ECE Department Univ. of Maryland, College Park

Transcription

1 EEE63 Part- Tr-basd Filtr Banks and Multirsolution Analysis ECE Dpartmnt Univ. of Maryland, Collg Park Updatd / by Prof. Min Wu. bb.ng.umd.du d slct EEE63); minwu@ng.umd.du md d M. Wu: EEE63 Advancd Signal Procssing Fall )

2 Dynamic Rang of Original and Subband Signals Can assign mor bits to rprsnt coars info Allocat rmaining bits, if availabl, to finr dtails via propr quantization) Figurs from Gonzalz/ Woods DIP 3/ book wbsit. M. Wu: EEE63 Advancd Signal Procssing [8]

3 Brif ot on Subband and Wavlt Coding Th octav dyadic ) frquncy partition can rflct th logarithmatic charactristics in human prcption p Wavlt coding and subband coding hav many similaritis.g. from filtr bank prspctivs) p Traditionally subband coding uss filtrs that hav littl ovrlap to isolat diffrnt bands Wavlt transform imposs smoothnss conditions on th filtrs that usually rprsnt a st of basis gnratd by shifting and scaling dilation ) of a mothr wavlt function Wavlt can b motivatd from ovrcoming th poor timdomain localization of short-tim FT Explor mor in Pro#. S PPV Book Chaptr M. Wu: EEE63 Advancd Signal Procssing [9]

4 Rviw and Exampls of Basis Rviw and Exampls of Basis Standard basis vctors Standard basis imags g 3 3 Exampl: rprsnting a vctor with diffrnt basis M. Wu: EEE63 Digital Imag Procssing Spring ) Lc Unitary Transform []

5 Tim-Frq or Spac-Frq) Intrprtations Invrs transf. rprsnts a signal as a linar combination of basis vctors Forward transf. dtrmins combination coff. by procting signal onto basis E.g. Standard Basis for data sampls); Fourir Basis; Wavlt Basis Figurs from Gonzalz/ Woods DIP / book wbsit. M. Wu: EEE63 Advancd Signal Procssing []

6 Rcall: Matrix/Vctor Form of DFT Rcall: Matrix/Vctor Form of DFT ) ) n nk W n z k { zn) } { k) } nk= W = xp{ /} ) ) k nk W k n z /4) n, k =,,, -, W = xp{ - / } ~ complx conugat of primitiv th root of unity by M.Wu a z T * ) ) Slids cratd z a a z z T T * * ) / 4 / / ) / / / ) ) ) ) ) ) ) z CP EEE63 S a z T / ) ) / ) / ) ) ) ) ) ) ) ) ) ) ) / ) ) / ) / ) / 4 / / ) / / / a a a z z UMC M. Wu: EEE63 Advancd Signal Procssing Fall'9) /4/9 [3] ) ) ) / ) ) / ) / z

7 From Kn Lam s DCT talk HK Polytch) UMCP EEE63 Slids crat d by M.Wu ) Exampl of -D DCT: = u= to u= to u= to 3 - zn) n Original signal u= - u= to 4 u= to 5 u= to 6 k) k u= to 7 Transform coff. Basis vctors M. Wu: EEE63 Advancd Signal Procssing Fall'9) Rconstructions /4/9 [4]

8 Haar Transform,),) x U MCP EEE63 Slids crat d by M.Wu ) Haar basis functions: indx by p, q) Scaling capturs info. at diffrnt frq. Translation capturs info. at diffrnt locations Transition at ach scal p is localizd according to q Haar transform H ~ orthogonal Sampl Haar function to obtain transform matrix Filtr bank rprsntation filtring and downsampling Rlativly poor nrgy compaction Equiv. filtr rspons dosn t hav good cutoff & stopband attnuation => Basis imags of -D Haar transform,),),) M. Wu: EEE63 Advancd Signal Procssing Fall'9) /4/9 [5]

9 Compar Basis Imags of DCT and Haar UMCP EEE63 Slids cratd by M.Wu ) S also: Jain s Fig.5. pp36 M. Wu: EEE63 Advancd Signal Procssing Fall'9) /4/9 [6]

10 M. Wu: EEE63 Advancd Signal Procssing Fall'9) /4/9 [9]

11 Comprssiv Snsing Downsampling as a data comprssion tool For bandlimitd signals. Considrd uniform sampling so far Mor gnral cas of sparsity in som domain E.g. non-zro coff. at a small # of frquncis but ovr a broad support of frquncy? How to lvrag such sparsity to gt rducd d avrag sampling rat? Can w sampl at non-qually spacd intrvals? How to dal with ral-world issus.g. approx. but not xactly spars? Rf: IEEE Signal Procssing Magazin: Lctur ots on Comprssiv Snsing 7); Spcial Issu on Comprssiv Snsing 8); EEE698A Fall 8 Graduat Sminar: UMD EEE63 Advancd Signal Procssing F') Discussions []

12 L vs. L Optimization for Spars Signal Fig. from Cands-Wakins SPM 8 articl) UMD EEE63 Advancd Signal Procssing F') Discussions []

