Class Note for ECE 482 at UA-Comp Visn Dig Image Proc(12)
Class Note for ECE 482 at UA-Comp Visn Dig Image Proc(12)
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This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Alabama - Tuscaloosa taught by a professor in Fall. Since its upload, it has received 45 views.
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Date Created: 02/06/15
Computer Vision amp Digital Image Processing Fourier Transform aaonca AcumvmerEnginSQring Dr D J Jackson La ure 1 Introduction to the Fourier transform Let fx be a continuous function of a real variable x The Fourier transform of fx denoted by fx is given by SUM Fu IrfXeXP12MdX where 15 Given Fu fx can be obtained by using the inverse Fourier transform 5quotFu x Emu expj2iwcdu Eemrica sconomaznonoonno Dr D J Jacisorv La ureEZ The Fourier transform continued These two equations called the Fouriertransform pair exist iffx is continuous and integrable and Fu is integrable These conditions are almost always satis ed in practice We are concerned with Jnctions fx which are real however the Fouriertransform ofa real function is generally complex So Fu Ru u where Ru and u denote the real and imaginary components of Fu respectively aaonca AcumvmerEnginSQring Dr D J Jackson La ure l The Fourier transform continued Expressed in exponential form Fu is Fu Fuemquot 39 Where FuR2u12u and W tan Ru The magnitude function Fu is called the Fourier spectrum of fx and pu is the phase angle Eemrica c Eumvma anoinoonng Dr D J Jacisorv La ureSA The Fourier transform continued The square of the spectrum Fu Fu2 R2u 1204 is commonly called the power spectrum or the spectral density of fx The variable u is often called the frequency variable This name arises from the expression ofthe exponential term expj27iux in terms of sines and cosines from Euler s formula exprj2zux cos2zux r jsin2mx The Fourier transform continued aaonca AcumvmerEnginSQring Dr D J Jackson La ure Interpreting the integral in the Fourier transform equation as a limit summation of discrete terms make it obvious that Fu is composed of an in nite sum of sine and cosine terms Each value of u determines the frequency of its corresponding sinecosine pair Eemrica sconomaznonoonno Dr D J Jacisorv La ure s Fourier transform example Fourier transform example continued Cunsiderthefuiiuwmg simpie mhehuh The Fewer transform is mrL nmt zmldx m l L Aexpp mwx A 1 TA an 1 TA Er1471 rm rm A n x x i W e 7 1mg 2 1a 1 W m summon This is a cumpiex mhehuh The Fewer spectrum is Imh swolm mom M m A pint of iFui iuuks We the fEIHEIng amumomworawnumn a DA mm mme amumouuworawnnnn V Di mm tutmxx The 2D Fourier transform The 2D Fourier transform continued The Fourier transform can be extended to 2 dimensions sum m J ffltwgtexptvnltmvygt1dxdy and the inverse transform SquotFuvgt fxy J fFuvexp27ruxvyr 1dv The ID Fourierspectrum is Fuv e R201 v 120w The phase angle is The powerspectrum is Ptuvgtlmvgti1 Kuv uv amumomworawnumn a DA mm mm amumouuworawnnnn V Di mm mm m Sample 2D function and its Fourier spectrum Example 2D Fourier transform amumomworawnumn a DA mm mm h my I f 1w Hipk mxwy m disemmzmxlaximpte mlay eh I a y rmh 712wquot A 71 1 71 i w z eth wtw 71 gimmeM simmerN lam M M 1 quotmums lFtuv AX1MltWXMltWY M M amumouuworawnnnn V Di mm mm a Example 2D functions and their spectra EImncampCmv Engnemng nr n J Jimquot um 13 The discrete Fourier transform Suppose a continuous function fx is discretized into a sequence fxu fXDAgtlt may qumx by taking N samples Ax units apart Let x refer to either a continuous or discrete value by saying f x f M xAx where xassumes the discrete values 0 1 N1 and fOf1fN1 denotes any N uniformly spaced samples from a corresponding continuous function EImnul CnnvmuEmlmng nr n J Jacksm huhEI1A Sampling a continuous function fx rm zAx fun JAx Ixn N e um X0 X1 X2 M XN 7 1 EImncampCmv Engnemng nr n J Jimquot um 15 The discrete Fourier transform