光電圖像處理：第五章 圖像復(fù)原

第五章圖像復(fù)原Ch5ImageRestoration圖像采集采集過程成像器材的固有缺陷；成像系統(tǒng)的散焦；成像設(shè)備與物體的相對(duì)運(yùn)動(dòng)；噪聲、外部干擾等。桶形畸變-負(fù)畸變枕形畸變-正畸變攝像機(jī)導(dǎo)致的幾何畸變圖像模糊FencingFitness運(yùn)動(dòng)模糊（產(chǎn)生效果圖）運(yùn)動(dòng)模糊（產(chǎn)生效果圖）Noisyimageexamples退化與進(jìn)化圖像復(fù)原(考古學(xué))主要內(nèi)容退化與復(fù)原概述圖像退化模型噪聲模型及去噪聲復(fù)原退化函數(shù)估計(jì)經(jīng)典圖像復(fù)原方法非線性復(fù)原方法圖像退化（Degradation）一、圖像退化與復(fù)原概述圖像復(fù)原為圖像退化的逆過程，是將退化的過程加以估計(jì)，建立退化數(shù)學(xué)模型，補(bǔ)償退化過程造成的失真。在圖像退化確知的情況下，圖像復(fù)原是有可能進(jìn)行的，這屬于反問題求解。(Inversionproblem)圖像復(fù)原（Restoration）成像過程中的“退化”，是指由于成像系統(tǒng)各種因素的影響，使得圖像質(zhì)量降低。系統(tǒng)（模型）退化(正問題)復(fù)原(反問題)觀測(cè)圖理想圖近似惟一性多解性復(fù)原結(jié)果理想結(jié)果一、圖像退化與復(fù)原概述圖像增強(qiáng)Imageenhancement圖像恢復(fù)Imagerestoration目的改善圖像質(zhì)量。改善圖像質(zhì)量。過程及評(píng)價(jià)未考慮退化模型，主觀過程；提供便于人眼觀察或機(jī)器識(shí)別。需要考慮退化模型，客觀過程；恢復(fù)原始圖像的最優(yōu)估值。實(shí)現(xiàn)手段空域（卷積,

Convolution）或頻域?yàn)V波?？沼?反卷積,Deconvolution)或頻域?yàn)V波

。圖像恢復(fù)與圖像增強(qiáng)的異同一、圖像退化與復(fù)原概述主要內(nèi)容退化與復(fù)原概述圖像退化模型噪聲模型及去噪聲復(fù)原退化函數(shù)估計(jì)經(jīng)典圖像復(fù)原方法非線性復(fù)原方法二、圖像退化模型激勵(lì)與響應(yīng)(Input/Output)

線性系統(tǒng)(LinearSystem)0f(x,y)H+g(x,y)n(x,y)二、圖像退化模型

移不變系統(tǒng)

離散脈沖→position-invariantsystem二、圖像退化模型

脈沖響應(yīng)(Impulsesresponse)Pointspreadfunction(PSF)

加性(Additivityproperty)

一致性(Homogeneityproperty)二、圖像退化模型Thesuperpositionintegralofthefirstkind(Fredholm)H←Position-invariant,

卷積積分(Theconvolutionintegral)二、圖像退化模型

空間域表示(Spatialdomain)

頻率域表示(Frequencydomain)

含噪聲線性退化模型(Lineardegradationmodelwithadditivenoise)二、圖像退化模型f(x,y)表示輸入圖像，g(x,y)是f(x,y)產(chǎn)生的一幅退化圖像，H表示退化函數(shù)。退化/復(fù)原的一體模型給定g(x,y)和H，怎樣獲得關(guān)于原始圖像的近似估計(jì)？退化模型向量空間表示

離散卷積形式:二、圖像退化模型退化模型向量空間表示

(1)矩陣形式其中，f，g

是M×N維向量，H是MN×MN維分塊循環(huán)矩陣。每行最后一項(xiàng)等于下一行最前一項(xiàng)；最后一行最后一項(xiàng)等于第一行第一項(xiàng)。每個(gè)Hj是由PSF的he(x,y)第j行而來，即H中的每塊是循環(huán)標(biāo)注的，所以這里的H也是循環(huán)矩陣。退化模型向量空間表示

