Series Expansion with Wavelets

Transcription

1 Series Expasio wih Waveles Advaced Sigal Processig 7 Berhard Reiisch Georg Teicheiser SERIES EXPANSION WITH WAVELETS... ADVANCED SIGNAL PROCESSING 7... INTRODUCTION... SIGNAL REPRESENTATION.... TIME DOMAIN REPRESENTATION.... FOURIERSERIES AND TRANSFORM SHORT TIME FOURIER TRANSFORM Tie ad requecy resoluio PIECEWISE FOURIER SERIES DESIRED FEATURES OF BASIS FUNCTIONS INTRODUCTION TO WAVELETS THE HAAR - WAVELET Orho-oraliy o he Haar Wavele Basis or L Space MULTI RESOLUTION ANALYSIS AXIOMATIC DEFINITION ORTHONORMAL COMPLEMENTS... 5 CONSTRUCTING THE SINC WAVELET... 6 ITERATED FILTER BANKS HAAR CASE SINC CASE ITERATED FILTER BANKS CONT WAVELET SERIES AND PROPERTIES LINEARITY SHIFT SCALING DYADIC SAMPLING AND TIME FREQUENCY TILING TIME LOCALIZATION FREQUENCY LOCALIZATION DECAY PROPERTIES MULTIDIMESIONAL WAVELETS... 8 PRACTICAL ASPECTS... 9 REFERENCES...

2 Iroducio The opic o his repor is Series Expasio wih Waveles. Waveles are a aily o basis ucios or sigals, eiher coiuous or discree i ie. Oe o he os ipora properies o waveles is he iie localizaio i ie ad requecy, as corary o ie represeaio o localizaio i requecy ad requecy represeaios o localizaio i ie. A he begiig he basics o sigal represeaio will be covered, soe exaples will be give ad heir properies cocerig ie ad requecy resoluio will be discussed. Based o his discussio, soe desired properies o basis ucios or sigals will be derived. The basic idea o waveles will he be discussed by usig he Haar Wavele he irs ad probably also he siples Wavele as a exaple. This will be doe, by prooig, ha he Haar Wavele is acually a basis or he space o ucios which are squared suable L space. I his sep, he basic idea o waveles, aely coarsely approxiaig he sigal ad he addig deails are preseed. This is called he Muliresoluio cocep, which properies will be highlighed, because o is iporace o he Wavele aalysis. The Haar Wavele isel is oe boudary case or he Wavele aalysis, where as he oher boudary is he Sic Wavele. The Sic - Wavele is also shorly iroduced ad is basic cosrucio seps are show. Sigal represeaio Sigals are pois i a vecor space, which has iiie diesios. As kow, oe eeds basis ucios which spa he vecor space o projec he sigal o. The choice o his basis ucios ad heir properies will be he ai opic o his repor. To recap, a sigal is give by k ϕ u, u k ϕ u, u ϕ k * ϕ u u du k k φ k are he basis ucios o which he sigal is projeced. Now a ew possibiliies o basis ucios ad heir properies will be discussed. Especially he iluece o he basis ucios o he localizaio i ie ad requecy. To clariy, we are alkig abou how well a coeicie o he sigal represeaio is localized i boh ie ad requecy or, i oher words, wha i ells us abou he requecy ad ie properies o he sigal.. Tie doai represeaio The ie doai represeaio is he represeaio oe obais whe easurig a sigal. I his represeaio he basis ucios are iiie shor ipulses: ϕ k δ kd ϕ [ ] δ k k As ca be easily see, he basis ucios are precisely localized i ie, bu do ell ayhig abou he requecy because you eed o look a how he sigal chages o say soehig abou he requecy.

