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1 3//6 Coprght otce: Most ages these sldes are Gozalez ad oods Pretce-Hall Note: ages are [spatall] ostatoar sgals. e eed tools to aalze the locall at dfferet resolutos e ca do ths the data doa or sutable trasfor doas A age prad ad a sste for buldg a par of approato ad predcto resdual prads The flter a be a sple lowpass Gaussa flter Gaussa approato ad Laplaca resdual prads A redudat represetato s obtaed Note: gra-level rage hstogras s e.g. [55] for approato prad [-5555] for resdual prad

2 3//6 - Subbad codg: oredudat represet. perfect recostructo f flters satsf ether g h g or g h h g h tap aubeches - borthogoal & orthooral flters perfect recostructo All flters are derved fro the prototpe sthess flter g o.e. the are borthogoal h g δ δ If the flters are also orthooral: g + g δ δ the ca all be derved fro a sgle prototpe flter - Subbad codg: - borthogoal flters ca be used as - separable flters for subbad codg of ages Four output ages: Approato a Vertcal detal dv Horzotal detal dh agoal detal d a dh dv d ages A wdow scree barel vsble the age geerates the horzotal bars dh ad the alasg effect dv. espte the alasg perfect recostructo ca be acheved. The process ca be terated

3 3//6 I the trasfor doa: ltatos of the FT Hz FT 55 Hz FT - case Magtude of FT does ot provde fo o teporal structure of sgal Applg the FT to overlapped wdowed segets of the put sgal we get a fucto of two varables te ad frequec: the Short Ter Fourer Trasfor STFT Good for pecewse-statoar sgals If statoart tervals becoe shorter ad shorter we eed to deal wth Heseberg's prcple of deterac 97: we ca ow ol the te tervals whch certa frequec bads est ages R. Polar avelet tutoral Parts ad 3 Note: ot the sae sgal as the prev. slde arrow te wdow good te resoluto poor freq. resoluto wde te wdow poor te resoluto good freq. resoluto Foral descrpto: STFT Gabor 94: where g s a wdow fucto that ust copl wth g t dt Te-frequec resoluto s: t g t dt ω G ω dω t ω g t dt G ω dω π t A Gaussa wdow g t ep ω ep σ ω / 4π σ σ G σ provdes the best precso pertted b Heseberg's Prcple. The STFT uses the sae wdow to aalse all frequeces. Sce hgh frequec sgals tpcall have a shorter durato tha low-frequec oes t aes sese to use: arrow te wdow aalse hgh frequec sgals wde te wdow aalse low frequec sgals * S f ω τ f t g t τ ep ωt dt CONTINUOUS AVELET TRANSFORM CT Grossa & Morlet 984: The wdow used the STFT becoes a wavelet fucto derved fro a other wavelet b traslato τ ad scalg s: τ s τ s s τ ad s are cotuous paraeters; s> dlates <s< copresses the fucto. s s equal to /frequec the Fourer doa The CT s a fucto of two varables defed as avelet fuctos ust be Adssble: oscllator zero ea value Regular: epoetall decag The the verse wavelet trasfor ests. * s τ f d f sτ

4 3//6 To copute the CT:. Start wth s sall s hgh frequec. Place the wavelet at the begg of the sgal 3. Multpl the sgal b the wavelet ad tegrate over 4. The result s the CT value for τ s the traslato-scale plae 5. Move the wavelet slghtl to the rght ad repeat 6. Repeat utl the wavelet reaches the ed of the sgal [CT values for the frst row the trasl.-scale plae are foud] 7. Repeat -6 for progressvel larger values of s [CT values foud the whole plae] Note: a - fucto s represeted o the traslato-scale plae the represetato s redudat. Le the Fourer case dscretzg τ ad s elds the avelet seres; f the sgal s dscrete the T results E.g.: s Ths frst wavelet should be as arrow as the hghest-frequec copoet the sgal If the sgal has a spectral copoet that correspods to the curret value of s the sgal wavelet product wll be large where such spectral copoet s located At larger scales e.g. s5 s the trasfors pcs up lowerfrequec copoets of the sgal Eaple 35 Hz

5 3//6 About resoluto: STFT plae vs. T plae Ever bo cotas a sgle value of the STFT ad of the T Boes have fte area; u area s π/4: deterac prcple All boes have the sae area T boes gve varable proportos to te ad scale The Haar trasfor of a NN age F s obtaed as T T H F H where H s a NN Haar atr that ca be derved a teratve fasho startg fro H Low scales / hgh frequeces have good te resoluto but poor frequec resoluto. Hgh scales / low frequeces have good frequec resoluto but poor te resoluto. the splest perfect recostructo - flter E.g. for N8 STFT has costat te to frequec resoluto boes are square The T aubeches 988 of a legth-m sequece s: f f M M where s a approato fucto at the referece scale e.g. are the wavelet fuctos detals for a set { } of scales. The Haar fuctos cota a sgle prototpe shape: a square wave ad ts egatve verso. The paraeters specf the wdth or scale of the shape ad ts posto or shft. I ths wa the represet ot ol the detals the sgal but also ther locatos space. The Haar trasfor s the splest wavelet trasfor. The verse T s f M + M For both the T ad IT a fast calculato algorth ests FT based o the subbad codg strateg

6 3//6 The - T ca be obtaed usg a separable approach. e defe: avelets of tpe easure test varatos alog the vertcal rows horzotal colus dagoal drectos. The the scaled ad traslated bases are defed: Ad the T of a age s H V } { / / V H } { H V f f The - IT s + + H V f Trasfor eaple three-level decoposto

7 3//6 Applcato eaple edge ehaceet Applcato eaple ose reoval Hard thresholdg t t > δ Hard t otherwse Soft thresholdg sg t t t > t δ δ Soft otherwse Applcato eaple copresso of database of fgerprts