(2)考慮噪聲，設(shè)n是M×N維噪聲向量，則退化模型為：退化模型向量空間表示

點(diǎn)擴(kuò)展函數(shù)(PSF/H)

噪聲模型(方差,性質(zhì))主要內(nèi)容退化與復(fù)原概述圖像退化模型噪聲模型及去噪聲復(fù)原退化函數(shù)估計(jì)經(jīng)典圖像復(fù)原方法非線性復(fù)原方法

噪聲及來源光學(xué)圖像的噪聲主要來源于圖像的獲取和傳輸兩個(gè)過程。三、噪聲模型及去噪聲復(fù)原圖像獲取的數(shù)字化過程，如圖像傳感器(CCD/CMOS)的質(zhì)量和環(huán)境條件;圖像傳輸過程中傳輸信道的噪聲干擾，如通過無線網(wǎng)絡(luò)傳輸?shù)膱D像會(huì)受到光或其它大氣因素的干擾。1.噪聲的概率密度函數(shù)三、噪聲模型及去噪聲復(fù)原高斯噪聲瑞利噪聲伽馬（愛爾蘭）噪聲指數(shù)分布噪聲均勻分布噪聲脈沖噪聲（椒鹽噪聲）

Gaussiannoise1.NoiseProbabilityDensityFunctionsRayleighnoisewhereErlang(Gamma)noise1.NoiseProbabilityDensityFunctionsExponentialnoisewherewhereUniformnoise1.NoiseProbabilityDensityFunctionswhereImpulse(Salt&Pepper)noise1.NoiseProbabilityDensityFunctionsExample–noisyimagesandtheirhistogramsDEMOExample–noisyimagesandtheirhistograms13452gaussianlocalvarpoissonsalt&pepperspeckleJ=imnoise(I,type,parameters)MATLABcodes2.PeriodicNoise3.EstimationofNoiseParameters

均值濾波(MeanFilters)

統(tǒng)計(jì)排序?yàn)V波器(Order-StatisticFilters)

自適應(yīng)濾波(AdaptiveFilters)空域?yàn)V波器分類Agenda…三、噪聲模型及去噪聲復(fù)原

算數(shù)均值濾波(Arithmeticmeanfilter)

幾何均值濾波(Geometricmeanfilter)

諧均值濾波(Harmonicmeanfilter)

逆諧均值濾波(Contraharmonicmeanfilter)1.均值濾波器1.MeanFilters

ArithmeticmeanfilterormovingaveragefilterGeometricmeanfilter：mn=sizeofmovingwindowDegradationmodel:Toremovethispart

Harmonicmeanfilter

ContraharmonicmeanfilterWorkswellforsaltnoisebutfailsforpeppernoisemn=sizeofmovingwindowPositiveQissuitableforeliminatingpeppernoise.NegativeQissuitableforeliminatingsaltnoise.Q=thefilterorderForQ=0,thefilterreducestoanarithmeticmeanfilter.ForQ=-1,thefilterreducestoaharmonicmeanfilter.1.MeanFilters

Example-MeanFilters

Example-MeanFilters

Example-ContraharmonicFilters

中值濾波器(Medianfilter)最大值濾波器(Maxfilter)最小值濾波器(Minfilter)中點(diǎn)濾波器(Midpointfilter)阿爾法截?cái)嗑禐V波器(Alpha-trimmedMeanFilter)2.Order-StatisticFiltersMedianfilterMaxfilterMinfilterMidpointfilterReduce“dark”noise(peppernoise)Reduce“bright”noise(saltnoise)2.Order-StatisticFiltersAlpha-trimmedMeanFilterwheregr(s,t)representtheremainingmn-dpixelsafterremovingthed/2highestandd/2lowestvaluesofg(s,t).Thefilterisusefulinsituationsinvolvingmultipletypesofnoisesuchasacombinationofsalt-&-pepper

andGaussiannoise.Formula:Notice:d~[0,mn-1],

d=0→arithmeticmeanfilter;

d=mn-1→medianfilter.Example-MedianFilteringrepeatedlyDEMOExample–Max&MinFilters

B=ordfilt2(A,5,ones(3,3))implementsa3-by-3medianfilter;B=ordfilt2(A,1,ones(3,3))implementsa3-by-3minimumfilter;B=ordfilt2(A,9,ones(3,3))implementsa3-by-3maximumfilter.forexampleExample-Order-StatisticFilters