3 . Fourierseries ad rasor This secio covers he diere ypes o Fourier expasio will be covered. Firs he Fourier Trasor will be discussed. The Fourier Trasor is oe o he os used ools i requecy aalysis. x[ ] π π π X e jω e jω dω Here he coposiio or a ie discree sigal is show. The ai drawback o his ehod o represeig sigals he bad liiaio o he sigals. The ore iuiive versio o requecy aalysis is he Fourier series. I his case he sigal does eed o be bad liied, bu i has o be periodic. The basis ucios i his case are iiie log sie ad cosie waves. k F[ k] e jπ k/ T This equaio shows he coposiio o a sigal, whe he coeicies o a Fourier series are give, ad uses he coplex oaio. To suarize: There are wo ehods or requecy aalysis. Bu each has is ow drawbacks. Whereas he Fourier rasor ca hadle sigals wih iie suppor, he sigals have o be bad liied. O he oher had he Fourier series ca represe sigals wih arbirary requecies, bu he sigals eed o be periodic, which eas hey eed o have iiie suppor. As ca easily see or he Fourier series expasio, he localizaio i ie is o exise because he basis ucios are deied ro - bis, bu he localizaio i requecy is iiiely sharp.

4 .3 Shor ie ourier rasor Up o ow we had represeaios o sigals which are eiher perecly localized i ie bu iiiely bad i requecy, or perecly localized i requecy bu iiiely bad i ie. Oe soehig i bewee is eeded. Oe coo proble is he deecio o chages i he specru. Whe hikig o a chage, orally he words beore ad aer coe o our ids, which iplies soe sor o ie resoluio. So o deec chages i he specru we eed a specru wih ie resoluio, e.g. he specru o he las secod o he ie sigal. The shor ie ourier rasor STFT is oe ool, which ca provide such ioraio. Whe applyig he STFT, he ie sigal will be widowed, ad a sadard Fourier Trasor will be copued over he widowed sigal. To ge STFT or he whole sigal, which resuls i a periodogra, he widow is shied over he ie sigal. I he op igure a sie sweep x[] ad he widow w[] are show, i he boo igure he resulig periodogra. O he x-axis is he ie idex ad he y-axis idicaes he requecy a ha ie..3. Tie ad requecy resoluio The aalysis o he ie ad requecy resoluio o he STFT is o ha easy as i he previous cases. As said above, ad as also see i he periodogra, he STFT is acually a ucio o ie ad requecy. So he quesio abou he accuracy o he STFT i ie ad requecy arises. The os ipora acor, which ilueces hese accuracies, is he shape o he widow ucio. I he widow is wider, oe ca copue a ore accurae Fourier rasor, i he widow is arrower, he ie localizaio is uch beer. Fro his iuiive approach i is easily see, ha oe ca o opiize boh requecy ad ie resoluio. Acually here exiss a equivale o he Heiseberg uceraiy priciple kow ro quau echaics. Whe akig a widow, oe ca copue is Fourier rasor ad urherore is disribuio oes i he ie ad requecy doai. For he discree

5 ie/requecy case he produc o he sadard deviaios has o be equal o or greaer ha a hal. σ ω σ Where σ ω is he sadard deviaio o he widow i requecy doai, ad σ is he sadard deviaio o he widow i ie doai. The value o ½ is jus reached, i oe uses a Gaussia widow. Oe oher ipora propery o he ie/requecy resoluio o he STFT is is equaliy over all requecies ad ie as see below. Whe alkig abou he ie resoluio, i is o course desired ha i is ivaria wih respec o he ie paraeer. O he oher had i is o always desired ha he requecy resoluio is idepede o he requecy. The adapaio o he requecy resoluio ad hece he ie resoluio oo o requecy is oe ajor advaage o he Wavele expasio..4 Piecewise Fourier series Aoher possibiliy o achieve iie ie ad requecy resoluio, would be o widow he ie doai sigal wih a recagle, coiue i periodically ad copue he Fourier series expasio o his ew sigal. Corary o he STFT, which copues he Fourier rasor o overlappig widows, his approach does copue reduda ioraio. Furherore oe ca represe arbirary sigals ro he L space, because he Fourier series expasio eables oe o represe bad uliied sigals, ad he way o cosrucig he sigal, which is goig o be used as ipu or he Fourier series expasio, guaraees ha here is always a periodic sigal, eve i he origial sigal is periodic. O he oher had here also soe ajor drawbacks: Because o he o-overlappig widows, he resul o his expasio is deeried by he legh ad he locaio o he widows. E.g. i oe was o represe a pure sie wih his ehod, he resul chages i he legh o he widow is o a uliply o he sie period ad he widow is placed o diere locaios o he origial sigal. This happes because o he discoiuiies a he widow borders.

