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19  Tools for fractal analysis

19.1   AtanH Arctangent variation

Author: Paulo Goncalves

Generates an arc-tangent trajectory






USAGE :



Ht=AtanH(N,h1,h2,shape);



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :




EXAMPLE: :

[Ht] = AtanH(1024,0,1,0.9,0.01) ;

19.2   FWT 1D Forward Discrete Wavelet Transform

Author: Bertrand Guiheneuf

This routine computes discrete wavelet transforms of a 1D real signal. Two transforms are possible : Orthogonal and Biorthogonal






USAGE :

[wt,index,length]=FWT(Input,NbIter,f1,[f2])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




INTRODUCTION  :

The discrete wavelet transform of Input is a projection on multiresolution Spaces. The number of scales NbIter tells how many convolutions are computed. Each convolution is followed by a downsampling of the output. For example, if the original signal size is 500, the resulting size of the projection after the first iteration is 250. Each iteration consists then in two convolution/downsampling steps. One is high-pass (H) and the other one is low-pass (L). Except for the last iteration, the low-pass output is used as the input of the next iteration. Thus, only the high-pass is stored in wt except at the last iteration where both the outputs are stored. This explains why the wti array dimension is equal to NbIter+1. The last index index(NbIter+1) is the index of firts element of the last low-pass projection. 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 unidimensional matrix. NbIter is the number of scales computed. It must be a positive integer greater than one and should be smaller than 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. length contains the dimension of each output.


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 contains all the informations for the reconstruction: wt(1) : size of the original signal wt(2) : Number of iterations wt(3) : Number of causal coefficients of the synthesis filter wt(4) : Number of anticausal coefficients of the synthesis filter then the Synthesis filter coefficients and finaly the wavelet coefficient are stored .


EXAMPLES :

a=rand(1,250); q=MakeQMF('daubechies',4); wt,wti,wtl = FWT(a,6,q); M=WTMultires(wt); plot(M(2,:)); Then to suppress the Lowest Frequency component and then reconstruction: for i=1:wtl(6), wt(wti(6)+i-1)=0; end; result=IWT(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 :

IWT, MakeQMF, MakeCQF, WTStruct, WTNbScales, WTMultires

19.3   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 :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




INTRODUCTION  :

The 2D discrete wavelet transform of Input is a projection on 2D multiresolution 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 resulting 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 projections 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 convolutions are kept, the LL output is always used as the input of the following 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 vector, 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 component 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

19.4   GeneWei Generalized Weierstrass function

Author: Paulo Goncalves

Generates a Generalized Weierstrass function






USAGE :



[x,Ht]=GeneWei(N,ht,lambda,tmax,randflag)



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :




EXAMPLE: :

[x,Ht] = GeneWei(1024,'abs(sin(16*t))',2,1,0) ;

19.5   IWT 1D Inverse Discrete Wavelet Transform

Author: Bertrand Guiheneuf

This routine computes inverse discrete wavelet transforms of a real signal. Two inverse transforms are possible : Orthogonal and Biorthogonal






USAGE :

[result]=IWT2D(wt,[f])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




INTRODUCTION  :

The wavelet transform is an invertible linear transform. This routines is the inverse transform. For details on the algorithm procedure, see FWT.


PARAMETERS  :

Input must be a real matrix. It's generally obtained with FWT but can be created "by hand". In that case, it's strongly recommended to decompose a null signal with FWT. f is the linear FIR filter to use for the reconstruction. It is only necessary if the analysis filter had been obtained with MakeCQF() and the reconstruction filter had not been passed to FWT. If not specified the filter given in the synthesis is used. (See FWT). result is the reconstructed signal. It has the same dimension as the original one.


ALGORITHM DETAILS  :

Convolutions are computed through discrete linear convolutions in time domain. No FFT is used. The signal is mirrored at its boundaries. .SH Example a=rand(1,256); q=MakeQMF('daubechies',4); wt,wti,wtl = FWT(a,8,q); wt=abs(wt); result=IWT(wt);.SH 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 :

FWT, MakeQMF, MakeCQF, WTMultires, WTStruct

19.6   IWT2D 2D Inverse Disrete Wavelet Transform

Author: Bertrand Guiheneuf

This routine computes inverse discrete wavelet transforms of a 2D real signal. Two inverse transforms are possible : Orthogonal and Biorthogonal






USAGE :

[result]=IWT2D(wt,[f])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




INTRODUCTION  :

The wavelet transform is an invertible linear transform. This routines is the inverse transform. For details on the algorithm procedure, see FWT2D.


PARAMETERS  :

Input must be a real matrix. It's generally obtained with FWT2D but can be created "by hand". In that case, it's strongly recommended to decompose a null signal with FWT2D. f is the linear FIR filter to use for the reconstruction. It is only necessary if the analysis filter had been obtained with MakeCQF() and the reconstruction filter had not been passed to FWT2D. If not specified the filter given in the synthesis is used. (See FWT2D). result is the reconstructed signal. It has the same dimensions as the original one.


ALGORITHM DETAILS  :

Convolutions are computed through discrete linear convolutions in time domain. No FFT is used. The signal is mirrored at its boundaries. .SH Example a=rand(256,256); q=MakeQMF('daubechies',4); wt,wti,wtl = FWT2D(a,8,q); wt=abs(wt); result=IWT2D(wt);.SH 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 :

FWT2D, MakeQMF, MakeCQF, WT2Dext, WT2DVisu



19.7   Koutrouvelis Stable Law parameters estimation (Koutrouvelis method)

Author: Lotfi Belkacem

This routine estimates parameters of a stable law using the Koutrouvelis (1985) method.






USAGE :

[alpha,beta,mu,gamma]=Koutrouvelis(data)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




EXAMPLE :

[proc1,inc1]=sim_stable(1,0,0,1,10000); generates a standard 1-stable motion (Cauchy process). [alpha,beta,mu,gamma]=Koutrouvelis(inc1); estimates parameters of the previous simutated 1-stable random variable inc1.


REMARQUE :

Skewness and location parameters are badly estimated with this methode.



19.8   McCulloch Stable law parameters estimation (McCulloch method)

Author: Lotfi Belkacem

This routine estimates parameters of a stable law using the Mc-Culloch (1985) method.






USAGE :

[param,sd_param]=McCulloch(data)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




EXAMPLE :

[proc1.5,inc1.5]=sim_stable(1.5,0,0,1,10000); generates a standard 1.5-stable motion. [param,sd_param]=McCulloch(inc1.5); estimates parameters of the previous simutated 1.5-stable random variable inc1.5 To visualize the estimates parameters or their sd-deviations use respectively param or sd_param. alpha=param(1), beta=param(2), mu=param(3), gamma=param(4). sd_alpha=sd_param(1), sd_alphabeta=sd_param(2), sd_alphamu=sd_param(3), sd_gamma=sd_param(4).


REMARQUE :

Skewness parameter and its sd-deviation estimations are not very accurate. Specially when the characteristic exponent is arround 2.



19.9   WT2DStruct Retrieve the Structure of a 2D DWT

Author: Bertrand Guiheneuf

This routine retrieve the structure informations contained in a 2D Wavelet Transform.






USAGE :

[ScIndex, ScLength]=WT2DStruct(wt)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




INTRODUCTION  :

This routine is used to retreive the structure information of a wavelet transform. It must be used in all routine that might work on a wavelet transform whose structure is not passed as an imput parameter. (That should be the case of all routines taking a Wavelet Transform as input parameter to minimize the input).see FWT2D.


PARAMETERS  :

Input must be a real matrix. It's generally obtained with FWT2D. It contains the wavelet transform. 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.


EXAMPLE :

a=rand(256,256); q=MakeQMF('daubechies',4); wt = FWT2D(a,8,q); (a few days pass...) wti, wtl=WT2DStruct(wt); wtl


SEE ALSO :

FWT2D, IWT2D, WT2Dext, WT2DVisu



19.10   WT2DVisu Visualise a 2D Multiresolution

Author: Bertrand Guiheneuf

This routine constructs a matrix that shows all the wavelet coefficients of a 2D matrix.






USAGE :

[V]=WT2DVisu(wt)


INPUT PARAMETER :

1




OUTPUT PARAMETER :

1




DESCRIPTION :




INTRODUCTION  :

This routine is used to display all the scales and all the frequency components of a wavelet transform.


PARAMETERS  :

wt must be a real matrix. It's generally obtained with FWT2D. V the wavelet coefficents.


EXAMPLE :

a=rand(256,256); q=MakeQMF('daubechies',4); wt = FWT2D(a,8,q); V=WT2DVisu(wt); viewmat(V);


SEE ALSO :

FWT2D, IWT2D, WT2Dext,

19.11   WT2Dext Extract a Projection from a 2D WT

Author: Bertrand Guiheneuf

This routine extracts a projection from the wavelet transform of a 2D matrix.






USAGE :

[V]=WT2Dext(wt, Scale, Num)


INPUT PARAMETER :

1




OUTPUT PARAMETER :

1




DESCRIPTION :




INTRODUCTION  :

At each scale, a wavelet transform contains 4 outputs (HL, HH, LH and HH at the last scale). This routine is used to extract a particular component at a desired scale.


PARAMETERS  :

wt must be a real matrix. It's generally obtained with FWT2D. It contains the wavelet transform coefficients. Num is 1,2,3, or 4 (at the last scale). Each number coresponds to a particular 2D frequency component.

1




EXAMPLE :

a=rand(256,256); q=MakeQMF('daubechies',4); wt = FWT2D(a,8,q); V=WT2Dext(wt,2,2); viewmat(V);


SEE ALSO :

FWT2D, IWT2D, WT2DVisu,

19.12   WTMultires Construct a 1D Multiresolution Representation

Author: Bertrand Guiheneuf

This routine constructs a matrix that shows the projections of the signal on each multiresolution subspace






USAGE :

[V]=WTMultires(wt)


INPUT PARAMETER :

1




OUTPUT PARAMETER :

1




DESCRIPTION :




INTRODUCTION  :

This routine is used to display all the scales of a wavelet transform. The projections are different from the wavelet coefficients as they represent "filtered" signals. Here each projection


PARAMETERS  :

wt must be a real matrix containing the wavelet coefficients but also misc informations such as the original signal dimension and the reconstruction filter. It's generally obtained with FW. V Is the matrix containing the projection of the signal (decomposed in wt) on each Multiresolution subspace. The Nbiter first ones are the projections on the details subspaces. The last one is the projection on the last trend subspace.


EXAMPLE :

x=0.1:0.005:1; s=(x.0.7) .* sin( x.(-2)) ; q1 q2=MakeCQF(1); wt = FWT(s,3,q1,q2); V=WTMultires(wt); plot(V');


SEE ALSO :

FWT, IWT, WTStruct,

19.13   WTStruct Retrieve a 1D Discrete Wavelet Structure.

Author: Bertrand Guiheneuf

This routine retrieves the structure informations contained in a 1D Wavelet Transform.






USAGE :

[ScIndex, ScLength]=WT2DStruct(wt)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




INTRODUCTION  :

This routine is used to retreive the structure information of a wavelet transform. It must be used in all routine that might work on a wavelet transform whose structure is not passed as an imput parameter. (That should be the case of all routines taking a Wavelet Transform as input parameter to minimize the input).


PARAMETERS  :

Input must be a real matrix. It's generally obtained with FWT. It contains the wavelet transform. index contains the indexes of the first coefficient of each output. The first "NbIter" indexes are the indexes (in wt) of the "high-pass" subspaces projections (Wj), the last one is the last "low-pass" projection (Vj); length contains the dimension of each output.


EXAMPLE :

a=rand(1,256); q=MakeQMF('daubechies',4); wt = FWT2D(a,8,q); wti, wtl=WTStruct(wt); wtl


SEE ALSO :

FWT2D, IWT2D, WT2Dext, WT2DVisu



19.14   alphagifs Holder function estimation using IFS

Author: Khalid Daoudi

Estimates the pointwise Holder exponents of a 1-D real signal using the GIFS method.






USAGE :

[Alpha, Ci]=wave2gifs(sig, limtype)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS :

1




ALGORITHM DETAILS :

The algorithm uses the GIFS method to estimate the Holder exponent at each point of a given signal. The first step of this method consists in computing the coefficients of the GIFS whose attractor is the given signal. In the second step, we replace each coefficient which absolute value is greater than 1 (resp. smaller than 1/2) by 1 (resp. by 1/2). We then perform the computation of the limit that yields the estimated Holder function using the chosen type of limit.


SEE ALSO: :

gifs and prescalpha


EXAMPLE: :

Synthesis of an fbm with exponent H=0.7 (of size 1024 samples) : x = fmblevinson(1024,0.7) ; Estimation of The Holder function : Alpha = alphagifs(x,'slope'); plot(Alpha)



19.15   bbch beneath-beyond concave hull

Author: Christophe Canus

This C_LAB routine determines the concave hull of a function graph using the beneath-beyond algorithm.






USAGE :

[rx,ru_x]=bbch(x,u_x)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The abscissa x and the function u_x to be regularized must be of the same size [1,N] or [N,1]. The abscissa rx and the concave regularized function ru_x are of the same size [1,M] with M<=N.


ALGORITHM DETAILS :

Standard beneath-beyond algorithm.


EXAMPLES :




MATLAB :



h=.3;beta=3;
N=1000;
% chirp singularity (h,beta)
x=linspace(0.,1.,N);
u_x=abs(x).^h.*sin(abs(x).^(-beta));
plot(x,u_x);
hold on;
[rx,ru_x]=bbch(x,u_x);
plot(rx,ru_x,'rd');
plot(x,abs(x).^h,'k');



SCILAB :



//



REFERENCES :

None.SH See Also linearlt (C_LAB routine).

19.16   binom binomial measure synthesis

Author: Christophe Canus

This C_LAB routine synthesizes a large range of pre-multifractal measures related to the binomial measure paradigm (deterministic, shuffled, pertubated, and mixing of two binomials: lumping and sum) and computes linked theoretical functions (partition sum function, Reyni exponents function, generalized dimensions, multifractal spectrum).






