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FWT2D(2) Scilab Function FWT2D(2)
NAME
FWT2D - 2D Forward Disrete Wavelet Transform
Author: Bertrand Guiheneuf
This routine computes discrete wavelet transforms of a 2D real signal. Two
transforms are possible : Orthogonal and Biorthogonal
Usage
[wt,index,length]=FWT2D(Input,NbIter,f1,[f2])
Input parameters
o Input : real matrix [m,n] Contains the signal to be decomposed.
o NbIter : real positive scalar Number of decomposition Levels
o f1 : Analysis filter
o f2 : real unidimensional matrix [m,n] Synthesis filter
Output parameters
o wt : real matrix Wavelet transform. Contains all the datas of the
decomposition.
o index : real matrix [NbIter,4] Contains the indexes (in wt) of the
projection of the signal on the multiresolution subspaces
o length : real matrix [NbIter,2] Contains the dimensions of each pro-
jection
Description
Introduction
The 2D discrete wavelet transform of Input is a projection on 2D multireso-
lution Spaces. The number of scales NbIter tells how many convolutions are
computed. Each convolution is followed by a downsampling of the signal in
both direction. For example, if the original matrix is (256,512), a result-
ing projection after the first iteration is (128,256). In 2D, there are 4
projections for each iteration corresponding to 2 projections in the row
directions and 2 in the column direction. In each direction, the 2 projec-
tions are obtained through the convolutions with a low-pass filter and its
associated high-pass filter. The projections are then HL HH LH LL where the
first letter represents the filter used for the row filtering and the
second letter is the filter used for column filtering. H is High-Pass
filter and L Low-pass filter. Except for the last level where the four con-
volutions are kept, the LL output is always used as the input of the fol-
lowing iteration. Two types of filters are available : Quadrature Mirror
Filters (Orthogonal) or Conjugate Quadrature Filters (Biorthogonal). Each
one allows perfect reconstruction of the signal but only CQF pairs can be
symetric. The advantage of QMF is that synthesis and reconstruction filters
are the same.
Parameters
Input must be a real matrix. All dimensions are allowed but for a 1D vec-
tor, FWT is best suited. NbIter is the number of scales computed. It must
be a positive integer greater than one and should be smaller then
log2(max(size(Input))) but this is not necessary. f1 is the linear FIR
filter used for the analysis and might be obtained with MakeQMF() or
MakeCQF() f2 is the linear FIR filter to use for the reconstruction. It is
only necessary if f1 has been obtained with MakeCQF(). wt is the wavelet
decomposition structure. The next two parametres must be used to read the
wavelet coefficients. index contains the indexes of the first coefficient
of each output. At each scale Scale, the output indexes are: index(Scale,1)
: HL index(Scale,2) : LH index(Scale,3) : HH index(Scale,4) : LL on the
last scale and 0 otherwise length contains the dimensions (height, width)
of each output at a given Iteration.
Algorithm details
Convolutions are computed through discrete linear convolutions in time
domain. No FFT is used. The signal is mirrored at its boundaries. The
wavelet structure (wt) is a vector and NOT a 2D matrix. It contains all the
informatiosn for the reconstruction: wt(1) : height of the original signal
wt(2) : width of the original signal wt(3) : Number of iterations wt(4) :
Number of causal coefficients of the synthesis filter wt(5) : Number of
anticausal coefficients of the synthesis filter then the Synthesis filter
coefficients and finaly the wavelet coefficient are stored .
Examples
a=rand(256,256); q=MakeQMF('daubechies',4); wt,wti,wtl = FWT2D(a,3,q);
V=WT2Dext(wt,1,2); viewmat(V); Then to suppress the Lowest Frequency com-
ponent and then reconstruction: index=0; for i=1:wtl(3,1), for
j=1:wtl(3,2), wt(wti(3,4)+index)=0; end; end; result=IWT2D(wt);
References
Meyer Y. : Wavelets, Algorithms & Applications, SIAM. Meyer Y. :
Ondelettes et Operateurs (I) : Hermann, Paris Daubechies I. : Ten Lectures
on Wavelets, CBMS-NSF Regional conference series in applied mathematics.
See Also
IWT2D, MakeQMF, MakeCQF, WT2Dext, WT2DVisu, WT2DStruct
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