Introduction to the Fourier transform. Computer Vision & Digital Image Processing. The Fourier transform (continued) The Fourier transform (continued)


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1 Introduction to th Fourir transform Computr Vision & Digital Imag Procssing Fourir Transform Lt f(x) b a continuous function of a ral variabl x Th Fourir transform of f(x), dnotd by I {f(x)} is givn by: I{ f ( x)} f ( x)xp[ jπux whr j Givn, f(x) can b obtaind by using th invrs Fourir transform: I { } f ( x) xp[ jπux du. + Dr. D. J. Jackson Lctur 8 Dr. D. J. Jackson Lctur 8 Ths two quations, calld th Fourir transform pair, xist if f(x) is continuous and intgrabl and is intgrabl. Ths conditions ar almost always satisfid in practic. W ar concrnd with functions f(x) which ar ral, howvr th Fourir transform of a ral function is, gnrally, complx. So, F ( R( + ji ( whr R( and I( dnot th ral and imaginary componnts of rspctivly. Exprssd in xponntial form, is: whr F ( R ( + I ( ϕ( tan I( R( jϕ ( Th magnitud function is calld th Fourir spctrum of f(x) ϕ( is th phas angl. Dr. D. J. Jackson Lctur 83 Dr. D. J. Jackson Lctur 8 Th squar of th spctrum, P( R ( + I ( is commonly calld th powr spctrum (or th spctral dnsit of f(x). Th variabl u is oftn calld th frquncy variabl. This nam ariss from th xprssion of th xponntial trm xp[jπux in trms of sins and cosins (from Eulr s formula): Intrprting th intgral in th Fourir transform quation as a limit summation of discrt trms mak it obvious that: is composd of an infinit sum of sin and cosin trms. Each valu of u dtrmins th frquncy of its corrsponding sincosin pair. xp[ jπux cos(πux) j sin(πux) Dr. D. J. Jackson Lctur 85 Dr. D. J. Jackson Lctur 86
2 Fourir transform xampl Considr th following simpl function. Th Fourir transform is: f(x) X x X f ( x)xp[ jπux xp[ jπux [ jπu [ jπu j πux jπux sin( πux ) πu X [ jπu jπux jπux jπux j πux Fourir transform xampl (continud) This is a complx function. Th Fourir spctrum is: sin( πux) πu sin( πux) X ( πux) jπux plot of looks lik th following: Dr. D. J. Jackson Lctur 87 Dr. D. J. Jackson Lctur 88 Th D Fourir transform Th Fourir transform can b xtd to dimnsions: I{ } th invrs transform I { } xp[ jπ ( ux + v dy. xp[ jπ ( ux + v dudv. Th D Fourir transform (continud) Th D Fourir spctrum is: F ( R ( + I ( Th phas angl is: ϕ( tan Th powr spctrum is: I( R( P( R ( + I ( Dr. D. J. Jackson Lctur 89 Dr. D. J. Jackson Lctur 8 Sampl D function and its Fourir spctrum Exampl D Fourir transform X xp[ jπ ( ux + v dy xp[ jπux xp[ jπvy dy j πux j πvy X Y [ [ jπu jπv [ jπu jπux sin( πux ) XY[ ( πux ) Y [ jπy jπux j πvy sin( πvy) [ ( πvy) sin( πux ) sin( πvy) Th spctrum is XY[ [ ( πux ) ( πvy) jπvy Dr. D. J. Jackson Lctur 8 Dr. D. J. Jackson Lctur 8
3 Exampl D functions and thir spctra Th discrt Fourir transform Suppos a continuous function, f(x), is discrtizd into a squnc {f(x ), f(x +Δx), f(x +Δx),.., f(x +[Δx)} by taking sampls Δx units apart Lt x rfr to ithr a continuous or discrt valu by saying f ( x) f ( x + xδx) whr x assums th discrt valus,,,  and {f(),f(),,f()} dnots any uniformly spacd sampls from a corrsponding continuous function Dr. D. J. Jackson Lctur 83 Dr. D. J. Jackson Lctur 8 Sampling a continuous function Th discrt Fourir transform pair Th discrt Fourir transform is givn by: f ( x)xp[ jπux / x for u,,, Th discrt invrs Fourir transform is givn by: f ( x) xp[ jπux / u for x,,, Th valus of u,,,in th discrt cas corrspond to sampls of th continuous transform at, Δ Δ, ( )Δu Δu and Δx