Fourier Transform: Overview. The Fourier Transform. Why Fourier Transform? What is FT? FT of a pulse function. FT maps a function to its frequencies

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1 orr Transform: Ovrvw Th orr Transform Wh T s sfl D T, DT, D DT T proprts Lnar ltrs Wh orr Transform? T hlps to analz Samplng artfacts Lnar ltrs Som ntrstng mag transformaton c proprts for pattrn matchng or classfcaton orr Transform of p T maps a fncton to ts frqncs p ω Anglar frqnc Contnos fncton ωt p t dt ωt cos ωt sn ωt T of a pls fncton What s T? Spatal: Pls rqnc: snc sn snc T dcomposs a fncton nto a wghtd sm of snsodal fnctons > W can rconstrct th orgnal fncton: ω ω t p t p dω π

2 Rprsntng T Dscrt Samplng T s compl Rprsntaton: Ral / Imagnar agntd / Phas agntd Samplng at low frqnc Samplng at hgh frqnc Phas -D Dscrt orr Transform Assmptons: Samplng crtron satsfd Sampld fncton rplcats to nfnt Samplng a rotatng whl Ovrsampld rotatng whl: orward DT : p p π Sam whl, ndrsampld: Invrs DT : p p π SUICIET SAPLIG RATE SAPLIG ARTIACTS

3 ISUICIET SAPLIG RATE YQUIST THEORE Th sampl frqnc mst b at last twc th hghst frqnc prsnt for a sgnal to b rconstrctd from a sampld vrson. -D Dscrt orr Transform Dcomposton nto snsodal fnctons p p, ν, ν p, p, ν π / ν / π / ν / Ral part of whr v π v rprsnts th frqnc a tan v, rprsnts th orntaton Bracwll, chap. D Pls T HORIZOTAL AD VERTICAL STUCTURES T -> pl wd strps: Vrtcal strctrs Half th ma frq. Sqar Pls D snc fncton Horzontal tt: Horzontal strctrs Ln spacng 3

4 PHASE AD AGITUDE PHASE AD AGITUDE agntd of th transform agntd of th transform Phas of th transform Phas of th transform SWITCHIG PHASE AD AGITUDE T s Shft Invarant Aftr shftng: agntd sta constant Phas changs Zbra phas Chtah magntd Chtah phas Zbra magntd Rotaton T of a rotatd mag also rotats Imag rplcaton do not rplcat for vr angl. Spctral Analss Th magntd of th DT captrs th man orntatons n th mag. 4

5 rqnc Scalng Spacal comprsson rqnc ncras T Intrpolaton. Compt DT. Add zros at both nds 3. Invrs DT Sprposton Rmovng os [ p p ] [ p ] [ p ] rqnc Ct Rconstrcton 5

6 ltplcaton In orr Doman ast orr Transform { 8,7,6,5,4,3,, } { 8,,6,,4,,, } {,7,,5,,3,, } { 8,6,4, } { A, B, C, D} { 8,,6,,4,,, } { A, B, C, D, A, B, C, D} Strtchng Thorm ltplcaton n orr Doman can spprss nwantd frqncs. Rmovng hgh frq smoothng { 7,5,3, } { P, Q, R, S} { 7,,5,,3,,, } { P, Q, R, S, P, Q, R, S} Strtchng Thorm {,7,,5,,3,, } { P, WQ, W R, W S, W P, W Q, W R, W S} wth W p-π /8 Shft Thorm Bracwll, Chap. ast orr Transform T whr f f ω f ω f ω vn ω odd vn and π / odd wth wth and ω n n f ω f ω ω πn ar DTs ovr / ponts from to -. ast orr Transform Snc ω ω and ω ω vn ω odd vn ω odd w can wrt W can compt an -pont DT b:. Comptng vn and odd for from..-,. Addng thm to obtan for m from..-. Total nmbr of rqrd mltplcatons s T n T n wth n n n n log / log C CODE OR THE D CASE DT vs T Comptatonal Complt n th D Cas f π / Ordnar orr Transform : O complt ast orr Transform : O log complt 6

