Preview. Digital Image Processing Unit 7: Image Restoration. Unit Outline. Preview (cont.)

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1 --6 Digital Procssing Unit 7: Rstoration Prviw Goal of imag rstoration Improv an imag in som prdfind sns Diffrnc with imag nhancmnt? Faturs rstoration v.s imag nhancmnt Objctiv procss v.s. subjctiv procss A prior knowldg v.s huristic procss A prior knowldg of th dgradation phnomnon is considrd Modling th dgradation and apply th invrs procss to rcovr th original imag //6 ajaybolar.wbly.com //6 ajaybolar.wbly.com Prviw (cont. argt Dgradd digital imag Snsor, digitizr, display dgradations ar lss considrd Spatial domain approach Frquncy domain approach Unit Outlin A modl of th imag dgradation / rstoration procss Nois modls Rstoration in th prsnc of nois only spatial filtring Priodic nois rduction by frquncy domain filtring Linar, position-invariant dgradations Estimating th dgradation function Invrs filtring //6 ajaybolar.wbly.com 3 //6 ajaybolar.wbly.com 4 Concpt of Rstoration rstoration is to rstor a dgradd imag back to th original imag whil imag nhancmnt is to manipulat th imag so that it is suitabl for a spcific application. Dgradation modl: f ( h( ( whr h( is a systm that causs imag distortion and ( is nois. //6 ajaybolar.wbly.com 5 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Nois modls Sourc of nois acquisition (digitization transmission Spatial proprtis of nois Statistical bhavior of th gray-lvl valus of pixls Nois paramtr corrlation with th imag Frquncy proprtis of nois Fourir spctrum Ex. whit nois (a constant Fourir spctrum //6 ajaybolar.wbly.com 6 Canara Enginring Collg

2 --6 Nois Modls Nois cannot b prdictd but can b approximatly dscribd in statistical way using th probability dnsity function (PDF Gaussian nois: Rayligh nois Erlang (Gamma nois p( z ( z / ( z a p( z b ( za / b b b a z ( z a p( z ( b! az for z a for z a for z for z //6 ajaybolar.wbly.com 7 Nois Modls (cont. Exponntial nois Uniform nois p( z a Impuls (salt & pppr nois az p( z b - a for a z b othrwis Pa for z a p( z Pb for z b othrwis //6 ajaybolar.wbly.com 8 PDF: Statistical Way to Dscrib Nois PDF tlls how much ach z valu occurs. Nois probability dnsity functions Noiss ar takn as random variabls Random variabls Probability dnsity function (PDF //6 ajaybolar.wbly.com 9 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. //6 ajaybolar.wbly.com Gaussian nois Math. tractability in spatial and frquncy domain Elctronic circuit nois and snsor nois p( z Not: p( z dz ( z / man varianc //6 ajaybolar.wbly.com Gaussian nois (PDF 7% in [(, (] 95% in [(, (] //6 ajaybolar.wbly.com Canara Enginring Collg

3 --6 Uniform nois Uniform Nois PDF Lss practical, usd for random numbr gnrator if a z b p( z b a othrwis a b Man: ( b a Varianc: //6 ajaybolar.wbly.com 3 //6 ajaybolar.wbly.com 4 Impuls (salt-and-pppr nosi Quick transint such as faulty switching during imaging Pa for z a p( z Pb for z b othrwis If ithr P a or P b is zro, it is calld unipolar. Othrwi it is calld bipoloar. In practical, impulss ar usually strongr than imag signals. Ex., a=(black and b=55(whit in 8-bit imag. //6 ajaybolar.wbly.com 5 Impuls (salt-and-pppr nosi PDF //6 ajaybolar.wbly.com 6 Dgradation with Additiv Nois st pattrn st for nois bhavior f ( ( Dgradd imags Its histogram: Original imag 55 Histogram //6 ajaybolar.wbly.com 7 //6 ajaybolar.wbly.com 8 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Canara Enginring Collg 3