6 .5 Desired Feaures o basis ucios Up o ow a ew ways o represeig sigals were iroduced. All o he have heir advaages ad drawbacks. Now he quesio arises, which properies are desirable ad which are o? Siple characerizaio I oe was o use sigal represeaios or pracical use, i is o course useul i i is siple o use Localizaio properies i boh, ie ad requecy doai O he oe had localizaio properies o he basis ucios should be well deied, o he oher had oe should be able o choose wha resoluio o use, ad o course hey should be as good as possible. Ivariace uder cerai operaios The resul o a rasoraio should o chage i soe operaios are applied o he origial sigal. This propery ca be relaxed a lile bi i a way, so ha i you apply a operaio o he origial sigal, he sae operaio is visible o he oupu o he rasoraio. E.g. i a ie shi is applied o he origial sigal, oe could o expec ha he oupu o a STFT is he sae as beore he ie shi, bu he resul is also shied i ie Soohess properies Soeies i is useul, ha basis ucios ca be derived or are coiuous. Moe properies Aoher hig which could be useul soeies are cerai oe properies. Especially zero oes ake copuaios uch easier. As ca be see, soe o hose properies colic wih each oher, so ha here is o opial basis or ucio spaces. However, i a applicaio is kow, oe ca choose which properies are eeded ad he derive or jus selec a useul se o basis ucios. 3 Iroducio o Waveles I his chaper he basic ideas o Waveles will be preseed ad discussed. Firs he idea o Waveles is iuiively iroduced usig he Haar Wavele, aer ha a proo is give ha he Haar Waveles are acually basis ucios or he L space, he cocep o uli resoluio aalysis is highlighed ad ially he Sic Wavele is iroduced o show he diereces o he Haar Wavele.

7 3. The Haar - Wavele The Haar Wavele is oe o he siples Wavele ucios. The oher wavele is bewee ad.5, ad - bewee.5 ad. The wavele is show i he picure below. To represe he whole space o squared suabel ucios, his oher wavele is scaled ad shied. / φ, φ I he above equaio φ, is he scaled ad shied versio o he oher wavele φ. As ca be easily see, idicaes he scale ad he shi. I his igure we see he oher wavele ad is shied versios i he iddle row, φ, i he op row ad φ, i he boo row. Wha was ow achieved by usig his way o cosrucig a basis or he L space? Oe o he os ipora higs is, ha he requecy ad ie resoluio is ow depedig o he requecy.

8 As ca be see i he igure, he higher he requecy, he beer he ie resoluio bu he requecy resoluio suers. Oe hig has o be highlighed: The area o his boxes which is he produc o he sadard deviaios always says he sae! Wha are he advaages o usig such a schee dyadic ilig o he ie/requecy plae? The wo ai reasos are: Firs, higs which have a high requecy ed o happe as so good ie localizaio is eeded, secod, low requecies ca be beer discriiaed, he relaive iial dierece bewee wo requecies which ca be diereiaed says he sae over all requecies. 3.. Orho-oraliy o he Haar Wavele Oe ajor requiree or basis is heir orho-oraliy. O course oe ca cosruc oorho-oral basis oo, bu orho-oral basis have soe ice properies, which akes i easy o use he. A se o basis ucios φ, is orho-oral, i i saisies ollowig cosrai: φ,, φ ', ' δ[ '] δ[ '] Tha eas, oly he ier produc o a wavele a he sae scale ad wih he sae shi is equal o, every ier produc o waveles wih diere scales ad/or shis have o be zero. Are Haar Waveles ulillig his cosrai? Yes, hey do. Whe sayig he sae scale, waveles are always shied, so ha hey do overlap. Thereore he projecio waveles i he sae scale ideical bu wih diere shis diere is always zero. Bu wha happes wih diere scales? I he igure above i is see ha he loger wavele is cosa over he suppor o he shorer oe, ad hereore he ier produc is he average o a Haar Wavele, which is o course equal o zero. Jus o hig has o be added: Because o he cosrucio rule, i is o possible, ha he jups o he shorer ad loger wavele are he soe posiio, so his reasoig is valid or he geeral case. 3.. Basis or L Space Above i was show ha he Haar Waveles are acually orho-oral o each oher, bu do hey also or a basis or he space o ucios i L? Luckily hey are. Oe iuiive