USAGE :

[varargout,[optvarargout]]=binom(p0,str,varargin,[optvarargin])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The binomial measure is completly characterized by its weight p0. This first parameter must be >0. and <1. (the case of p0=.5 corresponds to the Lebesgue measure). The second parameter str is a variable string used to determine the desired type of output. There are six suffix strings ('meas' for measure, 'cdf' for cumulative distribution function, , 'pdf' for probability density function, 'part' for partition sum function, 'Reyni' for Reyni exponent function , 'spec' for multifractal spectrum) for the deterministic binomial measure and a lot of possibly composed prefix strings for related measures ('shuf' for shuffled, 'pert' for pertubated, 'lump' for lumping , 'sum' for sum, 'sumpert' for sum of pertubated, and so on) which can be added to the first ones to form composed strings. For example, 'lumppertmeas' is for the synthesis of the lumping of 2 pertubated binomial pre-multifractal measures and 'sumspec' is for the computation of the multifractal spectrum of the sum of two binomials. Note that all combinaisons of strings are not implemented yet. When a string containing suffix string 'meas' is given as second input, a pre-multifractal measure mu_n (first output argument) is synthesized on the dyadic intervals I_n (second optional output argument) of the unit interval. In that case, the third input argument is a strictly positive real (integer) scalar n which contains the resolution of the pre-multifractal measure. The size of the output real vectors mu_n (and I_n if used) is equal to 2n (so be aware the stack size ;-)). This option is implemented for the deterministic ('meas'), shuffled ('shufmeas') and pertubated ('pertmeas') binomial, and also for the mixing (lumping or sum) of two deterministic ('lumpmeas' and 'summeas') or pertubated ('lumppertmeas' and 'sumpertmeas') binomial measures. When a string containing prefix 'shuf' is given as second input, the synthesis is made for a shuffled binomial measure. At each level of the multiplicative cascade and for all nodes of the corresponding binary tree, the weight is chosen uniformly among p0 and 1-p0. This option is implemented only for the binomial measure ('shufmeas'). When a string containing prefix 'pert' is given as second input, the synthesis is made for a pertubated binomial measure. In that case, the fourth input argument is a strictly positive real scalar epsilon which contains the pertubation around weights. The weight is an independant random variable identically distributed between p0-epsilon and p0+epsilon which must be >0., <1. This option is implemented only for the binomial measure ('pertmeas') and the mixing (lumping and sum) of two binomial measures ('lumppertmeas' and 'sumpertmeas'). When replacing suffix string 'meas' with suffix string 'cdf', respectively suffix string 'pdf', the cumulative distribution function F_n, respectively the probability density function p_n, related to this pre-multifractal measure is computed (first output argument). When string 'part' is given as second input, the partition sum function znq of multifractal measure is computed as sole output argument. In that case, the third input argument is a strictly positive real (integer) vector vn which contains the resolutions, and the fourth input argument is a real vector q which contains the measure exponents. The size of the output real matrix znq is equal to size(q)*size(vn). This option is implemented only for the binomial measure. When string 'Reyni' is given as second input, the Reyni exponents function tq (and the generalized dimensions Dq if used) of the multifractal measure is computed as first output argument (and second optional output argument if used). In that case, the third input argument is a real vector q which contains the measure's exponents. The size of the output real vector tq is equal to size(q)). This option is implemented only for the binomial measure. When a string containing suffix string 'spec' is given as second input, the multifractal spectrum f_alpha (second output argument) is synthesized on the Hoelder exponents alpha (first output argument). In that case, the third input argument is a strictly positive real (integer) scalar N which contains the number of Hoelder exponents. The size of both output real vectors alpha and f_alpha is equal to N. This option is implemented only for the binomial measure ('spec') and the mixing (lumping and sum) of two binomial measures ('lumpspec' and sumspec').


ALGORITHM DETAILS :

For the deterministic binomial, the pre-multifractal measure synthesis algorithm is implemented is a iterative way (supposed to run faster than a recursive one). For the shuffled or the pertubated binomial, the synthesis algorithm is implemented is a recursive way (to be able to pick up a i.i.d. r.v. at each level of the multiplicative cascade and for all nodes of the corresponding binary tree w.r.t. the given law). Note that the shuffled binomial measure is not conservative.


EXAMPLES :




MATLAB :



p0=.2;
n=10;
% synthesizes a pre-multifractal binomial measure
[mu_n,I_n]=binom(p0,'meas',n);
plot(I_n,mu_n);
% synthesizes the cdf of a pre-multifractal shuffled binomial measure
F_n=binom(p0,'shufcdf',n);
plot(I_n,F_n);
e=.19;
% synthesizes the pdf of a pre-multifractal pertubated binomial measure
p_n=binom(p0,'pertpdf',n,e);
plot(I_n,p_n);
vn=[1:1:8];
q=[-5:.1:+5];
% computes the partition sum function of a binomial measure
znq=binom(p0,'part',vn,q);
plot(-vn*log(2),log(znq));
% computes the Reyni exponents function of a binomial measure
tq=binom(p0,'Reyni',q);
plot(q,tq);
N=200;
q0=.4;
% computes the multifractal spectrum of the lumping of two binomial measures
[alpha,f_alpha]=binom(p0,'lumpspec',N,q0);
plot(alpha,f_alpha);



SCILAB :



p0=.2;
n=10;
// synthesizes a pre-multifractal binomial measure
[mu_n,I_n]=binom(p0,'meas',n);
plot(I_n,mu_n);
// synthesizes the cdf of a pre-multifractal shuffled binomial measure
F_n=binom(p0,'shufcdf',n);
plot(I_n,F_n);
e=.19;
// synthesizes the pdf of a pre-multifractal pertubated binomial measure
p_n=binom(p0,'pertpdf',n,e);
plot(I_n,p_n);
xbasc();
vn=[1:1:8];
q=[-5:.1:+5];
// computes the partition sum function of a binomial measure
znq=binom(p0,'part',vn,q);
mn=zeros(max(size(q)),max(size(vn)));
for i=1:max(size(q))
   mn(i,:)=-vn*log(2);
end
plot2d(mn',log(znq'));
// computes the Reyni exponents function of a binomial measure
tq=binom(p0,'Reyni',q);
plot(q,tq);
N=200;
q0=.4;
// computes the multifractal spectrum of the lumping of two binomial measures
[alpha,f_alpha]=binom(p0,'lumpspec',N,q0);
plot(alpha,f_alpha);



REFERENCES :

"Multifractal Measures", Carl J. G. Evertsz and Benoit B. MandelBrot. In Chaos and Fractals, New Frontiers of Science, Appendix B. Edited by Peitgen, Juergens and Saupe, Springer Verlag, 1992 pages 921-953. "A class of Multinomial Multifractal Measures with negative (latent) values for the "Dimension" f(alpha)", Benoit B. MandelBrot. In Fractals' Physical Origins and Properties, Proceeding of the Erice Meeting, 1988. Edited by L. Pietronero, Plenum Press, New York, 1989 pages 3-29. .SH See also sbinom, multim1d, multim2d, smultim1d, smultim2d (C_LAB routines). MFAS_measures, MFAS_dimensions, MFAS_spectra (Matlab and/or Scilab demo scripts).

19.17   contwt Continuous L2 wavelet transform

Author: Paulo Goncalves

Computes a continuous wavelet transform of a 1-D signal (real or complex). The scale operator is unitary with respect to the L2 norm. Two closed form wavelets are available: the Mexican Hat or the Morlet Wavelet (real or analytic). For arbitrary analyzing wavelets, numerical approximation is achieved using a Fast Mellin Transform.






USAGE :

[wt,scale,f,scalo,wavescaled]=contwt(x,[fmin,fmax,N,wvlt_length])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS :

1




ALGORITHM DETAILS :

The wavelet transform of x is computed via convolutions of dilated and translated versions of a single function called the "mother wavelet". The scales are given by the dilatation factor. As the scales are not absolute, the scale factor is determined through the specification of the minimum and maximum frequency of the decomposition considered as a time/frequency transform. The maximum frequency might not be greater than the Nyquist Frequency i.e. 0.5 as the wavelet at this scale would be undersampled (and therefore would create aliasing). The number of scales tells how many convolutions are computed. The bigger it is, the slower the transform is. The frequency (or scale) axis is geometrically sampled. The resulting scales and frequencies values can be obtained as output parameters. The meaning of the wavelet length parameter is manyfold. When non zero integer, it tells the routine to use a real Morlet Wavelet and gives its length at scale 1 (maximum frequency). When it is a positive imaginary integer, the analytic Morlet wavelet is used. If wvlt_length = 0, the Mexican Hat is used. The resulting wavelet transform is then real but has a quite poor frequency resolution. If wvlt_length is a real vector, it corresponds to the analyzing wavelet waveform in time at any arbitrary scale. Dilated and compressed version of it (according to the range [ fmin , fmax] are computed directly from wvlt_length using a Fast Mellin Transform. For all choices of wavelet, approximative reconstruction of the decomposed signal is possible (see icontwt).


SEE ALSO: :

icontwt, contwtmir and cwt


EXAMPLE: :

Signal synthesis 

x = morlet(0.1,128) ;
A Morlet (of size 2*8+1 samples ) wavelet transform 

[wtMorlet,scale,f,scaloMorlet] = contwt(x,0.01,0.5,128,8) ;
viewmat(scaloMorlet,[1 1 24]) ;
Compared with a Mexican hat wavelet transform 

[wtMex,scale,f,scaloMex] = contwt(x,0.01,0.5,128,0) ;
viewmat(scaloMex,[1 1 24]) ;

19.18   contwtmir Continuous L2 wavelet transform with mirroring

Author: Paulo Goncalves

Computes a continuous wavelet transform of a mirrored 1-D signal (real or complex). The scale operator is unitary with respect to the L2 norm. Two closed form wavelets are available: the Mexican Hat or the Morlet Wavelet (real or analytic). For arbitrary analyzing wavelets, numerical approximation is achieved using a Fast Mellin Transform.






USAGE :

[wt,scale,f,scalo,wavescaled]=contwtmir(x,[fmin,fmax,N,wvlt_length])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS :

1




ALGORITHM DETAILS :

The overall details of the algorithm are similar to those of contwt . The difference stems from the mirror operation applied to the signal before computing the wavelet transform to minimize border effects. At each scale j the analyzed signal is mirrored at its both extremities. The number of added samples at both sides is equal to scale(j)* wvlt_length (the half length of the analyzing wavelet at this particular scale). After convolution of the mirrored signal with the analyzing wavelet, the result is truncated to the actual size of the initial signal.


SEE ALSO: :

contwt, icontwt and cwt


EXAMPLE: :

Signal synthesis x = fbmlevinson(1024,0.8) ; Regular Wavelet transform [wt_nomirror,scale,f] = contwt(x,2(-6),2(-1),128,8) ; viewmat(abs(wt_nomirror),[1 1 24]) ; Compared with a mirrored wavelet transform [wt_mirror,scale,f] = contwtmir(x,2(-6),2(-1),128,0) ; viewmat(abs(wt_mirror),[1 1 24]) ;



19.19   contwtspec Continuous L2 wavelet based Legendre spectrum

Author: Paulo Goncalves

Estimates the multifractal Legendre spectrum of a 1-D signal from the wavelet coefficients of a L2 continuous decomposition






USAGE :



[alpha,f_alpha,logpart,tau] = contwtspec(wt,scale,Q[,FindMax,ChooseReg])



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

contwt, cwtspec, cwt, dwtspec, FWT


EXAMPLE: :



N = 2048 ; H = 0.7 ; Q = linspace(-4,4,11) ;
[x] = fbmlevinson(N,H) ;
[wt,scale] = contwtmir(x,2^(-8),2^(-1),16,8) ;
[alpha,f_alpha,logpart,tau] = contwtspec(wt,scale,Q,1,1) ;
plot(alpha,f_alpha),

19.20   cwt Continuous Wavelet Transform

Author: Bertrand Guiheneuf

This routine computes the continuous wavelet transform of a real signal. Two wavelets are available: the Mexican Hat or the Morlet Wavelet.






USAGE :

[wt,scales,freqs]=cwt(sig,fmin,fmax,nbscales,[wvlt_length])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The wavelet transform of sig is computed via convolutions of dilated and translated versions of a single function called the "wavelet". The scales are given by the dilatation factor. As the scales are not absolute, the scale factor is determined through the specification of the minimum and maximum frequency of the decomposition considered as a time/frequency transform. The maximum frequency might not be greater than the Nyquist Frequency i.e. 0.5 as the wavelet at this scale would be undersampled. The number of scales tells how many convolutions are computed. The bigger it is, the slower the transform is. The frequency (or scale) axis is logarithmicaly sampled. The resulting scales and frequencies values can be obtained as output parameters. The meaning of the wavelet length parameter is twofold. If non zero, it tells the routine to use a Morlet Wavelet and gives its length at scale 1 (maximum frequency). Otherwise (zero or not specified), the Mexican Hat is used. The resulting wavelet transform is then real but has a quite poor frequency resolution.


ALGORITHM DETAILS  :

Convolutions are computed through discrete linear convolutions in time domain. No FFT is used. The linear filters are obtained by a sampling of the wavelet after dilatation. The signal is mirrored at its boundaries.