4 --6 Dgradation with Additiv Nois (cont. Original imag f ( ( Dgradd imags Priodic nois Sourc: lctrical or lctromchanical intrfrnc during imag acquisition Charactristics Spatially dpndnt Priodic asy to obsrv in frquncy domain Procssing mthod Supprssing nois componnt in frquncy domain Histogram //6 ajaybolar.wbly.com 9 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. //6 ajaybolar.wbly.com Priodic Nois Priodic nois looks lik dots In th frquncy domain //6 ajaybolar.wbly.com (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Estimation of nois paramtrs Priodic nois Obsrv th frquncy spctrum Random nois with unknown PDFs Cas : imaging systm is availabl Captur imags of flat nvironmnt Cas : noisy imags availabl ak a strip from constant ara Draw th histogram and obsrv it Masur th man and varianc //6 ajaybolar.wbly.com Estimation of Nois Rstoration in th prsnc of Nois Only- Spatial Filtring W cannot us th imag histogram to stimat nois PDF. Whn only dgradation prsnt in an imag is noi Dgradation modl: f ( h( ( o rmov this part It is bttr to us th histogram of on ara of an imag that has constant intnsity to stimat nois PDF. Spatial filtring is th mthod of choic in situations whn only additiv nois is prsnt. Enhancmnt and Rstoration bcom almost indistinguishabl disciplins in this particular cas. //6 ajaybolar.wbly.com 3 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. //6 ajaybolar.wbly.com 4 Canara Enginring Collg 4

5 --6 Man Filtrs (Nois Rduction Spatial Filtrs Arithmtic or moving avrag filtr mn S xy Gomtric Man Filtr: Exampl Original imag by AWGN Gomtric S xy mn mn = siz of moving window Achivs smoothing comparabl to arithmtic, but it tnds to los lss imag dtail in th procss. //6 ajaybolar.wbly.com 5 arithmtic gomtric //6 ajaybolar.wbly.com 6 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. AWGN: Additiv Whit Gaussian Nois Harmonic and Contraharmonic Filtrs Harmonic mn S xy Contraharmonic S xy S xy Q = th filtr ordr Q Q Works wll for salt nois but fails for pppr nois Works wll for Gaussian Nois mn = siz of moving window Rducs salt and pppr noi Positiv Q is suitabl for liminating pppr nois. Ngativ Q is suitabl for liminating salt nois. Cannot do both simultanously //6 For Q =, th ajaybolar.wbly.com filtr rducs to an arithmtic man 7 filtr. For Q = -, th filtr rducs to a harmonic. Contraharmonic Filtrs: Exampl by pppr nois with prob. =. contraharmonic With Q =.5 by salt nois with prob. =. contraharmonic With Q=-.5 //6 ajaybolar.wbly.com 8 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Contraharmonic Filtrs: Incorrct Us Exampl by pppr nois with prob. =. by salt nois with prob. =. Ordr-Statistic Filtrs: Rvisit Original imag subimag Statistic paramtrs Man, Mdian, Min, Ma Etc. contraharmonic With Q=-.5 contraharmonic With Q=.5 Moving window //6 ajaybolar.wbly.com 9 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. //6 ajaybolar.wbly.com 3 Output imag Canara Enginring Collg 5

6 --6 Ordr-Statistics Filtrs Mdian filtr mdian g S xy Max filtr max S xy Min filtr min S xy Midpoint filtr max S Rduc dark nois (pppr nois Rduc bright nois (salt nois g g min //6 ajaybolar.wbly.com xy S xy 3 Mdian Filtr : How it works A mdian filtr is good for rmoving impul isolatd nois Dgradd imag Salt nois Pppr nois Moving window Pppr nois Mdian Sortd array Salt nois Filtr output Normally, impuls nois has high magnitud and is isolatd. Whn w sort pixls in th moving window, nois pixls ar usually at th nds of th array. hrfor, //6 it s rar that th ajaybolar.wbly.com nois pixl will b a mdian valu. 3 Mdian Filtr : Exampl by saltand-pppr nois with p a =p b =. Max and Min Filtrs: Exampl by pppr nois with prob. =. by salt nois with prob. =. 3 4 max filtr min filtr s mdian filtr //6 ajaybolar.wbly.com 33 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. //6 ajaybolar.wbly.com 34 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Alpha-trimmd Man Filtr Formula: mn d r S xy g whr g r ( rprsnt th rmaining mn-d pixls aftr rmoving th d/ highst and d/ lowst valus of. his filtr is usful in situations involving multipl typs of nois such as a combination of salt-and-pppr and Gaussian nois. //6 ajaybolar.wbly.com 35 Alpha-trimmd Man Filtr: Exampl by additiv uniform nois additionally by additiv salt-andpppr nois arithmtic gomtric //6 ajaybolar.wbly.com 36 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Canara Enginring Collg 6