9 explaaio is based o he coposiio o a sigal. A sigal is give by he su over all basis ucios weighed wih soe coeicies. Usig he cosrucio rule or Haar Waveles, oe could ake arbirary large ha eas he wavele ges arrower ad arrower ad hereore oe ca represe all ucios o L usig iiie such iiie sall Waveles. However, here is a ore oral proo, which will be preseed ow. This proo also preses he coceps o uli resoluio aalysis o sigals. To sar he proo, les cosider ucios which are cosa o a ierval [, ] ad have iie suppor o he ierval [, ]. I is obvious, ha i choosig boh ad large eough, every L ucio ca be approxiaed arbirarily well. Such piecewise cosa ucios are called ro ow o. This piecewise cosa ucios ca be represeed by a su o idicaor ucios ϕ :, ϕ, oherwise < The heigh o he idicaor ucios is scaled, so ha hey have ui or. This ucio is called he scalig ucio. As said above he piecewise cosa ucio ca be wrie as a su o idicaor ucios: Where N is jus a saig how wide he suppor ad how deailed he piecewise cosa ucio is, ad is he coeicie or he paricular scaled ad shied scalig ucio. N ϕ The key sep i his proo is he relaio bewee wo adjace iervals: [ [,, ad. The piecewise cosa ucio over hese wo iervals is: N N ϕ,, ϕ Jus he su o wo versios o he scalig ucio wih heir respecive coeicies. Bu here is aoher way o describe he ucio over hese wo iervals. Oe ca copue he average o he wo iervals ad he add he dierece bewee he average ad he origial ucio. To copue he average, he scalig ucio wih is used ha scalig ucio is as wide as boh iervals ogeher, o copue he dierece o he origial ucio, he Wavele o scale is used. Reeber, his special wavele has acually wo pars which direcly correspod o he old iervals. The aheaical expressio o he average ϕ,,

10 ád he dierece usig he wavele ucio, φ : Do ease he derivaio a lile bi, we iroduce coeicies or he Scalig ucio ad he wavele, ad d respecively: This resuls i ollowig expressio or our origial wo iervals: Jus he average over he wo iervals plus he dierece. To represe he whole ucio, he explaied echais is jus applied o all adjace pairs o he whole ucio. Aer doig his, ollowig represeaio is achieved: The piecewise cosa ucio is represeed as he piecewise cosa ucio wih loger cosa iervals acually wice as log, plus he error which is ade whe copuig he average. The secod equaio is jus he ore deailed represeaios wih all he scalig ucios ad waveles wih heir respecive coeicies. To sae agai, he origial ucio is decoposed io average coarse ad dierece deails. Now he ial sep will be applied: Decopose d, ha eas he old average i a ew average ad deails ad do his repeaedly. Fially is obaied. Where is a residual average ucio. However, i ca be show ha he or o his residual decreases whe i is decoposed urher. However his decoposiio does coribue o he basic idea o he proo, so i will be skipped. The residual is expressed as M ε i he righ represeaio. As he residual error ca be ade arbirary sall, ad ad ca be ade arbirary large, his shows ha he Haar Wavele is suicie o represe all ucios ro L. The scalig ucio is o eeded. The key o he proo ad also he key idea o waveles, is he decoposiio o a sigal i a coarse approxiaio ad he deails which ge los durig ha coarse approxiaio. This is, φ d,, d φ ϕ / /, / /, N N N N d d φ ϕ M M d d ε φ φ,,

