User manual MATHWORKS WAVELET TOOLBOX 4

[. . . ] Wavelet ToolboxTM 4 User's Guide Michel Misiti Yves Misiti Georges Oppenheim Jean-Michel Poggi How to Contact MathWorks Web Newsgroup www. mathworks. com/contact_TS. html Technical Support www. mathworks. com comp. soft-sys. matlab suggest@mathworks. com bugs@mathworks. com doc@mathworks. com service@mathworks. com info@mathworks. com Product enhancement suggestions Bug reports Documentation error reports Order status, license renewals, passcodes Sales, pricing, and general information 508-647-7000 (Phone) 508-647-7001 (Fax) The MathWorks, Inc. 3 Apple Hill Drive Natick, MA 01760-2098 For contact information about worldwide offices, see the MathWorks Web site. Wavelet ToolboxTM User's Guide © COPYRIGHT 1997­2010 by The MathWorks, Inc. The software described in this document is furnished under a license agreement. The software may be used or copied only under the terms of the license agreement. [. . . ] 'd' 'p' [ 1. 41421356] }; [ -1. 00000000] [ 0. 50000000] [ 0. 70710678] [0] [0] [] % Add a primal ELS to the lifting scheme. els = {'p', [-0. 125 0. 125], 0}; lsnew = addlift(lshaar, els); displs(lsnew); lsnew = {. . . 'd' 'p' 'p' [ 1. 41421356] }; [ -1. 00000000] [ 0. 50000000] [ -0. 12500000 0. 12500000] [ 0. 70710678] [0] [0] [0] [] % Transform the lifting scheme to biorthogonal % filters quadruplet. [LoD, HiD, LoR, HiR] = ls2filt(lsnew); % Visualize the two pairs of scaling and wavelet % functions. bswfun(LoD, HiD, LoR, HiR, 'plot'); 4-58 Lifting Method for Constructing Wavelets Illustrating LWT and integer LWT % Perform LWT at level 1 of a simple signal. x = 1:8; [cA, cD] = lwt(x, lsnew) cA = 1. 9445 cD = 0. 7071 0. 7071 0. 7071 0. 7071 4. 9497 7. 7782 10. 6066 % Perform % integer to integer LWT of the same signal. lshaarInt = liftwave('haar', 'int2int'); 4-59 4 Advanced Concepts lsnewInt = addlift(lshaarInt, els); [cAint, cDint] = lwt(x, lsnewInt) cAint = 1 cDint = 1 1 1 1 3 5 7 % Invert the two transforms. err = max(max(abs(x-ilwt(cA, cD, lsnew)))) err = 4. 4409e-016 errInt = max(max(abs(x-ilwt(cAint, cDint, lsnewInt)))) errInt = 0 Example 2. Two primal liftings starting from the Haar wavelet % Get Haar filters. [LoD, HiD, LoR, HiR] = wfilters('haar'); % Lift the Haar filters. twoels(1) = struct('type', 'p', 'value', . . . laurpoly([0. 125 -0. 125], 0)); twoels(2) = struct('type', 'p', 'value', . . . laurpoly([0. 125 -0. 125], 1)); [LoDN, HiDN, LoRN, HiRN] = liftfilt(LoD, HiD, LoR, HiR, twoels); % The biorthogonal wavelet bior1. 3 is obtained up to % an unsignificant sign. [LoDB, HiDB, LoRB, HiRB] = wfilters('bior1. 3'); samewavelet = . . . 4-60 Lifting Method for Constructing Wavelets isequal([LoDB, HiDB, LoRB, HiRB], [LoDN, -HiDN, LoRN, HiRN]) samewavelet = 1 % Visualize the two times two pairs of scaling and wavelet % functions. bswfun(LoDN, HiDN, LoRN, HiRN, 'plot'); 4-61 4 Advanced Concepts Frequently Asked Questions Continuous or Discrete Analysis? When is continuous analysis more appropriate than discrete analysis?To answer this, consider the related questions: Do you need to know all values of a continuous decomposition to reconstruct the signal s exactly?When the energy of the signal is finite, not all values of a decomposition are needed to exactly reconstruct the original signal, provided that you are using a wavelet that satisfies some admissibility condition (see [Dau92] pages 7, 24, and 27). In which case, a continuous-time signal s is characterized by the knowledge of the discrete transform . In such cases, discrete analysis is sufficient and continuous analysis is redundant. When the signal is recorded in continuous time or on a very fine time grid, both analyses are possible. It depends; each one has its own advantages: · Discrete analysis ensures space-saving coding and is sufficient for exact reconstruction. · Continuous analysis is often easier to