7 --6 Alpha-trimmd Man Filtr: Exampl (cont. by additiv uniform nois mdian filtr additionally by additiv salt-andpppr nois alphatrimmd with d = 5 //6 ajaybolar.wbly.com 37 Alpha-trimmd Man Filtr: Exampl (cont. arithmtic mdian filtr gomtric alphatrimmd with d = 5 //6 ajaybolar.wbly.com 38 Adaptiv Filtrs h filtrs discussd so far ar applid to an ntir imag without any rgard for how imag charactristics vary from on point to anothr h bhaviour of adaptiv filtrs changs dpnding on th charactristics of th imag insid th filtr rgion //6 ajaybolar.wbly.com 39 Adaptiv Filtr Gnral concpt: -Filtr bhavior dpnds on statistical charactristics of local aras insid mxn moving window - Mor complx but suprior prformanc compard with fixd filtrs Statistical charactristics: Local man: Nois varianc: ml mn S xy Local varianc: L ( m L mn S xy //6 ajaybolar.wbly.com 4 Adaptiv, Local Nois Rduction Filtr Purpos: want to prsrv dgs Concpt: Formula:. If is zro, No nois th filtr should rturn bcaus = f(. If L is high rlativ to, Edgs (should b prsrvd, th filtr should rturn th valu clos to 3. If L =, Aras insid objcts th filtr should rturn th arithmtic man valu m L ml //6 ajaybolar.wbly.com 4 L Adaptiv Nois Rduction Filtr: Exampl by additiv Gaussian nois with zro man and = using a 7x7 gomtric using a 7x7 arithmtic using a 7x7 adaptiv nois rduction filtr //6 ajaybolar.wbly.com 4 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Canara Enginring Collg 7

8 --6 Adaptiv Mdian Filtring h mdian filtr prforms rlativly wll on impuls nois as long as th spatial dnsity of th impuls nois is not larg h adaptiv mdian filtr can handl much mor spatially dns impuls noi and also prforms som smoothing for nonimpuls nois h ky insight in th adaptiv mdian filtr is that th filtr siz changs dpnding on th charactristics of th imag Adaptiv Mdian Filtring (cont h ky to undrstanding th algorithm is to rmmbr that th adaptiv mdian filtr has thr purposs: Rmov impuls nois Provid smoothing of othr nois Rduc distortion //6 ajaybolar.wbly.com 43 //6 ajaybolar.wbly.com 44 Adaptiv Mdian Filtr Purpos: want to rmov impuls nois whil prsrving dgs Algorithm: Lvl A: whr Lvl B: A= z mdian z min A= z mdian z max If A > and A <, goto lvl B Els incras window siz If window siz <= S max rpat lvl A Els rturn z xy B= z xy z min B= z xy z max If B > and B <, rturn z xy Els rturn z mdian z min = minimum gray lvl valu in S xy z max = maximum gray lvl valu in S xy z mdian = mdian of gray lvls in S xy z xy = gray lvl valu at pixl ( S max = maximum allowd siz of S xy //6 ajaybolar.wbly.com 45 Adaptiv Mdian Filtr: How it works Lvl A: A= z mdian z min A= z mdian z max If A > and A <, goto lvl B Dtrmin whthr z mdian is an impuls or not Els Window is not big nough incras window siz If window siz <= S max rpat lvl A Els rturn z xy Lvl B: z mdian is not an impuls Dtrmin B= z xy z min whthr z xy B= z xy z max is an impuls or not If B > and B <, z xy is not an impuls rturn z xy to prsrv original dtails Els rturn z mdian to rmov impuls //6 ajaybolar.wbly.com 46 Adaptiv Mdian Filtr: Exampl Priodic Nois Rduction by Frq. Domain Filtring by salt-and-pppr nois with p a =p b =.5 using a 7x7 mdian filtr using an adaptiv mdian filtr with S max = 7 Dgradd imag DF Priodic nois can b rducd by stting frquncy componnts corrsponding to nois to zro. Mor small dtails ar prsrvd //6 ajaybolar.wbly.com 47 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Band rjct filtr Rstord imag //6 ajaybolar.wbly.com 48 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Canara Enginring Collg 8

