnotdependonthelocationbeingestimatedandthereforepointestimatesforeverylocationaresimultaneouslygeneratedby

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

6、arcombinationoftheobserveddatavalues:TKk=wYwherewisaS×1vectorcontainingthecoefficientsofthelinearcombinationassociatedwithpositionxk.Thetechniqueisbasedontheassumptionthatthemeanandcovariancestructureofthedataisknown.Asbeforewesupposethatthedatahasbeenpreprocessedsotha

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

8、ocalculatetheposteriordistributionbecausetheprecisepriordistri-butionisunknown.However,thereissufficientinformationtocalculatetheestimatorwiththenicestpropertiesamongtherestictedchoiceofpurelylinearestimators.Moreprecisely,wischosentoachievetheminimumexpectedenergyofth

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