L. Yaroslavsky. Fundamentals of Digital Image Processing. Course

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1 L Yaoslavs Fudametals of Digital Image Pocessig Couse Lectue 10 Piciples of image estoatio 101 Mathematical models of imagig sstems Imagig sstems alwas have cetai techical limitatios i thei desig ad implemetatios ad geeate images that ae ot as pefect as the would be if thee wee o implemetatio limitatios Deviatios of eal images fom pefect oes ma be teated as distotios itoduced b imagig sstems to hpothetical pefect, o ideal sigals Coectio of these distotios is the pima goal of image pocessig Methods fo distotio coectio ae based o the caoical model of imagig sstems show i Fig 101 Fig 101 A caoical model of imagig sstems The model epesets image fomatio as a combiatio of sigal liea tasfomatios, poit-wise oliea tasfomatios ad stochastic tasfomatios that ae applied to a hpothetical pefect sigal a ( ad joitl detemie the sstem s output sigal b ( Liea tasfomatio ae specified i tems of the sstem poit spead fuctio o fequec espose Fequec espose of the ideal imagig sstem is assumed to be uifom fo all fequecies i a cetai base bad Fequec esposes of eal imagig usuall moe o less apidl deca o high fequecies This esults i image distotios such as image blu Poit-wise oliea ae specified i tems of the sstem tasfe fuctios It is assumed that the ideal sstem has a liea tasfe fuctio Deviatios of sstem tasfe fuctios fom liea oe cause distotios of ga scale oliea distotios Stochastic tasfomatios model sigal adom distotios ad cause adom itefeeces, o oise i output images Stochastic tasfomatios ae specified b statistical oise models discussed i Lect 9 The pocessig goal is estimatig the pefect sigal a ( give distoted sigal b ( poduced b the sstem This poblem is fequetl efeed to as the ivese poblem Sigal tasfomatios that ae applied to sstem s output sigals { b ( } to poduce a estimate â ( of the ideal sigal ae called sigal ecove o, i applicatio to image pocessig, image estoatio Simila poblem is image ecostuctio This tem is usuall efes to image fomatio i tasfom domai imagig such as tomogaph ad hologaph If the sstem does ot itoduce a adom distotios ad sstem s paametes such as poit spead fuctio ad tasfe fuctio ae ow, the ivese poblem has a tivial solutio: sigal b ( has to be subjected to tasfomatios ivese to those itoduced b the sstem Howeve, oe ca ot, i geeal, eglect adom o simila ucotolled distotios such as oud-off eos i digital pocessig I ealit, applig ivese tasfomatio ma esult i atifacts that ma eve be useless To solvig the ivese poblem i such cases, the statistical appoach that eplicitl accouts fo sigal adom distotios appeas to be the most appopiate Geeall, sigal ecove, image estoatio ad ecostuctio is idivisible pocedue that should accout fo ad coect all distotios Howeve, i pactice this pocess is divided ito sepaate steps caied out i the ode ivese to that distotio factos have i the model 10 Tasfom domai MSE optimal scala Wiee Filtes I this sectio we coside image estoatio fo a educed imagig sstem model that disegads poitwise oliea tasfomatio i imagig sstems ad teats image distotios as a combiatio of distotios caused b liea filteig ad of those caused b actio of adom itefeeces The desig of the optimal estoatio pocedue equies specifig a citeio fo evaluatig b be a set of N sigal samples ( 0,1,, N 1 at the output of the the estoatio qualit Let { } imagig sstem, { } Liea tasfomatio Poit-wise oliea tasfomatio Stochastic tasfomatio a ( b( a be a set of the sstem s iput sigal samples that model pefect, o ideal sigal