19.21   cwtspec Continuous L1 wavelet based Legendre spectrum

Author: Paulo Goncalves

Estimates the multifractal Legendre spectrum of a 1-D signal from the wavelet coefficients of a L1 continuous decomposition






USAGE :



[alpha,f_alpha,logpart] = cwtspec(wt,scale,Q[,FindMax,ChooseReg])



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

cwt, contwtspec, contwt, dwtspec


EXAMPLE: :



N = 2048 ; H = 0.7 ; Q = linspace(-4,4,11) ;
[x] = fbmlevinson(N,H) ;
[wt,scale] = cwt(x,2^(-8),2^(-1),16,8) ;
[alpha,f_alpha,logpart,tau] = cwtspec(wt,scale,Q,1,1) ;
plot(alpha,f_alpha),

19.22   cwttrack Continuous L2 wavelet based Holder exponent estimation

Author: Paulo Goncalves

Estimates the local or global Holder exponent of a 1-D signal from its L2 continuous wavelet transform ( output of contwt(mir) ). In some cases, the global Holder exponent can also be refered to as the long range dependance parameter






USAGE :



[HofT] = cwttrack(wt,scale,whichT,FindMax,ChooseReg,radius,DeepScale,Show)



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




ALGORITHM DETAILS :

The maxima line search follows the two steps:

1




SEE ALSO: :

cwttrack_all, contwtspec, contwt, dwtspec


EXAMPLE: :



N = 1024 ; 
[x] = GeneWei(N,[ones(1,N/2)*0.2 ones(1,N/2)*0.8],2,1,1) ;
[wt,scale] = contwtmir(x,2^(-8),2^(-1),64,8*i) ;
HofT_1 = cwttrack(wt,scale,N/4,1,1) 
HofT_2 = cwttrack(wt,scale,3*N/4,1,1)

19.23   cwttrack_all Continuous L2 wavelet based Holder function estimation

Author: Paulo Goncalves

Estimates the Holder function of a signal from its continuous wavelet transform (L2 contwt). cwttrack_all merely runs cwttrack as many times as there are time samples to be analyzed






USAGE :



[HofT,whichT] = cwttrack_all(wt,scale,FindMax,ChooseReg,radius,DeepScale,dT)



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

cwttrack


EXAMPLE: :



N = 2048 ; 
[x] = GeneWei(N,linspace(0,1,N),1.2,1,1) ;
[wt,scale] = contwtmir(x,2^(-6),2^(-1),64,8*i) ;
[HofT,whichT] = cwttrack_all(wt,scale,1,0,8,1,(N/64)) ;

19.24   dilate Dilation of a signal

Author: Paulo Goncalves

Computes dilated/compressed version of a signal using Fast Mellin transform.






USAGE :

[sscaled,mellin,beta] = dilate(s,a,[fmin,fmax,N])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS :

1




ALGORITHM DETAILS :

This algorith uses a Fast Mellin Transform (dmt) to diagonalize the Scale operator. The algorithm runs as follows

1




SEE ALSO: :

dmt, idmt


EXAMPLE: :

Signal synthesis 

x = morlet(0.1,32) ; 
plot(x)
Dilation by a factor 2 

[sscaled,mellin,beta] = dilate(x,2,0.01,0.5,256) ;
plot(sscaled(2:sscaled(1)))
Compression by a factor 2 

[sscaled,mellin,beta] = dilate(x,1/2,0.01,0.5,256) ;
plot(sscaled(2:sscaled(1)))

19.25   dimR2d Regularization dimension of the surface of a 2d function

Author: Francois Roueff

Computes the regularization dimension of the surface of a 2d function. Two kernels are available: the Gaussian or the Rectangle.






USAGE :

[dim, handlefig]=dimR(x,sigma,voices,Nmin,Nmax,kernel,mirror,reg,graphs)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :

This function is the same as dimR but adapted to 2d signals. For a more complete explanation of the regularization dimension, one can refer to: "A regularization approach to fractionnal dimension estimation", F. Roueff, J. Levy-Vehel, submitted to Fractal 98 conference. The regularized graphs of x are computed via convolutions of x with dilated versions of the kernel at different scales. The lengths of the regularized graphs are computed via convolutions of x with the derivatives of the dilated versions of the kernel. The regularization dimension is computed either via an automatic range regression or via a regression by hand on the loglog plot of the lengths versus scales. If sigma is strictly positive, an estimation of the lengths without noise is used for the regression. These lengths are displayed in red while those of the noisy signal are in black. They should seperate at fine scales. When one specifies the range regression, the loglog plot of the lengths versus scales appears. Above are either increments (when sigma is null) or a loglog plot of the noise prevalence in the lengths. One selects the scale range of the regression. In the case of noisefree signals, select a scale region with stable increments. In the case of a strictly positive sigma, select a scale region where the noise prevalence is not too close to 1 (0 in log10): it should correspond to an approximately linear region for the red estimations. The number of scales (voices) tells how many convolutions are computed. The bigger it is, the slower the computation is. The scale axis is geometrically sampled (i.e. its log is arithmetically sampled). The gaussian kernel should give a better result but the rectangle is faster. As a general rule, be careful of the size of the input signal and of the maximal size of the kernel (Nmax x Nmax) to avoid too long computing times.


SEE ALSO: :

cwttrack, cwtspec.


EXAMPLE: :

Signal synthesis 

x = GeneWei(100,0.6,2,1.0,0);
y = GeneWei(100,0.4,3,1.0,1);
w = x'*y;
mesh(w);
Dimension of the graph with a regression by hand 

[dim,H] = dimR2d(w,0,25,5,30,'gauss',0,1,0);
Close the figures 

close(H)

19.26   dmt Discrete Mellin transform of a vector

Author: Paulo Goncalves

Computes the Fast Mellin transform of a signal.






USAGE :

[mellin,beta] = dmt(s,[fmin,fmax,N])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS :

1




ALGORITHM DETAILS :

The fast Mellin transform can be simply interpreted as a FFT applied to a geometrically sampled vector.


SEE ALSO: :

idmt, dilate


EXAMPLE: :

Signal synthesis 

x = morlet(0.1,32) ;
plot(x)
Computation of the Mellin transform 

[mellin,beta] = dmt(x,0.01,0.5,128) ;
plot(beta,abs(mellin))

19.27   dwtspec Discrete wavelet based Legendre spectrum

Author: Paulo Goncalves

Estimates the multifractal Legendre spectrum of a 1-D signal from the wavelet coefficients of a discrete decomposition






USAGE :



[alpha,f_alpha,logpart] = dwtspec(wt,Q[,ChooseReg])



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

cwtspec, FWT, WTStruct, MakeQMF, flt, iflt


EXAMPLE: :



N = 2048 ; H = 0.3 ; Q = linspace(-4,4,11) ;
[x] = fbmlevinson(N,H) ;
qmf = MakeQMF('daubechies',2) ;
[wt] = FWT(x,log2(N),qmf) ;
[alpha,f_alpha,logpart,tau] = dwtspec(wt,Q,1) ;
plot(alpha,f_alpha),

19.28   fbmfwt Discrete wavelet based synthesis of a fBm

Author: Paulo Goncalves

Generates a 1/f Gaussian process from a discrete wavelet transform






USAGE :



[x] = fbmfwt(N,H,[noctave,Q,randseed]) ;



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




ALGORITHM DETAILS :

Generates a 1/f Gaussian process by inverting a discrete wavelet transform. Step 1: generates y a [1,N] i.i.d. standard Gaussian noise Step 2: computes the discrete wavelet coefficients y Step 3: weigth the wavelet coefficients y with the corresponding scale power law Step 4: invert the weighted discrete wavelet transform


SEE ALSO: :

fbmlevinson, synth2, FWT, MakeQMF


EXAMPLE: :



Q = MakeQMF('daubechies',4) ;
[x] = fbmfwt(1024,0.5,10,Q) ;
[wt,scale,f] = contwt(x,2^(-8),2^(-1),64,8) ;
[H] = cwttrack(wt,scale,0,1,1,8,1,1) ;

19.29   fbmlevinson Levinson synthesis of a fractional Brownian motion

Author: Paulo Goncalves

Generates a Fractional Brownian Motion (fBm) using Cholesky/Levinson factorization






USAGE :



[x,y,r] = fbmlevinson(N,H,[seed])



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




ALGORITHM DETAILS :

Generates a Fractional Brownian Motion using Levinson algorithm to triangularize the covariance matrix. R = E W * W' being the variance/covariance matrix of the fBm increment W[n ; shift] = X[n+shift] - X[n-shift], R = L*L', with L the lower left triangle matrix (Choleski or Levinson factorization). Then, we pose Z = L(-1) * W <=> W = L * Z with Rz the var/cov matrix of the process Z, Rz = E {Z * Z' }Rz = E {L(-1) * W * W' * (L(-1))' }Rz = L(-1) * R * (L(-1))' Rz = L(-1) * L * L' * (L(-1))' Rz = I (identity) Thus, Z is a white Gaussian noise with unit variance.


SEE ALSO: :

mbmlevinson


EXAMPLE: :



[x,y,r] = fbmlevinson(1024,0.8) ;

19.30   fft1d Operates a column-wise direct or inverse FFT

Author: Paulo Goncalves

Operates a column-wise direct or inverse FFT on a matrix






USAGE :



Y = fft1d(X,DirInv) ;



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

fft


EXAMPLE: :

Matrix synthesis: 

t = linspace( 0,1,128 ) ;
f0 = [4 8 16 32]
X = sin( 2*%pi*t(:)*f0 ) ;
Y = abs( fft1d( X , -1 ) ) ;
Y = [Y(65:128,:) ; Y(1:64,:)] ;
f = linspace(-64,63,128) ;
plot2d(f(ones(4,1),:)',Y) ;

19.31   findWTLM Finds local maxima lines of a CWT

Author: Paulo Goncalves

Finds the local maxima of a continuous wavelet transform






USAGE :



[maxmap] = findWTLM(wt,scale[,depth])



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

contwt, cwt


EXAMPLE: :



N = 2048 ; H = 0.3 ; Q = linspace(-4,4,11) ;
[x] = fbmlevinson(N,H) ;
[wt,scale] = cwt(x,2^(-6),2^(-1),36,0) ;
[maxmap] = findWTLM(wt,scale) ;
Vizualisation in Matlab: 

viewWTLM(maxmap,scale,wt) ,
axis([1024 - 64 1024 + 64 0 log2(max(scale))]) ,
Vizualisation in Scilab: Not implemeted yet!...



19.32   flt Fast Legendre transform

Author: Paulo Goncalves

Computes the Legendre transform of y y*(s) = sup_{x in X}[s.x - y(x)]






USAGE :



[u,s] = flt(x,y[,ccv])



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :




EXAMPLE: :

Function synthesis 

m0 = .55 ; m1 = 1 - m0 ;
m2 = .95 ; m3 = 1 - m2 ;
q = linspace(-20,20,201) ;
tau1 = - log2(exp(q.*log(m0)) + exp(q.*log(m1))) ;
tau2 = - log2(exp(q.*log(m2)) + exp(q.*log(m3))) ;
tau3 = min(tau1 , tau2) ;
Legendre Transforms 

[u1,s1] = flt(q,tau1) ;
[u2,s2] = flt(q,tau2) ;
[u3,s3] = flt(q,tau3) ;
Vizualisation - Matlab 

plot(s1,u1,'g',s2,u2,'b',s3,u3,'r') ; grid ;
legend('u(tau1(q))','u(tau2(q))','u(tau3(q))') ;
Vizualisation - Scilab 

plot2d(s3,u3,17) ; plot2d(s1,u1,18,'001') ; plot2d(s2,u2,19,'001') ;

19.33   gauss Gaussian window

Author: Paulo Goncalves

Returns a Gaussian window






USAGE :



Win = gauss(N[,A])



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

mexhat, morlet


EXAMPLE: :



t = linspace(-1,1,128) ;
Win1 = gauss(128,2) ;
Win2 = gauss(128,5) ;
Vizualisation - Matlab 

plot(t,win1,'b',t,win2,'r') ; 
legend('Gaussian window 1','Gaussian window 2')
Vizualisation - Scilab 

plot2d([t(:) t(:)],[Win1(:) Win2(:)],[17 19])

19.34   gifs2wave wavelet coefficients from new GIFS coefficients

Author: Khalid Daoudi

Computes the wavelet c0efficients of the synthetic 1-D real signal from its new GIFS coefficients.






USAGE :

[wt_new]=gifs2wave(Ci_new,wt,wt_idx,wt_lg)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS :




ALGORITHM DETAILS :




SEE ALSO: :

wave2gifs.


EXAMPLE: :

19.35   gifseg Replaces nodes of the diadic tree by a ceratin unique value.

Author: Khalid Daoudi

Replaces at each scale the left (resp. right) nodes of the diadic tree, associated to the GIFS coefficients, that belong to [cmin,cmax] by a ceratin unique value.






USAGE :

[Ci_new, marks, L]=gifseg(Ci,[cmin,cmax,epsilon])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS :




ALGORITHM DETAILS :




SEE ALSO: :

hist, wave2gifs.


EXAMPLE: :

19.36   holder2d holder exponents of a measures defined on 2D real signal

Author: Pascal Mignot - Bertrand Guiheneuf

This routine computes holder exponents of a measures defined on 2D real signal. Several measures and capacities are available.






USAGE :

[holder]=holder2d(Input,[Meas],[Res],[Ref],[RefMeas])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




INTRODUCTION  :

This routines computes holder exponents by regressing an analysing measure (in a log-log plot) at different scales. Given a pixel, one defines a (square) window around it. The window size is called the resolution. The specified measure (or capacity) is then evaluated on the set defined by the window. For example, in the case of the "sum" measure, at resolution 2, a 5x5 square center on a pixel p0 is extracted from the input image. The mean of the gray levels of the obtained pixels defines the measure at pixel p0 and resolution 2. The measure type is thus given by the input parameter Meas whereas the actual measure is obtained through the input signal. In the case of a simple measure analysis, the regression is computed with respect to the size of the window, this corresponds to comparing the analysing measure to the Lebesgue measure. Nevertheless, it is possible to compute the regression by comparison with a reference measure given by the last two parameters.


SEE ALSO :

19.37   icontwt Inverse Continuous L2 wavelet transform

Author: Paulo Goncalves

Computes the inverse continuous wavelet transform: reconstructs a 1-D signal from its wavelet coefficients. The scale operator is unitary with respect to the L2 norm.






USAGE :



[x_back]=icontwt(wt,f,wl_length)



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS :

1




ALGORITHM DETAILS :

The reconstruction algorithm Inverse Wavelet Transform , proceeds by convolving the wavelet coefficients (obtained from contwt ) by the synthesis wavelet. As we deal with continuous wavelet decomposition, the analyzing wavelet and its dual for reconstruction are the same (continuous basis). This operation is iterated at each analyzed scale j yielding N corresponding band-passed signal versions. The reconstructed signal is the scale weighting sum of these N vectors.