Comparisonof

DifferentFilteringresults:(c)Arithmeticmean(d)Geometricmean(e)Median(f)Alpha-trimmedAdaptive,LocalNoiseReductionFilterAdaptiveMedianFilter3.AdaptiveFilter3.AdaptiveFilterFilterbehaviordependsonstatisticalcharacteristicsoflocalareasinsidem×nmovingwindow.Morecomplexbutsuperiorperformancecomparedwith“fixed”Filters.

Statisticalcharacteristics:Localmean:Localvariance:Noisevariance:Adaptive,LocalNoiseReductionFilter①Ifsh2iszero,Nonoise thefiltershouldreturng(x,y)becauseg(x,y)=f(x,y).②IfsL2

ishighrelativetosh2,Edges(shouldbepreserved), thefiltershouldreturnthevalueclosetog(x,y).③IfsL2=sh2,Areasinsideobjects thefiltershouldreturnthearithmeticmeanvaluemL.Formula:Example-Adaptive,LocalNoiseReductionFilterAlgorithm:LevelA: A1=zmedian–zmin;

A2=zmedian–zmax;

IfA1>0andA2<0,gotolevelB. Elseincreasewindowsize. Ifwindowsize≤

SmaxrepeatlevelA. Elsereturnzxy.LevelB: B1=zxy–zmin B2=zxy–zmax

IfB1>0andB2<0,returnzxy.

Elsereturnzmedian.AdaptiveMedianFilter

zmin=minimumgraylevelvalueinSxy.zmax=maximumgraylevelvalueinSxy.zmedian=medianofgraylevelsinSxy.zxy=graylevelvalueatpixel(x,y).Smax

=maximumallowedsizeofSxy.wherePurpose:

wanttoremoveimpulsenoisewhilepreservingedgesLevelA:A1=zmedian–zmin;

A2=zmedian–zmax;

Else

windowisnotbigenough. increasewindowsize. Ifwindowsize≤

SmaxrepeatlevelA. Elsereturnzxy..

zmedianisnotanimpulse. B1=zxy–zmin; B2=zxy–zmax;

IfB1>0andB2<0,

zxyisnotanimpulse.

returnzxy

topreserveoriginaldetails.

Else returnzmedian

toremoveimpulse.AdaptiveMedianFilter:HowitworksIfA1>0andA2<0,gotolevelB.LevelB:DeterminewhetherzmedianisanimpulseornotDeterminewhetherzxyisanimpulseornotDEMOExample-AdaptiveMedianFilterfunctionf=adpmedian(g,Smax)Edge-preservingfiltering(EPS),suchasbilateralfiltering.Learnmore…

主要內(nèi)容退化與復(fù)原概述圖像退化模型噪聲模型及去噪聲復(fù)原退化函數(shù)估計(jì)經(jīng)典圖像復(fù)原方法非線性復(fù)原方法

圖像觀察(ImageObservation)

實(shí)驗(yàn)估計(jì)(byExperiment)

建模估計(jì)(EstimationbyModeling)估計(jì)方法Agenda…四、退化函數(shù)估計(jì)1.EstimationbyImageObservationf(x,y)*h(x,y)SubimageReconstructedSubimageDFTDFTRestorationprocessbyestimationOriginalimagef(x,y)(unknown)Degradedimageg(x,y)EstimatedtransferfunctionObservationThiscaseisusedwhenweknowonlyg(x,y)andcannotrepeattheexperiment!DFT2.EstimationbyExperimentDFTUsedwhenwehavethesameequipmentsetupandcanrepeattheexperiment.Atmosphericturbulence3.Estimationby