2 ad { â } be a set of estoed sigal samples Fo the sae of geealit, we will coside the set { a } as a ealizatio tae fom a statistical esemble o data base of iput sigals ad the set { } b as a ealizatio of a sigal esemble geeated b esembles ad of iput sigals ad of adom itefeeces Defie the estoatio pocedue pefomace measue as a squaed diffeece betwee estoed ad pefect sigals aveaged ove the available set of sigal samples ad ove statistical esembles ad We will call this measue mea squaed estoatio eo (MSE The estoatio pocedue R b ˆ that miimizes this diffeece: { } { } a { aˆ } N { } { } 1 ag mi a ˆ a (101 R b aˆ 0 will be efeed to as to MSE-optimal filteig Fo the implemetatio of the MSE-optimal filteig, we will estict ouselves to liea filteig I geeal, liea filteig of a discete sigal ma be descibed as multiplicatio of a vecto of iput B b a filte mati H : b sigal samples { } A ˆ H B, (10 whee ˆ { } A â is a vecto of filte output sigal samples Fo sigals of N samples, a geeal vecto filte mati H has dimesios N N Specificatio of such a filte equies detemiig N filte coefficiets, ad the filteig itself equies pefomig N opeatios pe N sigal samples I image pocessig, the computatioal compleit of both detemiatio of filte coefficiets ad of the filteig ma be become too high because of high dimesioalit of image aas Fast tasfoms that ma be computed fo O( N log N opeatios allow to adicall decease the filte desig ad the implemetatio compleit Theefoe i what follows we will coside ol scala filteig i a domai of othogoal tasfoms that ca be computed with fast algoithms This class of filtes ma be descibed b the equatio: ˆ 1 Τ Η Τ Β, (103 d 1 whee, Τ ad Η diag, ( 0,1,, N 1 is a diagoal filte mati Such a scala filteig implies the followig elatioship betwee filte output ad iput sigal samples { ˆ α } Τ A ˆ ad { } Τ B : Τ ae, coespodigl, diect ad ivese othogoal tasfoms, { } ˆ α (104 I the assumptio of othogoalit of the tasfom Τ, oe ca, b vitue of the Paceval elatioship (Eq 18, modif the filte optimalit coditio defied b Eq 101 i the followig wa: { ˆ α } N { } 1 N ag mi ˆ α α 0 { } 1 ag mi α 0 (105 B computig deivatives ove sought vaiables ad equalig them to zeo, oe ca obtai fom Eq 105 that optimal scala filte coefficiets ma be foud as coss-coelatio coefficiets betwee spectal coefficiets ad α of the iput ad pefect sigals:

3 (106 ( We will efe to MSE optimal liea filtes ad, i paticula, to MSE optimal scala filtes defied b Eq 106 as to Wiee filtes This ame gives a cedit to Nobet Wiee fo his pioee wos i the theo of statistical methods of sigal estoatio 1 I ode to implemet optimal scala Wiee filte, oe should theefoe ow coss-coelatio α betwee filte iput sigal ad pefect sigal spectal coefficiets ad powe { ( } { } spectum ( of the iput sigal i the selected basis The statistical appoach we adopted that assumes aveagig of the estoatio eo ove statistical esembles of pefect sigals ad of filte iput sigals implies that these statistical paametes should be measued i advace fo these esembles o ove the data bases 81 Empiical Wiee filtes fo image deoisig Coside a additive sigal idepedet oise model i which filte iput sigal samples{ b } ae of sigal idepedet zeo mea obtaied as a sum of pefect sigal samples { } adom oise: a ad samples { } b a + (107 I spectal domai, the same elatioship holds fo sigal ad oise spectal coefficiets: α + ν, (108 whee { ν } T{ } ad Fo this model oe ca obtai that [ α + ν ] ( ( + ( + α ν α ν [ ] (109 + (1010 because fo zeo mea oise 0 Theefoe scala Wiee filte fo as suppessig additive sigal idepedet oise is defied though its coefficiets { } + (1011 Oe ca give to this fomula a clea phsical itepetatio Defie sigal-to-oise atio as: SNR (101 1 It will be just to give also a cedit to Ade N Kolmogoov who developed the simila theo fo discete sigals