SEE ALSO: :

contwt, contwtmir


EXAMPLE: :

Signal synthesis 

x = morlet(0.1,64) ;
t = 1:129 ;
A Morlet (of size 2*8+1 samples ) wavelet transform 

[wtMorlet,scale,f,scaloMorlet] = contwt(x,0.01,0.5,128,8) ;
viewmat(scaloMorlet,1:129,f,[1 1 24]) ;
Reconstruction with the same synthesis wavelet 

[x_back] = icontwt(wtMorlet,f,8) ;
plot([t(:) t(:)],[x(:) x_back(:)]) ;

19.38   idmt Inverse Discrete Mellin transform

Author: Paulo Goncalves

Computes the Inverse Fast Fourier-Mellin transform of a signal.






USAGE :

[x,t] = idmt(mellin,beta,[M])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :

The Inverse Fourier-Mellin transform can be viewed as an Inverse Fast Fourier Transform which result is assumed geometrically sampled. To recover the initial time signal, a Discrete Inverse Fourier Transform is applied to this geometrically Fourier representation. Important The Inverse Fourier-Mellin transform is correct only if the direct Fourier-Mellin transform has been computed from fmin to 0.5 cycles per sec.


SEE ALSO: :

dmt, dilate


EXAMPLE: :

Signal synthesis 

x = morlet(0.1,32) ; 
plot(x)
Computation of the Mellin transform 

[mellin,beta] = dmt(x,0.01,0.5,128) ;
plot(beta,abs(mellin))
Computation of the Inverse Mellin transform 

[y,t] = idmt(mellin,beta,65) ;
plot(t,abs(x-y))

19.39   integ Approximate 1-D integral

Author: Paulo Goncalves

Approximate 1-D integral. integ(y,x) approximates the integral of y with respect to the variable x






USAGE :



SOM = integ(y[,x])



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

integ2d


EXAMPLE: :

Cumulative Normal Distribution 

sigma = 1 ; N = 100 ;
x = logspace(log10(0.001),log10(3),N/2) ;
x = [ -fliplr(x) x ] ;
y = 1/sqrt(2*pi) * exp( -(x.^2)./2 ) ;
plot(x,y) 
for n = 1:N
  PartialSom(n) = integ( y(1:n),x(1:n) ) ;
end
Vizualisation Matlab 

plot(x,PartialSom,x,PartialSom,'or') 
grid ; xlabel('x') ; title('\int_{-\infty}^{x} g(u) du')
Vizualisation Scilab 

xbasc()
plot2d(x,PartialSom,-1)

19.40   isempty Checks if a matrix is empty

Author: Paulo Goncalves

isempty True for empty matrix. isempty(x) returns %T if x is an empty array and %F otherwise. An empty array has no elements, that is prod(size(X))==0.






USAGE :



isempty(x)



INPUT PARAMETERS :

1




SEE ALSO: :

all

19.41   lambdak k's lambda functions for pseudoAW

Author: Paulo Goncalves

Computes the parametrizing function lambdak defining the Affine Wigner distributions.






USAGE :

[y] = lambdak(u,k)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

pseudoAW


EXAMPLE: :

x = linspace(-10,10,101) ; y0 = lambdak(x,-1) ; y1 = lambdak(x,2) ; plot(y0) plot(y1)



19.42   lepskiiap lepskii adaptive procedure

Author: Christophe Canus

This C_LAB routine is an implementation of the Lepskii's adaptive procedure. This algorithm selects the "best" estimator which balances the bias-variance tradeoff in a sequence of noisy and biased estimators theta_hat_j of a non-random parameter theta with the assumption that when j increases, bias b_j increases as variance sigma2_j decreases.






USAGE :

[K_star,j_hat,I_c_j_min,I_c_j_max,E_c_j_hat_min,E_c_j_hat_max]= lepskiiap(theta_hat_j,[sigma2_j,K])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The sequence of variances sigma_j must be stricly positive, decreasing when j increases and of the same size than theta_hat_j. When no sequence of variances is given as input or when it is uniformly equal to 0, the algorithm computes the sequence of variances as sigma2_j=1./j. The default value for epsilon is 1./[1:J]. The confidence constant K must be >=1. For the meaning of the ouput parameters, see next section.


ALGORITHM DETAILS  :

Define the sequence of confidence intervals I_c_j=[theta_hat_j-K*sigma_j,theta_hat_j+K*sigma_j], the sequence of their decreasing intersections E_c_j and j_hat as the largest index j such as that E_c_j is non void. The best estimator with respect to the Lepskii's adaptive procedure is selected as theta_hat_j_hat in E_c_j_hat. The two parameters to be handled are the sequence of variances sigma2_j and the confidence constant K. sigma2_j can be any sequence dominating the estimator variance. Choosing a smaller K speeds up the selection and results to smaller j_hat.


EXAMPLES :




MATLAB :



T=33;
% linear model
f_t=linspace(0,1,T);
% jump for t=floor(3/4*T)
f_t(floor(3/4*T):T)=2*f_t(floor(3/4*T):T);
% Wiener process
W_t=randn(1,T);
sigma=.1;
Y_t=f_t+sigma*W_t;
subplot(2,1,1);
plot(f_t);hold on;plot(Y_t);
title('White noise model Y(t)');
xlabel('index: t');
ylabel('Y(t)=f(t)+\sigma W(t)');
% estimation for t=t_0=floor(T/2)
t_0=floor(T/2)+1;
Y_t=f_t+sigma*W_t;
for t=1:floor(T/2)
  f_hat_t(t)=mean(Y_t(t_0-t:t_0+t));
end
% Lespkii's adaptive procedure
[K_star,t_hat,I_c_t_min,I_c_t_max,E_c_t_hat_min,E_c_t_hat_max]=lepskiiap(f_hat_t,.005*1./[1:floor(T/2)],2);
% plot and disp results
plot(t_0,Y_t(t_0),'k*');
plot(t_0-t_hat,Y_t(t_0-t_hat),'kd');
plot(t_0+t_hat,Y_t(t_0+t_hat),'kd');
subplot(2,1,2);
plot(f_hat_t);
hold on;
plot(I_c_t_max,'r^');
plot(I_c_t_min,'gV');
title(['estimator \theta_t(t_0) vs. index t with t_0=',num2str(floor(T/2)+1)]);
xlabel('index: t');
ylabel('estimator: \theta_t(t_0)');
plot(t_hat,E_c_t_hat_min,'ko');
plot(t_hat,E_c_t_hat_max,'ko');
disp(['linear estimation of f_t for t=t_0=',num2str(t_0)]);
disp(['selected index t=',num2str(t_hat)]);
disp(['estimated f_t_0 in [',num2str(E_c_t_hat_min),',',num2str(E_c_t_hat_min),']']);



SCILAB :



//



REFERENCES :

To be published..SH See Also monolr (C_LAB routine).

19.43   linearlt linear time legendre transform

Author: Christophe Canus

This C_LAB routine the Legendre transform of a function using the linear time Legendre transform algorithm.






USAGE :

[s,u_star_s]=linearlt(x,u_x)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The abscissa x and the function u_x to be transformed must be of the same size [1,N] or [N,1]. The abscissa s and the Legendre conjugate function u_star_s are of the same size [1,M] with M<=N.


ALGORITHM DETAILS :

The linear time Legendre transform algorithm is based on the use of a concave regularization before slopes' computation.


EXAMPLES :




MATLAB :



x=linspace(-5.,5.,1024);
u_x=-1+log(6+x);
plot(x,u_x);
% looks like a Reyni exponents function, isn't it ?
[s,u_star_s]=linearlt(x,u_x);
plot(s,u_star_s);



SCILAB :



//



REFERENCES :

None..SH See Also bbch (C_LAB routine).

19.44   mbmlevinson Levinson synthesis of a multifractional Brownian motion

Author: Paulo Goncalves

Generates a Multi-Fractional Brownian Motion (mBm) using Cholesky/Levinson factorization






USAGE :



[x,y,r] = mbmlevinson(N,H,[seed])



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




ALGORITHM DETAILS :

For each time n, a fbm process with constant Holder exponent H[n/] is synthesized over N points (see fbmlevinson). Only the sample at rank n is kept. As a result of this computationally expensive procedure, only small sample sizes of mBms can be generated (typically less than 1024 samples).


SEE ALSO: :

mbmlevinson


EXAMPLE: :



[x,y,r] = mbmlevinson(512,AtanH(512,2,1,0.5)) ; 
plot(x) ;

19.45   mcfg1d Continuous large deviation spectrum estimation on 1d measure

Author: Christophe Canus

This C_LAB routine estimates the continuous large deviation spectrum on 1d measure.






USAGE :

[alpha,fgc_alpha,[pc_alpha,epsilon_star,eta,alpha_eta_x]]= mcfg1d(mu_n,[S_min,S_max,J],progstr,ballstr,N,epsilon,contstr,adapstr,kernstr,normstr,I_n])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The continuous large deviation spectrum (alpha,fgc_alpha) is estimated for J sizes eta_j and for the precision vector epsilon by taking into account the resolution of the 1d measure mu_n. The minimum size S_min sets the equivalent size eta_1 in the unit interval at which the first spectrum is estimated. eta_1 is equal to S_min*eta_n where eta_n is related to the resolution of the 1d measure (eta_n=N_n{-1}when all intervals are of equal size else it is max(|I_n|{-1}). It must be >=1. The default value for S_min is 1. The maximum size S_max sets the equivalent size eta_J in the unit interval at which the last spectrum is estimated. eta_J is equal to S_max*eta_n. It must be >=S_min. The default value for S_max is 1. The number of scales J sets the number of computed spectra. The bigger J is, the slower the computation is. It must be >=1. The default value for J is 1. The scale progression string progstr specifies the type of scale discretization. It can be 'dec' for decimated, 'log' for logarithmic or 'lin' for linear scale. The default value for progstr is 'dec'. The ball string ballstr specifies the type of ball B_eta(x). It can be 'asym' for asymmetric, 'cent' for centered or 'star' for three times bigger asymmetric ball. The default value for ballstr is 'asym'. The number N sets the discretization of the Hoelder exponents interval. They are linearly spaced between alpha_eta_min and alpha_eta_max which are the minimum and maximum values of the coarse grain Hoelder exponents at size eta. The bigger N is, the slower the computation is. It must be >=1. The default value for N is 100. The precision vector epsilon sets the precisions at which the spectrum is estimated. It must be of size [1,N] or [N,1]. When no precision vector is given as input or when it is uniformly equal to 0, the algorithm determines the optimal precisions vector epsilon_star. The default value for epsilon is zeros(1,N). The continuous string contstr specifies the definition of continuous spectrum. It can be equal to 'hnokern' for definition without precision and kernel or 'hkern' for definition with precision and kernel. The default value for contstr is 'hkern'. The precision adaptation string adapstr specifies the local adaptation of the precision w.r.t. the Hoelder exponents alpha. It can be equal to 'maxdev' for maximum deviation or 'maxadaptdev' for maximum adaptive deviation. The default value for adapstr is 'maxdev'. The kernel string kernstr specifies the kernel. It can be equal to 'box' for boxcar, 'tri' for triangle, 'mol' for mollifier, 'epa' for epanechnikhov or 'gau' for gaussian kernel. The default value for kernstr is 'gau'. The normalization string normstr specifies the type of pdf's normalization conducted before double log-normalization. It can be equal to 'nonorm' for no normalization conducted, 'suppdf' for normalization w.r.t the supremum of pdf's, 'infsuppdf' for normalization w.r.t the infimum and the supremum of pdf's. The default value for normstr is 'suppdf'. The intervals vector I_n can be useful when the intervals on which the pre-multifractal 1d measure is defined are not of equal size (not implemented yet). The pdf of the coarse grain Hoelder exponents matrix or vector pc_alpha, the optimal precisions matrix or vector epsilon_star, the sizes vector eta and the coarse grain Hoelder exponents matrix or vector alpha_eta_x can be obtained as outputs parameters.


ALGORITHM DETAILS  :

The coarse Hoelder exponents are estimated on each point x of the unit interval discretization by summing interval measures into a sliding window of size eta containing x (which corresponds to ball B_eta(x)). The probability density function pc_alpha is obtained by integrating horizontal sections.


EXAMPLES :




MATLAB :



% synthesis of pre-multifractal binomial measure: mu_n
% resolution of the pre-multifractal measure
n=10; 
% parameter of the binomial measure
p_0=.4; 
% synthesis of the pre-multifractal beiscovitch 1d measure 
mu_n=binom(p_0,'meas',n);  
% continuous large deviation spectrum estimation: fgc_alpha  
%  minimum size, maximum size & # of scales
S_min=1;S_max=8;J=4;
% # of hoelder exponents, precision vector
N=200;epsilon=zeros(1,N); 
% estimate the continuous large deviation spectrum
[alpha,fgc_alpha,pc_alpha,epsilon_star]=mcfg1d(mu_n,[S_min,S_max,J],'dec','cent',N,epsilon,'hkern','maxdev','gau','suppdf'); 
% plot the continuous large deviation spectrum
plot(alpha,fgc_alpha); 
title('Continuous Large Deviation spectrum');  
xlabel('\alpha');
ylabel('f_{g,\eta}^{c,\epsilon}(\alpha)');



SCILAB :



// computation of pre-multifractal besicovitch measure: mu_n 
// resolution of the pre-multifractal measure 
n=10; 
// parameter of the besicovitch measure 
p_0=.4; 
// synthesis of the pre-multifractal besicovitch 1d measure 
[mu_n,I_n]=binom(p_0,'meas',n); 
// continuous large deviation spectrum estimation: fgc_alpha  
// minimum size, maximum size & # of scales
S_min=1;S_max=8;J=4;
// # of hoelder exponents, precision vector
N=200;epsilon=zeros(1,N); 
// estimate the continuous large deviation spectrum
[alpha,fgc_alpha,pc_alpha,epsilon_star]=mcfg1d(mu_n,[S_min,S_max,J],'dec','cent',N,epsilon,'hkern','maxdev','gau','suppdf'); 
// plot the Continuous Large Deviation spectrum 
plot2d(a,f,[6]); 
xtitle(["Continuous Large Deviation spectrum";" "],"alpha","fgc(alpha)");



REFERENCES :

To be published..SH See Also mch1d, fch1d, fcfg1d, cfg1d (C_LAB routines). MFAG_continuous, MFAG_epsilon, MFAG_eta, MFAG_epsilon_eta (Matlab and/or Scilab functions).

19.46   mdfl1d Discrete Legendre spectrum estimation on 1d measure

Author: Christophe Canus

This routine estimates the discrete Legendre Spectrum on 1d measure.






