Notes on the Fourier Transform

Transcription

1 BE/EECS 56 6 FT Note Note on the Fourier Tranform Definition. The Fourier Tranform FT relate a function to it frequenc omain equivalent. The FT of a function i efine b the Fourier integral: G F{ } e iπ for R. There are a variet of eitence criteria an the FT oen t eit for all function. For eample the function co/ ha an infinite number of ocillation a an the FT integral can t be evaluate. If the FT oe eit then there i an invere FT relationhip: iπ F { G } G e Uniquene: Given the eitence of the invere FT it follow that if the FT eit it mut be unique. That i for a function form a unique pair with it FT: G Caveat. An eception to the uniquene propert i a cla of function calle male or null function. An eample i the continuou function f {. Thi function an other like it have the ame Fourier tranform a f : F. Thu the uniquene eit onl for a function plu or minu arbitrar null function. In practice thee function are not realizable energle an thu for the purpoe of thi cla we will aume that the FT i unique. Alternate FT Definition. In the above erivation the i the frequenc parameter. There i another common FT efinition that ue a raian frequenc parameter ω: iω G ω F{ } e G ω / π with an invere FT of: g F G ω ω G ω e i { } ω π Unit. If ha unit of Q then will have unit of ccle/q or Q -. Pleae note that uner our efinition of the FT thi i not an angular frequenc with unit of raian/q but jut plain Q -. Pleae alo keep in min that i the ine of variation for eample we can have repreent a velocit that varie a a function of patial location. The function ha unit cm/ but ha unit cm an G ha unit of cm/ but ha unit of cm -. Eample: Time econ Ditance cm Temporal Frequenc - Hz ccle/ Spatial Frequenc cm - ccle/cm

2 BE/EECS 56 6 FT Note Smmetr Definition. We firt ecompoe ome function in to even an o component e an o repectivel a follow: e o [ + ] [ ] thu e + o an e e an o o A function i Hermitian Smmetric Conjugate Smmetric if: Re{ } e an Im{ } o thu e + io g * Smmetr Propertie of the FT. There are everal relate propertie:. If i real then G i Hermitian mmetric e.g. G G*-.. If i real an even G i real an even. 3. If i real an o G i imaginar an o. 4. If i real G can be efine trictl b non-negative frequencie. 5. If i imaginar then G i Anti-Hermitian mmetric e.g. G -G*-. Proof of. G e iπ [ e + o [coπ i in π] co i evenin i o e co π + e + o i o co π i o i e in π i e o in π E + i io co i even in in i o i - o E io E + io G * Q.E.D. Comment. One intereting conequence of the mmetr propertie i that if i real the onl one-half of the Fourier tranform i necear to pecif the function thi follow from propert. above. ore pecificall i trictl etermine b G for all non-negative frequencie.

3 BE/EECS 56 6 FT Note 3 Comment on negative frequencie. Conier a real-value ignal imagine a voltage on a wire or the oun preure againt our earrum the Fourier tranform of thee i completel pecifie b the non-negative frequencie e.g. G- G*. We can argue that we have the concept of a frequenc ocillation/econ but it oen t reall make phical ene to talk about poitive or negative frequencie. In thi cae we coul argue that the having poitive an negative frequencie i merel a mathematical convenience. Are there cae where negative frequencie have meaning? Conier the bit in a rill it can turn clockwie or counter clockwie an ifferent rotational rate. Here poitive an negative frequencie have phical meaning the irection of rotation. A we hall ee there are cae in meical imaging where thi itinction i important for eample the magnetic moment in RI i a cae where the ign inicate the irection of preceion. Convolution Definition. The convolution operator i efine a: The convolution operator commute: g * h h - g * h h h h * - The elta function δ. The Dirac elta or implue function i a mathematical contruct that i infinitel high in amplitue infinitel hort in uration an ha unit area: δ { an δ ot propertie of δ can eit onl in a limiting cae e.g. a a equence of function δ or uner an integral. Some important propertie of δ : g n - δ with continuou at δ a a with continuou at a δ a a with continuou at F{ δ } Delta function propertie. Firt two are technicall onl efine uner the integral but we ll till talk about them. Similarit tretching δ a δ a Prouct/Sifting δ a a δ a Sifting δ a a 3