4 The obtai: SNR ( SNR which meas that scala Wiee filte weight coefficiets ae defied, fo each sigal spectal coefficiet, b the sigal-to-oise atio fo this coefficiet The lowe is sigal-to-oise atio fo a paticula sigal spectal compoet, the lowe will be the cotibutio of this compoet to the filte output sigal I ode to implemet scala Wiee filte oe have to measue i advace powe specta ad of pefect sigals ad of oise i the selected basis Noise powe spectum ma be ow fom the specificatio cetificate of the imagig device Othewise it ma be measued i ois iput sigals usig methods descibed i Sect 71 As fo the pefect sigal powe spectum, it is most fequetl ot ow Howeve, oe ca, usig Eq 1010, attempt to estimate it fom the powe spectum ( of iput ois sigals as ( (1014 The latte has to be estimated fom the obseved sigal spectum Deote this estimate as This empiical estimate made b aveagig ove available ealizatio of iput images ma, whe used as a eplacemet fo (, give egative values fo some spectal coefficiets because of the limited depth of the aveagig Sice powe specta ca ot assume egative values, the followig modified spectum estimatio ma be adopted: ( ( ma ν ; α ( I this wa we aive at the filte: ma ; 0 (1016 We will efe to this filte as to the empiical Wiee filte If the imagig sstem oise is ow to be white oise with vaiace filte taes the fom: σ ma ; 0 (1017 σ, the empiical Wiee As a zeo ode appoimatio to the iput images powe spectum, powe spectum of a sigle iput image subjected to filteig ma be used I this case empiical Wiee filte weight coefficiets ae foud as: σ ma ; 0 (1018 Note that such a empiical Wiee filte is adaptive because its weight coefficiets deped o the spectum of the image to which it will be applied

5 Weight coefficiets of scala Wiee filtes assume values i the age betwee zeo ad oe A vesio of the empiical Wiee filte of Eq 1018 with bia weight coefficiets: 1, if Th (1019 0, othewise whee Th is a ejectig theshold is called the ejectig filte As it follows fom Eq 1018, the ejectig theshold has a value of the ode of magitude of the oise vaiace σ Rejectig filtes elimiate fom the iput images specta all compoets fo which sigal-to-oise atio is lowe the a cetai theshold A vesio of the empiical Wiee filte ad of the ejectig filtes that ae implemeted with wavelet tasfom image decompositio is ow as wavelet shiage filteig Empiical Wiee filteig accodig to Eq 1018 a a wavelet jago is called soft thesholdig Rejectig filteig accodig to Eq 1019 is called had thesholdig Fig 10 shows flow diagam of sigal deoisig b the wavelet shiage Iput Low pass filteig ad dowsamplig + Soft/had + thesholdig - Itepolatio Output Itepolatio Low pass filteig ad dowsamplig Itepolatio Soft/had thesholdig Itepolatio Low pass filteig ad dowsamplig Itepolatio Fig 10 Wavelet shiage: sigal deoisig i wavelet tasfom domai Descibed empiical Wiee filtes fo sigal deoisig ae paticulal ve efficiet if sigal ad/o oise specta ae well cocetated ad ae sepaated i the tasfom domai A tpical eample of such a situatio is filteig of aow bad oise, whose spectum has ol a few compoets i the tasfom domai Figs 103 though 105 illustate eamples of such a aow bad oise filteig Fig 103 demostates filteig peiodical oise patte i a image Such itefeeces fequetl appea i images digitized b fame gabbes fom aalog video Left colum i Fig 103 shows iput ad filteed images Right colum shows aveaged specta of iput ad output image ows Oe ca cleal see aomalous peas of oise spectum i iput image spectum that ae elimiated i the output image spectum afte applig empiical Wiee filteig Note that the filteig is caied out i this eample as 1-D ow-wise filteig i DFT domai This tpe of itefeeces ca also be successfull filteed i Walsh tasfom domai