11 also he basic propery o he uli resoluio aalysis, which will be covered i he ex chaper. U V Z L R 4 Muli resoluio Aalysis The uli resoluio cocep direcly ollows ro he cosrucio o waveles. Repeaedly applyig he decoposiio o a ie sigal i a coarse approxiaio ad he dierece bewee he ie sigal ad coarse approxiaio leads o a represeaio which jus eeds successive deails o represe L sigals. I will be show, ha he coarse approxiaio ad he deails o add are orhogoal subspace o he space o he ie sigal. 4. Axioaic Deiiio Eclosed Subspaces KV V V V V K Upward Copleeess U L R V Z Dowward Copleeess I V {} Z Scale Ivariace V V Shi Ivariace V V Z Exisace o a orho-oral basis To ake higs a bi clearer, his axios will be explaied sarig ro he op. Whe alkig abou eclosed subspaces as deied, i is ipora o oe ha sigals which ca be represeed by V ca also be represeed by V - as V is subse o V -. Upward Copleeess eas, ha all Subspaces cobied resul i he L space, which eas such a uli resoluio approach ca acually represe hose sigals. Dowward copleeess eas, ha here is o error le. Scale Ivariace is kid o clear. I you scale a ucio which ca be represeed by V by i ca he be represeed by V. I is also he sae wih shi ivariace: Shied versios o a ucio are sill represeed by he sae subspace. Fially his subspaces have o be spaed soehow, so here has o be a basis or he. As oe ca oralize basis, i is o eeded ha hose basis are orho-oral ro he begiig. 4. Orhooral Coplees I he las secio he axios which are eeded or uli resoluio aalysis were deied ad discussed. Bu how does his i ogeher wih waveles ad he proo which was discussed above? As said above, V is a subspace o V -. Tha eas, i you subsrac V o V -, here has o be soehig le. Acually his is he orho-oral coplee o V i V -. I will be called W. Tha eas, i V ad W are cobied, he resul is V -. V V W

12 Now he spaces V are said o be he spaces o he scalig ucios ad W are he spaces o he waveles. By repeaig he process o spliig up V - io is subspace V ad is orhogoal coplee W or all, we ge L R Z This is equivale wih he resul we go above, whe provig ha he Haar Wavele is acually able o represe all ucios o he L space. W M 5 Cosrucig he Sic Wavele While discussig he basic properies o waveles, he Haar Wavele was used as a exaple. Oe hig which was o covered i he discussio abou waveles up o ow was heir requecy ad ie resoluio, excep ha he wavele cosrucio iroduces he dyadic ilig o he ie/requecy plae. The acual size o he box was o discussed. Thikig abou he Haar Wavele, which is basically build up ro wo recagular sigals, i is iuiively see ha i will have a good localizaio i ie bu a bad oe i requecy. The opposie o he specru, good localizaio i requecy bu a bad oe i ie, is reached whe usig he Sic wavele. The Haar Wavele uses a scalig ucio which has liied suppor i ie, he Sic wavele uses a scalig ucio which has liied suppor i requecy, hece is bad liied. As he ae already suggess, his is he Sic ucio. si π ϕ π The space V will iclude he ucios which are bad liied o [-π, π], V- he ucios which are bad liied o [-π, π]. Usig he kowledge gaied i he uli resoluio secio, W will he be he be he ucios which are bad liied o [-π, -π] ad [π, π]. As ca be easily see, cobiig V ad W resuls i V -. So wha is he wavele ucio? As already discussed, V is a subspace o V -, so ucios i V ca be represeed by ucios i V -. As ϕ is i V, i ca be represeed by ucios i V - : ϕ g g [ ] ; g [ ] ϕ [ ] ϕ, ϕ The ucio g ad is ourier rasor G is he characerisic ucio o a uliresoluio aalysis ad he key sep i cosrucio waveles, whe sarig wih a scalig ucio. Wihou proo i is show, how o cosruc g ad he he oher wavele. φ g [ ] Z g [ ] g [ ] ϕ