13 Exampl: Tomography problm Logan-Shpp phantom tst imag Sampling in th frquncy plan Along radial lins with 5 sampls on ach Minimum nrgy rconstruction Rconstruction by minimizing total variation Slid sourc: by Dikpal Rddy, EEE698A, UMD EEE63 Advancd Signal Procssing F') Discussions []

14 U MCP EEE63 Slids crat d by M.Wu 4) A Clos Look at Wavlt Transform Haar Transform unitary Orthonormal Wavlt Filtrs Biorthogonal Wavlt Filtrs M. Wu: EEE63 Advancd Signal Procssing Fall'9) /4/9 [33]

15 Construction of Haar Functions by M.Wu /4) Uniqu dcomposition of intgr k p, q) k =,, - with = n, <= p <= n- q =, for p=); <= q <= p for p>).g., k= k= k= k=3 k=4,),),),),) powr of q = for p=); <= q <= p for p>) k= p +q rmaindr UMC CP EEE63 Slids cratd h k x) = h p,q x) for x [,] h h x ) h, x ) for [, ] p / q- q- for x p p p / q- q x) h ) for x p, q x p for othr x [,] x k p,),),),),) x M. Wu: EEE63 Advancd Signal Procssing Fall'9) /4/9 [34]

16 Mor on Wavlts ) 4) Slids crat d by M.Wu MCP EEE63 U Linar xpansion of a function via an xpansion st Form basis functions if th xpansion is uniqu Orthogonal basis on-orthogonal basis Cofficints ar computd with a st of dual-basis Discrt Wavlt Transform Wavlt xpansion givs a st of -paramtr basis functions and xpansion cofficints: scal and translation M. Wu: EEE63 Advancd Signal Procssing Fall'9) /4/9 [7]

17 Mor on Wavlts ) UMCP EEE63 Slids crat d by M.Wu 4) st gnration wavlt systms: Scaling and translation of a gnrating wavlt mothr wavlt ) Multirsolution conditions: Us a st of basic xpansion signals with half width and translatd in half stp siz to rprsnt a largr class of signals than th original xpansion st th scaling function ) Rprsnt a signal by combining scaling functions and wavlts M. Wu: EEE63 Advancd Signal Procssing Fall'9) /4/9 [8]

18 Orthonormal Filtrs Equiv. to procting input signal to orthonormal basis U MCP EEE63 Slids crat d by M.Wu ) Enrgy prsrvation proprty Convnint for quantizr dsign MSE by transform domain quantizr is sam as rconstruction MSE in imag domain Shortcomings: cofficint xpansion Linar filtring with -lmnt input & M-lmnt filtr +M-)-lmnt output +M)/ aftr downsampl Lngth of output pr stag grows ~ undsirabl for comprssion Solutions to cofficint xpansion Symmtrically xtndd input circular convolution) & Symmtric filtr M. Wu: EEE63 Advancd Signal Procssing Fall'9) /4/9 [36]

19 ) UMCP EEE63 Slids crat d by M.Wu Solutions to Cofficint Expansion Circular convolution in plac of linar convolutionol Priodic xtnsion of input signal Problm: artifacts by larg discontinuity at bordrs Symmtric xtnsion of input Rduc bordr artifacts not th signal lngth doubld with symmtry) Problm: output at ach stag may not b symmtric F U it h IEEE From Usvitch IEEE Sig.Proc. Mag. 9/) M. Wu: EEE63 Advancd Signal Procssing Fall'9) /4/9 [37]

20 Solutions to Cofficint Expansion cont d) U MCP EEE63 Slids crat d by M.Wu ) Symmtric xtnsion + symmtric filtrs o cofficint xpansion and littl artifacts Symmtric filtr or asymmtric filtr) => linar phas filtrs no phas distortion xcpt by dlays) Problm Only on st of linar phas filtrs for ral FIR orthogonal wavlts Haar filtrs:, ) &,-) do not giv good nrgy compaction Rf: rviw EEE63 discussions on FIR prfct rconstruction Qudratur Mirror Filtrs QMF) for -channl filtr banks. M. Wu: EEE63 Advancd Signal Procssing Fall'9) /4/9 [38]

21 Biorthogonal Wavlts From Usvitch IEEE Sig.Proc. Mag. 9/) U MCP EEE63 Slids crat d by M.Wu ) Biorthogonal Basis in forward and invrs transf. ar not tth sam but giv ovrall prfct rconstruction PR) rcall EE63 PR filtrbank o strict t orthogonality for transf. filtrs so nrgy is not prsrvd But could b clos to orthogonal filtrs prformanc Advantag Covrs a much broadr class of filtrs including symmtric filtrs that t liminat i cofficint i xpansion Commonly usd filtrs for comprssion 9/7 biorthogonal symmtric filtr Efficint implmntation: Lifting approach rf: Swldn s tutorial) M. Wu: EEE63 Advancd Signal Procssing Fall'9) /4/9 [39]

22 Smoothnss Conditions on Wavlt Filtr 4) Ensur th low band cofficints obtaind by rcursiv filtring can provid a smooth approximation of th original signal d by M.Wu UMCP EEE63 Slids crat From M. Vttrli s wavlt/filtr-bank papr M. Wu: EEE63 Advancd Signal Procssing Fall'9) /4/9 [4]