Wavelets and Multiresolution. Processing. Multiresolution analysis. New basis functions called wavelets. Good old Fourier transform

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1 Digital Image Processig d ed. Wavelets ad Multiresolutio Processig Preview ood old Fourier trasform A trasform were te basis fuctios are siusoids ece localied i frequecy but ot localied i time; i trasform domai te temporal iformatio is lost New basis fuctios called wavelets Varyig frequecy ad limited duratio; a attempt to reveal bot te frequecy ad temporal iformatio i te trasform domai Multiresolutio aalysis Sigal image represetatio at multiple resolutios R. C. oale & R. E. Woods

2 Digital Image Processig d ed. Bacgroud Images Coected regios of similar teture combied to form differet obects Small sied or low i cotrast obects: eamied at ig resolutio Large sied or ig i cotrast: eamied at a coarse view Several resolutios eeded to distiguised betwee differet obects Matematically images are -D arrays of itesity values of locally varyig statistics resultig from edges omogeous regios etc. see Fig. 7. R. C. oale & R. E. Woods

3 Digital Image Processig d ed. istograms computed i local squares R. C. oale & R. E. Woods

4 Digital Image Processig d ed. Image Pyramids Creatio by iterated approimatios ad iterpolatios predictios Te approimatio output is tae as te iput for te et resolutio level For a umber of levels P two pyramids: approimatio aussia ad predictio residual Laplacia are created approimatio filters could be: averagig low-pass aussia R. C. oale & R. E. Woods

5 Startig image is 5 5; covolutio erel approimatio erel is 5 5 aussia Differet resolutios are appropriate for differet image obects widow vase flower etc. R. C. oale & R. E. Woods Digital Image Processig d ed. aussia ad Laplacia Pyramids J J 3 J J

6 Digital Image Processig d ed. Subbad Codig Te image is decomposed ito a set of bad-limited compoets subbads. Two-cael perfect recostructio system; two sets of alf-bad filters Aalysis: low-pass ad ig-pass ad Sytesis: g low-pass ad g igpass R. C. oale & R. E. Woods

7 Digital Image Processig d ed. Digital Image Processig d ed. R. C. oale & R. E. Woods Brief remider of -trasform properties Brief remider of -trasform properties Dowsampled ad upsampled by a factor of two sequeces i -domai [ ] [ ] [ ] / / / up up dow dow Dowsamplig followed by upsamplig [ ][ ] [ ] ˆ ˆ Z

8 Digital Image Processig d ed. Digital Image Processig d ed. R. C. oale & R. E. Woods Subbad Codig Subbad Codig [ ] [ ] [ ] [ ] [ ] [ ] [ ] det were [ ] Two PR coditios : ˆ PR : - free perfect recostructio - For error ˆ Rearaged terms : ˆ Sytesis brac : Aalysis brac : / / / / / / / / Y Y m m

9 Digital Image Processig d ed. Digital Image Processig d ed. R. C. oale & R. E. Woods Subbad Codig Subbad Codig {} coditios : More restrictive {} coditios : ; ; : It ca be also sow tat : Odd ideed terms cacel trasform : By iverse - PR coditio trasforms to : Te first det - : det Sice det det pas aalysis ad sytesis filters te low - Biortogoality : Tae te product of : or for trasform of Tae iverse - ad let Let omit te delay term pure delayi.e. det a is For FIR filters det! i m i m g g i i g g g g g g g g P P g g - a g g a a i i m m m m m m δ δ δ δ δ δ δ ortogoal Biortogoal

10 Digital Image Processig d ed. Families of alf-bad PR filters For eac family of filters a prototype filter is desiged ad te oter filters are computed from te prototype. QMF quadrature-mirror filters CQF cougate-quadrature filters K is te legt umber of coefficiets filter taps g i g K g K i {} i R. C. oale & R. E. Woods

11 Digital Image Processig d ed. Separable filterig i -D images Vertical followed by oriotal filterig Four output subbads: approimatio ad vertical oriotal ad diagoal detail subbads R. C. oale & R. E. Woods

12 Digital Image Processig d ed. Eample: 8-tap ortoormal filter desiged by Daubecies R. C. oale & R. E. Woods

13 Digital Image Processig d ed. Eample: four bad subbad filterig of a image Aliasig is preseted i te vertical ad oriotal detail subbads. It is due to te dow-samplig ad will be caceled durig te recostructio stage. R. C. oale & R. E. Woods