9 --6 Band Rjct Filtrs Us to liminat frquncy componnts in som bands Notch Rjct Filtrs A notch rjct filtr is usd to liminat som frquncy componnts. Priodic nois from th prvious slid that is Filtrd out. //6 ajaybolar.wbly.com 49 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. //6 ajaybolar.wbly.com 5 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Notch Rjct Filtr: Dgradd imag DF Notch filtr (frq. Domain Exampl: Dgradd by Priodic Nois Dgradd imag DF (no shif (s //6 from Rafal C. Gonzalz and Richard E. ajaybolar.wbly.com 5 Wood, Digital Procssing, nd Edition. Nois Rstord imag DF of nois Nois Rstord imag //6 ajaybolar.wbly.com 5 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Estimation of Dgradation Modl Dgradation modl: Estimation by Obsrvation Original imag (unknown Dgradd imag or f ( h( ( F( N( f( f(*h( Purpos: to stimat h( or Why? If w know xactly h(, rgardlss of noi w can do dconvolution to gt f( back from. Mthods:. Estimation by Obsrvation. Estimation by Exprimnt 3. Estimation by Modling //6 ajaybolar.wbly.com 53 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Estimatd ransfr function Gs( H ( H s( Fˆ ( his cas is usd whn w know only and cannot rpat th xprimnt! s DF Subimag G s ( ( ˆ ( F s Obsrvation //6 ajaybolar.wbly.com f 54 s g s Rstoration procss by stimation DF Rconstructd Subimag ˆ ( Canara Enginring Collg 9

10 --6 Estimation by Exprimnt Usd whn w hav th sam quipmnt st up and can rpat th xprimnt. Rspons imag from Input impuls imag th systm DF DF A ( A ( A Systm A DF //6 ajaybolar.wbly.com 55 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Estimation by Modling Usd whn w know physical mchanism undrlying th imag formation procss that can b xprssd mathmatically. Original imag Mild turbulnc k =. Svr turbulnc k =.5 Low turbulnc k =.5 Exampl: Atmosphric urbulnc modl H ( 5/ 6 k ( u v //6 ajaybolar.wbly.com 56 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Estimation by Modling: Motion Blurring Assum that camra vlocity is ( x(, y( h blurrd imag is by f ( x x(, y y( dt whr = xposur tim. j ( uxvy dxdy f ( x x(, y y( dt f ( x x (, y y ( j ( uxvy j ( uxvy dxdy dxdydt //6 ajaybolar.wbly.com 57 Estimation by Modling: Motion Blurring (cont. F( F( f ( x x (, y y ( j ( ux ( vy ( j ( ux ( vy ( dt dt j ( uxvy hn w gt, th motion blurring transfr function: j ( ux ( vy ( For constant motion ( x(, y( ( at, b j ( uavb dt dxdydt dt sin( ( ua vb ( ua vb j ( uavb //6 ajaybolar.wbly.com 58 Motion Blurring Exampl For constant motion sin( ( ua vb ( ua vb j ( uavb Rstoration Modl f( Dgradation Modl Rstoration Filtr f( Original imag Motion blurrd imag a = b =., = //6 ajaybolar.wbly.com 59 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Unconstraind Invrs Filtr Psudo-invrs Filtr Constraind Winr Filtr dmos/dmo5blur_invfiltr/ //6 ajaybolar.wbly.com 6 Canara Enginring Collg