USAGE :

[alpha,f_alpha]=mdfl1d(mu_n,N,n)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The discrete Legendre spectrum f_alpha is estimated on the finite finer resolution of the pre-multifractal 1d measure mu_n. The three steps of the estimatation are:

1




ALGORITHM DETAILS  :

The discrete partition function is estimated by coarse-graining masses mu_n into non-overlapping boxes of increasing diameter (box method). If nu_n is a power of 2, 2n corresponds to the coarser scale. The reyni exponents are estimated by least square linear regression. The Legendre transform of the mass exponent function is estimated with the linear-time Legendre transform.


SEE ALSO :

mdzq1d,mdzq2d,reynitq,linearlt,mdfl2d.

19.47   mdfl2d Discrete Legendre spectrum estimation on 2d measure

Author: Christophe Canus

This routine estimates the discrete Legendre spectrum on a pre-multifractal 2d measure.






USAGE :

[alpha,fl_alpha]=mdfl2d(mu_n,N,n)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS :

The discrete Legendre spectrum fl_alpha is estimated on the finite finer resolution of the 2d measure mu_n. The three steps of the estimation are:

1




ALGORITHM DETAILS  :

The discrete partition function is estimated by coarse-graining masses mu_n into non-overlapping boxes of increasing diameter (box method). If nux_n and nuy_n are power of 2, 2n corresponds to the coarser scale. The Reyni exponents are estimated by least square linear regression. The Legendre transform of the mass exponent function is estimated with the linear-time Legendre transform.


SEE ALSO :

mdznq1d,mdznq2d,reynitq,linearlt,mdfl1d.

19.48   mdznq1d Discrete partition function estimation on 1d measure

Author: Christophe Canus

This routine computes the discrete partition function on a pre-multifractal 1d measure.






USAGE :

[mznq]=mdznq1d(mu_n,n,q)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The discrete partition function mznq is computed on the pre-multifractal 1d measure mu_n. The vector of resolutions n and the vector of exponents q sets the size of the output real matrix mznq to size(q)*size(n).


ALGORITHM DETAILS  :

The discrete partition function mznq is computed by coarse-graining masses mu_n into non-overlapping boxes of increasing diameter (box method). If nu_n is a power of 2, n corresponds to the resolution.


SEE ALSO :

mdzq2d,reynitq,linearlt,mdfl1d,mdfl2d.

19.49   mdznq2d Discrete partition function estimation on 2d measure

Author: Christophe Canus

This routine computes the discrete partition function on a pre-multifractal 2d measure.






USAGE :

[mznq]=mdznq2d(mu_n,n,q)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The discrete partition function mznq is computed on the pre-multifractal 2d measure mu_n. The vector of resolutions n and the vector of exponents q sets the size of the output real matrix mznq to size(q)*size(n).


ALGORITHM DETAILS  :

The discrete partition function mznq is computed by coarse-graining masses mu_n into non-overlapping boxes of increasing diameter (box method). If nux_n and nuy_n are power of 2, n corresponds to the resolution.


SEE ALSO :

mdznq1d,reynitq,linearlt,mdfl1d,mdfl2d.

19.50   mexhat Mexican hat wavelet

Author: Paulo Goncalves

Computes a Mexican Hat wavelet (seconde derivative of the gaussian).






USAGE :

[wavelet,alpha] = mexhat(nu)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

morlet, contwt


EXAMPLE: :

wavelet synthesis 

wavelet1 = mexhat(0.05) ; 
wavelet2 = mexhat(0.2) ;
plot(wavelet1) ; pause
plot(wavelet2)

19.51   monolr monovariate linear regression

Author: Christophe Canus

This C_LAB routine provides six different algorithms to proceed linear regression on monovariate data: least square, weighted least square, penalized least square, multiple least square, maximum likelyhood and Lepskii's adaptive procedure least square, in one sole routine.






USAGE :

[a_hat,[b_hat,y_hat,e_hat,sigma2_e_hat,optvarargout]= monolr(x,y,[lrstr,optvarargin])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The abscissa x and the ordinate y to be regressed with must be of the same size [1,J] or [J,1]. The linear regression string lrstr specifies the type of linear regression used. It can be 'ls' for least square, 'wls' for weighted least square, 'pls' for penalized least square, 'mls' for multiple least square (that is for j varying from 1 to J), 'ml' for maximum likelyhood, 'lapls' for Lepskii's adaptive procedure least square. The default value for lrstr is 'ls'. The weights w or the sequence of variances sigma2_j must be strictly positive and of size [1,J] or [J,1]. For the meaning of the variable optional input parameters sigma2_j and K, see lepskiiap (Lepskii's Adaptive Procedure) C_LAB routine's help. The number of iterations I must be >=2. The variance of the normal weights s must be strictly positive. If multiple least square, maximum likelyhood or Lepskii's adaptive procedure least square is chosen, the estimated slope a_hat and the ordinate at the origin b_hat are vectors of size [1,J], resp. the regressed ordinates y_hat and the residuals e_hat vectors are of size [1,(J+2)*(J-1)/2] (as they contains results for multiple linear regression, be aware of that when vizualising them :-), see examples), otherwise there are scalars, resp. vectors of size [1,J]. For maximum likelyhood, multiple least square linear regressions are proceeded in order to obtain variance estimates. Then maximum likelyhood linear regression is proceeded (corresponding results are found in a_hat(1), b_hat(1), y_hat(1:J), e_hat(1:J) and sigma2_e_hat(1), see examples). For the meaning of the variable optional output parameters K_star, j_hat, I_c_j_min, I_c_j_max, E_c_j_max, and E_c_j_max, see lepskiiap (Lepskii's Adaptive Procedure) C_LAB routine's help.


ALGORITHM DETAILS :

For the details of the Lepskii's adaptive procedure, see lepskiiap (Lepskii's Adaptive Procedure) C_LAB routine's help.


EXAMPLES :




MATLAB :



J=32;
x=1+linspace(0,1,J);
% Wiener process
W=randn(1,J); 
epsilon=.1;
y=x+epsilon*W;
% least square
[a_hat,b_hat,y_hat,e_hat,sigma2_e_hat]=monolr(x,y);
plot(x);hold on;plot(y);plot(y_hat,'kd');
plot(epsilon.*W);hold on;plot(e_hat); 
title('least square');
disp('type return');
pause;
clf;
% weighted least square
epsilon=linspace(.05,.5,J);
y=x+epsilon.*W;
[a_hat,b_hat,y_hat,e_hat,sigma2_e_hat]=monolr(x,y,'wls',1./epsilon);
plot(x);hold on;plot(y);plot(y_hat,'kd');
plot(epsilon.*W);hold on;plot(e_hat); 
title('weighted least square');
disp('type return');
pause;
clf;
% penalized least square
[a_hat,b_hat,y_hat,e_hat,sigma2_e_hat]=monolr(x,y,'pls',30);
plot(x);hold on;plot(y);plot(y_hat);
title('penalized least square');
disp('type return');
pause;
clf;
% multiple least square
[a_hat,b_hat,y_hat,e_hat,sigma2_e_hat]=monolr(x,y,'mls');
plot(x);hold on;plot(y)
start_j=0;
hold on;
for j=2:J
  plot([1:j],y_hat(start_j+1:start_j+j),'k');
  disp(['estimated slope a_hat =',num2str(a_hat(j))]);
  disp('type return');
  pause;
  start_j=start_j+j;
  j=j+1; 
end
clf
% maximum likelyhood
[a_hat,b_hat,y_hat,e_hat,sigma2_e_hat]=monolr(x,y,'ml');
plot(x);hold on;plot(y_hat(1:J),'kd');
plot(epsilon.*W);hold on;plot(e_hat(1:J));
clf;
% Lespkii's adaptive procedure
epsilon=.01;
y(1:16)=x(1:16)+epsilon*W(1:16);
y(16:32)=2*x(16:32)+epsilon*W(16:32);
[a_hat,b_hat,y_hat,e_hat,sigma2_e_hat,K_star,j_hat,I_c_j_min,I_c_j_max,E_c_j_hat_min,E_c_j_hat_max]=monolr(x,y,'lapls');
plot(a_hat);
hold on;
plot(I_c_j_max,'r^');
plot(I_c_j_min,'gV');
title('LAP: estimator vs. index');
xlabel('index: j');
ylabel('estimator: \theta_j');
plot(j_hat,E_c_j_hat_min,'ko');
plot(j_hat,E_c_j_hat_max,'ko');



SCILAB :



//



REFERENCES :

To be published..SH See Also lepskiiap (C_LAB routine).

19.52   morlet Morlet wavelet

Author: Paulo Goncalves

Computes a Morlet wavelet.






USAGE :

[wavelet,alpha] = morlet(nu,[N,analytic])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

mexhat, contwt


EXAMPLE: :

wavelet synthesis 

wavelet1 = morlet(0.1,64) ;
wavelet2 = morlet(0.1) ;
plot(wavelet1) ; pause 
plot(wavelet2)

19.53   mtlb_diff Difference and approximate derivative

Author: Paulo Goncalves

Difference and approximate derivative. If x is a matrix, the differences are computed columnwise.






USAGE :



[y] = mtlb_diff(x[,order])



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :




EXAMPLE: :

Matrix synthesis: 

N = 100 ;
t = 0:N-1 ;
x = sin(2*%pi*0.05*t) ;
approximate 1-st order derivative 

y = mtlb_diff(x) ;
plot2d([t(:) t(:)] , [x(:) [y(:);0]]) ;

19.54   mtlb_fftshift Move zeroth lag to center of spectrum

Author: Paulo Goncalves

Move zeroth lag to center of spectrum. Shift FFT. For vectors MTLB_FFTSHIFT(X) returns a vector with the left and right halves swapped. For matrices, MTLB_FFTSHIFT(X) swaps the upper and the lower halves.






USAGE :



y = mtlb_fftshift(x) ;



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

fft


EXAMPLE: :

Matrix synthesis: 

t = linspace( 0,1,128 ) ;
x = sin( 2*%pi*t*16 ) ;
SpectX = abs( fft( x,-1 ) ) ;
FFT-SHIFT 

xsetech([0 0 0.5 1]) ; plot2d( Freq,SpectX ) ; 
SwapSpectX = mtlb_fftshift( SpectX ) ;
Freq = linspace( -0.5,0.5,128 ) ;
xsetech([0.5 0 0.5 1]) ; plot2d( Freq,SwapSpectX )

19.55   mtlb_fliplr Flip matrix in left/right direction

Author: Paulo Goncalves

Flip matrix in left/right direction. For matrices, mtlb_fliplr(x) returns x with row preserved and columns flipped in the left/right direction.






USAGE :



y = mtlb_fliplr(x) ;



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

flipud, fftshift


EXAMPLE: :

Matrix synthesis: 

x = [1 2 3 ; 4 5 6]
Left - Right flip 

FlipX = mtlb_fliplr(x)

19.56   mtlb_flipud Flip matrix in up/down direction

Author: Paulo Goncalves

Flip matrix in up/down direction. For matrices, fliplr(x) returns x with columns preserved and rowa flipped in the up/down direction.






USAGE :



y = mtlb_flipud(x) ;



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

fliplr, fftshift


EXAMPLE: :

Matrix synthesis: 

x = [1 4 ; 2 5 ; 3 6]
Up - Down flip 

FlipX = mtlb_flipud(x)

19.57   mtlb_hilbert Hilbert transform of a signal

Author: Paulo Goncalves

Hilbert transform of a signal. mtlb_hilbert(x) is the Hilbert transform of the real part of vector X. The real part of the result is the original real data; the imaginary part is the actual Hilbert transform.






USAGE :



y = mtlb_hilbert(x) ;



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

fft


EXAMPLE: :

Matrix synthesis: 

t = linspace( -1,1,128 ) ;
X = cos( 2*%pi*t*16 ) ;
SpectX = abs( fft( X , -1 ) ) ;
SpectX = fftshift( SpectX ) ;
Hilbert Transform 

AnalyticX = mtlb_hilbert(X) ;
SpectAnalyticX = abs( fft( AnalyticX , -1 ) ) ;
SpectAnalyticX = fftshift( SpectAnalyticX ) ;
Freq = linspace( -0.5,0.5,128 ) ;
xsetech([0 0 0.5 1]) ; plot2d( Freq,SpectX ) ; 
xsetech([0.5 0 0.5 1]) ; plot2d( Freq,SpectAnalyticX )

19.58   mtlb_isreal Check is an rarry is real

Author: Paulo Goncalves

mtlb_isreal True for real array. mtlb_isreal(x) returns %T if all elements in x have zero imaginary part and %F otherwise.






USAGE :



mtlb_isreal(x)



INPUT PARAMETERS :

1




SEE ALSO: :

isempty, real, imag

19.59   mtlb_log2 Base 2 logarithm.

Author: Paulo Goncalves

Base 2 logarithm. y = mtlb_log2(x) is the base 2 logarithm of the elements of x.






USAGE :



y = mtlb_log2(x) ;



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

log, exp

19.60   mtlb_mean rithmetic mean.

Author: Paulo Goncalves

Arithmetic mean. y = mtlb_mean(x) is the arithmetic mean of the elements of x.






USAGE :



y = mtlb_mean(x) ;



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

sum

19.61   mtlb_rem Remainder after division

Author: Paulo Goncalves


SYNTAX :

leftover = mtlb_rem(x,q)



PARAMETERS :




DESCRIPTION :

Remainder after division. MTLB_REM(x,y) is x - y.*int(x./y) if y ~= 0. The input x and y must be real arrays of the same size, or real scalars.

19.62   multim1d multinomial 1d measure synthesis

Author: Christophe Canus

This C_LAB routine synthesizes a large range of pre-multifractal measures related to the multinomial 1d measure (deterministic, shuffled, pertubated) and computes linked theoretical functions (partition sum function, Reyni exponents function, generalized dimensions, multifractal spectrum).






