Discrete Fourier Transform. Ajit Rajwade CS 663

Transcription

1 Discrete orier Trasform Ajit Rajwade CS 663

2 orier Trasform Yo have so far stdied the orier trasform of a 1D or D cotios aalog fctio. The fctios we deal with i practical sigal or image processig are however discrete. We eed a aalog of the orier trasform of sch discrete sigals.

3 Towards the Discrete orier Trasform Give a cotios sigal ft, we will cosider sigal gt, obtaied by samplig ft at eqally-spaced discrete istats. So gt is the poitwise prodct of ft with a implse trai s ΔT t give by: s T t t g t f t s T t T The implse fctios here are the Kroecker delta fctios ot the Dirac delta fctios.

4 Towards the Discrete orier Trasform T T T S S t s t f t g G / 1 where * T T dt T t t T dt t S t G / 1 / 1 This shows that the orier trasform Gμ of the sampled sigal gt is a ifiite periodic seqece of copies of μ which is the orier trasform of ft. The periodicity of Gμ is give by the samplig period ΔT. Also thogh gt is a sampled fctio, its orier trasform is cotios. Check the book for the derivatio for this

5 Gμ Gμ Gμ

6 Towards the Discrete orier Trasform Let s ow try to derive Gμ directly i terms of gt istead of μ. G g t f f t t f t t exp T exp T exp jt g texp jt dt jt dt jt dt

7 Towards the Discrete orier Trasform We have see earlier that the orier trasform of gt is periodic with period ΔT, so it is of iterest to characterize oe period. Cosider that we take some eqally spaced samples of Gμ over oe period from μ = 0 to 1/ΔT. This gives s the form μ = /ΔT where = 0,1,,-1.

8 Towards the Discrete orier Trasform Plggig these vale for μ we ow have: G d 1 0 f exp j / This is called the Discrete orier Trasform of the seqece {f } where = 0,1,,-1. Give the seqece { d } where = 0,1, -1, we ca recover {f } sig: f f d exp j / This is the iverse discrete forier trasform IDT.

9 Cosider: G f Towards the Discrete orier d d 1 0 Trasform f exp j / exp j / It ca be proved that plggig i the expressio for f ito the expressio for d yields the idetity d = d. Also plggig i the expressio for d ito the expressio for f yields the idetity f = f.

10 Towards the Discrete orier Trasform I some books, the followig expressios are sed: d f exp j / 1 1 f d exp j / 0 Note that the above expressios ca be writte i the followig matrix vector format: f U Vectors of size by 1 d U d T f U trs ot to be a orthoormal by matrix called the discrete orier basis matrix or DT matrix. The sqare root of i the deomiator is reqired for U to be orthoormal, else it wold be proportioal to a orthoormal matrix.

11 domai real complex domai, ;,,, 1 / 1 1 / 1 1 / 0 11 / 11 / 01 / 1 0 / 1 0 / 0 0 / 1 / 1 11 / 1 0 / 1 / 1 11 / 1 0 / 1 / 0 01 / 0 0 / e e e e e e e e e e e e e e e e e e f f f d j j j d j j j d j j j d d d j j j j j j j j j d d R C C C R U f U f f U U f d d T Vectors of size by 1 U is a orthoormal by matrix called the discrete orier basis matrix or DT matrix. The sqare root of is reqired for U to be orthoormal, else it wold be proportioal to a orthoormal matrix. Notice i the last eqality how the sigal f is beig represeted as a liear combiatio of colm vectors of the DT matrix. The coefficiets of the liear combiatio are the discrete orier coefficiets!

12 Samplig i time ad freqecy Remember that the fctio gt was created by samplig ft with a period of ΔT. Ad the spacig betwee the samples i the freqecy domai to get the DT from Gμ is 1/ ΔT sice μ = /ΔT where = 0,1,,-1. Likewise the rage of freqecies spaed by the DT is also iversely proportioal to ΔT.

13 DT properties Liearity: af+bg = af + bg Periodicity: = + k for iteger k, Ad so is the iverse DT sice f = f+k. The DT is i geeral complex. Hece it has a magitde ad a phase.

