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# IMAGE PROCCOMPU VIS EEL 6562

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This 9 page Class Notes was uploaded by Brandon Douglas on Friday September 18, 2015. The Class Notes belongs to EEL 6562 at University of Florida taught by Staff in Fall. Since its upload, it has received 5 views. For similar materials see /class/206838/eel-6562-university-of-florida in Electrical Engineering at University of Florida.

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Date Created: 09/18/15

EEL 6562 Image Processing amp Computer Vision Notes on Frequency Domain Theory 1 Fourier Transform of Continuous Functions 11 De nitions Forward transform Fw v e m we WWm dzdy Reverse transform Fu v is in general a complex function so it has a rectangular form FWU RWU 11 Where Ru v and 1u v are real functions of u and v Fu 2 also has a polar form V F v lFWa UHBPMW Where 1 WW UH 32 v 12W and gtu v tanil Power spectrum Puv lFuvl2 Fuv Fuv R2u v 12uv 12 Properties Linearity am y bye y i am v barn v Spatial translation J39QMUEOJHyo fx I0yy0 iFuav5 Frequency translation fx yej27ru awry gt L Fu 7 no 1 7 vo 871 gm y i j27ru Fu v Proof 127rurvy axn x axn 100 100 Fuve dudv Fu v ginej27rwvy du dv 00 700 x Fu v j27runej27rwvygt du dv j27ru Fu1ej27r ydudv mf x y i 1M v vw y i 7mm mm v ax by i ma vbgt Laplacian Scaling Separable functions Jammy m y i F0 v mu Fyltvgt Proo FWWX1hmhm mmwmh 1 1 WWW flWW my fyygt5 quot y fxltxgtei m dz dy f xki m dx fyye 12quotvy dy FEW FM 1 3 Convolution Linear convolution in continuous spatial domain is de ned as fxygxy jo fo a ghiayi dsdt Convolution is commutative and distributive over addition x 2 996 2 996 2 x 2 x 2 996 2 MI 21 x 2 996 2 x 2 MI 2 The Dirac delta function is de ned as the function 695 y that satis es 6xjydA l 1f0039 R R 0 otherW1se The Dirac delta function is the identity element for convolution x 2 50621 60621 x 2 x 2 The convolution theorem x y if we y i F0 mam v x y we 2 i Fm v Gm v 14 Common transform pairs Some common Fourier transform pairs so y L 1 27r025727r20212y2 L 5u2v2202 Proof oo oo Fu v 271172572780212y2e j2quotwvydxdy 700 700 2 2 00 00 2 2 e itUT 27quot 72572780212y2gt5 127 vyego duly oo 700 2 z 00 00 2 2 2 v 2 e r 27r025727r a 2 ij27ruxm 700 700 2 2 2 U2 72 a y 7127Wym dxdy Working With the rst exponent after the integral signs 1 1 7271302952 7 j27rux 112 7 471304952 j47r02ux u2 722 27117295 ju2 a The second exponent after the integral signs is similarly manipulated 2 U2 00 00 2 V 2 FW 1 57 20 27r025 if07wzx u effigm y v dxdy 700 700 Perform the following substitution r 27r02xju s 27r02yjv dr 2711725195 513 271172513 2 U2 00 00 1 T2 52 Fuv 5 20quot 25 20quot drds 700 00 27m The integral of a two dimensional Gaussian distribution is exactly unity oo 00 T2 52 25 20quot drds 1 00 700 7117 u2 U2 Fu v 5 20quot sin 7rua sin 7rvb Rectabx y L ab 7 7 7r ua 7rvb where 1 if S 12 and S bZ 0 otherwise Rectabx y 3 cos27ru0x 27rv0y L 6u u0v 20 6u 7 u0v 7 v0 1 sin27ru0x 27rv0y L ji 6u u0v 20 7 6u 7 no 1 7 v0 2 Fourier Transform of Discrete Functions 21 De nitions For functions that are either discrete and nite or discrete and periodic we use the discrete Fourier transform DFT 71 N71 2 fx ye7j27ruxMvyN y0 Mg Fuv 3 H o as Inverse transform 1 M71 N71 Y My 7 7W Z Z Fltuvgte WMWNgt u0 110 De ne