USAGE :

[varargout,[optvarargout]]=multim1d(b,p,str,varargin,[optvarargin])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The multinomial 1d measure is completly characterized by its base b and its weights p(i) (i=1 to b). The first parameter b must be >1. The second parameter must be a vector of size equal to b. The weights p(i) must be >0., <1. and their sum must be =1. (the case of p(i)=1/b corresponds to the Lebesgue measure) (i=1 to b). The third parameter str is a variable string used to determine the desired type of output. There are six suffix strings ('meas' for measure, 'cdf' for cumulative distribution function , 'pdf' for probability density function, 'part' for partition sum function, 'Reyni' for Reyni exponent function , 'spec' for multifractal spectrum) for the deterministic multinomial measure and two prefix strings for related measures ('shuf' for shuffled , 'pert' for pertubated) which can be added to the first ones to form composed strings. For example, 'shufmeas' is for the synthesis of a shuffled multinomial 1d pre-multifractal measure. Note that all combinaisons of strings are not implemented yet. When a string containing suffix string 'meas' is given as third input, a pre-multifractal measure mu_n (first output argument) is synthesized on the b-adic intervals I_n (second optional output argument) of the unit interval. In that case, the fourth input argument is a strictly positive real (integer) scalar n which contains the resolution of the pre-multifractal measure. The size of the output real vectors mu_n (and I_n if used) is equal to bn (so be aware the stack size ;-)). This option is implemented for the deterministic ('meas'), shuffled ('shufmeas') and pertubated ('pertmeas') multinomial 1d measure. When a string containing prefix 'shuf' is given as third input, the synthesis is made for a shuffled multinomial measure. At each level of the multiplicative cascade and for all nodes of the corresponding binary tree, the vector of weights p is shuffled. This option is implemented only for the multinomial 1d measure ('shufmeas'). When a string containing prefix 'pert' is given as third input, the synthesis is made for a pertubated multinomial measure. In that case, the fifth input argument is a strictly positive real scalar epsilon which contains the pertubation around weights. The weights are independant random variables identically distributed between p(i)-epsilon and p(i)+epsilon which must be >0., <1. (i=1 to b). This option is implemented only for the multinomial 1d measure ('pertmeas'). When replacing suffix string 'meas' with suffix string 'cdf', respectively suffix string 'pdf', the cumulative distribution function F_n, respectively the probability density function p_n, related to this pre-multifractal measure is computed (first output argument). When string 'part' is given as third input, the partition sum function znq of multifractal measure is computed as sole output argument. In that case, the fourth input argument is a strictly positive real (integer) vector vn which contains the resolutions, and the fifth input argument is a real vector q which contains the measure exponents. The size of the output real matrix znq is equal to size(q)*size(vn). This option is implemented only for the multinomial 1d measure. When string 'Reyni' is given as third input, the Reyni exponents function tq (and the generalized dimensions Dq if used) of the multifractal measure is computed as first output argument (and second optional output argument if used). In that case, the fourth input argument is a real vector q which contains the measure's exponents. The size of the output real vector tq is equal to size(q)). This option is implemented only for the multinomial 1d measure. When string 'spec' is given as third input, the multifractal spectrum f_alpha (second output argument) is synthesized on the Hoelder exponents alpha (first output argument). In that case, the fourth input argument is a strictly positive real (integer) scalar N which contains the number of Hoelder exponents. The size of both output real vectors alpha and f_alpha is equal to N. This option is implemented only for the multinomial 1d measure.


ALGORITHM DETAILS :

For the deterministic multinomial, the pre-multifractal measure synthesis algorithm is implemented is a iterative way (supposed to run faster than a recursive one). For the shuffled or the pertubated multinomial, the synthesis algorithm is implemented is a recursive way (to be able to pick up a i.i.d. r.v. at each level of the multiplicative cascade and for all nodes of the corresponding binary tree w.r.t. the given law). In the case of the pertubated multinomial, the weights of each node are normalised by their sum for the measure to remain conservative. Note that the shuffled multinomial 1d measure is not conservative.


EXAMPLES :




MATLAB :



b=3;
p=[.1 .3 .6];
n=8;
% synthesizes a pre-multifractal multinomial 1d measure
[mu_n,I_n]=multim1d(b,p,'meas',n);
plot(I_n,mu_n);
% synthesizes the cdf of a pre-multifractal shuffled multinomial 1d measure
F_n=multim1d(b,p,'shufcdf',n);
plot(I_n,F_n);
e=.09;
% synthesizes the pdf of a pre-multifractal pertubated multinomial 1d measure
p_n=multim1d(b,p,'pertpdf',n,e);
plot(I_n,p_n);
vn=[1:1:8];
q=[-5:.1:+5];
% computes the partition sum function of a multinomial 1d measure
znq=multim1d(b,p,'part',vn,q);
plot(-vn*log(2),log(znq));
% computes the Reyni exponents function of a multinomial 1d measure
tq=multim1d(b,p,'Reyni',q);
plot(q,tq);
N=200;
% computes the multifractal spectrum of a multinomial 1d measure
[alpha,f_alpha]=multim1d(b,p,'spec',N);
plot(alpha,f_alpha);



SCILAB :



b=3;
p=[.1 .3 .6];
n=8;
// synthesizes a pre-multifractal multinomial 1d measure
[mu_n,I_n]=multim1d(b,p,'meas',n);
plot(I_n,mu_n);
// synthesizes the cdf of a pre-multifractal shuffled multinomial 1d measure
F_n=multim1d(b,p,'shufcdf',n);
plot(I_n,F_n);
e=.09;
// synthesizes the pdf of a pre-multifractal pertubated multinomial 1d measure
p_n=multim1d(b,p,'pertpdf',n,e);
plot(I_n,p_n);
xbasc();
vn=[1:1:8];
q=[-5:.1:+5];
// computes the partition sum function of a multinomial 1d measure
znq=multim1d(b,p,'part',vn,q);
mn=zeros(max(size(q)),max(size(vn)));
for i=1:max(size(q))
   mn(i,:)=-vn*log(2);
end
plot2d(mn',log(znq'));
// computes the Reyni exponents function of a multinomial 1d measure
tq=multim1d(b,p,'Reyni',q);
plot(q,tq);
N=200;
// computes the multifractal spectrum of a multinomial 1d measure
[alpha,f_alpha]=multim1d(b,p,'spec',N);
plot(alpha,f_alpha);



REFERENCES :

"Multifractal Measures", Carl J. G. Evertsz and Benoit B. MandelBrot. In Chaos and Fractals, New Frontiers of Science, Appendix B. Edited by Peitgen, Juergens and Saupe, Springer Verlag, 1992 pages 921-953. "A class of Multinomial Multifractal Measures with negative (latent) values for the "Dimension" f(alpha)", Benoit B. MandelBrot. In Fractals' Physical Origins and Properties, Proceeding of the Erice Meeting, 1988. Edited by L. Pietronero, Plenum Press, New York, 1989 pages 3-29.


SEE ALSO :

binom, sbinom, multim2d, smultim1d, smultim2d (C_LAB routines). MFAS_measures, MFAS_dimensions, MFAS_spectra (Matlab and/or Scilab demo scripts).

19.63   multim2d multinomial 2d measure synthesis

Author: Christophe Canus

This C_LAB routine synthesizes a large range of pre-multifractal measures related to the multinomial 2d measure (deterministic, shuffled, pertubated) and computes linked theoretical functions (partition sum function, Reyni exponents function, generalized dimensions, multifractal spectrum).






USAGE :

[varargout,[optvarargout]]=binom(bx,by,p,str,varargin,[optvarargin])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The multinomial 2d measure is completly characterized by its abscissa base bx, ordonate base by and its weights p(i) (i=1 to bx*by). The first two parameters bx and by must be >1. The third parameter must be a vector of size equal to bx*by. The weights p(i) must be >0., <1. and their sum must be =1. (the case of p(i)=1/(bx*by) corresponds to the Lebesgue measure) (i=1 to bx*by). The fourth parameter str is a variable string used to determine the desired type of output. There are six suffix strings ('meas' for measure, 'cdf' for cumulative distribution function , 'pdf' for probability density function, 'part' for partition sum function, 'Reyni' for Reyni exponent function , 'spec' for multifractal spectrum) for the deterministic multinomial measure and two prefix strings for related measures ('shuf' for shuffled , 'pert' for pertubated) which can be added to the first ones to form composed strings. For example, 'shufmeas' is for the synthesis of a shuffled multinomial 2d pre-multifractal measure. Note that all combinaisons of strings are not implemented yet. When a string containing suffix string 'meas' is given as fourth input, a pre-multifractal measure mu_n (first output argument) is synthesized on the bx-adic and by-adic intervals I_nx and I_ny (second and third optional output argument) of the unit square. In that case, the fifth input argument is a strictly positive real (integer) scalar n which contains the resolution of the pre-multifractal measure. The size of the output real matrix mu_n is equal to bxn*byn and the one of the output real vectors I_nx and I_ny (if used) is equal to bxn and byn (so be aware the stack size ;-)). This option is implemented for the deterministic ('meas'), shuffled ('shufmeas') and pertubated ('pertmeas') multinomial 2d measure. When a string containing prefix 'shuf' is given as fourth input, the synthesis is made for a shuffled multinomial measure. At each level of the multiplicative cascade and for all nodes of the corresponding binary tree, the vector of weights p is shuffled. This option is implemented only for the multinomial 2d measure ('shufmeas'). When a string containing prefix 'pert' is given as fourth input, the synthesis is made for a pertubated multinomial measure. In that case, the fifth input argument is a strictly positive real scalar epsilon which contains the pertubation around weights. The weights are independant random variables identically distributed between p(i)-epsilon and p(i)+epsilon which must be >0., <1. (i=1 to bx*by). This option is implemented only for the multinomial 2d measure ('pertmeas'). When replacing suffix string 'meas' with suffix string 'cdf', respectively suffix string 'pdf', the cumulative distribution function F_n, respectively the probability density function p_n, related to this pre-multifractal measure is computed (first output argument). When string 'part' is given as fourth input, the partition sum function znq of multifractal measure is computed as sole output argument. In that case, the fifth input argument is a strictly positive real (integer) vector vn which contains the resolutions, and the sixth input argument is a real vector q which contains the measure exponents. The size of the output real matrix znq is equal to size(q)*size(vn). This option is implemented only for the multinomial 2d measure. When string 'Reyni' is given as third input, the Reyni exponents function tq (and the generalized dimensions Dq if used) of the multifractal measure is computed as first output argument (and second optional output argument if used). In that case, the fifth input argument is a real vector q which contains the measure's exponents. The size of the output real vector tq is equal to size(q)). This option is implemented only for the multinomial 2d measure. When string 'spec' is given as fourth input, the multifractal spectrum f_alpha (second output argument) is synthesized on the Hoelder exponents alpha (first output argument). In that case, the fifth input argument is a strictly positive real (integer) scalar N which contains the number of Hoelder exponents. The size of both output real vectors alpha and f_alpha is equal to N. This option is implemented only for the multinomial 2d measure.


ALGORITHM DETAILS :

For the deterministic multinomial, the pre-multifractal measure synthesis algorithm is implemented is a iterative way (supposed to run faster than a recursive one). For the shuffled or the pertubated multinomial, the synthesis algorithm is implemented is a recursive way (to be able to pick up a i.i.d. r.v. at each level of the multiplicative cascade and for all nodes of the corresponding binary tree w.r.t. the given law). In the case of the pertubated multinomial, the weights of each node are normalised by their sum for the measure to remain conservative. Note that the shuffled multinomial 2d measure is not conservative.


EXAMPLES :




MATLAB :



bx=2;
by=3;
p=[.05 .1; .15 .2; .24 .26];
n=5;
% synthesizes a pre-multifractal multinomial 2d measure
[mu_n,I_nx,I_ny]=multim2d(bx,by,p,'meas',n);
mesh(I_nx,I_ny,mu_n);
% synthesizes the cdf of a pre-multifractal shuffled multinomial 2d measure
F_n=multim2d(bx,by,p,'shufcdf',n);
mesh(I_nx,I_ny,F_n);
e=.049;
% synthesizes the pdf of a pre-multifractal pertubated multinomial 2d measure
p_n=multim2d(bx,by,p,'pertpdf',n,e);
mesh(I_nx,I_ny,p_n);
vn=[1:1:8];
q=[-5:.1:+5];
% computes the partition sum function of a multinomial 2d measure
znq=multim2d(bx,by,p,'part',vn,q);
plot(-vn*log(2),log(znq));
% computes the Reyni exponents function of a multinomial 2d measure
tq=multim2d(bx,by,p,'Reyni',q);
plot(q,tq);
N=200;
% computes the multifractal spectrum of a multinomial 2d measure
[alpha,f_alpha]=multim2d(bx,by,p,'spec',N);
plot(alpha,f_alpha);



SCILAB :



bx=2;
by=3;
p=[.05 .1; .15 .2; .24 .26];
n=5;
// synthesizes a pre-multifractal multinomial 2d measure
[mu_n,I_nx,I_ny]=multim2d(bx,by,p,'meas',n);
plot3d(I_nx,I_ny,mu_n);
// synthesizes the cdf of a pre-multifractal shuffled multinomial 2d measure
F_n=multim2d(bx,by,p,'shufcdf',n);
plot3d(I_nx,I_ny,F_n);
e=.049;
// synthesizes the pdf of a pre-multifractal pertubated multinomial 2d measure
p_n=multim2d(bx,by,p,'pertpdf',n,e);
plot3d(I_nx,I_ny,p_n);
xbasc();
vn=[1:1:8];
q=[-5:.1:+5];
// computes the partition sum function of a multinomial 2d measure
znq=multim2d(bx,by,p,'part',vn,q);
mn=zeros(max(size(q)),max(size(vn)));
for i=1:max(size(q))
   mn(i,:)=-vn*log(2);
end
plot2d(mn',log(znq'));
// computes the Reyni exponents function of a multinomial 2d measure
tq=multim2d(bx,by,p,'Reyni',q);
plot(q,tq);
N=200;
// computes the multifractal spectrum of a multinomial 2d measure
[alpha,f_alpha]=multim2d(bx,by,p,'spec',N);
plot(alpha,f_alpha);



REFERENCES :

"Multifractal Measures", Carl J. G. Evertsz and Benoit B. MandelBrot. In Chaos and Fractals, New Frontiers of Science, Appendix B. Edited by Peitgen, Juergens and Saupe, Springer Verlag, 1992 pages 921-953. "A class of Multinomial Multifractal Measures with negative (latent) values for the "Dimension" f(alpha)", Benoit B. MandelBrot. In Fractals' Physical Origins and Properties, Proceeding of the Erice Meeting, 1988. Edited by L. Pietronero, Plenum Press, New York, 1989 pages 3-29. .SH See also binom, sbinom, multim1d, smultim1d, smultim2d (C_LAB routines). MFAS_measures, MFAS_dimensions, MFAS_spectra (Matlab and/or Scilab demo scripts).