14 Clarificatio abot DT We have see earlier that the orier trasform Gμ of the sampled versio gt of aalog sigal ft is cotios. We also saw earlier that we take some eqally spaced samples of Gμ over oe period from μ = 0 to 1/ΔT. This way the DT ad the discrete sigal both were vectors of elemets. Obtaiig the DT give the sigal or the sigal give its DT is a efficiet operatio owig to the orthoormality of the discrete forier matrix why efficiet? Becase for a orthoormal matrix, iverse = traspose. Why do t we take more tha samples i the freqecy domai? If we did, the aforemetioed comptatioal efficiecy wold be lost. The iverse trasform wold reqire a matrix psedo-iverse which is costly. Also the colms of the orthoormal matrix U T size x costitte a basis: i.e. ay vector i -dimesioal space ca be iqely represeted sig liear combiatio of the colms of that matrix. If yo took more tha samples i the freqecy domai, that iqeess wold be lost as U T wold ow have size x where >.

15 Covoltio of discrete sigals Discrete eqivalet of the covoltio is: f * h 1 m0 f m h m De to the periodic atre of the DT ad IDT, the covoltio will also be periodic. The above formla represets oe period of that periodic resltat sigal.

16 Covoltio of discrete sigals The covoltio theorem from cotios sigals has a extesio to the discrete case as well: f * h H Therefore discrete covoltio ca be implemeted sig prodct of DTs followed by a IDT. Bt i doig so, oe has to accot for periodicity isses to avoid wrap-arod artifacts see ext slide.

17 Covoltio of discrete sigals Cosider the discrete covoltio: f * h 1 l0 f l h l To covolve f with h, yo eed to 1 flip h abot the origi, traslate the flipped sigal by a amot, ad 3 compte the sm i the above formla. Steps ad 3 are repeated for each vale of. The variable rages over all itegers reqired to completely slide h over f. If h ad f have size, the has to rage p to -1.

18 Covoltio of discrete sigals I other words, the resltat sigal mst have legth of -1. The covoltio ca be implemeted i the time/spatial domai ad ATLAB has a rotie called cov which does the job for yo! Now imagie yo tried to implemet the covoltio sig a DT of -poit sigals followed by a -poit IDT. Owig to the assmed periodicity, yo wold get a desirable wrap-arod effect. See ext slide for a illstratio.

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20 Covoltio of discrete sigals How does oe deal with this codrm? If f has legth ad h has legth K, the yo shold zero-pad both seqeces so that they ow have legth at least +K-1. The compte +K-1 poit DT for both, mltiply the DTs ad compte the +K-1 poit IDT.

21 Covoltio of discrete sigals: methods Cosider yo wat to covolve sigal f havig N elemets with sigal h havig K elemets. Yo ca se the cov rotie i ATLAB which by defalt prodces a sigal of N+K-1 elemets for yo. It takes care of zero-paddig iterally for yo. The other eqivalet alterative is to: Apped f with K-1 zeros. Apped h with N-1 zeros. Compte the N+K-1 poit DT of f ad h sig fft. ltiply the two DTs poit-wise. Compte the IDT of the reslt this gives yo a sigal with N+K-1 elemets. I both cases, yo may wish to extract the first N elemets of the resltat sigal ote that the trailig K-1 elemets may ot be zeros!.

22 Why covoltio sig orier trasforms? The time complexity of comptig the covoltio of a sigal of legth with aother of legth is O. With a DT compted aively, it wold remai O the complexity of mltiplyig a x matrix with a x 1 vector. f U d U d T f

23 Why covoltio sig orier trasforms? Bt there s a smarter way of doig the same which comptes the DT i O log time. That is called the ast orier Trasform, discovered by Cooley ad Tkey i It is based o a divide ad coqer algorithmic strategy. The same strategy ca be sed to compte the IDT i O log time. Hece covoltio ca ow be compted i O log time.

24 Towards the ast orier Trasform / exp where / exp j W W f j f K Let K K K K K K W f W f W f Eve idices Odd idices K K K K K W W f W f K K W W

25 Towards the ast orier Trasform K K K K K W W f W f K K K K K K K K odd eve K odd eve K K odd K K eve W K K j W W W W K W K K W W f W f / exp as 1 0,1,..., for ad 1 0,1,..., for, Defie The -poit DT comptatio for is split p ito two halves. The first half reqires two /-poit DT comptatios oe for eve, oe for odd. The secod half follows directly withot ay additioal trasform evalatios oce eve ad odd are compted. Now eve ad odd ca be frther split p recrsively.