ModelingThephysicalcharacteristicsofatmosphericturbulence——amodelbyHufnagelandStanley(1964)where，kisaconstantthatdependsonthenatureoftheturbulence.3.Estimationby

ModelingwhereTisthedurationoftheexposure，x0(t)，y0(t)arethetime-varyingcomponentsofmotioninthex-andy-directions,respectively.MotionbetweentheimageandthesensorFFT：Theblurredimage:3.Estimationby

ModelingF(u,v)TransferfunctionForuniformlinearmotion,i.e.3.Estimationby

ModelingH(u,v)Example—MotionBlurringMATLABfunction:H=fspecial('motion',20,45);MotionBlur=imfilter(I,H,'replicate');DEMO主要內(nèi)容退化與復(fù)原概述圖像退化模型噪聲模型及去噪聲復(fù)原退化函數(shù)估計(jì)經(jīng)典圖像復(fù)原方法非線性復(fù)原方法基本原理由退化模型得：最小均方誤差準(zhǔn)則：非約束情況（無條件滿足）：五、經(jīng)典圖像復(fù)原方法有約束情況：令Q為的線性算子，有約束最小二乘復(fù)原就是要使得最小。這類最小化問題，可用Lagrange算子來處理，設(shè)為拉格朗日乘子，尋找使下面的準(zhǔn)則函數(shù)最小的（即約束條件）：基本原理五、經(jīng)典圖像復(fù)原方法基本原理，得到：由極值條件：式中：注：適當(dāng)選擇這一參數(shù)，使約束條件滿足，即可求得最佳估計(jì)。不同Q對(duì)應(yīng)不同的復(fù)原方法五、經(jīng)典圖像復(fù)原方法逆濾波(Inversefiltering)維納濾波(Wienerfilter)

約束最小平方濾波(Constrainedleastsquaresfilter)

幾何均值濾波(FilterGeometricMeanFilter)復(fù)原方法分類Agenda…五、經(jīng)典圖像復(fù)原方法EstimationofinitialimageinFTdomain1.InverseFilteringConsideringnoiseFrequencylimitedinversefilterRThetransferfunctionofTurbulenceExample-InverseFilteringwithfrequencylimitedItalsoiscommonlyreferredtoastheminimummeansquareerror

(MMSE)filterortheleastsquareerror

(LSE)filter.5.8WienerFilteringTheerrormeasurewhereE(.)istheexpectedvalueoftheargument.TheWienerfilter(1942)minimizesthemeansquareerror(MSE)betweentheestimatedrandomprocessandthedesiredprocess.NorbertWiener(1894-1964)2.WienerFilteringwhere

g=1,thestandardWienerfilter.Otherwise,theparametricWienerfilter.Hardtoestimateg

=1,thestandard

Wiener2.WienerFilteringPractically,kischosenmanuallytoobtainedthebestvisualresult!Approximatedformula

Wienerfilterformula

Signal-to-noise-ratio(SNR)2.WienerFilteringMeansquareerror(MSE)SNRinthespatialdomainSignalsNoisesExample-WienerFilteringComparisonoftheWienerfilteringandInverseFiltering.Example-WienerFilteringAWGNsh2=650AWGNsh2=65AWGNsh2=0.0065約束最小平方濾波(Constrainedleastsquaresfiltering)，也叫正則濾波(Regularizedfiltering),是一種有約束條件下的迭代復(fù)原技術(shù)。方法簡介補(bǔ)充：正則化(Regularization/Normalization)，是指在線性代數(shù)理論中，不適定問題通常是由一組線性代數(shù)方程定義的，而且這組方程組通常來源于有著很大的條件數(shù)的不適定反問題。大條件數(shù)意味著舍入誤差或其它誤差會(huì)嚴(yán)重地影響問題的結(jié)果。

3.ConstrainedLeastSquaresFilteringDegradationmodelObjective:tofindtheminimumofacriterionfunctionSubjecttotheconstraintwhere3.ConstrainedLeastSquaresFilteringwritteninamatrixformwhere3.ConstrainedLeastSquaresFilteringgisadaptivelyadjustedtoachievethebestresult.P(u,v)=theFouriertransformofp(x,y)——SmoothingConstrained