13 For he sic case hese ucios are give by: Ad ially he sic wavele I he igure below, he scalig ucio ad he wavele are show. oherwise e e G g j j / / si ] [ π π ω ω ω π π / cos3 / / si π π π φ

14 6 Ieraed iler baks Uil ow we have cosruced waveles by usig orhooral ailies o ucios by scalig ad shiig. This coiuous ie approach is direcly based o he axios o uliresoluio aalysis. Fro ow o we will derive waveles by usig discree ie ilers. These ilers ca be ieraed uder cerai codiios ad will also lead o coiuous ie waveles. We will have a recap o he Haar ad Sic case which ca be see as liis ro ieraed iler baks. This cosrucio ca be geeralized. The key propery o such discree ie ilers is he regulariy codiio. Discree ilers ca be called regular i hey coverge o a scalig ucio wih soe degree o regulariy piecewise sooh, coiuous, or derivable. 6. Haar case We sar ow wih he discree Haar ilers. We kow ha he lowpass averages wo eighborig saples ad he highpass builds he dierece o he. The iler bak or his is give by he basis ucios g ad g. Kow we sar o ierae he iler bak o he lowpass chael see Figure. Figure : Ieraed iler bak We kow will derive ro his corucio a equivale iler bak by usig resuls ro ulirae sigal processig. Filerig by g ollowed by upsaplig by wo is he sae o upsaplig by wo, ollowed by ilerig by g upsapled versio o g. By doig so we ca rasor he ieraed iler bak io oe equivale o a iler which is show i Figure. Figure : Trasored iler bak This is a size 8 discree Haar rasor o sucessive blocks o 8 saples. By ieraio o he lowpass ad highpass chaels we will coe o wo ilers Figure 3:

15 Figure 3: Ieraed iler equaios Fro he iler equaios we ca see, ha g is a lowpass iler ad g is a badpass iler. We also ca see a ieresig propery. I we icrease he ieraios deph he legh o he iler also icreases expoeially ad o he oher side he coeicies go o zero. I we ow deie he coiuous ie ucio wih g ad g i his way: Figure 4: Coiuous ie ucios These ucios are piecewise cosa ad heir leghs reai bouded. 6. Sic case As a couerpar or he Haar case we ow have a look a he sic case. I his case he ilers i he iler bak are ideal low- ad highpass ilers. The ipulse respose or he lowpass iler is: si π / g [ ] π / The perec high pass iler ca be cosruced by odulaig - ad shiig by oe. For he ipulse respose we ge wih use o he lowpass iler: g ] g [ ] [ The we rasor boh ipulse resposes ad we will ge: Figure 5: Fourier rasor o g [] Figure 6: Figure 5: Fourier rasor o g [] We ow will do he sae as i he Haar case. We cosider he ieraed iler bak wih ideal ilers. For isace upsaplig he iler resposes by wo will lead o a iler wih a saller badwidh ad so o or urher ieraios. I a relaed way he raser ucio o he highpass iler ca be cosruced ad leads o a badpass iler. Now we have our iler we ca eulae he Haar cosrucio or he low- ad highpass ipulse resposes g [] ad g [] which are he ieraed ilers. The we ca deie a φ as we have see i Figure 4. I he sic case we are ieresed i he Fourier rasor o his scalig ucio ad we will obai i i i i i / i jω / jω / si ω / Φ ω G e e i ω / where he ieraed raser ucio ca be rewrie as i i jω jω jω j ω jω G e G e G e... G e M ω G e