26 Towards the ast orier Trasform The -poit DT comptatio for is split p ito two halves. The first half reqires two /-poit DT comptatios oe for eve, oe for odd. The secod half follows directly withot ay additioal trasform evalatios oce eve ad odd are compted. Now eve ad odd ca be frther split p recrsively. This gives: T 1 c costat T T / T... / 4 / T / T k T / k k O log There is also a fast iverse orier trasform which works qite similarly i O log time. The speedp achieved by the fast forier trasform over a aïve DT comptatio is rather dramatic for large!

27 ATLAB implemetatio The ast orier Trasform is implemeted i ATLAB directly there are the roties fft ad ifft for the iverse.

28 D-DT Give a D discrete sigal image fx,y of size W 1 by W, its DT is give as: d W1 1 1W 1, v f x, yexp j x / W1 vy/ W WW 1 x0 y0 f W1 1 1W 1 x, y, vexp j x / W1 vy/ W WW 1 0 v0

29 D-DT comptatio d Give a D discrete sigal image fx,y of size W 1 by W, its DT is give as: W1 1 1W 1, v f x, yexp jvy/ W exp jx / W1 WW 1 x0 y0 Compte row-wise T, followed by a colm-wise T. Overall complexity is W 1 W logw 1 W. f W1 1 1W 1 x, y, vexp j x / W1 vy/ W WW 1 0 v0

30 ATLAB implemetatio The T for D arrays or images is implemeted i ATLAB roties fft ad ifft.

31 D-DT properties Liearity: af+bg,v = af,v + bg,v where a ad b are scalars. Periodicity:,v = + k 1 W 1,v =,v + k W = + k 1 W 1,v + k W for iteger k 1 ad k. Ad so is the iverse DT sice fx,y = fx + k 1 W 1,y = fx,y + k W = fx + k 1 W 1,y + k W for iteger k 1 ad k.

32 D-DT: agitde ad phase The phase carries very critical iformatio. If yo retai the magitde of the orier trasform of a image, bt chage its phase, the appearace drastically chages.

33 agitde of T of Salma Kha, ad phase of P V Sidh agitde of T of P V Sidh, ad phase of Salma Kha

34 D-DT properties Traslatio: Note that traslatio of a sigal does ot chage the magitde of the orier trasform oly its phase. / / exp,, / / exp,,, W yv W x j y x f v v W vy W x j v v y y x x f

35 D-DT properties The orier trasform of a rotated image yields a rotated versio of its orier trasform If yo cosider coversio to polar coordiates, the we have: cos si, si cos, cos si, si cos,,, v v v y x y x f v v y x f si, cos si, cos v r y r x si, cos, si, cos,,, v r r f v v y x f

36 D-DT visalizatio i ATLAB I the formla for the DT, the freqecy rage from = 0 to = -1 actally represets two half periods back to back meetig at / see ext slide. It is more coveiet to istead view a complete period of the DT istead. or that prpose we visalize -/ istead of i 1D, ad -W 1 /,v-w / istead of,v i D. Thereby freqecy = 0 or = v = 0 ow occrs at the ceter of the displayed orier trasform. Note that -W 1 /,v-w / = fx,y-1 x+y,v, so the ceterig operatio is easy to implemet.

37 Oe fll period Two half periods back to back

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39 D-DT visalizatio i ATLAB To visalize the DT of a image, we visalize the magitde of the DT. The orier trasform is first cetered as metioed o the previos slide. De to the large rage of magitde vales, the DT magitdes are visalized o a logarithmic scale, i.e. we view log,v +1 where the 1 is added for stability.

40 Withot ceterig the fft fim = fftim; absfim = logabsfim+1; imshowabsfim1,[-1 18]; colormap jet; colorbar; The 0,0 freqecy lies at the top left corer v With ceterig the fft fim = fftshiftfftim; absfim = logabsfim+1; imshowabsfim1,[-1 18]; colormap jet; colorbar; v The 0,0 freqecy lies i the Ceter

41 Viewig the rotatio property of the DT

42 D covoltio I ATLAB, D covoltio ca be implemeted sig the rotie cov. This ca be very expesive if the sigals yo wish to covolve aother with, are of large size. Hece oe resorts to the covoltio theorem which holds i D as well.