19.64   nextpowQ Rounds a number to the up-nearest power of an integer

Author: Paulo Goncalves

Rounds a number x to the up-nearest power of an integer Q






USAGE :



[xup2Q,powQ] = nextpowQ(x[,Q])



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

log, log2

19.65   oscillsing Oscillating Singularity synthesis

Author: Paulo Goncalves

Generates oscillating singularities located in the interval [0 .. 1]






USAGE :

[x,Fj,Fs] = oscillsing(alpha,beta,sing_pos,N,show) ;


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :




EXAMPLE: :

[x,Fj,Fs] = oscillsing([1/2 1 2],[1 2 4],[-0.5 0 0.5],256,1) ;

19.66   prescrib Generation of signals with prescribed Holder function

Author: Khalid Daoudi

Using the GIFS method, this routine generates a continous function with prescribed Holder function, while interpolating a set of point.






USAGE :

[x,y]=prescrib(Interp_pts, Holder_funct, nbr_iter)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS :

1




ALGORITHM DETAILS :

Generalized Iterated Functions Systems (GIFS) are a generalization of the usual IFS. This generalization consists in allowing the contarations to change at each step (scale) of the attractor generation process. We also allow their number and their support to change. Here, we use the GIFS to construct continuous function with prescribed local regularity. More precisely, if H(t) is the prescribed Holder function, then for each j=1,...,nbr_iter-1, and for each k=0,...,pow(m,j)-1, the GIFS coefficient c_kj is definied as : c_kj = pow(m,H(k*pow(m,-j))), where m+1 is the number of interpolation points. The resulting attractor is the graph of a continuous function F such that the Holder exponent of F, at each point t, is H(t). Moreover, if {(t_i, y_i), i=1,...,m+1}is the set of interpolation points, then F(t_i)=y_i for each i=1,...,m+1.


SEE ALSO: :

gifs and alphagifs


EXAMPLE: :

I = [0 0 1 0]; [x,y] = prescrib(I,'abs(sin(3*pi*t))',10); plot(x,y) [x,y] is the graph of a continuous function F which interpolates {(0,0); (0.5 1); (1,0)}and such that the Holder exponent of F, at each point t, is abs(sin(3*pi*t)).

19.67   pseudoAW Pseudo affine Wigner distribution

Author: Paulo Goncalves

Computes a Pseudo Affine Wigner Distributions of a 1-D signal (real or complex).






USAGE :

[tfr,scale,f,wt] = pseudoAW(x,K,[wave,smooth,fmin,fmax,N]);


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS :

1




ALGORITHM DETAILS :

A pseudo affine Wigner distribution requires to compute a continuous wavelet transform first. For each time, the corresponding column of the wavelet transform is affine convolved (generalized affine convolution defined through function lambda_K(u) ) with itself.


SEE ALSO: :

contwt, cwt and lambdak


EXAMPLE: :

Signal synthesis 

x = morlet(0.35,32)+morlet(0.1,32) ;
K = -1 pseudo affine Wigner distribution with no time smoothing 

[tfr,scale,f,wt] = pseudoAW(x,-1,12*i,0,0.01,0.5,128) ;
viewmat(tfr,1:length(x),f,[1 0 .5]) ;
K = -1 time smoothed pseudo affine Wigner distribution 

[tfr,scale,f,wt] = pseudoAW(x,-1,12*i,3,0.01,0.5,128) ;
viewmat(tfr,1:length(x),f,[1 0 0]) ;

19.68   regdim Estimate the regularization dimension of a 1d or 2d sample.

Author: Francois Roueff

Computes the regularization dimension of the graph of a 1d or 2d sampled function. Two kernels are available: the Gaussian or the Rectangle.






USAGE :

dim = regdim(x,sigma,voices,Nmin,Nmax,kernel,mirror,reg,graphs)



INPUT PARAMETERS :




OUTPUT PARAMETERS :




DESCRIPTION :

For a more complete explanation of the regularization dimension, one can refer to: "A regularization approach to fractionnal dimension estimation", F. Roueff, J. Levy-Vehel, submitted to Fractal 98 conference. The regularized graphs of x are computed via convolutions of x with dilated versions of the kernel at different scales. The lengths of the regularized graphs are computed via convolutions of x with the derivatives of the dilated versions of the kernel. The regularization dimension is computed either via an automatic range regression or via a regression by hand on the loglog plot of the lengths versus scales. If sigma is strictly positive, an estimation of the lengths without noise is used for the regression. These lengths are displayed in red while those of the noisy signal are in black. They should seperate at fine scales. When one specifies the range regression, the loglog plot of the lengths versus scales appears. Above are either increments (when sigma is null) or a loglog plot of the noise prevalence in the lengths. One selects the scale range of the regression. In the case of noisefree signals, select a scale region with stable increments. In the case of a strictly positive sigma, select a scale region where the noise prevalence is not too close to 1 (0 in log10): it should correspond to an approximately linear region for the red estimations. The number of scales (voices) tells how many convolutions are computed. The bigger it is, the slower the computation is. The scale axis is geometrically sampled (i.e. its log is arithmetically sampled). The gaussian kernel should give a better result but the rectangle is faster.


SEE ALSO: :

cwttrack, cwtspec.


EXAMPLE: :

//          1D: 
// Signal synthesis 
x = GeneWei(1024,0.6,2,1.0,0); 
plot(x);
//Dimension of the graph with a regression by hand
dim = regdim(x,0,128,10,500,'gauss',0,1,1);

//          2D
// Signal synthesis 
z = GeneWei(200,0.6,2,1.0,0);
y = GeneWei(200,0.4,3,1.0,0);
w = z'*y;
plot3d(linspace(0,1,200),linspace(0,1,200),w);
//Dimension of the graph with a regression by hand
dim = regdim(w,0,25,10,50,'gauss',0,1);

19.69   reynitq Reyni exponents estimation

Author: Christophe Canus

This routine estimates the Reyni exponents on a partition function.






USAGE :

[tq,[Dq]]=reynitq(znq,n,q)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The mass exponents tq and the generalized dimensions Dq (if used) are computed on the partition function znq. The input real matrix znq must be of height size(q) and of width size(n). The output real vectors tq and Dq (if used) are of size size(q).


ALGORITHM DETAILS  :

The mass exponent function tq by least mean square linear-fit.


SEE ALSO :

mdzq1d,mdzq2d,linearlt,mdfl1d,mdfl2d.

19.70   sbinom stochastic binomial measure synthesis

Author: Christophe Canus

This C_LAB routine synthesizes two types of pre-multifractal stochastic measures related to the binomial measure paradigm (uniform law and lognormal law) and computes linked multifractal spectrum.






USAGE :

[varargout,[optvarargout]]=sbinom(str,varargin,[optvarargin])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The first parameter str is a variable string used to determine the desired type of output. There are four suffix strings ('meas' for measure, 'cdf' for cumulative distribution function , 'pdf' for probability density function, 'spec' for multifractal spectrum) and a two prefix strings for the type of stochastic measure ('unif' for uniform and 'logn' for lognormal) which must added to the first ones to form composed. For example, 'unifmeas' is for the synthesis of a uniform law binomial pre-multifractal measure and 'lognspec' is for the computation of the multifractal spectrum of a lognormal binomial measure. When a string containing suffix string 'meas' is given as second input, a pre-multifractal measure mu_n (first output argument) is synthesized on the dyadic intervals I_n (second optional output argument) of the unit interval. In that case, the third input argument is a strictly positive real (integer) scalar n which contains the resolution of the pre-multifractal measure. The size of the output real vectors mu_n (and I_n if used) is equal to 2n (so be aware the stack size ;-)). This option is implemented for the uniform law ('unifmeas') and the lognormal law ('lognmeas') binomial measures. When a string containing prefix 'unif' is given as second input, the synthesis or the computation is made for a uniform law binomial measure. In that case, the optional fourth input argument is a strictly positive real scalar epsilon which contains the pertubation around weight .5. The weight is an independant random variable identically distributed between epsilon and 1-epsilon which must be >0., <1. The default value for epsilon is 0. When a string containing prefix 'logn' is given as second input, the synthesis or the computation is made for a lognormal law binomial measure. In that case, the optional fourth input argument is a strictly positive real scalar sigma which contains the standard deviation of the lognormal law. When replacing suffix string 'meas' with suffix string 'cdf', respectively suffix string 'pdf', the cumulative distribution function F_n, respectively the probability density function p_n, related to this pre-multifractal measure is computed (first output argument). When a string containing suffix string 'spec' is given as second input, the multifractal spectrum f_alpha (second output argument) is synthesized on the Hoelder exponents alpha (first output argument). In that case, the third input argument is a strictly positive real (integer) scalar N which contains the number of Hoelder exponents. The size of both output real vectors alpha and f_alpha is equal to N. This option is implemented for the uniform law ('unifspec') and the lognormal law ('lognspec') binomial measures.


ALGORITHM DETAILS :

For the uniform and lognormal law binomial, the synthesis algorithm is implemented is a recursive way (to be able to pick up a i.i.d. r.v. at each level of the multiplicative cascade and for all nodes of the corresponding binary tree w.r.t. the given law). Note that the lognormal law binomial measure is not conservative.


EXAMPLES :




MATLAB :



n=10;
% synthesizes a pre-multifractal uniform Law binomial measure
[mu_n,I_n]=sbinom('unifmeas',n);
plot(I_n,mu_n);
s=1.;
% synthesizes the cdf of a pre-multifractal lognormal law binomial measure
F_n=sbinom('logncdf',n,s);
plot(I_n,F_n);
e=.19;
% synthesizes the pdf of a pre-multifractal uniform law binomial measure
p_n=sbinom('unifpdf',n,e);
plot(I_n,p_n);
N=200;
s=1.;
% computes the multifractal spectrum of the lognormal law binomial measure
[alpha,f_alpha]=sbinom('lognspec',N,s);
plot(alpha,f_alpha);



SCILAB :



n=10;
// synthesizes a pre-multifractal uniform Law binomial measure
[mu_n,I_n]=sbinom('unifmeas',n);
plot(I_n,mu_n);
s=1.;
// synthesizes the cdf of a pre-multifractal lognormal law binomial measure
F_n=sbinom('logncdf',n,s);
plot(I_n,F_n);
e=.19;
// synthesizes the pdf of a pre-multifractal uniform law binomial measure
p_n=sbinom('unifpdf',n,e);
plot(I_n,p_n);
N=200;
// computes the multifractal spectrum of the lognormal law binomial measure
[alpha,f_alpha]=sbinom('lognspec',N,s);
plot(alpha,f_alpha);



REFERENCES :

"A class of Multinomial Multifractal Measures with negative (latent) values for the "Dimension" f(alpha)", Benoit B. MandelBrot. In Fractals' Physical Origins and Properties, Proceeding of the Erice Meeting, 1988. Edited by L. Pietronero, Plenum Press, New York, 1989 pages 3-29. "Limit Lognormal Multifractal Measures", Benoit B. MandelBrot. In Frontiers of Physics, Landau Memorial Conference, Proceeding of the Tel-Aviv Meeting, 1988. Edited by Errol Asher Gotsman, Yuval Ne'eman and Alexander Voronoi, New York Pergamon, 1990 pages 309-340.


SEE ALSO :

binom, multim1d, multim2d, smultim1d, smultim2d (C_LAB routines). MFAS_measures, MFAS_dimensions, MFAS_spectra (Matlab and/or Scilab demo scripts).

19.71   sgifs Semi Generalized IFS generation

Author: Khalid Daoudi

This routine generates stochastical Semi-Generalized Iterated Functions Systems (SGIFS) attractors.






USAGE :

[x, y, Ci]=sgifs(Interp_pts, coefs, nbr_iter,law_type,var)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS :

1




ALGORITHM DETAILS :

Semi-Generalized Iterated Functions Systems (SGIFS) are a generalization of the usual IFS. This generalization consists in allowing the contarations to change at each step (scale) of the attractor generation process. Here, we use GIFS to construct stocastical SGIFS. More precisely, at each scale j, the GIFS coefficients c_kj, for k even (resp. odd), are a random variable of law law_type, of mean coefs(1) (resp. coefs(2)) and of variance 1/pow(j,var) Moreover, if {(t_i, y_i), i=1,...,n+1}is the set of interpolation points, then any realisation of the attractor is the graph of a continuous function F such that : F(t_i)=y_i for each i=1,...,n+1.


SEE ALSO: :

fif, alphagifs and prescrib


EXAMPLE: :

I = [0 0 1 0]; coefs = [.3 -.9]; [x,y,Ci] = sgifs(I,coefs,10,'uniform',1); plot(x,y)



19.72   sim_stable Generation of stable random processes

Author: Lotfi Belkacem

This routine generates a stable random process and its increments using the Chambers, Mallows and Stuck (1976) algorithm.






USAGE :

[proc,inc]=sim_stable(alpha,beta,mu,gamma,size)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




EXAMPLES :




EXAMPLE 1 :

[proc,inc]=sim_stable(2,0,0,1.4142136,5000); generates a standard stable random process with alpha=2, beta=0 (symmetric), mu=0 and gamma=1.4142 which coinside with a standard gaussian process (Brownian motion). To visualize the process or the increments use plot(proc) or plot(inc).


EXAMPLE 2 :

[proc,inc]=sim_stable(1.5,0,0,1,5000); generates a standard 1.5-stable motion

19.73   smultim1d multinomial 1d measure synthesis -

Author: Christophe Canus

This C_LAB routine synthesizes two types of pre-multifractal stochastic measures related to the multinomial 1d measure (uniform law and lognormal law) and computes linked multifractal spectrum.