6 Iput image 14 Colum wise aveaged powe spectum alog ows Filteed image "Filteed" powe spectum Fig 103 Filteig peiodic itefeeces Fig 104 shows et aothe eample of empiical Wiee filteig aow-bad itefeeces Badig oise i the iitial image show i Fig 104 is chaacteistic fo imagig sstems with mechaical scaig This paticula image was poduced b a atomic foce micoscope Badig oise adoml chages image dc compoet (ow-wise mea value i the diectio of scaig Uppe ight plot i Fig 104 shows ow-wise mea values, o ow-wise image Rado tasfom, (hoizotal coodiate as a fuctio of the ow umbe (vetical coodiate Taig as a estimate of pefect mea ow-wise values a mea value ove all ows (show with a staight lie i the bottom ight plot i Fig 104, oe ca estimate badig itefeece o eve ow b subtactig this estimate fom the obseved mea values Subtactig the foud values fom all piels i the coespodig ow elimiates the oise Wiee filteig of wide bad oise ad especiall white oise is less efficiet Whe filteig of white oise, Wiee filte teds to weae low eeg sigal spectal compoets Howeve usuall these compoets ae eactl the compoets that ae the most impotat because the ca ifomatio about sigal chages such as at edges i images Moeove, Wiee filteig covets iput white oise ito output coelated oise though with a educed vaiace As oe ca see fom Eq 1011, i case of the itesive iput white oise, powe spectum of the esidual oise is oughl popotioal to the sigal powe spectum which meas that the esidual oise becomes, statisticall, sigal alie This ma hampe subsequet image aalsis I paticula, it is well ow that huma visio is moe sesitive to coelated oise that to white oise of the same itesit Theefoe Wiee filteig fo image deoisig ma eve wose images

7 Iitial image Filteed image Fig 104 Filteig badig oise Fig 105 gives a eample of compute simulatio of Wiee filteig of a test image with additive white oise I the simulatio, oe ca implemet the ideal Wiee filte because both sigal ad oise specta ae ow The empiical Wiee filte was implemeted i this eample accodig to Eq 1018 As oe ca see fom the figue, ideal Wiee filteig does impove image qualit Fo the empiical Wiee filte, impovemet is much less appeciable The diffeece image betwee iitial ois image ad the esult of the empiical Wiee filteig (estoatio eo shows that the filteig, alog with oise suppessio, destos image edges 8 Image debluig, ivese filtes ad apetue coectio Coside ow a imagig sstem model that accouts also fo sigal liea tasfomatios i imagig sstems Suppose that the liea tasfomatio uit ca be modeled as a scala filte with λ i a selected basis Fo DFT basis, this assumptio is just, to the accuac of filte coefficiets { } bouda effects, fo shift ivaiat liea filteig (see Ch 5 I this case coefficiets { } λ ae samples of imagig sstem fequec espose I ideal imagig sstems, the should all be equal to uit I ealit, the deca with the fequec ide, which esults, i paticula, i image blu Pocessig aimed at coectig this tpe of distotios is fequetl efeed to as image debluig

8 Test ois image Ideal Wiee filteed image Empiical Wiee filteed image Restoatio eo Fig 105 Ideal Wiee ad Empiical Wiee filteig fo image deoisig Fo such sstems, we have i the tasfom domai: λ α + ν, (100 Usig Eqs 106, oe ca obtai that the scala Wiee image estoatio filte is defied i this case b the equatio: 1 SNR, (101 λ 1 + SNR whee SNR is sigal-to-oise atio at the output of the liea filte uit of the imagig sstem model: λ SNR (10

9 Coespodigl, the geeal empiical Wiee filte, empiical Wiee filte with zeo ode appoimatio to pefect sigal spectum ad the ejectig filte fo apetue coectig ad image debluig will be i this case as follows: ad 1 ma ; 0 ; (103 λ 1 σ ma ; 0 (104 λ 1, if Th λ (105 0, othewise All these filtes ma be teated as two filtes i cascade: the filte with coefficiets iv 1 (106 λ usuall called the ivese filte ad sigal deoisig filtes descibed b Eqs Ivese filtes compesate weaeig sigal fequec compoets i the imagig sstem while deoisig filtes pevet fom ecessive amplificatio of oise ad pefom what is called egulaizatio of ivese filtes As oe ca see fom Eq 10, weight coefficiets of deoisig filtes fo small { λ } fall faste tha weight coefficiets of the ivese filte gow Oe of the most immediate applicatios of ivese filtes is coectig distotios caused b fiite size of apetues of image sesos, image discetizatio devices ad image displas We will efe to this pocessig as to apetue coectio Let a image seso ad discetizatio device is a aa of light sesitive elemets with a squae ( d ( d apetue of size d d (Fig 8-7 The fequec espose of the idividual seso elemets is H ( d d / ( f, f ep( iπ f ( d d / ( d ( πf d / ( d si πf d ( d d / d ep / si ( d ( πf d ( d πf d ( d d / ( iπf d ( d ( d ( πf d sic πf d ( sic, (107 o, i dimesioless coodiates { f f f f } both coodiates, H ( d ( d ( f, f sic( f d / sic( πf d /,, whee is the discetizatio iteval i π (108