43 D covoltio The covoltio theorem applies to D-DTs as well: f * h, v, v H, v Cosider a image f of size W 1 x W which yo wat to covolve with a kerel k of size K 1 x K sig the DT method. The yo shold symmetrically zero-pad f ad k so that they acqire size W 1 +K 1-1 x W +K -1. Compte the DTs of the zero-padded images sig the T algorithm, poit-wise mltiply them ad obtai the IDT of the reslt. Extract the cetral W 1 x W portio of the reslt for the fial aswer.

44 Image ilterig: reqecy domai Yo have stdied image filters of varios types: mea filter, Gassia filter, bilateral filter, patchbased filter. The former two are liear filters ad the latter two are t. Liear filters are represeted sig covoltios ad hece have a freqecy domai iterpretatio as see o the previos slides.

45 Image ilterig: reqecy domai Hece sch filters ca also be desiged i the freqecy domai. I certai applicatios, it is i fact more ititive to desig filters directly i the freqecy domai. Why? Becase yo get to desig directly which freqecy compoets to weake/elimiate ad which oes to boost, ad by how mch.

46 Low pass filters Edges ad fie textres i images cotribte sigificatly to the high freqecy cotet of a image. Whe yo smooth/blr a image, these edges ad textres are weakeed or removed. Sch filters allow oly the low freqecies to remai itact ad are called as low pass filters. I the freqecy domai, a ideal low pass filter ca be represeted as follows: H, v 1, if 0 otherwise v D Note: we are assmig 0,0 to be the ceter lowest freqecy. reqecies otside a radis of D from the ceter freqecy are elimiated.

47 Low pass filters

48 Low pass filters To apply sch a filter to a image f to create a filtered image g, we do as follows: H, v 1, if 0 otherwise G, v v, v H, v D D is a desig parameter of the filter ofte called the ctoff freqecy. This is called the ideal low pass filter as it completely elimiates freqecies otside the chose radis.

49 Notice the rigig artifacts arod the edges! Low pass filters

50 Low pass filters The rigig artifacts ca be explaied by the covoltio theorem. The correspodig spatial domai filter is called the jic fctio. The jic is a D ad circlarly symmetric versio of the sic fctio see also the ext slide. Ay cross sectio of the jic is basically a sic fctio. wiki/sombrero_fctio

51 A image ca be regarded as a weighted sm of Kroecker delta fctios implses. Covolvig with a sic fctio meas placig copies of the sic fctio at each implse. The cetral lobe of the sic fctio cases the blrrig ad the smaller lobes give rise to the ripple artifacts or rigig.

52 Low pass filters: other types These rigig artifacts ca be qite desirable. Hece the ideal low pass filter is replaced by other types of low pass filters which weake bt do ot totally elimiate the higher freqecies. or example: H, v 1 1 v / D,, D filter parameters Btterworth filter: D = ctoff freqecy, = order of the filter H, v exp v /, filter parameters Gassia filter

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54 Low pass filters: Btterworth As D icreases, the Btterworth filter allows more freqecies to pass throgh withot weakeig see previos slide. As the order icreases, the Btterworth filter weakes the higher freqecies more aggressively why? which actally icreases rigig artifacts see previos slide.

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56 Low pass filters: Gassia The spatial domai represetatio for the Gassia LP is basically a Gassia. I fact, the followig are orier trasform pairs: g 1 x x exp G exp 1/ Notice from the eqatios that a spatial domai Gassia with high stadard deviatio correspods to a freqecy domai Gassia with low ct off freqecy! The extreme case of this is a costat itesity sigal Gassia with very high stadard deviatio ifiite as sch whose orier trasform is a implse at the origi of the reqecy plae. Aother extreme case is a spatial domai sigal which is jst a implse its orier trasform has costat amplitde everywhere.

57 High pass filter It is a filter that allows high freqecies ad elimiates or weakes the lower freqecies. The eqatio for the freqecy respose of a ideal high pass filter is give as: H, v 0, if v D 1otherwise I fact, give a LP, a HP ca be costrcted from it sig: H HP, v 1 H, v LP

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60 Rigig artifacts for a Ideal HP! Rigig artifacts goe!

