CS 548: Computer Vision Image Transformation: 2D Fourier Transform and Sampling Theory. Spring 2016 Dr. Michael J. Reale

Size: px

Start display at page:

Download "CS 548: Computer Vision Image Transformation: 2D Fourier Transform and Sampling Theory. Spring 2016 Dr. Michael J. Reale"

Irene Wilson
6 years ago
Views:

1 CS 548: Comptr Vision Imag Transformation: D Forir Transform and Sampling Thory Spring 016 Dr. Michal J. Ral

2 FOURIER TRANSFORM OF SAMPLED FUNCTION EXAMPLE

3 Sampling Exampl Say w hav th fnction blow, ft, with 4 sampls takn ΔT = 1 apart Imags from Gonzalz-Woods Digital Imag Procssing

4 Compt Forir Transform Lt s gt 4 sampls of th Forir Transform = 0: = 1: 3 F0 F1 x0 f x x f x / f 0 1 x / 4 4 F f 1 4 M 1 x0 f 3 / f x f 3 x/ M Rmmbr: W can s Elr s formla hr 3 = : = 3: F 1 0 F 3 3 Not: W s ALL th vals of ft to compt on val of F

5 Compting Invrs Forir Transform Taking for sampls of F, rtriv sampls of ft 1 0 / 1 M M x F M x f F F f x = 0: Th vals for f1, f, and f3 can b comptd th sam way

6 D FOURIER TRANSFORM

7 Rviw: 1D Continos Forir Transform Pair Forward: F f t t dt Invrs: f t F t d

8 Rviw: 1D Discrt Forir Transform Pair Forward: Invrs: 1 0 / M x M x x f F 1 0 / 1 M M x F M x f = 0, 1,,, M-1 x = 0, 1,,, M-1

9 Rviw: Elr s Formla W can rplac th componnt with Elr s Formla: cos sin which givs s: x cos M sin x/ M x M

10 D Implss A D continos impls is dfind by: if t z 0 t, z 0 othrwis whr t and z ar continos variabls W also hav this constraint: t, z dtdz 1

11 D Impls Sifting Proprty Th D impls also has th sifting proprty: f t, z t, z dtdz f 0,0 And, for an impls at t 0, z 0 : f t, z t t, z z dtdz f t, z

12 Discrt D Implss W also hav th discrt vrsion of a D impls, givn by: x, y 1 0 if x y 0 othrwis

13 Discrt D Impls Sifting Proprty Th discrt vrsion also has th sifting proprty: x y f x, y x, y f 0,0 And gnrally for implss at x 0, y 0 : x y f x, y x x0, y y0 f x0, y0

14 D Continos Forir Transform Pair Forward: F, f t, z t z dtdz Invrs: f t, z F, t z dd t,z continos spatial domain μ,ν continos frqncy domain

15 D Discrt Forir Transform Pair Forward: whr M = width, N = hight Invrs: Sam M and N from Forward Transform x,y discrt spatial domain,v discrt frqncy domain / /,, M x N y N vy M x y x f v F / /, 1, M N v N vy M x v F MN y x f

16 SAMPLING THEORY AND ALIASING

17 Propr Introdction to Sampling Lt s assm w hav a 1D continos fnction ft Lt s also say th Forir transform of this fnction is Fμ Undr what conditions can w niqly rcovr ft from Fμ if w startd with sampls of ft?

18 Band-Limitd Fnction If Fμ = 0 otsid of finit intrval [-μ max, μ max ], thn th original fnction ft is a band-limitd fnction [-μ max, μ max ] band of frqncis Imags from Gonzalz-Woods Digital Imag Procssing

19 Fnctions and Transforms f t original fnction ~ f t sampld fnction F Forir transform of f t ~ F Forir transform of ~ f t

20 Forir Transform of Sampld Fnction Rmmbr th transform of a sampld fnction: ~ F 1 T n F Basically, w gt an infinit, priodic sqnc of copis of Val of 1/ΔT dtrmins sparation btwn copis n T F Imags from Gonzalz-Woods Digital Imag Procssing

21 Rcovring ft To rcovr ft, w nd to gt Fμ To gt Fμ, w nd to pll ot a clan ~ copy of Fμ from F I.., nd on complt priod

22 Problm with Sampling Sampling distanc ΔT gos p distanc 1/ΔT btwn priods gos down I.., sampls farthr apart priods of transforms closr togthr Bcas ft is band-limitd, complt priod is μ max in siz If 1/ΔT is too small, priods will start to ovrlap!