10 d (d d (d Seso s apetue Fig106 Aagemet of light sesitive elemets i image seso aas The discete fequec espose is the: λ ( d ( d ( π λ / N sic πd ( N, s λλs sic λs d /, (109 ( d ( d whee d d / ( ( If the image displa device has also a squae apetue of size d d, the oveall imagig discete sstem fequec espose is: ( d ( ( d ( λ, s λλs sic( πd λ / N sic( πd λ / N sic( π d λs / N sic( πd λs / N (1030 ( d ( Paametes d, d ad ae imagig sstem desig paametes that ma be ow sstem s cetificate The ca be used fo coectig image distotios b pocessig images i compute Fig 8-8 illustates a eample of the apetue coectio of a ai photogaph Right image i this figue is obtaied b applig to the left image a ivese filte fo the sstem s fequec espose ( d ( defied b Eq 100 with d d 1 Iitial ad apetue coected images Fig 107 Apetue coectio: iitial (left ad apetue coected (ight images

11 L Yaoslavs Fudametals of Digital Image Pocessig Couse Summa Lectue 10 Piciples of image estoatio Caoical model of imagig sstems cosists of a liea filte ad poit-wise olieait i cascade ad a geeato of adom itefeeces Image estoatio assumes a vitual ideal imagig sstem ad is aimed at covetig images geeated b a eal imagig sstem ito a image that is as close, i tems of a cetai qualit citeio, to the vitual ideal image as possible Tasfom domai scala filteig Geeal liea filteig of a digital sigal ca be descibed as a mati multiplicatio of a sigal vecto b a filte mati Computatioal compleit of the geeal filteig pe output piel is O ( N, whee N is the size of the image aa Scala filteig is implemeted as poit-wise modificatio of sigal spectal coefficiets i a basis of a cetai othogoal tasfom Povided the tasfom with fast tasfom algoithm is used, computatioal compleit of the scala filteig is O( log N pe output piel o lowe MSE optimal Wiee filteig: N { } { } { } 1 aˆ ˆ ag mi a a R b aˆ 0 MSE optimal scala Wiee filteig: N { } 1 α ag mi α { } 0 ( MSE optimal Wiee filte fo image deoisig: + SNR 1+ SNR Empiical Wiee filtes fo image deoisig (white additive oise model: σ ma ; 0 ; Soft thesholdig: σ ma ; 0 1, if Th Rejectig filte (had thesholdig: 0, othewise MSE optimal scala Wiee filte fo image debluig: 1 SNR λ 1 + SNR Empiical Wiee filtes fo image debluig: 1 σ Soft thesholdig: ma ; 0 λ Rejectig filte: 1, if Th λ 0, othewise

12 Questios fo self-testig 1 Descibe the caoic model of imagig sstems Fomulate the image estoatio poblem ad eplai wh it is usuall implemeted as a multi-stage pocess 3 What is tasfom domai scala filteig? 4 Fomulate the piciple of MSE optimal filteig 5 Deive optimal scala Wiee filtes fo image deoisig 6 Deive optimal scala Wiee filtes fo image debluig ad show its elatio to ivese filteig 7 Eplai what is empiical Wiee filte ad give eamples of implemetatios of the empiical Wiee filte 8 Whe Wiee filteig ca be efficietl used fo image estoatio ad whe it fails? 9 What is image seso apetue coectio ad how ca it be implemeted?