61 Rigig artifacts for a Ideal HP! Rigig artifacts goe!

62 High pass filters Note that ay gradiet-based operatio o images is essetially a type of high-pass filter. This icldes first order derivative filters or the Laplacia filter. Why? Cosider the first order derivative filter i y directio. g x, y G, v I x, y 1 I x, y I, v1 e jv / G,0 0

63 High pass filters Cosider the Laplacia filter:, 4 4, 4 /, 4 1, 1, 1, 1,, / / / / v I e e e e v G y x I y x I y x I y x I y x I y x g j v j j v j 0 0,0 G

64 Boostig high freqecies I some applicatios, yo either wish to weake the higher freqecies or the lower freqecies. or example, i image sharpeig applicatios yo wish to boost the edges or textres basically higher freqecies. I that case yo wish to perform operatios of the followig form: g x, y 1 1 kh HP, v, v

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66 Notch filters There are applicatios where images are preseted with periodic oise patters e.g. images scaed from old ewspapers see ext slide. Usally, the amplitdes of the orier compoets of atral images are see to decay with freqecy. The orier trasform of images with periodic patters show atral peaks like ordiary atral images. Sch images ca be restored sig otch filters, which basically weake or elimiate these atral freqecy compoets or ay freqecy compoets specified by the ser.

67 The yellow ellipses idicate atral peaks i the orier trasform of the image with the sperimposed iterferig patter. These are atral becase typical images do ot have sch orier trasforms. Therefore oe ca apply a otch filter to remove the iterferig patter.

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69 Cortesy:

70 Cortesy:

71 Notch filters The expressio for a otch filter freqecy respose is give as follows Q = mber of freqecy compoets to sppress:,, ;,, 1 R v v H v H i i Q i N NR 1otherwise 0 if,, ;, R v v R v v H i i i i NR Ideal otch reject filter / exp 1,, ;, R v v R v v H i i i i NR Gassia otch reject filter

72 Algorithm for freqecy domai filterig Cosider yo have to filter a image f of size H x W sig oe of the filters described i these slides. Zero pad the image symmetrically to create a ew image f of size H x W. Compte the orier trasform of f ad ceter it sig fftshift. Let the orier trasform be,v. Desig a freqecy domai filter H,v of size H x W with the zero freqecy at idex H,W of the D filter array. Compte the prodct,vh,v. Compte the iverse orier trasform of the prodct after applyig ifftshift to do the effect of fftshift. Cosider oly the real part of the iverse orier trasform ad extract the cetral portio of size H x W. That gives yo the fial filtered image. Note that the zero paddig is importat. To derstad the differece, see the reslts o the ext slide with ad withot zero-paddig.

73 Origial image Effect of Gassia LP with appropriate zero paddig Effect of Gassia LP withot appropriate zero paddig zero the border artifacts!

74 Iterpretig the DT of Natral Images I a few cases, oe ca gess some strctral properties of a image by lookig at its orier trasform. ost atral images have stroger low freqecy compoets as compared to higher freqecy compoets. This is geerally tre, althogh its ot a strictly mootoic relatioship. or example, how does the DT of a vertical edge image look like?

75 There is a sic fctio alog the axis!

76 Strog horizotal ad vertical edges = strog resposes alog ad v axes i the orier plae! A strog edge i directio θ i XY plae = a strog respose i the directio perpediclar to θ i UV plae

77 Iterpretig the DT or a image or ay D array with orier trasform,v, 0,0 is proportioal to the average vale of the image itesity why? The DT of a o-zero costat itesity image is basically a implse at the zero freqecy. What is the height of the implse?

78 DT limitatios A strog respose at freqecy,v i.e. a large magitde for,v idicates the presece of a sisoid with freqecy,v ad with large amplitde somewhere i the image. I geeral, the orier trasform caot ad does ot tell s where i the image the sisoid is located read sectio 1 of or that, yo eed to compte separate orier trasforms over smaller regios of the image Short time orier trasform that will tell yo which regios cotaied a particlar sisoidal compoet.

79 with orier: Hybrid images Hybrid images are a sperpositio of a image J 1 with strog low freqecy compoets ad weak higher freqecy compoets, ad a image J with stroger higher freqecy compoets ad weak lower freqecy compoets. J 1 = apply LP with some ctoff freqecy D o a image. J = apply HP with some ctoff freqecy D o aother image. The appearace of the hybrid image chages with viewig distace!

80 Tiger whe see p-close, Cheetah whe see from far Eistei from earby, arily oroe from far away or if yo sqit! Refer to: pdf paper from a graphics joral!