数字图像处理第五章3(2)资格考试认证

1/1数字图像处理-第五章3(2)-资格考试认证

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(酉矩阵):n阶复方阵U的n个列向量是U空间的一个标准正交基，则U是酉矩阵(UnitaryMatrix)。一个简洁的充分必要判别准则是：方阵U的共扼转置乘以U等于单位阵，则U是酉矩阵。酉矩阵的逆矩阵与其伴随矩阵相等。

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轾0x2+y2x2y22犏ee22犏thematrixis犏1.Bute=ee2,

-1犏2ee犏臌0轾1e犏轾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.

设一维离散序列fx,x0,1,2,

,N1,以12为中心反折,形成

N至1的序列,与原序列合并形成2N的偶序列。此时傅立叶变换的核函数为ej2uxN转变为ecos2x1u2N这时的变换就叫余弦变换1j2xu2N2

按傅立叶变换性质,虚部为0不进行运算,核函数等价于

因此余弦正变换：Fufxcos2x1u2Nx0为保证每行正交向量模=1,对上式进行归一化处理，N1

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

Fuaufxcos2x1u2Nx0N1

1当u0时Nau2当u0时N余弦变换采纳矩阵表示为FCCf其中核矩阵C中元素为Cu,xaucos2

x1u2N

直流系数DC(u=0时),沟通系数AC(其他)

c1cC=2...cn

余弦变换是正交变换，即0,lkcl,ck=1,lk

由于余弦变换是傅立叶变换的特例，傅立叶反变换的核矩阵即是W阵的共轭矩阵，对于余弦变换共轭矩阵即等于本身，因此fCTFC

5.2.2、二维余弦变换思想：如何形成二维偶函数？先水平做对折镜象，然后再垂直做对折镜象。偶对称偶函数：fx,yf1x,yfx,yfx,1yf1x,1yN1M1

当x,y0时当x0y0当x0y0当x