Forward transform M71N71 Fltuvgt Z 2 16mm W 10 y0 Inverse transform 1 M71 N71 7 fuss ivy 9531 W 1 FuavWN WM Fu v is in general a complex function so it has a rectangular form Fm v Rm v mu v Where Ru v and 1u v are real functions of u and v Fu 2 also has a polar form V Fm v Fltuvgte1 ltuv gt Where Wu vl Rm v mm WM tan l and 22 Properties Power spectrum PM WWW FuavFuav RQWW 12 Periodicity Fu v Fu pM v qN for all integers p and q DFT of the complex conjugate WW 3 Fua iv The above DFT pair implies that for real x y the DFT is conjugate sym metric Fu v F7u 7v Conjugation in frequency domain m4 72 mu v DC term MilNil Flt00gt Z 2 my 10 y0 Also MilNil FM2aN2 Z Z xay ilf N M HIM20 21 41 0 0 s lta E Z Flt0N2gt 7 fltxygtlt71gty ac y o o Linearity afxy 590621 mu v ham v So far we have tacitly assumed that fx y is a spatially nite image and that its region of support is 0 S x S M 7 1 0 S y S N 7 1 De ne the periodic extension of fx y NW 3 x mod My mod N The DFT has a dual interpretation 0 The DFT consists of frequency samples of the continuous Fourier trans form of a nite image 0 The DFT corresponds to the Fourier series expansion of a periodic signal Both interpretations are useful and we can switch back and forth as long as we are careful Spatial translation x x0 yo FWW Wife W yo Frequency translation fxyW12u x W voy Fu 7 no 1 7 vo Proof M71 71 Fm Z Z fltzygtWW 10 y0 Letru7u0andsv7v0 71 N71 Fu 7 no 1 7 v0 Z 2 x y ngfuiuo WIS7110M 10 y0 M71 N71 7 7u an 11y 71oy 7 fx y W EWN 0 WM WM 10 y0 M71 N71 7 w 2 WW WW W W a o 0 lta A special case of frequency translation Separable functions fxy71xy Fu 7 M2 v 7 N2l fxxfy aw Fuav FxuFyv 23 Convolution and correlation Linear discrete convolution f9vy99vy Z Z f3t9xisyit Circular discrete convolution M71N71 N fzy 9xy Z Zf3t xisyit 700 t7ocgt 50 addition t0 Both forms of convolution de ned above are commutative and distribute over x 2 90621 996 2 x 2 x 2 996 2 MI 21 x 2 996 2 x 2 MI 2 x 2 90621 996 2 x 2 x 2 90 y WI 21 x 2 90 y x 2 We 2 The unit sample function 695 y is de ned as 1 if 0 6ltxygt 19 y convolution 0 otherwise The unit sample function is the identity element for both forms of discrete and x 2 50621 60621 aw aw x 2 506 606 x 2 x 2 tion The discrete form of the convolution theorem applies only to circular convolu fx 2 90 2 F v GM v fxygxy lt gt WFu1 Gu 2 However the theorem given below gives us a way to calculate the linear convo lution using circular convolution Suppose fx y is A X B and hxy is C X D Suppose further that we zero pad fx y and hxy out to P X Q and call the new functions fexy and heme y respectively Then if P 2 A C 7 1 and Q 2 B D 7 1 the results of circular and linear convolution match fexy gexy fexy gexy for 03 x S P7 1 and OS yS Q71 Correlation MilNil f96y 996yA Z Zfquot3t9x3yt 50 t0 Relationship between convolution and correlation 1 x 2 0 996 2 W fquot 7y 996 2 If fxy and gxy are sufficiently zero padded so that linear and circular convolution correspond then hm e we 2 FltuvgtGltuvgt F06 y9x 2 F v 0 0W U 24 Example 1 Find the frequency response of the lter with the following mask nu man an Way fx1y1fx 1217 1 Cu v FuvW12uW v Fu Frequency response Cu v Fu v WWW WXZWE ej27ruMvN eij27ruMvN 2 cos 27ruM vN Huv Magnitude and phase of frequency response mm M 2l mamM vNm an v o

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