11 --6 Invrs Filtr From dgradation modl: F( N( aftr w obtain, w can stimat F( by th invrs filtr: N( F( Nois is nhancd whn is small. o avoid th sid ffct of nhancing noi w can apply this formulation to frq. componnt ( with in a radius D from th cntr of. In practical, th invrs filtr is not popularly usd. //6 ajaybolar.wbly.com 6 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Invrs Filtring Limitations: Evn if th dgradation function is known th undgradd imag cannot b rcovrd xactly bcaus N( is th random function which is not known. If th dgradation function has or small valu th ratio asily dominats th stimat F( On approach to gt rid of or small valu problm is to limit th filtr frquncy to th valu nar th origin. //6 ajaybolar.wbly.com 6 Invrs Filtr: Exampl WIENER FILERING Invrs filtring has no xplicit provision for handling nois Original imag Rsult of applying th full filtr Rsult of applying th filtr with D =4 h winr filtr incorporats both dgradation function, and statistical charactristics of nois in th rstoration procss. Objctiv of th winr filtr is to find th stimat of un imag f, such that th man squar rror is minimum. Blurrd imag Du to urbulnc H ( Rsult of applying th filtr with D =7 Rsult of applying th filtr with D =85 //6.5( u v ajaybolar.wbly.com 63 5/ 6 h winr filtr is an optimum filtr Conditions ( Nois and imag ar uncorrlatd ( On or th othr has zro man (3 Gray lvls in ˆ f ar linar function of gray lvls in g //6 ajaybolar.wbly.com 64 Norbrt Winr ( //6 ajaybolar.wbly.com h rnownd MI profssor Norbrt Winr was famd for his absnt-minddnss. Whil crossing th MI campus on day, h was stoppd by a studnt with a mathmatical problm. h prplxing qustion answrd, Norbrt followd with on of his own: "In which dirction was I walking whn you stoppd m?" h askd, prompting an answr from th curious studnt. "Ah," Winr dclard, "thn I'v had my lunch Ancdot of Norbrt Winr 65 Winr Filtr: Minimum Man Squar Error Filtr Objctiv: optimiz man squar rror: E ( f fˆ Winr Filtr Formula: whr * ˆ H ( S ( f F( S f ( H ( S ( * H ( (, G u v S ( / S f ( H ( (, G u v H ( H ( S ( / S f ( = Dgradation function S ( = Powr spctrum of nois S f ( = Powr spctrum of th undgradd imag //6 ajaybolar.wbly.com 66 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Canara Enginring Collg

12 --6 Winr Filtr: Minimum Man Squar Error Filtr Approximation of Winr Filtr = dgradation function H ( = complx conjugat of = H ( S η ( = N( = powr spctrum of th nois S f ( = F( = powr spctrum of undgradd imag Sinc Sη( = N( and S f ( = F( ar sldom known, th Winr filtr is frquntly approximatd Winr Filtr Formula: S ( / S f ( No problm with zros unlss and Sη( ar both zro Approximatd Formula: Difficult to stimat Whn nois is zro, Winr filtr = invrs filtr //6 ajaybolar.wbly.com 67 K Practically, K is chosn manually to th bst visual rsult! //6 ajaybolar.wbly.com 68 Winr Filtring Exampl ADVANAGES:.h winr filtr dos not hav zro valu problm..h rsult by winr filtr is mor closr to th original imag than invrs filtr. motion blurrd imag dblurrd imag aftr winr filtring (K=. //6 ajaybolar.wbly.com 69 //6 ajaybolar.wbly.com 7 Exampl (Con Winr Filtr: Exampl Original imag Rsult of th full invrs filtr Rsult of th invrs filtr with D =7 K=. K=. //6 ajaybolar.wbly.com K=. 7 Blurrd imag Du to urbulnc Rsult of th full Winr filtr //6 ajaybolar.wbly.com 7 Canara Enginring Collg

13 --6 Winr Filtr: Exampl (cont. Original imag Rsult of th invrs filtr with D =7 Exampl: Winr Filtr and Motion Blurring dgradd by motion blur + AWGN =65 Rsult of th invrs filtr Rsult of th Winr filtr Blurrd imag Du to urbulnc Rsult of th Winr filtr //6 ajaybolar.wbly.com 73 =35 =3 Not: K is chosn manually //6 ajaybolar.wbly.com 74 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Constraind Last Squars Filtr Dgradation modl: Writtn in a matrix form f ( h( ( g Hf η Objctiv: to find th minimum of a critrion function C M N f ( x y Subjct to th constraint g Hfˆ η W gt a constraind last squar filtr * H ( P( whr whr w w P( = Fourir transform of p( = 4 w //6 ajaybolar.wbly.com 75 Constraind Last Squars Filtr: Exampl Constraind last squar filtr * H ( P( is adaptivly adjustd to achiv th bst rsult. Rsults from th prvious slid from th constraind last squar filtr //6 ajaybolar.wbly.com 76 Constraind Last Squars Filtr: Exampl (cont. dgradd by motion blur + AWGN =65 Rsult of th Constraind Last squar filtr Rsult of th Winr filtr Constraind Last Squars Filtr:Adjusting Dfin r g Hfˆ It can b shown that ( r r r W want to adjust gamma so that. Spcify an initial valu of r η a whr a = accuracy factor. Comput r =35 //6 ajaybolar.wbly.com 77 =3 3. Stop if is satisfid Othrwis rturn stp aftr incrasing if r η a or dcrasing if Us th nw valu of to rcomput r η a * H ( P( //6 ajaybolar.wbly.com 78 Canara Enginring Collg 3