4 BE/EECS 56 6 FT Note 4 Convolution * δ δ * * δ a δ a * a Fourier Tranform Theorem. There are man Fourier tranform propertie an theorem. Thi i a partial lit. Aume that F { } G F { h } H an that a an b are contant: Linearit F { a + bh } ag + bh Similarit tretching Shift F { a} G a a F{ a} G e iπa Convolution F { * h } G H Prouct F { h } G * H Comple oulation F i π g e } G { oulation F{ coπ } [ G + G + ] Raleigh Power F{ inπ } G h * G [ G G ] i + Cro Power g H * Ai Reveral F{ } G Comple Conjugation F{ g * } G * Autocorrelation F { * g * } G G * G Revere Relationhip F{ G } Differentiation { } iπg F oment i F{ } G π DC Value G Some common FT pair: g G δ δ coπ [ δ + δ + ] inπ rect { inc triangle { < < [ δ δ ] i + inπ inc π rect inc 4

5 BE/EECS 56 6 FT Note 5 gn { e π e π < iπ e + π e for > ; otherwie iπ + π J π comb rect / π comb The comb function comb. The ampling or comb function i a train of elta function: n comb δ n The Fourier tranform of comb i: F { comb } comb Proof. iπn F{ comb } F δ n e F n n The RHS of the above epreion can be viewe a the eponential Fourier erie repreentation of a perioic function F with perio an α n for all n. Recall the Fourier erie epreion are: F iπn α e whereα F e. n n iπn Now let G rectf one perio of F an thu α n F e iπn G e n iπn m F G m. Now oberve that F { G } One function that atifie thi relationhip i G δ. Thu one poible Fourier tranform of comb i: F δ m comb m B uniquene of the Fourier tranform thi i the unique Fourier tranform of comb. n 5

6 BE/EECS 56 6 FT Note 6 Sampling an replication b comb. The comb function can be ue to ample or etract value of a continuou function. Sampling with perio can be one a: n δ n δ n n n. comb δ n n B the tretching an ifting propertie of the elta function. A function can be replicate with perio b convolving with a comb function: n [ * δ n ] [ * n ] *comb δ n B the tretching an convolution propertie of the elta function. n n Sampling Theor. When manipulating real object in a computer we mut firt ample the continuou omain object into a icretize verion that the computer can hanle. A ecribe above we can ample a function at frequenc f / uing the comb function: n g comb n δ n. The Fourier tranform i: G G * comb G * G * m m δ m G mf δ m mf Thu ampling in one omain lea to replication of the pectrum in the other omain. The pectrum i perioic with perio f. Tpicall onl frequencie le than f / can be repreente in the icrete omain ignal. An component that lie outie of thi pectral region f f / reult in aliaing the mi-aignment of pectral information. / G Original Spectrum G Replicate Spectra -f/ Aliaing f/ 6

7 BE/EECS 56 6 FT Note 7 The Whittaker-Shannon ampling theorem tate that a ban limite function with maimum frequenc ma can be full repreente b a icrete time equivalent provie the ampling frequenc atifie the Nquit ampling criterion: f ma If thi i the cae then the original pectrum can be etracte b filtering an b uniquene of the FT the original ignal can be recontructe. To recontruct the original ignal we appl a recontruction filter H rect / f rect : Gˆ G H G rect G if there i no aliaing In the omain thi reult in inc interpolation: ˆ g * inc n n δ n *inc n inc If the Nquit criterion i met then g ˆ. n n gn Info from all ample contribute to thi point D Linear Stem Conier a tem S[. ] with an input function f an an output or repone function S[f]. Thi tem i linear if an onl if: S [ α f + βf ] αs[ f ] + βs[ f ] αg + βg for all α β f an f. ore generall the uperpoition of an arbitrar et of input function will iel a net repone that i the uperpoition of repone to each input function alone. 7

8 BE/EECS 56 6 FT Note 8 Aitionall if an input i cale e.g. b α then the output will alo be cale b the ame amount. Bae on the ifting propertie of the elta function we know that f f δ f * δ which i the uperpoition of an infinite number of weighte an hifte elta function. Bae on linearit the output of thi tem S[f] i: g S f δ f S[ δ ] The tem operate on function of an f i a contant caling factor. S[δ- ] i a pecial function know a the impule repone an can i efine a: [ δ ] h ; S i the repone to an impule locate at an g f h ; i known a the uperpoition integral. Thi repreentation for the output i vali for an linear tem. Now conier a tem that i hift invariant or time invariant for function of time. We efine a tem a being hift invariant if an onl if: -a S[f-a] for all g an a. For linear hift invariant tem the impule repone can be written a: The uperpoition integral then become: [ δ ] h ; h h ; S f h f * h the convolution of the input function with the impule repone h. For linear hift invariant tem or linear time-invariant tem onl we can then conier the Fourier omain equivalent: G FH Where H i known a the tranfer function or tem pectral repone. 8