pair The discrete Fourier transform is given by 1 m F04 ZkaXPFJWN m for u0 1 N1 The discrete inverse Fourier transform is given by m fX Z FMeXPJZWm 0 forx0 1 N1 The values of u0 1 N1in the discrete case correspond to samples of the continuous transform at 0 Au 2Au N 1Au Au and Ax are related by Au1N Ax 1an amp can anan nrn J Jacksm huhEI1E The 2D discrete Fourier transform In the 2D case 1 mm Fuv7Zfxyexpr27r1uMvyN MN MW for u0 aM 1 and v0 aN1 MriNri fxy z FuvexpJ2nmMvym1 Fawn for x0 aM 1 and y0 aN1 The discrete function fxy represents samples of the continuous function at fx0xAx y0yAy D Au1MAX andAv1NAy EImncampCmv Engnemng nr n J Jimquot um 11 The 2D discrete Fourier transform continued For the case when NM such as in a square image mm FUNZZfayexpa irmvyW and mm fxyZZ FuvexpJ2nmvym Note each expression in this case has a 1N term The grouping of these constant multiplier terms in the Fourier transform pair is arbitrary EImnul CnnvmuEmlmng nr n J Jacksm huhe111 Discrete Fourier transform example n up Mr u on on m luv u Cunsreer samplrng at x0 5 x1 75 x51 u and x5125 Here AX 25 and X ranges trprn u gt 3 Discrete Fourier transform example continued The four corresponding Fourier transform terms are exp12mm tut elitetramp l 22 l 1 225 l 1 377pm Ira 77th amumouuworawnumn a DA 4mm tptunx a amumomworawmm t7 DA 4mm tptmx an Discrete Fourier transform example continued The Fourier spectrum is then F0 325 F1242 142 2 J34 F2142 o42 2 14 F3 242 142 2 434 Properties of the 2D Fourier transform The dynamic range ofthe Fourier spectra is generally higher than can be displayed A common technique is to display the function Dow c10g1lFuv where c is a scaling factor and the logarithm function performs a compression ofthe data c is usually chosen to scale the data into the range ofthe display device 0255 typically 1256 for 256 graylevel MATLAB image amumouuworawnumn a DA 4mm tptunx ll amumomworawmm t7 DA 4mm tptmx n Separability Separability continued The discrete transfurm parr can be wrrtten ln separable turms gear 12mm m y are 12mm furu v Ell N71 H my emszm pwexmzmrm furx y um er Set my urfgtlt y can be uptarnee ln 2 steps by sueeesswe applreatruns at the D Fuurrer transfurm ur rts rnverse The ZrDtransfurm can be expressed as H Fuv FEM epozmm Where mm N EVw whimNJ Graphically the prueess is as follows up n t n n t n n t mm m f 4 Fxv 4 Fuv m MW lt gt Wm lt gt m m m amumouuworawnumn a DA 4mm tptunx 2x amumomworawmm t7 DA 4mm tptmx n Translation Translation continued The translation prupemes ufthe Fuunertransfurm parrare fltXrygtltPZirxvnygtNltgtFuunrvivngt and fgtltgtltnryyngt FHrVgteXP 7rgtN Where the double arruvvmdicates a correspondence between a tuneran and its Fuunertransfurm urvree verse Multiplying m bythe exponential and takingthe transform results m a smnurme urrgrn urme frequency plane in the paint uu vu Fur our purpose uuvuNZ Therefore expmrzmnanny fgtcyrlW lt2 FurN2er2 seme urrgrn ufthe Fuuriertransfurm mm ean be moved in the eemer er the eurrespunmng NXN simply by multiplying my by 4 Wbefure taking we transform Nme Tm dues not affect the magmmde er the Fuurrer transform amumomvuorawnumn a DA mmquot mm a amumomworawmNw V Di mmquot mm as Matlab example Matlab example continued Creace data m rhe Les zzemsuzar in x m yensa rxmaze Compute the zen dlscrece Fourler transform Fiic2lfl Compute the Fourler spectrum Fspecrri rclreal m mmmaetn Azr Construct a mum factor based on the dynam 31192 of the Spectrum rspeeemxnex lnax lFspecEl 1 Compute n the scaled ats n 255 UDEUH speccl mxl w w neemrspeeer flgurelll pm as an wage a subset of n maBElDLSS745674HDulurmaplgtavlzs lH amumomvuorawnumn a DA mmquot mm 27 amumomworawmNw V Di mmquot mm H Example image and complete scaled Fourier spectrum plot Example image and partial scaled Fourier spectrum plot with shifted fxy amumomvuorawnumn a DA mmquot mm 25 amumomworawmNw V Di mmquot Mm m
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