14 --6 Constraind Last Squars Filtr:Adjusting (cont. * H ( P( R( r M N MN m MN M x y M N x y MN η r ( N ( m x y ( For computing For computing η MN m //6 ajaybolar.wbly.com 79 r Constraind Last Squars Filtr: Exampl Original imag Blurrd imag Du to urbulnc Us corrct nois paramtrs Us wrong nois paramtrs (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. //6 ajaybolar.wbly.com 8 Rsults from constraind last squar filtrs Corrct paramtrs: Initial = -5 Corrction factor = -6 a =.5 = -5 Wrong nois paramtr = - Gomtric Man filtr his filtr rprsnts a family of filtrs combind into a singl xprssion * * ˆ H ( H ( F( S ( S f ( = th invrs filtr = th Paramtric Winr filtr =, = th standard Winr filtr =, <.5 Mor lik th invrs filtr =, >.5 Mor lik th Winr filtr Anothr //6 nam: th spctrum ajaybolar.wbly.com qualization filtr 8 Gomtric ransformation hs transformations ar oftn calld rubbr-sht transformations: Printing an imag on a rubbr sht and thn strtch this sht according to som prdfin st of ruls. A gomtric transformation consists of basic oprations:. A spatial transformation : Dfin how pixls ar to b rarrangd in th spatially transformd imag.. Gray lvl intrpolation : Assign gray lvl valus to pixls in th spatially transformd imag. //6 ajaybolar.wbly.com 8 Gomtric ransformation : Algorithm ( f to b rstord Distortd imag g. Slct coordinat ( in f to b rstord. Comput x r( 4. gt pixl valu at By gray lvl intrpolation y s( 3. Go to pixl ( y 5. stor that valu in pixl f( in a distortd imag g 5 3 ( //6 ajaybolar.wbly.com 83 Spatial ransformation o map btwn pixl coordinat ( of f and pixl coordinat (x,y of g x r( y s( For a bilinar transformation mapping btwn a pair of Quadrilatral rgions x r( c x c y c xy c y s( c x c y c xy c 5 o obtain r( and s(, w nd to know 4 pairs of coordinats ( and its corrsponding ( which ar calld tipoints ( x, ( //6 ajaybolar.wbly.com 84 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. Canara Enginring Collg 4

15 --6 Gray Lvl Intrpolation: Narst Nighbor Sinc ( may not b at an intgr coordinat, w nd to Intrpolat th valu of Exampl intrpolation mthods that can b usd:. Narst nighbor slction. Bilinar intrpolation 3. Bicubic intrpolation Gomtric Distortion and Rstoration Exampl Original imag and tipoints ipoints of distortd imag Distortd imag Rstord imag Us narst nighbor intpolation //6 ajaybolar.wbly.com 85 (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. //6 ajaybolar.wbly.com (s from Rafal C. Gonzalz and Richard E. Wood, Digital 86 Procssing, nd Edition. Gomtric Distortion and Rstoration Exampl (cont. Original imag and tipoints ipoints of distortd imag Exampl: Gomtric Rstoration Original imag Gomtrically distortd imag Us th sam Spatial rans. as in th prvious xampl Distortd imag Rstord imag Rstord imag Us bilinar intpolation //6 ajaybolar.wbly.com (s from Rafal C. Gonzalz and Richard E. Wood, Digital Procssing, nd Edition. 87 Diffrnc btwn //6 abov imags ajaybolar.wbly.com (s from Rafal C. Gonzalz 88 and Richard E. Wood, Digital Procssing, nd Edition. Canara Enginring Collg 5