23 Th Sampling Thorm Sampling Thorm To compltly rcovr a continos, bandlimitd fnction from a st of sampls, yo mst sampl at a rat gratr than th Nyqist Rat μ max 1 T max μ max = maximm frqncy ΔT = sampling distanc

24 Ovr, Critically, and Undr Sampld Ovr sampld Critically sampld Undr sampld Imags from Gonzalz-Woods Digital Imag Procssing

25 Aliasing Rvisitd Aliasing Whn highr frqncis ovrlap with lowr frqncis, corrpting th signal Alias = fals idntity high frqncis prtnding to b low frqncis Casd by ndr-sampling sampling at lss than th Nyqist rat Rslts in a varity of imag artifacts blockinss, aggd lins, fals highlights, tc.

26 How to Rcovr th Original Fnction Assming priods sparatd nogh: F H ~ F whr: H T 0 max max othrwis Thn yo can s th rcovrd Fμ in th invrs Forir Transform: f t F t d

27 How to Rcovr th Original Fnction Imags from Gonzalz-Woods Digital Imag Procssing

28 Can W Compltly Avoid Aliasing with Imag Data? Alas, no.

29 Why Not? With imag data, w ar daling with a finit nmbr of sampls. In othr words, or sampld fnction is limitd in dration Limitd dration hr, in spatial domain Band-limitd in frqncy domain Any tim w limit th dration of th fnction, w introdc infinit frqncy componnts in th frqncy domain.

30 Limiting th Dration of a Fnction If w want to limit th dration of a fnction ft, w can mltiply that fnction by a fnction ht: h t t T othrwis

31 A Similar Fnction This fnction looks a lot lik this simpl fnction th Boxcar fnction: Imags from Gonzalz-Woods Digital Imag Procssing

32 Gtting th Forir Transform of a Boxcar Fnction sin / / / / W W AW A A A dt A dt t f F W W W W W W t W W t t t kt kt t k k d k d k dt dt dt d k kdt d kt k sbstittion in a hrry: sin bcas:

33 Th Forir Transform of a Boxcar Fnction sin W F AW AWsinc W W Fμ gos off to infinity in both dirctions: Imags from Gonzalz-Woods Digital Imag Procssing

34 Bringing Convoltion Back Rmmbr: h t f t H F Hμ xtnds to infinity, and w r sliding it across Fμ convoltion THUS: No fnction of finit dration spatial can b band-limitd frqncy

35 Always Hav Aliasing with Dration- Limitd Fnctions Finit dration not band-limitd cannot st pll ot on priod of Fμ to charactriz whol fnction

36 Rdcing Aliasing Smooth ft BEFORE sampling Attnats highr frqncis not as prominnt in Fμ Procss known as anti-aliasing NOTE: In graphics, anti-aliasing is ffctivly blrring aftr sampling sd to covr p artifacts

37 Exampl of 1D Aliasing ΔT = 1, bt sparation of sampls > Givs fals imprssion of anothr sin wav! Imags from Gonzalz-Woods Digital Imag Procssing

38 D Sampling Thorm Sam as th 1D Thorm, only now thr ar two rqird sampling rats on for ach dimnsion: 1 T max 1 Z max whr: ΔT and ΔZ = sparation btwn sampls μ max and ν max = max. frqncis in ach dimnsion

39 Two Typs of Imag Aliasing Spatial Aliasing Singl imag aliasing ffcts i.., aggd dgs, blocky apparanc, tc. Tmporal Aliasing Fram rat tmporal sampling rat too low E.g., wagon whl ffct

40 D Aliasing Exampl Sid Lngth of Sqars in pixls = p p = 16 p = 6 Imags from Gonzalz-Woods Digital Imag Procssing p = p = looks normal, bt isn t!

Basd on Imag:MoirGrid.png.. Licnsd ndr Pblic domain via Wikimdia Commons - http://commons.

41 Moiré Pattrns Moiré Pattrns Can happn whn sampling scns with rpating priodic or narly priodic componnts Mor gnral than sampling artifacts "Moiré grid" by Fibonacci talk - Own work. Basd on Imag:MoirGrid.png.. Licnsd ndr Pblic domain via Wikimdia Commons - il:moir%c3%a9_grid.svg

42 FOURIER SPECTRUM AND PHASE ANGLE

43 Translation Th Forir Transform has th following translation proprtis:,, 0 0 / / 0 0 v v F y x f N y v M x / / ,, N v y M x v F y y x x f

44 Changing Offst of Transform Data Sinc 4 complt priods mt in middl, oftn want to shift th data ovr so that yo gt on complt priod 0 = M/ and v 0 = N/ 4 back-to-back Priods mt hr

45 Changing Offst of Transform Data So: / / / / / / 0 0 1, sin cos,,,,,, 0 0 y x y x y x y x N Ny M Mx N y v M x y x f y x f y x f y x f y x f y x f v v F

Forir Transform in Polar Form Sinc D DFT is complx, can xprss in polar form: whr: http://i.tlgraph.co.k/mltimdia/archiv/01786/polar-bar-cb_1786691i.

