數(shù)字圖像處理第五章3(2)資格考試認證

1/1數(shù)字圖像處理-第五章3(2)-資格考試認證

Chapter5DiscreteImageTransform5.1FundamentalConcept5.2CosineTransform5.3RectangularWaveTransform5.4Principle-ComponentAnalysisandK-LTransform5.5WaveletTransform

5.1FundamentalConcept5.1.1One-DimensionalDiscreteLinearTransform

Definition.ifxisanN1vectorandTisanNNmatrix,thenyTxdefinesalineartransformofthevectorx.ThematrixTisalsocalledthekernalmatrixofthetransform.Example:therotationofavectorinatwo-dimensionalcoordinatesystem.y1cosysin2sinx1xcos2

Inversion:theoriginalvectorcanberecoveredbytheinversetransformxT1yprovidedthatTisnonsingular.

5.1.21DdiscreteorthogonaltransformUnitarymatrix(酉矩陣):n階復(fù)方陣U的n個列向量是U空間的一個標準正交基，則U是酉矩陣(UnitaryMatrix)。一個簡潔的充分必要判別準則是：方陣U的共扼轉(zhuǎn)置乘以U等于單位陣，則U是酉矩陣。酉矩陣的逆矩陣與其伴隨矩陣相等。

5.1.21Ddiscreteorthogonaltransform

Unitarytransform:yTxIfTisaunitarymatrix,thenT1T*,andTT*T*TI。Orthogonaltransform:IfTisarealtransform,thentheunitarytransformisanorthogonalone.T1T,TTI。

Orthogonalbasis:eachlineoftheorthogonalmatrixTiscalleditsorthonormalbasis.ThismeansthatanyN-by-1sequencecanbeviewedasrepresentingavectorfromtheorigintoapointinN-dimensionalspace.Theorthonormalbasisareorthogonaltoeachother.

Insummary,aunitarylineartransformgeneratesy,avectorofNtransformcoefficients,eachofwhichiscomputedastheinnerproductoftheinputvectorxwithoneoftherowsofthetransformmatrixT.Theforwardtransform:Theinversetransform:

yTxxT1y

5.1.3Two-DimensionalDiscreteLinearTransform

ThegenerallineartransformthattakestheNNmatrixFintothetransformedNNmatrixGisGu,vx,y,u,vFx,yx0y0N1N1

0u,vN1

isthekernalfunctionofthetransform,whichisaN2N2blockmatrixhavingNrowsofNblocks,eachofwhichisanNNmatrix.Theblocksareindexedbyu,vandtheelementsofeachblockbyx,y.

Separatable:Ifthekernalfunctioncanbeseparatedintotheproductofrowwiseandcolumnwisecomponentfunctions.Forsome(u,v),x,y,u,vVcy,vVrx,uthenthetransformiscalledseparable.Itmeansthatitcanbecarriedoutintwosteps__N1Gu,vVcy,vfx,yVrx,ux0y0GTc'FTr'N1

Vr(xu)

Vc(yv)Tr

Example:2Dfunctione,xandytakes0,1.1輊0x2+y2x2y22犏ee22犏thematrixis犏1.Bute=ee2,

-1犏2ee犏臌0輊1e犏輊02犏whichisequalto犏e.1e犏犏e2臌犏臌

x2+y22

Symmetric:Ifthetwocomponentfunctionsareidentical,thetransfromisalsocalledsymmetric.N1GVy,vfx,yVx,uTFTx0y0ItisaunitarytransformifTisaunitarymatrix,calledthekernalmatrixN1

ofthetransform.TheinversetransformisFT1GT1T*GT*

OrthogonalTransformations:Aunitarymatrixwithrealelementsisorthogonal.F=T'GT'IfTisasymmetricmatrix,asisoftenthecase,thentheforwardandinversetransformsareidentical,sothatG=TFTandF=TGT

5.1.4BasisFunctionsAndBasisImages

TherowsofthekernalmatrixofaunitarytransformareasetofbasisinN-dimensionalvectorspace.TT*INormallytheentiresetisderivedfromthesamebasicfunctionform.Theinversetwo-dimensionaltransformcanbeviewedasreconstructingtheimagebysummingasetofproperlyweightedbasisimages.Fx,y'u,v,x,yGu,vu0v0N1N1

Eachelementinthetransformmatrix,G,isthecoefficientbywhichthecorrespondingbasisimageismultipliedinthesummation.

Eachbasismatrixischaracterizedbyahorizontalandaverticalspatialfrequency.Thematricesshownherearearrangedlefttorightandtoptobottominorderofincreasingfrequencies.

5.2CosineTransform5.2.1OnedimensionalDiscreteCosineTransformAsweknow,whenf(x)isanevenfunction,Fouriertransformisonlyreal.HowabouttheFouriertransformiff(x)isnot.

設(shè)一維離散序列fx,x0,1,2,

,N1,以12為中心反折,形成

N至1的序列,與原序列合并形成2N的偶序列。此時傅立葉變換的核函數(shù)為ej2uxN轉(zhuǎn)變?yōu)閑cos2x1u2N這時的變換就叫余弦變換1j2xu2N2

按傅立葉變換性質(zhì),虛部為0不進行運算,核函數(shù)等價于

因此余弦正變換：Fufxcos2x1u2Nx0為保證每行正交向量模=1,對上式進行歸一化處理，N1

FCf111f0F01246F18888888f1real(e)real(e)real(e)real(e)f2F2F3f3

Fuaufxcos2x1u2Nx0N1

1當u0時Nau2當u0時N余弦變換采納矩陣表示為FCCf其中核矩陣C中元素為Cu,xaucos2

x1u2N

直流系數(shù)DC(u=0時),溝通系數(shù)AC(其他)

c1cC=2...cn

余弦變換是正交變換，即0,lkcl,ck=1,lk

由于余弦變換是傅立葉變換的特例，傅立葉反變換的核矩陣即是W陣的共軛矩陣，對于余弦變換共軛矩陣即等于本身，因此fCTFC

5.2.2、二維余弦變換思想：如何形成二維偶函數(shù)？先水平做對折鏡象，然后再垂直做對折鏡象。偶對稱偶函數(shù)：fx,yf1x,yfx,yfx,1yf1x,1yN1M1

當x,y0時當x0y0當x0y0當x