interpret, since its redundancy tends to reinforce the traits and makes all information more visible. This is especially true of very subtle information. [. . . ] Class EDWTTREE (parent class: DTREE) Fields dtree dwtMode wavInfo Parent object DWT extension mode Structure (wavelet information) Fields Description wavInfo wavName Lo_D Hi_D Lo_R Hi_R Wavelet Name Low Decomposition filter High Decomposition filter Low Reconstruction filter High Reconstruction filter Methods edwttree merge plot recons split Constructor for the class EDWTTREE. Split (decompose) the data of a terminal node. Running This Example The following figure is obtained using the example ex1_edwt, selecting the De-noise option in the Tree Action menu and clicking the node 0. B-28 Advanced Use of Objects The approximations are labeled in yellow and the details are labeled in red. The title of the figure contains the DWT extension mode used ('sym' in the present example) and the name of the de-noising method. B-29 B Object-Oriented Programming B-30 Index A Index adding a new wavelet 5-2 algorithms decomposition 4-23 discrete wavelet transform (DWT) 4-19 fast wavelet transform (FWT) 4-19 filters 4-19 for biorthogonal 4-28 lifting wavelet transform (LWT) 4-55 Mallat 4-19 polyphase 4-57 rationale 4-28 reconstruction 4-30 stationary wavelet transform (SWT) 4-45 analysis 4-173 biorthogonal 4-78 case study 2-35 continuous coefficients 4-12 continuous or discrete 4-62 discrete coefficients 4-12 illustrated examples 2-2 local and global 4-14 multiscale 2-35 one-dimensional wavelet packet 3-7 orthogonal algorithm 4-28 and wavelet families 4-73 basis 4-62 dbN wavelets 4-76 filters 4-19 redundant 4-13 time-scale using redundant representation instead 4-13 translation invariant 4-45 two-dimensional wavelet packet 3-22 wavelet packet 3-2 See also transforms approximations coefficients discrete wavelet transform 4-23 definition 4-17 notation 4-3 wavelet decomposition 4-6 axes view A-13 B bases. See analysis, wavelet packets besttree function 4-167 binning density estimation 4-120 regression estimation 4-125 biorthogonal quadruplets 4-53 biorthogonal wavelets 4-78 definition 4-78 See also analysis border distortion boundary value replication 4-35 periodic extension 4-35 periodic padding 4-36 periodized wavelet transform 4-45 smooth padding 4-36 symmetric extension 4-35 symmetrization 4-35 zero-padding 4-35 breakdowns peak 2-33 proximal slopes 2-19 rupture 2-17 second derivative 2-21 variance 4-112 C centfrq function 4-68 Index-1 Index chirp signal example analysis 4-146 coefficients approximation fast wavelet transform 4-23 coloration A-21 detail fast wavelet transform 4-23 coiflets definition 4-77 Coloration Mode color coding A-2 controlling A-7 controlling the colormap A-6 colored AR(3) noise example 2-13 complex frequency B-spline wavelets 4-88 complex Gaussian wavelets 4-87 complex Morlet wavelets 4-87 complex Shannon wavelets 4-89 compressing images fingerprint example 1-27 true compression 4-136 compression ddencmp function 3-4 difference with de-noising 4-116 energy ratio 4-118 methods 4-132 norm recovery 4-118 number of zeros 4-119 predefined strategies 4-128 procedure wavelet packets 3-5 wavelets 4-115 retained energy 4-118 thresholding strategies 4-132 true 4-136 using wavelet packets 3-26 D Daubechies wavelets definition 4-74 de-noising basic model one-dimensional 4-102 two-dimensional 4-111 default values 3-4 fixed form threshold 4-105 methods 4-132 minimax performance 4-105 noise size estimate 4-107 nonwhite noise 4-107 predefined strategies 4-128 procedure wavelet packets 3-5 wavelets 4-103 SURE estimate 4-105 using SWT 2-D analysis example 1-24 variance adaptive 4-112 white noise 4-101 de-noising images 2-D wavelet analysis and 2-D stationary wavelet analysis 1-21 two-dimensional procedure 4-111 de-noising signals wavelet analysis 1-18 decimation. [. . . ]