9 BE/EECS 56 6 FT Note 9 Note on the D Fourier Tranform Definition. The D Fourier Tranform FT relate a function to it frequenc omain equivalent. The FT of a function i efine b the D Fourier integral: G u v There i alo an invere FT relationhip: i π u+ v F{ } e i π u+ v F { G u v} G u. v e uv Uniquene: Given the eitence of the invere FT it follow that if the FT eit it mut be unique. That i for a function form a unique pair with it FT: G u v Given to me b a tuent I m ure I houl cite where thi came from but I on t know if ou know pleae tell me D FT in Polar Coorinate. We conier a pecial cae where the functional form of i eparable in polar coorinate that i rθ g R rg Θ θ. Since g Θ θ i perioic in θ it ha a Fourier erie repreentation: It can be hown that F inθ Θ θ a n e. n g inθ n inφ { g r e } i e D R πg R r J n πrρ rr where the part uner the integral in known a the Hankel tranform of orer n an J n i the n th orer Beel function of the firt kin: 9

10 BE/EECS 56 6 FT Note J π Derivation of the Hankel tranform relationhip relie on e D FT in polar form i: π i ainϕ nϕ n a e ϕ. π n inφ { g R r gθ θ } an i e n e iπ u+ v iπrρ co θ φ G ρ φ F πg r J πrρ rr For the pecial cae of circular mmetr of g that i rθ g R r then: G ρ φ G ρ π g R r J πrρ rr R which i alo a circularl mmetric function. The invere relationhip i the ame: g R r π G ρ J πrρ ρρ n. Thu the Smmetr Propertie of the FT. If i real then Guv i Hermitian Smmetric that i Guv G*-u-v. If i real an even that i -- then Guv i alo real an even. Finall a ecribe above if we have a real an circularl mmetric function rθ g R r the Gρφ Gρ a real an circularl mmetric function. The elta function δ. The elta function in two i equal the to prouct of two D elta function δ δ δ. In a manner imilar to the D elta function the D elta function ha the following efinition: an δ { an δ otherwie ot propertie of δ can be erive from the D elta function. There i alo a polar coorinate verion of the D elta function: δ δ r / πr.

11 BE/EECS 56 6 FT Note Fourier Tranform Theorem. Let a an b are non-zero contant an F{} Guv an F{h} Huv. Linearit F { a + bh } ag u v + bh u v agnification F { g a b } u v iπ ua+ vb Shift { } ab G F a b G u v e iπ a+ b Comple oulation F{ e } G u a v b Convolution/ultiplication { ** h } a b ** h h F G u v H u v Separabilit F { h } G u v ** H u v g g F{ } F D { g } F D { g } Power G u G v G u v uv h * Ai Reveral F { } G * u v Some common D FT pair: g G u v δ u v δ a b i π ua+ e δ vb πr e co rect π π e e coπ πρ π u πv e e e δ u + δ u + δ v G u v H * u v uv π [ ] rect δ uinc v rect a rect b u v inc inc r rect r { r > ab J πρ jinc ρ a ρ r r J circ r rect { πρ 4jincρ r > ρ r comb combcomb ρ b combuv combucombv

12 BE/EECS 56 6 FT Note The comb function in D comb. The D ampling or comb function i efine a combcombcomb an ha the D FT F{comb} combuv. Formall the D comb function i efine a: n m comb δ n m In a manner imilar to the D cae we can prove that Fourier tranform of the D comb function i alo a D comb function a given in the above table. Sampling Theor in D. In a manner imilar to ampling in D ampling in D can be moele a multipling a function time the D comb function. With ample pacing of an in the an irection the ample function i: g comb δ n m n m n m δ n m n m δ n m n m The icrete omain equivalent i g nm n m g n m. In the Fourier omain the reult i: G u v G u v ** comb u v G u v ** n m n m G u δ u Thu ampling in one omain lea to replication of the pectrum in the other omain. Spacing of the replicate pectra i //. The Whittaker-Shannon ampling theorem in D tate that a ban limite function with maimum frequencie ma an ma can be full repreente b a icrete time equivalent provie the ampling frequenc atifie the Nquit ampling criterion: an n n v ma Uner thee circumtance there i no pectral overlap or aliaing the original pectrum an b uniquene of the FT the original ignal can be recontructe. v To recontruct the original ignal we appl a recontruction filter H u v rect urect v. Gˆ u v G u v H u v G u vrect urect v m ma m G u v if there i no aliaing

13 BE/EECS 56 6 FT Note 3 In the omain thi correpon to inc interpolation in D inc gˆ g ** inc inc inπ : π n m δ n m n m **inc n m inc n m inc n n inc inc m m n m g n m inc The lat line emontrate how the original continuou ignal can be retrieve from the icrete ample verion of. 3

14 BE/EECS 56 6 FT Note 4 Eample of Fourier Tranform: D ata in Low patial freq High patial freq image omain ata image omain ata image omain D ata in Low patial freq High patial freq Fourier omain ata Fourier omain ata Fourier omain 4

