Image Enhancement in the Spatial Domain

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Christina Short
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1 s=tr Eampl PR3.: Eponntials of th form -ar a a positi constant ar sfl for constrcting smooth gra-ll transformation fnctions. Constrct th transformation fnctions haing th gnral shaps shown in th following figrs. Th constants shown ar inpt paramtrs and or proposd transformations mst incld thm in thir spcifications. s=tr s=tr A B D A B L r a a Gnral form of th fnction: a A L A L r C b s T r In Figr a: soling for a : Thn: s T r A.693 r L L r c ar A al ln a.693 L

2 Eampl PR3.: s=tr B s=tr D B b Gnral form of th fnction: In Figr b: s T r L B a B B.693 r L L C r b ar ar s T r B B B Thn: al ln a.693 L L c r c Gnral form of th fnction: s T r D C.693 r L C

3 Eampl PR3.3: Propos a st of gra-ll-slicing transformations capabl of prodcing all th indiidal bit plans of an 8-bit monochrom imag. Considr mod to b th modlar diision rslting th rmaindr of th intgr diision. 7 fnction can b writtn 55 for T r othrwis For Bit plan 7 SB: th following r For Bit plan 6: T r 55 for 6 othrwis mod r 7 For Bit plan 5: T r 55 for 5 othrwis mod r 6

4 Eampl PR3.3: 55 T r 55 T r 55 T r 55 T r 55 T r For Bit plan 4: For Bit plan 3: For Bit plan : For Bit plan : For Bit plan : for 4 othrwis for 3 mod r mod r othrwis for mod r othrwis for othrwis for othrwis mod r mod r 5 4 3

5 Eampl PR3.3: Considr Bit plan 6: Tr for mod r othrwis 5 s 5 s=tr ot that all th s corrsponding to bit 6 ar scald to 55 and all th bits corrsponding to s ar scald down to r 3= = 9= 5=

6 Eampl 3PR3.4: a What ffct wold stting to zro th lowr-ordr bit plans ha on th histogram of an imag in gnral? b What wold b th ffct on th histogram if w st to zro th highrordr bit plans instad? a Rmoing th low ordr bit plans wold man th loss of som high frqnc dtails. Frthrmor th imag histogram will b mor spars as compard with th all 8-bit plan cas. This is bcas thr will b no componnt rprsnting intrmdiat pil als sch as and 9345 tc. Instad thr will b and 8 and 6 tc. This wold cas th hight som of th rmaining histogram paks to incras in gnral. Tpicall lss ariabilit in gra ll als will rdc contrast All th bits incldd of th LSBs rmod

7 Eampl 3PR3.4: b What wold b th ffct on th histogram if w st to zro th highr-ordr bit plans instad? b Rmoing th high ordr bit plans wold man th loss of som r important DC componnts awa from th imag. Th maning of this is that th imag is mch darkr and a lot of th low frqnc componnts will b lost All th bits incldd SB rmod

8 Eampl 4PR3.6: Sppos that a digital imag is sbctd to histogram qalization. Show that a scond pass of histogram qalization will prodc actl th sam rslt as th first pass? Lt n b th total nmbr of pils and lt n r b th nmbr of pils in th inpt imag with intnsit al r. Thn th histogram qalization transformation is: Sinc r pil and no othrs with al r k is mappd to al s k it follows that n sk = n rk. A scond pass of histogram qalization wold prodc als k according to th transformation: k r k r k k n n n n r T s thn n n bt n n s T r s k s k k k k r k k s n n s T Which shows that a scond pass of histogram qalization wold ild th sam rslt as th first pass.

