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

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1 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 finds a st of cofficints a k for r spatial frqnc and orintation in th 2D orir domain spannd b th 2D frqnc ariabls k k. Ths cofficints ma b comptd b th orir Transform: Compl ponntials ar th Eignfnctions of linar sstms Th orir transform is a linr opration: Each is a compl cofficint which dfins th magnitd and phas of a sinsoid basis fnction. Sris pansion of transcndntal fnctions

2 2 Th Discrt orir Transform i m n f Inrs orir Transform rconstrction i f orir Transform How to intrprt a orir Spctrm 45 dg. fma f in ccls/imag Low spatial frqncis High spatial frqncis Log powr spctrm ij R Th orir Transform cofficints ar compl ald: i f i i sin cos cos cos sin sin ij R Th orir Transform cofficints ar compl: Ral componnt: cos f R Imaginar componnt: sin f J ij R Rprsntation in trms of magnitd and phas agnitd spctrm i * 2 2 I R Phas spctrm tan R I * Powr spctrm Th Discrt orir Transform * Th orir Transform of a ral-ald imag is conjgat-smmtric Thrfor th magnitd spctrm is n smmtric and th phas spctrm is odd smmtric ot that th orir spctrm is oftn r-arrangd for displa sch that th zro-frqnc componnt is in th cntr.

3 Strips of th zbra crat high nrg was gnrall along th -ais; grass pattrn is fairl random casing scattrd low frqnc nrg Sorc: atlab 7 Docmntation Comptr Vision - A odrn Approach - St: Pramids and Ttr - Slids b D.A. orsth Dmo atlab cod im = imrad'was.bmp'; = fft2im; %fft2x rtrns th two-dimnsional discrt orir transform DT of X comptd with a fast orir transform T algorithm. Th rslt Y is th sam siz as X. = fftshift; %fftshiftx rarrangs th otpts of fft fft2 and fftn b moing th zrofrqnc componnt to th cntr of th arra. It is sfl for isalizing a %orir transform with th zro-frqnc componnt in th middl of th spctrm Srfac plot of ral %anglx: Phas angl figr; sbplot3imagscim; colormap gra ais imag ais off titl'original imag'; sbplot32imshowlog+.*conj ; titl'log+ *'; sbplot33imshowangl ; titl'phas'; imspctim; %ISPECT - Plots imag amplitd spctrm aragd or all orintations. showsrfral; %SHOWSUR - shows paramtric srfac in a connint wa 3

4 DT captrs priodicit and dirctionalit DT captrs priodicit and dirctionalit DT captrs priodicit and dirctionalit Visalising indiidal orir componnts To gt som sns of what basis lmnts look lik w plot a basis lmnt or rathr its ral part as a fnction of for som fid. W gt a fnction that is constant whn + is constant. Th magnitd of th ctor gis a frqnc and its dirction gis an orintation. Th fnction is a sinsoid with this frqnc along this dirction and it is constant prpndiclar to this dirction. Lngth of is proportional to frqnc and inrsl proportional to walngth Th orir cofficints rprsnt th original imag as a linar combination of sch sinsoids. Th cofficints ar compl ald. W can add a conjgat pair of compl ponntials to obtain a ral-ald cosin fnction. i i 4

5 Adding a conjgat pair of compl ponntials ilds a ralald cosin i i Hr and ar largr than in th prios slid. i i And largr still... atlab dmo im = imrad'pat.png'; %forir rconstrction frqcompim 5; frqcompim 5.; i % frqcomp displas: % * Th imag. % * Th orir transform spctrm of th imag with a conjgat % pair of orir componnts markd with rd dots. % * Th sin wa basis fnction that corrsponds to th orir % transform pair markd in th imag abo. % * Th rconstrction of th imag gnratd from th sm of th sin % wa basis fnctions considrd so far. figr; imspctim 5; %% ISPECT - Plots imag amplitd spctrm aragd or all orintations. i 2 6 5

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8 ow an analogos sqnc of imags bt slcting orir componnts in dscnding ordr of magnitd

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11 orir Transform agnitd Phas orir transform of a ral fnction is compl difficlt to plot isaliz instad w can think of th phas and magnitd of th transform Comptr Vision - A odrn Approach - St: Pramids and Ttr - Slids b D.A. orsth This is th magnitd transform of th chtah pic Comptr Vision - A odrn Approach - St: Pramids and Ttr - Slids b D.A. orsth This is th phas transform of th chtah pic Comptr Vision - A odrn Approach - St: Pramids and Ttr - Slids b D.A. orsth This is th magnitd transform of th zbra pic Comptr Vision - A odrn Approach - St: Pramids and Ttr - Slids b D.A. orsth Comptr Vision - A odrn Approach - St: Pramids and Ttr - Slids b D.A. orsth

12 Rconstrction with zbra phas chtah magnitd This is th phas transform of th zbra pic Comptr Vision - A odrn Approach - St: Pramids and Ttr - Slids b D.A. orsth Phas and agnitd Rconstrction with chtah phas zbra magnitd Imag with chtah phas and zbra magnitd Imag with zbra phas and chtah magnitd Comptr Vision - A odrn Approach - St: Pramids and Ttr - Slids b D.A. orsth A simpl ttr dscriptor Randomizing th phas agnitd of th orir Transform f A f f i 2 j f f f agnitd of th orir Transform ncods nlocalisd information abot dominant orintations and scals in th imag. 2

13 Statistics of Scn Catgoris an-mad nironmnts Spctral signatr of man-mad nironmnts Olia t al 99 Olia & Torralba orir imag nhancmnt atral nironmnts Spctral signatr of natral nironmnts Look at mford s work Dr forchris modls Town 3

14 Corrlation iltring Conoltion This is calld cross-corrlation dnotd iltring an imag Rplac ach pil b a 2 wightd combination of H its nighbors. 3 4 Th filtr krnl or mask is th prscription for th wights in th linar combination. Conoltion: lip th filtr in both dimnsions bottom to top right to lft Thn appl cross-corrlation H H Slid crdit: K. Graman Slid crdit: K. Graman Conoltion s. Corrlation Corrlation Conoltion atlab: filtr2 imfiltr ot th diffrnc! atlab: con2 ot If H-- = H thn corrlation conoltion. Shift Inariant Linar Sstm Shift inariant: Oprator bhas th sam rwhr i.. th al of th otpt dpnds on th pattrn in th imag nighborhood not th position of th nighborhood. Linar: Sprposition: h * f + f2 = h * f + h * f2 Scaling: h * k f = k h * f Slid crdit: K. Graman Slid crdit: K. Graman 4

15 Proprtis of conoltion Linar & shift inariant Commtati: f * g = g * f Associati f * g * h = f * g * h Idntit: nit impls =. f * = f Diffrntiation: or krnls smallr or qal to 55 T followd b mltiplication is gnrall fastr than plicit conoltion 5