7 7 -D T Complt / / / /,,, f f π ν π ν π ν W can compt a two-dmnsonal T b. prformng a on-dmnsonal T for ach colmn of f,,. prformng a on-dmnsonal T for ach row on th rsltng vals. Ths rqrs a total of on dmnsonal transforms complt log O ltrs A blac bo transformng an mag Lnar ltrs: Dfnton Dos not dpnd on mag locaton IJIJ I I How to dfn sch a ltr? Wth th mpls rspons. O L L O O L L O Convolton Wghtd pl sm wthn a nghborhood Convolton oprator: I**H ** Orgnal Imag I Comptd Pl Convolton Krnl as Pls Rspons Smoothng b Avragng Constant Krnl Convolton rnl..... whr :,, H I H I R v v v j j v j v v j

8 Transfr ncton Convolton and orr Transform A convolton n spatal doman s a mltplcaton n orr doman - D Convolton :p t *p p τ p t t τ dτ [ ] ϖt p t*p t p τ p t τ dτ dt ** ϖt p t τ dt p τ dτ ω τ ω t p p dτ ω τ ω t p d p p ω ω p τ Gassan Smoothng A Gassan T s a Gassan σ σ g, p / σ πσ 6363 Gassan Krnl Its orr Transform Gassan Blr VS Avragng Convolton & orr T can compt a convolton: It s asr to ndrstand a convolton rnl n frqnc doman 8

TEXTURE CLASSIICATIO Tmplat atchng o r s t d l d s P o n d s V l l a g W a t r

USIG A GAUSSIA O VARIACE TO SOOTH Problm: Hgh frqncs lad to trobl wth samplng.

9 TEXTURE CLASSIICATIO Tmplat atchng o r s t d l d s P o n d s V l l a g W a t r SUBSAPLIG ARTIACTS SAPLIG WITHOUT SOOTHIG Partclarl notcabl n hgh frqnc aras, sch as on th har. Imags sampld at vr othr pl agntd spctra of ths mags SOOTHIG AS LOW-PASS ILTERIG SAPLIG USIG A GAUSSIA O VARIACE TO SOOTH Problm: Hgh frqncs lad to trobl wth samplng. Solton: Spprss hgh frqncs bfor samplng b. mltplng DT of th sgnal wth somthng that spprsss hgh frqncs. convolvng wth a low-pass fltr Imags sampld at vr othr pl agntd spctra of ths mags 9

LOSS O DETAILS BUT OT ARTIACTS o alasng bt

orr Transform n Short Comptaton: Wth ast orr

comptaton Lnar ltrs dsgn Corrlaton: tmplat

Convolton Consdr th followng mas : What wold

10 LOSS O DETAILS BUT OT ARTIACTS o alasng bt dtals ar lost as hgh frqncs ar progrssvl rmovd. orr Transform n Short Comptaton: Wth ast orr Transform Complt: O log Applcatons: Convolton comptaton Lnar ltrs dsgn Corrlaton: tmplat matchng Ttr Classfcaton Ercss: Whch s whch? Convolton Consdr th followng mas : What wold gv convolvng wth A constant wht mag? An mag wth onl horzontal lns? A blac mag, cpt a sngl wht pl? An blac mag, cpt a 5 b 5 wht sqar? or Ercss Yo can tr n ImagJ: Load an mag Dplcat t Procss/ltr/Gassan Blr Procss/T/T or Invrs T Compar orgnal and blrd T

The Fourier Transform

The Fourier Transform /9/ Th ourr Transform Jan Baptst Josph ourr 768-83 Effcnt Data Rprsntaton Data can b rprsntd n many ways. Advantag usng an approprat rprsntaton. Eampls: osy ponts along a ln Color spac rd/grn/blu v.s.