9 Eampl 4PR3.6: Gin an imag th following histograms and CDF Tr graphs can b obtaind Tr=CDF of th first pass Tr=CDF of th nd pass ~idntit transformation ot that th Idntit Transformation dos not chang th histogram of th inpt imag Histogram of th first pass 6 5 Histogram of th nd pass ~niforml distribtd

10 Eampl 6PR3.: Th thr imags shown blow wr blrrd sing th sqar araging masks of sizs n=3 5 and 45 rspctil. Th rtical bars on th lowr part of a and c ar blrrd bt clar sparation ists btwn thm. Howr th bars ha mrgd in in imag b in spit of th fact that th mask that prodcd th imag is significantl smallr than th mask prodcd imag c.eplain this. ot that th rtical bars ar 5 pils wid and pils apart. Original imag a b c

11 Eampl 6PR3.: ot that th rtical bars ar 5 pils wid and pils apart. a b c Th rason wh th mask with siz n=5 prodcing mrgd niform rgion arond th bars is bcas of th sizs of th bars and th sparation of th bars in th horizontal dirction. Th width of ach bar is 5 pils and ach bar is sparatd b pils. In sch an nironmnt as th mask mos in th horizontal dirction thr will b 5 black and light gra pils in ach row at a tim. This will proid th sam arag al for ach pil in th rgion prodcing a mrgd niform gra ll. Howr this will not b th sam in 3 and 45 pil masks!!

12 Eampl 7PR3.4: In a gin application an araging mask is applid to inpt imags to rdc nois and thn a Laplacian mask is applid to nhanc small dtails. Wold th rslt b th sam if th ordr of ths oprations wr rrsd? Laplacian opration and araging can b prssd b th following 33 masks: Laplacian g f f f f 4 f 9 Araging h f 9 i i Araging ask -4 Laplacian ask Both of th two oprators ar mltipling th pils in a 33 nighborhood with constant nmbrs and prform th addition. Thrfor ths oprations ar linar oprations. Th ordr of two linar oprations dos not mattr. Th rslt wold b sam in an ordr.

13 Imag Enhancmnt in th Frqnc Domain Eampl 8PR3.5 :Show that th Laplacian opration is isotropic inariant to rotation. Yo will nd th following qations rlating coordinats aftr ais rotation b an angl q. cosq sinq sinq cosq Laplacian oprator is dfind as: For th rotatd Laplacian oprator: Gin that: If w show that th right sids of th first qations ar qal than th Laplacian opration is rotation inariant. W start with

14 Imag Enhancmnt in th Frqnc Domain Eampl 8PR3.5 : Taking th partial driati of this prssion again with rspct to ilds: Rpat th sam opration for : Taking th partial driati of this prssion again with rspct to ilds: Adding th prssions for th scond driatis: Both sids ar qal Hnc Laplacian is rotational inariant.

15 Eampl 9PR4.7: What is th sorc of narl priodic bright points in th horizontal ais of th spctrm in th following figr. Th narl priodic bright points in th frqnc spctrm corrsponds to th priodic bars as wll as th rpating bos lttrs and circls in th horizontal dirction.

16 Eampl PR4.6 -a: Pro th alidit of th following Eqation. [ f ] F [.] dnots th Forir Transform π F f F f π π f π π cos π sinπ alwas zro or - dpnding on th addition of +. If + is n thn othrwis -. f π

17 Eampl PR4.6 -b: Pro th alidit of th following Eqations. F f ] [ f f F f Similarl it can b shown that : ] [ f F This is th Translation proprt of th D Forir transform: Whn = = and = = thn F f F f ] [ ] [ f F

18 Eampl PR4.9: Considr th imags shown blow. Th imag on th right is obtaind b a mltipling th imag on th lft b - + ; b compting th DFT; c taking th compl congat of th transform; d compting th inrs DFT; and mltipling th ral part of th rslt b - +. Eplain mathmaticall wh th imag on th right appars as it dos. Th compl congat simpl changs to in th inrs transform so th imag on th right is gin b: *] [ F F Which simpl mirrors f abot th origin ths prodcing th imag on th right F f

19 Eampl PR4.4: Sppos that o form a low pass filtr that arags th for immdiat nighbors of a point bt clds th point itslf. a Find th qialnt filtr H in th frqnc domain. b Show that H is a lowpass filtr. a Th spatial arag is gin b: Thn sing th following proprt: f F Whr th H is th filtr fnction. W gt th following transfr fnction:

20 Eampl PR4.4: Sppos that o form a low pass filtr that arags th for immdiat nighbors of a point bt clds th point itslf. a Find th qialnt filtr H in th frqnc domain. b Show that H is a lowpass filtr. a Th H Filtr fnction can b cntrd b: b Considr on ariabl for conninc. As rangs from to th al of cosπ[-] starts at - paks at whn = th cntr of th filtr and thn dcrass to - again whn =. Ths w s that th amplitd of th filtr dcrass as a fnction of distanc from th origin of th cntrd filtr which is th charactristic of a lowpass filtr. A similar argmnt is asil carrid ot whn considring both ariabls simltanosl.

