Wavelet Toolbox

Lifting Functions

The lifting functions of the Wavelet Toolbox are organized into five groups:

• Lifting Schemes
• Biorthogonal Quadruplets of Filters and Lifting schemes
• Usual Biorthogonal Quadruplets
• Lifting Wavelet Transform (LWT)
• Laurent Polynomials and Matrices

Lifting Schemes

Function Name	Description
lsinfo	Information about lifting schemes
displs	Display a lifting scheme
addlift	Add primal or dual elementary lifting steps to a lifting scheme

Biorthogonal Quadruplets of Filters and Lifting schemes

These functions connect lifting schemes to biorthogonal quadruplets of filters and associated scaling and wavelet function pairs.

Function Name	Description
liftfilt	Apply elementary lifting steps on quadruplet of filters
filt2ls	Transform a quadruplet of filters to a lifting scheme
ls2filt	Transform a lifting scheme to a quadruplet of filters
bswfun	Compute and plot biorthogonal "scaling and wavelet" functions

Usual Biorthogonal Quadruplets

These functions provide some basic lifting schemes associated with some usual orthogonal or biorthogonal ("true") wavelets and the "lazy" one. These schemes can be used to initialize a lifting procedure.

Function Name	Description
wavenames	Provides usual wavelet names available for LWT
liftwave	Provides lifting scheme associated with a usual wavelet
wave2lp	Provides Laurent polynomials associated with a usual wavelet

Lifting Wavelet Transform (LWT)

These functions contain the direct and inverse lifting wavelet transform (LWT) M-files for both 1-D and 2-D signals. LWT reduces to the polyphase version of the DWT algorithm with zero-padding extension mode and without extra-coefficients.

Function Name	Description
lwt	1-D lifting wavelet transform
ilwt	Inverse 1-D lifting wavelet transform
lwtcoef	Extract or reconstruct 1-D LWT wavelet coefficients
lwt2	2-D lifting wavelet transform
ilwt2	Inverse 2-D lifting wavelet transform
lwtcoef2	Extract or reconstruct 2-D LWT wavelet coefficients

Laurent Polynomials and Matrices

These functions permit an entry to representation and calculus of Laurent polynomials and matrices.

Function Name	Description
laurpoly	Constructor for the class of Laurent polynomials
laurmat	Constructor for the class of Laurent matrices

The lifting directory and the two object directories @laurpoly and @laurmat contain many other M-files.

Examples of Lifting Methods

These two simple examples illustrate the basic lifting capabilities of the Wavelet Toolbox. For more examples, see Wavelets in Action: Examples and Case Studies and the demos provided with the Wavelet Toolbox.

Example 1: .   A primal lifting starting from Haar wavelet

• % Start from the Haar wavelet and get the corresponding 
  % lifting scheme.
  lshaar = liftwave('haar');
  
  % Visualize the obtained lifting scheme.
  displs(lshaar);
  
  lshaar = {...                        
  'd'             [ -1.00000000]  [0]  
  'p'             [  0.50000000]  [0]  
  [  1.41421356]  [  0.70710678]  []   
  };                                   
  
  % Add a primal ELS to the lifting scheme.
  els = {'p',[-0.125 0.125],0};
  lsnew = addlift(lshaar,els);
  displs(lsnew);
  
  lsnew = {...                                     
  'd'             [ -1.00000000]              [0]  
  'p'             [  0.50000000]              [0]  
  'p'             [ -0.12500000  0.12500000]  [0]  
  [  1.41421356]  [  0.70710678]              []   
  };                                               
  
  % 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');
  
  
  
  

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    4.9497    7.7782   10.6066
  
  cD =
  
      0.7071    0.7071    0.7071    0.7071
  
  % Perform integer to integer LWT of the same signal.
  lshaarInt = liftwave('haar','int2int');
  lsnewInt = addlift(lshaarInt,els);
  [cAint,cDint] = lwt(x,lsnewInt)
  
  cAint =
  
       1     3     5     7
  
  cDint =
  
       1     1     1     1
  
  % 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 = 
  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');
  
  
   
  

Lifting Background

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