14 Digital Image Processig d ed. aar Trasform oldest ad simplest wavelets T F were F is a N N image matri; is a N N trasform matri ad T is te resultig trasform coefficiets. cotais te aar basis fuctios To geerate we defie te iteger suc tat pq N N p / [] p / q / q.5 / oterwie []. Tey are defied over te cotiuous closed iterval [] for... N p < q.5 / p < q / p p q were p q or for p ad q p p for matri. were N p : Te i - t row of te aar matri cotais te elemets of 4 4 ; i for / N / N / N... N / N. Te aar basis fuctios defie two - tap FIR filter ba. Te coefficiets of correspodig QMF aalysis filters are i te rows of R. C. oale & R. E. Woods

15 Digital Image Processig d ed. aar Fuctios i a Discrete Wavelet Trasform aar Fuctios i a Discrete Wavelet Trasform R. C. oale & R. E. Woods

16 Digital Image Processig d ed. Multiresolutio Epasios I Multiresolutio Aalysis MRA a scalig fuctio is used to create a series of approimatios of a fuctio image eac differig by a factor of two. Additioal fuctios wavelets are used to ecode te differece betwee adacet approimatios. Series Epasios Liear combiatio of epasio fuctios: Basis fuctios ad fuctio space: Dual fuctios: V f ϕ { ϕ } spa ~ ~ ~ ϕ α { ϕ } suc tat α ϕ f f d Cases: ortoormal basis biortogoal basis overcomplete epasios frames ad tigt frames R. C. oale & R. E. Woods

17 Digital Image Processig d ed. Multiresolutio Epasios Scalig Fuctios Set of fuctios {ϕ } formed by iteger traslatios ad biary scalig of a prototype fuctio ϕ : traslatio positio parameter ; scale parameter scalig fuctios ca be made to spa te space of all measurable square-itegrable fuctios deoted by L R by fiig te scale to te resultig set spas a subspace of L R if f V te ϕ spa { } V ϕ α ϕ f / ϕ R. C. oale & R. E. Woods

18 Digital Image Processig d ed. Te aar Scalig Fuctio Ay V epasio fuctio ca be represeted by V epasio fuctios i.e. V V. ϕ ϕ ϕ R. C. oale & R. E. Woods

19 Digital Image Processig d ed. MRA Requiremets MRA requiremets : Te scalig fuctio is ortogoal to its iteger traslates MRA requiremet : Te subspaces spaed by te scalig fuctio at low scales are ested witi tose spaed at iger scales V... V V V V... V MRA requiremet 3: Te oly fuctios tat is commo to all V is f MRA requiremet 4: Ay fuctio ca be represeted wit arbitrary precisio R. C. oale & R. E. Woods

20 Digital Image Processig d ed. Dilatatio Equatio Uder te MRA requiremets te epasio fuctios of subspace V ca be epressed as a weigted sum of te epasio fuctios of subspace V. ϕ ϕ ϕ settig bot ϕ α ϕ ad ϕ to we obtai te so - called dilatatio equatio ϕ / ϕ R. C. oale & R. E. Woods

21 Digital Image Processig d ed. Wavelet Fuctios Defie a wavelet fuctio ψ tat togeter wit its iteger traslates ad biary scaligs spas te differece betwee ay two adacet scalig subspaces ψ W if f spa / ψ { ψ } W f α ψ V V W We call W ortogoal complemet of V i V. ϕ ψ l R. C. oale & R. E. Woods

22 Digital Image Processig d ed. Wavelet Fuctios Te space of measurable square-itegrable fuctios ca be epressed as L L L R R R V V... W W W W W W W W W... Ay wavelet fuctio ca be epressed as a weigted sum of sifted doubleresolutio scalig fuctios ψ were ψ ψ ϕ ϕ R. C. oale & R. E. Woods

23 Digital Image Processig d ed. aar Wavelet Fuctio Coefficiets aar scalig filter : ϕ ϕ / aar wavelet filter : aar wavelet fuctio : ψ ψ / ; ψ <.5 < elsewere /.5 f f f a d f a ϕ ψ f d 8 8 ϕ ψ R. C. oale & R. E. Woods

