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％ Atomizer Main Directory, Version .802 里面信号含有分解去噪合成过程的代码 ％---------------------------------------

Also one can extend Atomizer to his own dictionary by supplying
the fast analysis/synthesis routines in a certain fashion
(See V. Customizing for detail).

In order to apply Atomizer to images, one needs to convert 2-d images
(n-by-n matrix) into 1-d objects (n^2-by-1 matrix), and design the fast
analysis/synthesis operators for his dictionary in the same fashion.

IV.  Getting Started

One should first install the software (See VI. Access and
Installation), and get the article "Atomic Decomposition by Basis
Pursuit" by Chen, Donoho and Saunders
(http:/www-stat.stanford.edu/~Atomizer) to have to overview of
the methodology in Atomizer. One can then start on Atomizer by
studying and trying out the demo scripts in the Scripts/Demos
subdirectory. Hopefully one can then be ready to analyze some of the
datasets in the Datasets/ subdirectory using the atomic decomposition
routines in Atomizer. It should also be very helpful for one to study
and try the scripts in Scripts/BP to reproduce figures in Chen and
Donoho.

V.   Customizing

V.1 Tuning optimzation paramters

BP_Interior: Our implementation of basis pursuit is based on a
primal-dual log barrier interior point method. Since it takes a huge
amount of computation ($n^3.5$) to achieve exact optimality,
BP_Interior only attemps to calculate "approximate" optimal solutions
within affordable time and storage.  The accuracy of BP_Interior is
controlled by several parameters. MaxBPIter controls the maximum
number of interior point iterations. FeaTol and OptTol are two
stopping criterions for BP_Interior; once the current iteration
achieves the amount of feasibility defined by FeaTol and the amount
primal-dual gap defined by OptTol, BP_Interior will stop. Inside each
BP_Interior iteration, a linear system is solved by conjugate
gradients method. The accuracy of the conjugate gradients solver is
controlled by CGAccuracy; MaxCGIter defines the maximum number of
conjugate gradients iterations. delta and gamma are two perturbation
parameters. We suggest that one should set OptTol, FeaTol and
CGAccuracy at the same level and set delta and gamma to one tenth of
that level.  According to our experience, the level 10^(-1) would give
a decent approximation and the level 10^(-3) would give a very good
approximation.

MOF: We use a conjugate gradients solver to calculate the generalized
inverse in MOF. The accuracy of the conjugate gradients solver is
controlled by CGAccuracy (default = 10 ^ -5); When dictionary matrices
are poorly conditioned, one might want to increase the maximum number of
conjugate gradients iterations defined by MaxCGIter.

MP: The number of atoms admitted by matching pursuit is controlled by
two paramters. natom defines the maximum number of atoms admitted; MP
stops once the coefficient of the selected atom gets below a fraction
(defined by frac) of the orignal signal energy. Also the
implementation of MP assumes that dictionary atoms are normalized by
$l^2$ norms; otherwise one has to change the way MP calculates
coefficients for atoms.

BOB: One can change the entropy in the best orthogonal basis algorithm
by changing the parameter entropy and par. The default entropy is
the $l^1$ entropy.

The denoising routines can be tuned in similar ways as their
counterparts discussed above, except they have a thresholding
parameter $\lambda$. For BPDN_Interior and MOFDN, $\lambda$ is the
penalization (or regulaization) parameter. For MPDN, $\lambda$ plays
the same role as frac in MP. The default values for $\lambda$ for all
the denoising routines are set to be $\sqrt{2 m \log(m)}$ where $m$ is
the cardinality of the dictionary. There are good reasons that we
choose such $\lambda$ for BPDN_Interior and MPDN, one shouldn't change
that for usual purpose.

V.2 Merging dictionaries

One can merge any of the implemented dictionaries to form a new mega 
dictionary by using MakeList.
e.g.
	dict1 = 'WP' ; par11 = D; par12 = qmf; par13 = 0;
	dict2 = 'DCT'; par21 = 2; par22 = 0;   par23 = 0;
One can use:
	dict = MakeList(dict1, dict2);
	par1 = MakeList(par11, par21);
	par2 = MakeList(par12, par22);
	par3 = MakeList(par13, par23);
to represent the merged dictionary

V.3 Adding new dictionaries

One can add a new dictionary by supplying the routines for the analysis 
and synthesis operator in the following format:
	Analysis:
		function c = Fast"NAMEOFDICT"Analysis(x, par1, par2, par3)
	Synthesis:
		function x = Fast"NAMEOFDICT"Synthesis(c, par1, par2, par3)
e.g.
	If one wants to add a new dictionary with parameters
		NameOfDict = 'SHIFT'
		par1 = Amount of SHIFT
	he needs to supply the following two matlab functions:
	
	FastSHIFTAnalysis.m
		function c = FastSHIFTAnalysis(x, par1, par2, par3)
		n = length(x);
		c = [x(par1+1:n); x(1:par1)];
	FastSHIFTSynthesis
		function x = FastSHIFTSynthesis(c, par1, par2, par3)
		n = length(c);
		x = [c(n-par1+1:n); c(1:n-par1)];


VI.  Access and Installation

Atomizer is freeware and may be obtained via:
        
    * World-Wide-Web:
    
        http://www-stat.stanford.edu/~Atomizer
        
        Your WWW browser may be used to peruse the documentation.
        
Please send comments or questions to [e-mail]



VII. Appendix

Here is the list of dictionaries implemented in Atomizer with their
specification.

Dictionaries normalized by l^2 norm

        o Dirac
               	NameOfDict = 'DIRAC'
        o Discrete Cosine Transform
               	NameOfDict = 'DCT'
               	par1 = overcompleteness (integer)
        o Discrete Sine Transform
               	NameOfDict = 'DST'
               	par1 = overcompleteness (integer)
        o Periodized-Orthogonal Wavelet
               	NameOfDict = 'WAVE'
               	par1 = L, coarsest level
               	par2 = qmf
        o Periodized-Biothogonal-Symmetric Wavelet
               	NameOfDict = 'PBS'
               	par1 = L, coarsest level
               	par2 = qmf
               	par3 = dqmf
        o Stationary PO Wavelet
               	NameOfDict = 'STAT'
               	par1 = D, degree of finest frequency partition
               	par2 = qmf
        o Spline (Stationary PBS Wavelet)
               	NameOfDict = 'SP'
               	par1 = D, degree of finest frequency partition
               	par2 = qmf
               	par3 = dqmf
        o Wavelet Packet
               	NameOfDict = 'WP'
               	par1 = D, degree of finest frequency partition
               	par2 = qmf
        o Cosine Packet
               	NameOfDict = 'CP'
               	par1 = D, depth of finest time splitting
	o Multi-Duration-Cosines
		NameOfDict = 'MCD'
		par1 = the depth, 0<=D<=log2(n)
		par2 = overcompleteness (integer)

Dictionaries normalized by total variation

        o HeaviSide
               	NameOfDict = 'HeaviSide'
	o Jump (Smoothed Heaviside)
		NameOfDict = 'JUMP'
		par1 = 	the decaying speed, 0 < par1 < 1, where
			small values correspond to faster decaying
	o POTV (PO Wavelet normalized by Total Variation)
		NameOfDict = 'JUMP'
               	par1 = L, coarsest level
               	par2 = qmf