16 ad we will ed up wih a copac versio o he Fourier rasor: ω si ω / i i i i jω / Φ ω M k e i k ω / So kow les have a look o he diere pars o his orula. The righ par expoeial ad si-ucio is jus a phase acor ad ierpolaio ucio ad i we le he uber o ieraios i grow his par becoes early oe or ay iie requecy ω. So le s have a look o he irs par i he brackes. The produc is a i π periodic ucio bewee π ad π. So i we le i grow, we will ed up wih a perec lowpass iler ro π ad π which is he sic scalig ucio. For he wavele ucio we ca do i siilarly ad we will ed up wih kow resuls. This sees o be a very cubersoe way o derive resuls we already have kow. Bu we have gaied a ore geeral cosrucio ad we have see he key or his aalysis is he produc i he brackes. For regulariy aalysis we have o ivesigae his iiie produc. 6.3 Ieraed iler baks co. Now we will use he derivaio o he Haar ad sic waveles usig ieraed iler baks o ake a ore geeral cosrucio or wavele bases. Oce agai we sar wih our wo ilers g [] ad g [], which are low- ad highpass ilers. For he ieraed iler bak we will ierae he o he lowpass chael brach o iiiy. As we have doe his i he wo cosrucios above we will ow reorulae our paricular ilers by heir ieraed represeaios by usig ulirae ideiies. So we will ed up wih ollowig raser ucios or our ilers: The we will associae he ieraed ilers wih he ie ucios o obai he scalig ucio ad he waveles as ollows: These ucios had o be rescaled ad oralized or regulariy reasos. I Figure 7 we ca our ieraios o a legh 4 iler. We ca see he scalig ucio φ is a piecewise cosa approxiaio ad i every ieraio we halve he ierval.

17 Figure 7: Ieraed iler equaios For he Fourier doai we ca do he sae as we have doe i he sic case. For he ieraed schee we have o ivesigae he liis o he scalig ad wavele ucios. As we have igured ou i he sic case we oly ocus our ieres o he er i he brackes because he secod par coverges o oe. For he Fourier doai he lii will be Ad i urs ou ha hese wo ucios are really scalig ucios which ca be obaied as liis o discree ie ieraed ilers. The exiseces o hese liis are very criical codiios. We ca easily say ha hese liis exis i he ilers are regular. Tha eas ha regular ilers leads, hrough ieraio o a scalig ucio. The key or he regulariy aalysis is he behavior o he iiie produc. There are covergece quesios ad i covergece exis we have also aswer i wha sese he covergece exis ad which properies ca be ulilled. I his repor we will o dig deeply io hese discussios. There are several approaches o ivesigae his behavior which gives us suicie codiios. Soe o he are also ecessary bu also o resricive. I geeral we ca say ha his par is sill a ope opic or research. 7 Wavele series ad properies The purpose o his secio is o have a look o he wavele cosrucio ad oo he properies. The ollowig will be a eueraio o soe geeral properies. Firs we review ad deie he proble as ollows F[, ] ψ, Z F[, ] ψ,,, ψ,, d

18 where ψ is he oher wavele ad is ay ucio i he L R space. 7. Lieariy I we suppose a operaor T which is deied as T[] F[,] ψ,, he or ay a,b R T[ a b g ] at bt g Thereore we ca say ha he wavele operaor is liear so ar uprove bu i ca be ollowed ro he lieariy o he ier produc. 7. Shi Firs we have a look o he shi propery o he Fourier rasor where a ucio rasors io Fω ad a ie shi o he ucio -τ rasors io exp-jωτ*fω. Wha happes ow wih he wavele series. The ucio ad is coeicies are wrie as ad F[,]. I ow is shied by τ, we ivesigae he ucio -τ: F F '[, ] ψ τ, ' / [, ] ψ τ d d I we ow rewrie he shi as a uliple o - τ he we ca igure ou ollowig rasor propery: τ Z or τ k F k, k Z ' ' [, k], ' < 7.3 Scalig Oce agai we irs have a look o he scalig propery o he rare Fourier rasor which is give by a ucio ad is rasor Fω. I we ow scale wih a paraeer a so ha a he he Fourier rasor is /a*fω/a. For he wavele expasio we ca wrie F '[, ] ψ a, d ' / F [, ] / a ψ d a Siilarly as i he shi case we ca rewrie he scalig acor as a power o wo ad we will coe o ollowig propery k a, k Z k k F k, [ ]