46 Forir Transform in Polar Form Sinc D DFT is complx, can xprss in polar form: whr: v v F v F,,, v I v R v F,, arctan, v R v I v Forir or frqncy spctrm Magnitd Phas Angl

47 Forir Spctrm and Phas Angl Forir Spctrm Insnsitiv to imag translation! It is affctd by rotation, howvr Phas Angl Givs littl intitiv information Contains VERY important shap information!

48 Powr Spctrm Powr spctrm: P, v F, v R, v I, v

49 DC Componnt If w look at th val for F,v whn both frqncy componnts ar zro, w can s that it is proportional to th avrag val of fx,y: F0,0 M 1 N 1 x0 y0 f x, y MN 1 MN M 1 N 1 x0 y0 f x, y MN f x, y Which mans: F 0,0 MN f x, y F0,0 is somtims calld th DC componnt DC crrnt zro frqncy Usally vry larg Which is why w wold want to s an intnsity transform, lik th log transform

50 DFT IN OPENCV

51 DFT in OpnCV An xcllnt xplanation of how to s OpnCV s DFT fnctionality can b fond hr: rt_forir_transform/discrt_forir_tran sform.html Th following slids ar basd on th information from th abov link

52 Stp Lt s say w hav an imag I. For prformanc rasons, th imag shold b incrasd in siz and paddd with zros: Mat paddd; int m = gtoptimaldftsiz I.rows ; int n = gtoptimaldftsiz I.cols ; // on th bordr add zro pixls copymakbordri, paddd, 0, m - I.rows, 0, n - I.cols, BORDER_CONSTANT, Scalar::all0;

53 Mak Inpt into Complx Nmbrs Sinc w will b doing th DFT in-plac, w nd to trn or inpt imag paddd into a plan of complx nmbrs: Mat plans[] = {Mat_<float>paddd, Mat::zrospaddd.siz, CV_3F}; Mat complxi; mrgplans,, complxi;

54 Call DFT! complxi will b ovrriddn with F vals: dftcomplxi, complxi;

55 Split into Sparat Plans and Gt Magnitd and Phas Angl splitcomplxi, plans; plans[0] = RalF plans[1] = ImaginaryF To gt th magnitd and phas angl: Mat phasangl; Mat magi; carttopolarplans[0], plans[1], magi, phasangl;

56 Us Logarithmic Intnsity Transform for Display Prposs Us Log Intnsity Transform: magi += Scalar::all1; logmagi, magi; Crop th spctrm, if it has an odd nmbr of rows or colmns magi = magirct0, 0, magi.cols & -, magi.rows & -;

57 R-arrang Vals to Adst Origin of Magnitd Imag int cx = magi.cols/; int cy = magi.rows/; Mat q0magi, Rct0, 0, cx, cy; // Top-Lft - Crat a ROI pr qadrant Mat q1magi, Rctcx, 0, cx, cy; // Top-Right Mat qmagi, Rct0, cy, cx, cy; // Bottom-Lft Mat q3magi, Rctcx, cy, cx, cy; // Bottom-Right Mat tmp; q0.copytotmp; q3.copytoq0; tmp.copytoq3; // swap qadrants Top-Lft with Bottom-Right q1.copytotmp; q.copytoq1; tmp.copytoq; // swap qadrant Top-Right with Bottom-Lft

58 Normaliz Magnitd and Phas Angl Imags for Display Prposs normalizmagi, magi, 0, 1, cv::norm_minmax; normalizphasangl, phasangl, 0, 1, cv::norm_minmax;

59 Exampl: Simpl Box Inpt Imag Spctrm Magnitd Phas Angl

60 Exampl: Simpl Box Translatd Inpt Imag Spctrm Magnitd Phas Angl sam, bcas translation dosn t affct th spctrm

61 Exampl: Simpl Box Rotatd Inpt Imag Spctrm Magnitd Phas Angl

62 Anothr Exampl Inpt Imag Spctrm Magnitd Phas Angl

63 OpnCV: Invrs DFT idftinptarray src, OtptArray dst

64 Exampl: Rconstrctd with Magnitd = 1 Using phas angl only Shap information prsrvd

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

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