ar rlatd by Δu/( Δx) Dr. D. J. Jackson Lctur 85 Dr. D. J. Jackson Lctur 86 Th D discrt Fourir transform In th D cas: M M x y for u M and v  M u v xp[ jπ ( ux / M + vy / ) xp[ jπ ( ux / M + vy / ) for x M and y  Th discrt function f(x, rprsnts sampls of th continuous function at f(x +xδx, y +yδ Δu/(MΔx) and Δv/(Δ Th D discrt Fourir transform (continud) For th cas whn M (such as in a squar imag) x y u v xp[ jπ ( ux + v / xp[ jπ ( ux + v / ot ach xprssion in this cas has a / trm. Th grouping of ths constant multiplir trms in th Fourir transform pair is arbitrary. Dr. D. J. Jackson Lctur 87 Dr. D. J. Jackson Lctur 88
4 Discrt Fourir transform xampl Discrt Fourir transform xampl (continud) Th four corrsponding Fourir transform trms ar 3 ) f ( x)xp[ x [ f () + f () + f () + f (3) [ ) f ( x)xp[ jπ / x jπ / jπ [ [ + j j3π / Considr sampling at x.5, x.75, x., and x 3.5 Hr Δx.5 and x rangs from 3 F ( ) [ + j F ( 3) [ + j Dr. D. J. Jackson Lctur 89 Dr. D. J. Jackson Lctur 8 Discrt Fourir transform xampl (continud) Th Fourir spctrum is thn ) 3.5 ) [( / ) + (/ ) ) [(/ ) + ( / ) 3) [( / ) + (/ ) / / / 5 5 Proprtis of th D Fourir transform Th dynamic rang of th Fourir spctra is gnrally highr than can b displayd common tchniqu is to display th function [ ) D ( c log + v whr c is a scaling factor and th logarithm function prforms a comprssion of th data c is usually chosn to scal th data into th rang of th display dvic, [55 typically ([56 for 56 graylvl MTLB imag) Dr. D. J. Jackson Lctur 8 Dr. D. J. Jackson Lctur 8 Sparability Sparability (continud) Th discrt transform pair can b writtn in sparabl forms x for v,,, xp[ jπux / u xp[ jπux / for x,y,,, So, or f(x, can b obtaind in stps by succssiv applications of th D Fourir transform or its invrs. y v xp[ jπvy / xp[ jπvy / Th D transform can b xprssd as f(x, x x, xp[ jπux / whr x, xp[ jπvy / y Graphically, th procss is as follows,,,,,, Row transforms Multiply by x, Column transforms , , , Dr. D. J. Jackson Lctur 83 Dr. D. J. Jackson Lctur 8
5 Translation Th translation proprtis of th Fourir transform pair ar xp[ jπ ( ux + v / u v f ( x x, y xp[ jπ ( ux + v / whr th doubl arrow indicats a corrspondnc btwn a function and its Fourir transform (or vic vrsa) Multiplying f(x, by th xponntial and taking th transform rsults in a shift of th origin of th frquncy plan to th point (u,v ). Translation (continud) For our purposs, u v /. Thrfor, xp[ jπ ( u x + v / jπ ( x+ ( ) x+ y x+ y ( ) u /, v / ) So, th origin of th Fourir transform of f(x, can b movd to th cntr of th corrsponding x simply by multiplying f(x, by () x+y bfor taking th transform ot: This dos not affct th magnitud of th Fourir transform Dr. D. J. Jackson Lctur 85 Dr. D. J. Jackson Lctur 86 Matlab xampl Matlab xampl (continud) %Crat data for th tst fzros(8); for x:6 for y:6 f(x,8; % Prform a translation shift on f(x, for x:8 for y:8 f(x,f(x,*(()^(x+); % Comput th D discrt Fourir transform Ffft(f); % Comput th Fourir spctrum Fspctsqrt(ral(F).^+imag(F).^); % Construct a scaling factor basd on % th dynamic rang of th spctrum FspctMXmax(max(Fspct)); % Comput D, th scald data D(56/(log(+FspctMX)))*log(+Fspct); figur(); % Plot, as an imag, a subst of D imag(d(56:7,56:7));colormap(gray(56)); Dr. D. J. Jackson Lctur 87 Dr. D. J. Jackson Lctur 88 Exampl imag and complt, scald Fourir spctrum plot Exampl imag and partial, scald Fourir spctrum plot (with shiftd f(x,) Dr. D. J. Jackson Lctur 89 Dr. D. J. Jackson Lctur 83
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