24 Digital Image Processig d ed. Digital Image Processig d ed. R. C. oale & R. E. Woods Wavelet Series Epasios Wavelet Series Epasios f d f c d c f ψ ϕ ψ ϕ Cosider te ortogoal case c are referred as approimatio coefficiets ad d are referred as detail coefficiets. Eample: < oterwise f

25 Digital Image Processig d ed. Digital Image Processig d ed. R. C. oale & R. E. Woods Discrete Wavelet Trasform DWT Discrete Wavelet Trasform DWT Trasform pair forward ad iverse trasform applicable for discrete sigals or sampled sigals W M W M f f M W f M W ψ ϕ ψ ϕ ψ ϕ ψ ϕ ere is a discrete variable: M - Normally M J ; ; J- ad.. - W ϕ ad W ψ are referred as approimatio ad detail coefficiets respectively.

26 Digital Image Processig d ed. Digital Image Processig d ed. R. C. oale & R. E. Woods Cotiuous Wavelet Trasform CWT Cotiuous Wavelet Trasform CWT Trasform a cotiuous fuctio ito a igly redudat fuctio of two variables traslatio τ ad scale s. ω ω ω τ ψ τ τ ψ ψ ψ τ ψ τ ψ ψ τ τ ψ d C ds d s s W C f s s f s W s s s Ψ were were

27 Digital Image Processig d ed. Compariso betwee Fourier Trasform ad Cotiuous WT R. C. oale & R. E. Woods

28 Digital Image Processig d ed. Fast Wavelet Trasform FWT I brief Aalysis filter ba Sytesis filter ba R. C. oale & R. E. Woods

29 Digital Image Processig d ed. A Two-stage Two-scale FWT: Aalysis R. C. oale & R. E. Woods

30 Digital Image Processig d ed. A Two-stage Two-scale FWT: Sytesis R. C. oale & R. E. Woods

31 Digital Image Processig d ed. Eample: Two-scale FWT by aar R. C. oale & R. E. Woods

32 Digital Image Processig d ed. Eample: Two-scale FWT by aar R. C. oale & R. E. Woods

33 Digital Image Processig d ed. Wavelet Pacets Te Wavelet Trasform decomposes a fuctio ito a series of logaritmically related frequecy bads. Low frequecies are grouped ito arrow bads wile te ig frequecies are grouped ito wider bads. Wavelet Pacets are geeraliatio tat allows greater cotrol over te time-frequecy plae partitioig. Cosider te two-scale filter ba as a biary tree. Te wavelet coefficiets are at te odes of te tree. Te root ode cotais te igest-scale approimatio coefficiets i.e. te sampled sigal itself. Eac ode cotais coefficiets represetig differet subspaces subspace aalysis tree. R. C. oale & R. E. Woods

34 Digital Image Processig d ed. WT: Filter Ba Aalysis Tree ad Spectrum Splittig R. C. oale & R. E. Woods

35 Digital Image Processig d ed. Wavelet Pacets Wavelet pacets are covetioal wavelet trasforms i wic te details are also iteratively filtered. Subscripts i te figure sow te scale ad a strig of A s ad D s ecodig te pat from te paret to te ode. Te pacet tree almost triples te umber of decompositios. R. C. oale & R. E. Woods

36 Digital Image Processig d ed. Wavelet Pacets: Filter Structure ad Spectrum Splittig R. C. oale & R. E. Woods

37 Digital Image Processig d ed. Wavelet Pacets Te frequecy selectivity ca be icreased toward ig frequecies R. C. oale & R. E. Woods

38 Digital Image Processig d ed. -D Wavelet Pacets I -D separable implemetatio of wavelet pacets leads to a quad-tree structure. Te frequecy spectrum is divided ito four areas. Te low-frequecy area is i te cetre. R. C. oale & R. E. Woods

39 Digital Image Processig d ed. Portio of -D Wavelet Pacet Tree R. C. oale & R. E. Woods

40 Digital Image Processig d ed. Wavelet Pacet Decompositio R. C. oale & R. E. Woods

41 Digital Image Processig d ed. Optimal Wavelet Pacet Basis Searc Etropy-based criteria for searcig te optimal te best wavelet pacet decompositio A additive cost fuctio for comparig differet decompositios R. C. oale & R. E. Woods

42 Digital Image Processig d ed. Optimal Wavelet Pacet Basis Optimal Wavelet Pacet Basis R. C. oale & R. E. Woods