19 7.4 Dyadic saplig ad ie requecy ilig Whe we have a look o he series expasio i is ipora o locae he basis ucios i he ie requecy plae. The saplig i ie, a a speciic scale akes place wih a period whe we ulill ψ, ψ, O he oher side requecy is he iverse o scale, we id whe he wavele is ceered aroud ω he Ψ, ω is ceered aroud ω /. This leads o dyadic saplig o he ie requecy plae. I Figure 8 we ca such a ie requecy plae where he scale axis is logarihic ad i relaio o his we have a wavele represeaio where we have a liear scale axis Figure 9. Figure 8: Dyadic saplig logarihic Figure 9: Dyadic saplig liear 7.5 Tie localizaio I geeral localizaio is he ajor advaage o he wavele represeaios. For he ie localizaio we suppose ha we ieresed i he sigal aroud a speciic ie. Now we wa o kow which values o F[,] are relaed o he sigal ad which values o F[,] carry ioraio o he sigal. Firs suppose a wavele which suppors Ψω i he ierval [-, ]. The Ψ, ω is i he rage o [-, ] ad urher ore Ψ, ω is suppored [-, ]. A a speciic scale, he wavele coeicies wih idex saisy ha - ca be rewrie - I Figure we ca which regio carries ioraio abou he scale i he wavele F[,].

20 Figure : Tie localizaio 7.6 Frequecy localizaio Siilarly as we have doe i he ie localizaio we kow ieresed which regio i F[,] carries soe ioraio abou he requecy localizaio. Firs we have a look a he Fourier / j ω rasor o ψ, ψ which is / Ψ ω e. The we ca wrie F[,] as F[, ] ψ, d / j F[, ] ω ω ω π F Ψ e ω d To arrow he regio o ieres we say he waveles vaishes i he Fourier doai ouside a speciic regio [ω i,ω ax ] he coeicies eed o o vaish bu os o he eergy should be i here. A a speciic scale he suppor o Ψ, ω will be [ω i /,ω ax / ]. So we ca say ha a requecy copoe aroud a speciic requecy ω ilueces he wavele series a scale i ωi ωax ω is saisied ad we ca rewrie or he scale ωi log ω log ω ω ax I Figure we ca see which values will be ilueced ro a speciic requecy. Figure : Frequecy localizaio

21 7.7 Decay properies The Fourier series ca be used o characerize he regulariy o a sigal, which ca be doe by lookig o he decay o he rasor coeicies. The drawback o he Fourier aalysis is ha we oly coe dow o a global regulariy o he sigal. This is aoher advaage o he wavele rasor. Whe we look here o he coeicies o he rasor we ca esiae a local regulariy o he sigal. 7.8 Mulidiesioal waveles For applicaios like iage copressio, we ca exed he heories o oe diesioal iler baks ca be exeded o uliple diesios. A direc way o cosruc ulidiesioal waveles is o use esor producs ad heir oe diesioal couerpars. This will lead us o oe scalig ucio wih hree diere oher waveles. The scale chages will ow represeed i a arix which oers several advaages bu also resricios o he arix. The lik o iler baks ca be easily oud bu he ask is uch harder icoplee cascade srucures. 8 Pracical aspecs This secio should give a shor overview where o sar i alab ad wha possible applicaios are or isace i iage processig. I alab here is whole oolbox which deals wih wavele ipleeaios. For a shor overview have a look o he waveeu. There are several possibiliies o use ad diere approaches. The oly suggesio we ca give here is o sar a his poi ad work hrough opics which you are ieresed i. 9 Reereces [] Waveles ad Subbad Codig, Mari Veerli, Jelea Kovacevic, ISBN [] Waveles prakische Aspeke, Markus Graber, VO6 [3] AK Copuergraik Bildverarbeiug ud Musererkeug WS 6/7