21 Eampl 8-PR4.9: Dri th frqnc domain filtr that corrsponds to th Laplacian oprator in th spatial domain. Considr thlaplacian mask gin. Thn Whr Th H Filtr fnction can b cntrd b:

Eampl 8-PR4.: Th two forir spctra shown ar of th sam imag. Th spctrm of th lft corrsponds to th original imag and th spctrm on th right was obtaind aftr th imag was paddd with zros.

22 Eampl 8-PR4.: Th two forir spctra shown ar of th sam imag. Th spctrm of th lft corrsponds to th original imag and th spctrm on th right was obtaind aftr th imag was paddd with zros. a Eplain th diffrnc of th orall contrast. b Eplain th significant incras in signal strngth along th rtical and th horizontal as of th spctrm shown on th right. a Padding with zro incrass th siz bt rdcs th arag gra ll of imag. Th arag gra ll of th paddd imag is lss than th original imag. F in th paddd imag is lss than F of th original imag. All th othrs awa from th origin ar lss in th paddd imag than th original imag. This prodcs a narrowr rang of als hnc a lowr contrast spctrm in th paddd imag. b Padding with s introdcs significant discontinitis at th bordrs of th original imag. This procss introdcs strong horizontal and rtical dgs whr th imag nds abrptl and thn contins with o als. Ths sharp transitions corrspond to th strngth of th spctrm along th horizontal and rtical as.

Imag Rstoration Rstoration in th prsnc of ois: Onl-Spatial Filtring Eampl -PR5.: Gin th two sbimags blow. Th sb imag on th lft is th rslt of sing arithmtic man filtr of siz 33.

23 Imag Rstoration Rstoration in th prsnc of ois: Onl-Spatial Filtring Eampl -PR5.: Gin th two sbimags blow. Th sb imag on th lft is th rslt of sing arithmtic man filtr of siz 33. Th othr sbimag is th rslt of sing th gomtric man filtr of th sam siz. a Wh th sbimag obtaind with gomtric filtring is lss blrrd. Hint o can start or analsis with -D stp dg profil of th imag. b Eplain wh th black componnts on th right imag ar thickr. a Lts considr th mathmatical prssions of th arithmtic and gomtric man filtrs: fˆ mn s t S g s t fˆ s ts g s t mn

24 Imag Rstoration Rstoration in th prsnc of ois: Onl-Spatial Filtring Eampl -PR5. : a If w tak a rogh stimat of -D Stp fnction profil bfor th filtring: Arithmtic man filtring: Bfor filtring: Aftr filtring: Gomtric man filtring: Bfor filtring: Aftr filtring: fˆ mn s t S g s t fˆ g s t mn s ts

25 Imag Rstoration Rstoration in th prsnc of ois: Onl-Spatial Filtring Eampl -PR5. : a Th rslting -D profils clarl indicats that th arithmtic man filtr prodcs smoothrblrrd transition and b Th gomtric filtring incrass th thicknss of th black componnts. Arithmtic man filtring: Bfor filtring: Aftr filtring: Gomtric man filtring: Bfor filtring: Aftr filtring:

Computer Vision. Fourier Analysis. Computer Science Tripos Part II. Dr Christopher Town. Fourier Analysis. Fourier Analysis

Computer Vision. Fourier Analysis. Computer Science Tripos Part II. Dr Christopher Town. Fourier Analysis. Fourier Analysis Comptr Vision Comptr Scinc Tripos Part II Dr Christophr Town orir Analsis An imag can b rprsntd b a linar combination of basis fnctions: In th cas of 2D orir analsis: orir Analsis orir Analsis Th transform