Restoration.WegetaconstrainedleastsquarefilterExample–ConstrainedLeastSquaresFilteringComparisonoftheWienerfilteringandConstrainedLeastSquaresFiltering.“Residual”vectorAdjust

g

sothatwhere,aisanaccuracyfactor.13.ConstrainedLeastSquaresFilteringMonotonicallyincreasingfunction(Hunt,1973)Stepsforadjustingg1.Specifyaninitialvalueofg.2.Compute.otherwisereturnstep2afterincreasinggif ordecreasinggifUsethenewvalueofgtorecompute3.Stopifissatisfied1StepsforadjustinggForcomputingStepsforadjustinggForcomputing

Discussion(1)g=0→

InverseFiltering

(2)g

=1→WienerFiltering

3.ConstrainedLeastSquaresFilteringExample–ConstrainedLeastSquaresFilteringIterativeestimationoftheoptimumConstrainedLeastSquaresFilteringwithdifferentnoiseparameters.

Wiener濾波與約束最小二乘濾波的比較(1)

都屬于有約束復(fù)原，頻域恢復(fù)公式類似。(2)后者在圖像恢復(fù)時(shí)，不需要知道圖像和噪聲的自相關(guān)矩陣。(3)后者帶有平滑約束，可得到更符合人眼視覺效果的平滑圖像，并且在噪聲較大的情況下比Wiener濾波的效果明顯要好。3.約束最小二乘濾波Thefilterrepresentsafamilyoffilterscombinedintoasingleexpression.a=1

theinversefiltera=0theParametricWienerfiltera=0,b=1thestandardWienerfilterb=1,a<0.5Moreliketheinversefilterb=1,a>0.5MoreliketheWienerfilterAnothername:thespectrumequalizationfilter(譜均衡濾波器)4.GeometricMeanFilter主要內(nèi)容退化與復(fù)原概述圖像退化模型噪聲模型及去噪聲復(fù)原退化函數(shù)估計(jì)經(jīng)典圖像復(fù)原方法非線性復(fù)原方法最大后驗(yàn)復(fù)原(如：R-L算法)最大熵復(fù)原同態(tài)濾波復(fù)原六、非線性復(fù)原方法1.同態(tài)濾波（HomomorphicFilter）目的：正常圖象是在均勻光強(qiáng)度情況下獲得的圖象，實(shí)際上光照射是不均勻，或光強(qiáng)范圍動(dòng)態(tài)太大。方法：為解決光照不均勻的影響，可用同態(tài)濾波來解決。模型原理：隨空間位置不同的光強(qiáng)分量。特點(diǎn)：緩慢變化，頻率集中在低頻部分。景物反射到眼睛的圖象。特點(diǎn)：包含景物各種信息，高頻分量豐富。1.同態(tài)濾波（HomomorphicFilter）①對(duì)原圖取對(duì)數(shù)得到如下兩個(gè)加性分量：②頻域表示為：1.同態(tài)濾波（HomomorphicFilter）③設(shè)計(jì)一個(gè)濾波器H(u,v)進(jìn)行濾波處理.④傅立葉反變換，即IFFT。⑤求指數(shù)結(jié)果，得到復(fù)原結(jié)果。LNFFTH(u,v)IFFTEXP1.同態(tài)濾波（HomomorphicFilter）H(u,v)設(shè)計(jì)壓縮照度分量i增強(qiáng)反射分量rD01.同態(tài)濾波（HomomorphicFilter）同態(tài)濾波：是把頻率過濾和灰度變換結(jié)合起來的一種圖像處理方法，它依靠圖像的照度/反射率模型作為頻域處理的基礎(chǔ)，利用壓縮亮度范圍和增強(qiáng)對(duì)比度來改善圖像的質(zhì)量。

同態(tài)系統(tǒng)：是將非線性問題，轉(zhuǎn)化為線性問題處理。即對(duì)非線性混雜信號(hào)，做某種