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MWMP
Multiwavelet MATLAB Package
Author: Vasily Strela
strela@math.dartmouth.edu
http://math.dartmouth.edu/~strela
MWMP is a library of Matlab 5 routines for multiwavelet analysis. Its aim
is to give researchers an opportunity to try multiwavelets in practice.
MWMP can be used for comparison of scalar and multiple filters
in image compression and signal denoising.
MWMP offers a variety of builtin scalar- and multi- filters (all filters are
considered to be matrix). Several types of prefilters are included.
For descriptions of available filters and their properties see functions
coef and coef_prep. New sets of coefficients can be easily added.
In addition to routines implementing preprocessing and discrete multiwavelet
transform (in 1 and 2 dimensions) the package contains a simple compression
function and several scripts for signal denoising via thresholding.
For description of multiwavelet thresholding methods see reference [SW] below.
There also is a routine which plots multiscaling and multiwavelet functions
and a routine which computes the transition operator.
All scripts in MWMP start with a help section describing input and output
parameters and giving an example of usage. Often there are references
to the literature from the list below. Function example1D demonstrates
how to use the transform functions. More complicated examples
can be found in scripts simpcomp2D (image compression via retaining given
number of largest coefficients) and th2D (denoising via thresholding).
To install the package on a UNIX machine:
1. Download the archive MWMP.tar.gz from
http://math.dartmouth.edu/~strela/MWMP
2. Uncompress the archive
3. Create a directory and move uncompressed archive there
4. tar -xvf MWMP.tar
5. Modify your Matlab path to include the directory
To install the package on a DOS machine:
1. Download the archive MWMP.zip from
http://math.dartmouth.edu/~strela/MWMP
2. Create a directory and move compressed archive there
4. pkunzip MWMP.zip
5. Modify your Matlab path to include the directory
All files are also available in ASCII format at
http://math.dartmouth.edu/~strela/MWMP/MWMP-SOURCE
To view the list and short descriptions of all available routines,
start Matlab 5 and type
help MWMP_list
To obtain more information on a specific _function_ (including description
of input and output parameters, example of usage), type
help _function_
In order to speed up the computations we recommend to compile routines
in to mex files using Matlab function mcc.
Please send your questions, comments, and corrections to Vasily Strela
strela@math.dartmouth.edu
Acknowledgments:
This software was partially developed while Vasily Strela was with the
Statistics Section, Imperial College of Science, Technology and Medicine
supported by EPSRC grant GR/L11182. Author is very grateful for this support.
Thanks to Fritz Keinert for providing precise coefficients for 'la8'
and 'bi7'.
LIST OF LITERATURE:
[CL] C. K. Chui and J. A. Lian, "A study of orthonormal multiwavelets",
Texas A&M University CAT Report 351 (1995).
[D] I. Daubechies, "Ten Lectures on Wavelets", SIAM, Philadelphia (1992).
[DJ] D. L. Donoho and I. M. Johnstone, "Ideal spatial adaptation by
wavelet shrinkage", Biometrica 81 (1994) 425-455.
[DS] T. R. Downie and B. W. Silverman, "The discrete multiple wavelet
transform and thresholding methods", IEEE Trans. on SP, to appear.
[GHM] J. S. Geronimo, D. P. Hardin, and P. R. Massopust, "Fractal functions and
wavelet expansions based on several functions", J. Approx. Theory
78 (1994) 373-401.
[HR] D. P. Hardin and D. W. Roach, "Multiwavelet prefilters I:
Orthogonal prefilters preserving approximation order p <= 2",
preprint (1997).
[HSS] C. Heil, G. Strang, and V. Strela, "Approximation by translates of
refinable functions", Numerische Mathematik 73 (1996) 75-94.
[J] Q. Jiang, "On the regularity of matrix refinable functions",
SIAM J. Math. Anal., to appear.
[Se] I. Selesnick, "Cardinal Multiwavelets and the Sampling Theorem",
prerprint (1998).
[STT] L.-X. Shen, H. H. Tan, and J. Y. Tham, "Symmetric-antisymmetric
orthonormal multiwavelets and related scalar wavelets",
preprint (1997).
[SS] G. Strang and V. Strela, "Short wavelets and matrix dilation equations",
IEEE Trans. on SP 43 (1995) 108-115.
[S] V. Strela, "A note on Construction of biorthogonal multi-scaling
functions", in Contemporary Mathematics, A. Aldroubi and
E. B. Lin (eds.), AMS (1998) 149-157.
[SHSTH] V. Strela, P. Heller, G. Strang, P. Topiwala, and C. Heil,
"The application of multiwavelet filter banks to signal and
image processing", IEEE Trans. on Image Proc. (1998).
[SW] V. Strela and A. T. Walden, "Signal and Image Denoising via Wavelet
Thresholding: Orthogonal and Biorthogonal, Scalar and Multiple Wavelet
Transforms", Imperial College, Statistics Section,
Technical Report TR-98-01 (1998).
[SW1] V. Strela and A. T. Walden, "Orthogonal and biorthogonal multiwavelets
for signal denoising and image compression", SPIE Proc. 3391
AeroSense 98, Orlando, Florida, April 1998.
[TS] R. Turcajova and V. Strela, " Smooth Hermite spline multiwavelets",
in preparation.
[XGHS] X.-G. Xia, J. S. Geronimo, D. P. Hardin, and B. W. Suter,
"Design of prefilters for discrete multiwavelet transforms",
IEEE Trans. on SP, 44 (1996) 25-35.
附注：对于2-D的对小波变换，首先要进行预滤波，将信号变成可以进行多小波变换的格式，
然后再进行多小波变换，这时可以将分解的图画出，从图中可以看到分解后的频带分
布，如果要将分解后的信号进行重构的话，要先对分解后的子频带进行后滤波。然后
才能进行重构。
进行多小波变换的一个主要问题在于预滤波的设计。如果预滤波的设计不好会影响多
小波变换的效果。
如果对多小波进行平衡处理的话，不要进行预滤波，它只要对信号变成它所能够处理
的格式，这种处理方法只要取单位矩阵I就可以了。
这里的平衡多小波变换没有点问题。
'clbal'平衡多小波可能有问题。'clghm'平衡多小波也可能有问题。（平衡化有问题）
注意：要再添加一个多小波，那最优的多小波，它有最小的时频分辨力。
这样，对于去躁与压缩主要在于对分解后的频带进行处理。