15 BE/EECS 56 6 FT Note 5 5

16 BE/EECS 56 6 FT Note 6 6

17 BE/EECS 56 6 FT Note 7 rectrec Guv incuincv caling magnification propert caling magnification propert hifting propert moulation Image AbFourier RealFourier ImaFourier 7

18 BE/EECS 56 6 FT Note 8 incinc Guv recturectv ampling pattern with Δ Δ In the Fourier tranform we have the replication pattern with pacing /Δ /Δ ampling pattern with Δ < Δ In the Fourier tranform we have the replication pattern with pacing /Δ > /Δ ampling pattern with Δ << Δ Thi ha aliaing in the v irection Image Data Fourier Data 8

19 BE/EECS 56 6 FT Note 9 Aliaing in the Spatial Domain.5 unit -.75 unit - In thi eample the frequenc i wept from a.5 to.75 ccle/unit itance a goe from -5 to 5. At location zero the frequenc i.5 ccle/unit. In the upper plot we ee the original ignal. In the lower plot the ignal i ample with a pace in of Δ unit f unit - which mean that all frequencie higher than f /.5 unit - anthing to the right of will be aliae. Inee a the frequenc continue to go higher the apparent frequenc get lower. The apparent frequenc i f f i where f i i the local frequenc. In the econ eample below we eten thi to two imenion. Here we have a linear variation of frequencie in both an that i the component of the frequencie varie from.5 to.75 ccle/unit an the component of the frequencie varie from.5 to.75 ccle/unit. The upper image i the original ignal an the lower image i the ignal ample at Δ Δ unit. The ahe line mark the +/-.5 unit - line the correpon the Nquit limit. Onl the pectral component in the central bo can be repreente. In the attache pectral plot the blue ot correpon to elta function of the true frequencie. The green ot are the pectral replicant in the cae of ampling. Onl the upper left quarant ha no aliaing. The upper left an lower right appear to have the ame frequenc but the latter i aliae. The upper right an lower left alo appear to have the ame frequenc but from ifferent aliaing mechanim aliaing of the component an -component repectivel. 9

20 BE/EECS 56 6 FT Note v v u u v v u u v v u u v v u u

21 BE/EECS 56 6 FT Note D Linear Stem Conier an imaging tem with an input image I that goe through ome tem S[. ] an prouce an output image I e.g. I S[I ]. Several propertie that we are interete in are: Linearit which a two part uperpoition an caling. Thu a tem i linear if an onl if: S[ α + βh ] αs[ ] + βs[ h ] For a linear tem we can efine an impule repone a the output of a tem for an impule locate at poition : h ; S[ δ ] an in general we can efine the output given ome input image I uing the uperpoition integral: I I h ; Space Invariance. A tem i pace invariant if an onl if: I -a-b S[I -a-b] for all a b an I. Alternatel a tem i pace invariant if an onl if the impule repone can be epree in term of the hift of the elta function: h S[ δ ]. Thu h S[ δ ] i all that i neee to pecif the tem. The uperpoition integral become: I I h I ** h where ** inicate D convolution.

22 BE/EECS 56 6 FT Note Eample of a D Imaging Stem. Here we conier a pinhole imaging tem: I thi tem linear? Auming that the aperture i open or cloe an that light alwa travel in traight line through the hole then e the tem i linear. We houl then be able to etermine the impule repone. Let firt conier two ifferent magnification factor one for the object an one for the pinhole aperture. For object magnification we imagine the pinhole in infinitel mall: For a hift of in the input we will get a hift of input ource magnification term a b. a b in the output plane. Thu we efine an a Conier then a tem for a elta function at poition : h ; S[ δ ] Cδ where the elta function appear cale b C an at location. I thi tem pace invariant? No the above epreion cannot be written a a function of an. Now given an input image I we can till etermine the output image uing the uperpoition integral remember the tem i till linear:

23 BE/EECS 56 6 FT Note 3 3 ; I C I C I C h I I δ δ The output i a cale an magnifie verion of the input image. For the pinhole magnification imagine that the pinhole ha a raiu of R then the raiu in output plane woul be R a a + b ieling an aperture magnification function of a b a m +. Now uppoe we ha ome aperture function a thi will now be magnifie in the output plane. Thu the impule repone will take on a form imilar to: ; m m Ca h an the ouput image will take on the form: ** m m a I C m m a I C m m a I C I Here the output image i the convolution of the cale an magnifie verion of the input image an the pinhole function. Though thi tem i not pace invariant we were can till able to write the output in the form of a convolution though not a convolution of the input image but with a magnifie verion of the input image.