USAGE :

[varargout,[optvarargout]]=sbinom(b,str,varargin,[optvarargin])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The stochastic multinomial 1d measure is completly characterized by its base b. This first parameter must be >1. The second parameter str is a variable string used to determine the desired type of output. There are four suffix strings ('meas' for measure, 'cdf' for cumulative distribution function q, 'pdf' for probability density function, 'spec' for multifractal spectrum) and a two prefix strings for the type of stochastic measure ('unif' for uniform and 'logn' for lognormal) which must added to the first ones to form composed. For example, 'unifmeas' is for the synthesis of a uniform law multinomial 1d pre-multifractal measure and 'lognspec' is for the computation of the multifractal spectrum of a lognormal multinomial 1d measure. When a string containing suffix string 'meas' is given as second input, a pre-multifractal measure mu_n (first output argument) is synthesized on the b-adic intervals I_n (second optional output argument) of the unit interval. In that case, the third input argument is a strictly positive real (integer) scalar n which contains the resolution of the pre-multifractal measure. The size of the output real vectors mu_n (and I_n if used) is equal to bn (so be aware the stack size ;-)). This option is implemented for the uniform law ('unifmeas') and the lognormal law ('lognmeas') multinomial 1d measures. When a string containing prefix 'unif' is given as second input, the synthesis or the computation is made for a uniform law multinomial 1d measure. In that case, the optional fourth input argument is a strictly positive real scalar epsilon which contains the pertubation around weight .5. The weight is an independant random variable identically distributed between epsilon and 1-epsilon which must be >0., <1. The default value for epsilon is 0. When a string containing prefix 'logn' is given as second input, the synthesis or the computation is made for a lognormal law multinomial 1d measure. In that case, the optional fourth input argument is a strictly positive real scalar sigma which contains the standard deviation of the lognormal law. When replacing suffix string 'meas' with suffix string 'cdf', respectively suffix string 'pdf', the cumulative distribution function F_n, respectively the probability density function p_n, related to this pre-multifractal measure is computed (first output argument). When a string containing suffix string 'spec' is given as second input, the multifractal spectrum f_alpha (second output argument) is synthesized on the Hoelder exponents alpha (first output argument). In that case, the third input argument is a strictly positive real (integer) scalar N which contains the number of Hoelder exponents. The size of both output real vectors alpha and f_alpha is equal to N. This option is implemented only for the lognormal law ('lognspec') multinomial 1d measures.


ALGORITHM DETAILS :

For the uniform and lognormal law multinomial 1d, the synthesis algorithm is implemented is a recursive way (to be able to pick up a i.i.d. r.v. at each level of the multiplicative cascade and for all nodes of the corresponding binary tree w.r.t. the given law). Note that the lognormal law multinomial 1d measure is not conservative.


EXAMPLES :




MATLAB :



n=10;
% synthesizes a pre-multifractal uniform Law multinomial 1d measure
[mu_n,I_n]=smultim1d(b,'unifmeas',n);
plot(I_n,mu_n);
s=1.;
% synthesizes the cdf of a pre-multifractal lognormal law multinomial 1d measure
F_n=smultim1d(b,'logncdf',n,s);
plot(I_n,F_n);
e=.19;
% synthesizes the pdf of a pre-multifractal uniform law multinomial 1d measure
p_n=smultim1d(b,'unifpdf',n,e);
plot(I_n,p_n);
N=200;
s=1.;
% computes the multifractal spectrum of the lognormal law multinomial 1d measure
[alpha,f_alpha]=smultim1d(b,'lognspec',N,s);
plot(alpha,f_alpha);



SCILAB :



n=10;
// synthesizes a pre-multifractal uniform Law multinomial 1d measure
[mu_n,I_n]=smultim1d(b,'unifmeas',n);
plot(I_n,mu_n);
s=1.;
// synthesizes the cdf of a pre-multifractal lognormal law multinomial 1d measure
F_n=smultim1d(b,'logncdf',n,s);
plot(I_n,F_n);
e=.19;
// synthesizes the pdf of a pre-multifractal uniform law multinomial 1d measure
p_n=smultim1d(b,'unifpdf',n,e);
plot(I_n,p_n);
N=200;
// computes the multifractal spectrum of the lognormal law multinomial 1d measure
[alpha,f_alpha]=smultim1d(b,'lognspec',N,s);
plot(alpha,f_alpha);



REFERENCES :

"A class of Multinomial Multifractal Measures with negative (latent) values for the "Dimension" f(alpha)", Benoit B. MandelBrot. In Fractals' Physical Origins and Properties, Proceeding of the Erice Meeting, 1988. Edited by L. Pietronero, Plenum Press, New York, 1989 pages 3-29. "Limit Lognormal Multifractal Measures", Benoit B. MandelBrot. In Frontiers of Physics, Landau Memorial Conference, Proceeding of the Tel-Aviv Meeting, 1988. Edited by Errol Asher Gotsman, Yuval Ne'eman and Alexander Voronoi, New York Pergamon, 1990 pages 309-340.


SEE ALSO :

binom, sbinom, multim1d, multim2d, smultim2d (C_LAB routines). MFAS_measures, MFAS_dimensions, MFAS_spectra (Matlab and/or Scilab demo scripts).

19.74   smultim2d multinomial 2d measure synthesis -

Author: Christophe Canus

This C_LAB routine synthesizes two types of pre-multifractal stochastic measures related to the multinomial 2d measure (uniform law and lognormal law).






USAGE :

[varargout,[optvarargout]]=sbinom(bx,by,str,varargin,[optvarargin])


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :




PARAMETERS  :

The stochastic multinomial 2d measure is completly characterized by its abscissa base bx, ordonate base by. These first two parameters must be >1. The third parameter str is a variable string used to determine the desired type of output. There are four suffix strings ('meas' for measure, 'cdf' for cumulative distribution function q, 'pdf' for probability density function, 'spec' for multifractal spectrum) and a two prefix strings for the type of stochastic measure ('unif' for uniform and 'logn' for lognormal) which must added to the first ones to form composed. For example, 'unifmeas' is for the synthesis of a uniform law multinomial 2d pre-multifractal measure and 'lognspec' is for the computation of the multifractal spectrum of a lognormal multinomial 2d measure. When a string containing suffix string 'meas' is given as third input, a pre-multifractal measure mu_n (first output argument) is synthesized on the bx-adic and by-adic intervals I_nx and I_ny (second and third optional output argument) of the unit square. In that case, the fourth input argument is a strictly positive real (integer) scalar n which contains the resolution of the pre-multifractal measure. The size of the output real matrix mu_n is equal to bxn*byn and the one of the output real vectors I_nx and I_ny (if used) is equal to bxn and byn (so be aware the stack size ;-)). This option is implemented for the uniform law ('unifmeas') and the lognormal law ('lognmeas') multinomial 2d measures. When a string containing prefix 'unif' is given as third input, the synthesis or the computation is made for a uniform law multinomial 2d measure. In that case, the optional fourth input argument is a strictly positive real scalar epsilon which contains the pertubation around weight .5. The weight is an independant random variable identically distributed between epsilon and 1-epsilon which must be >0., <1. The default value for epsilon is 0. When a string containing prefix 'logn' is given as third input, the synthesis or the computation is made for a lognormal law multinomial 2d measure. In that case, the optional fifth input argument is a strictly positive real scalar sigma which contains the standard deviation of the lognormal law. When replacing suffix string 'meas' with suffix string 'cdf', respectively suffix string 'pdf', the cumulative distribution function F_n, respectively the probability density function p_n, related to this pre-multifractal measure is computed (first output argument).


ALGORITHM DETAILS :

For the uniform and lognormal law multinomial 2d, the synthesis algorithm is implemented is a recursive way (to be able to pick up a i.i.d. r.v. at each level of the multiplicative cascade and for all nodes of the corresponding binary tree w.r.t. the given law). Note that the lognormal law multinomial 2d measure is not conservative.


EXAMPLES :




MATLAB :



n=5;
bx=2;
by=3;
% synthesizes a pre-multifractal uniform Law multinomial 2d measure
[mu_n,I_nx,I_ny]=smultim2d(bx,by,'unifmeas',n);
mesh(I_nx,I_ny,mu_n);
s=1.;
% synthesizes the cdf of a pre-multifractal lognormal law multinomial 2d measure
F_n=smultim2d(bx,by,'logncdf',n,s);
mesh(I_nx,I_ny,F_n);
e=.19;
% synthesizes the pdf of a pre-multifractal uniform law multinomial 2d measure
p_n=smultim2d(bx,by,'unifpdf',n,e);
mesh(I_nx,I_ny,p_n);



SCILAB :



n=5;
bx=2;
by=3;
// synthesizes a pre-multifractal uniform Law multinomial 2d measure
[mu_n,I_nx,I_ny]=smultim2d(bx,by,'unifmeas',n);
mesh(I_nx,I_ny,mu_n);
s=1.;
// synthesizes the cdf of a pre-multifractal lognormal law multinomial 2d measure
F_n=smultim2d(bx,by,'logncdf',n,s);
mesh(I_nx,I_ny,F_n);
e=.19;
// synthesizes the pdf of a pre-multifractal uniform law multinomial 2d measure
p_n=smultim2d(bx,by,'unifpdf',n,e);
mesh(I_nx,I_ny,p_n);



REFERENCES :

"A class of Multinomial Multifractal Measures with negative (latent) values for the "Dimension" f(alpha)", Benoit B. MandelBrot. In Fractals' Physical Origins and Properties, Proceeding of the Erice Meeting, 1988. Edited by L. Pietronero, Plenum Press, New York, 1989 pages 3-29. "Limit Lognormal Multifractal Measures", Benoit B. MandelBrot. In Frontiers of Physics, Landau Memorial Conference, Proceeding of the Tel-Aviv Meeting, 1988. Edited by Errol Asher Gotsman, Yuval Ne'eman and Alexander Voronoi, New York Pergamon, 1990 pages 309-340.


SEE ALSO :

binom, sbinom, multim1d, multim2d, smultim1d (C_LAB routines). MFAS_measures, MFAS_dimensions, MFAS_spectra (Matlab and/or Scilab demo scripts).

19.75   sortup Sorts the elements of an array in increasing order

Author: Paulo Goncalves

Sorts the elements of an array in increasing order






USAGE :



[yup,kup] = sortup(x[,how])



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

sort


EXAMPLE: :



[y,x] = sort(rand(4,4)) ; 
x
xSortAll = sortup(x,'*')
xSortCol = sortup(x,'c') 
xSOrtRow = sortup(x,'r')

19.76   stable_cov Covariation of two jointly symmetric Alpha-Stable random variables

Author: Lotfi Belkacem

This routine estimates the covariation of two jointly symmetric alpha-stable random variables.






USAGE :

[cov]=stable_cov(data1,data2)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :

The covariation of two jointly symmetric alpha stable random variables is defined only for alpha between 1 and 2. It designed to replace the covariance when the latter is not defined (alpha<2). Unfortunately, it lacks some of the desirable properties of the covariance (not symmetric, ...). It is however, a useful quantity and appears naturally in many settings, for example, in the context of linear regression..SH Example for two given signals S1 and S2, cov=stable_cov(S1,S2); estimates the covariation of S1 on S2.

19.77   stable_sm Spectral measure of a bivariate Stable random vector

Author: Lotfi Belkacem

This routine estimates a normalized spectral measure of a bivariate stable random vector.






USAGE :

[theta,sm]=stable_sm(data1,data2)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




EXAMPLE :

for two given signals S1 and S2, [theta,sm]=stable_sm(S1,S2); estimates the normalized spectral measure of the data vector (S1,S2). To visualize it use plot(theta,sm).



19.78   stable_test stable law conformicy test

Author: Lotfi Belkacem

This routine tests the stability property of a signal.






USAGE :

[param,sd_param]=stable_test(maxr,data)


INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




DESCRIPTION :

The stability test consists on estimating parameters of a fited alpha-satble law at different level of resolution. the variable is said to be stable if the charateristic exponent alpha remains approximatively constant at different resolution, and the scale parameter follows a scaling law with exponent (1/alpha)-1. .SH Example under scilab type: [proc1_5,inc1_5]=sim_stable(1.5,0,0,1,20000); [param,sd_param]=stable_test(7,inc1_5); alpha=param(:,1); m=(1:7)'; lnm=log(m); plot2d(m,alpha,1,'111','alpha',[1,0,7,2]); gamma=param(:,4); lngamma=log(gamma); plot(lnm,lngamma); [a,b,sig]=reglin(lnm',lngamma'); slope=a th_slope=1/1.5-1

1




 :

19.79   strfbm Structure Function of a Brownian Field

Author: B. Pesquet-Popescu (ENS-Cachan)

Creates the structure function of an isotropic fBm






USAGE :



[Y] = strfbm(x,y,H)



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

synth2


EXAMPLE: :



x = 1:128 ;
y = 1:128 ;
[Y] = strfbm(x,y,0.8) ;

19.80   symcori Symmetrization of a periodic 2D correlation field

Author: B. Pesquet-Popescu (ENS-Cachan)

Symmetrization of a periodic 2D correlation field






USAGE :



Ss = symcori(S) ;



INPUT PARAMETERS :

1




OUTPUT PARAMETERS :

1




SEE ALSO: :

synth2, strfbm


EXAMPLE: :




 
 
 	

19.81   synth2   Stationary Increments 2D Process


Author: B. Pesquet-Popescu (ENS-Cachan)





Incremental Fourier synthesis method for processes with stationary increments of order (0,1) and (1,0)















USAGE :










[B] = synth2(M,H,core)







INPUT PARAMETERS :



1








OUTPUT PARAMETERS :



1








REFERENCES :




L.M. Kaplan, C.C. J Kuo : IEEE Tran. on IP, May 1996 (extended version).





SEE ALSO: :




fbmlevinson, fbmcwt, fbmfwt, mbmlevinson





EXAMPLE: :










[B] = synth2(128,0.8,'strfbm') ;
viewmat(B) 


 
 
 	

19.82   viewWTLM   Vizualises the local maxima lines of a CWT


Author: Paulo Goncalves





Displays the local maxima of a continuous wavelet transform 















USAGE :










viewWTLM(maxmap[,scale,wt])







INPUT PARAMETERS :



1








SEE ALSO: :




findWTLM, viewmat, contwt, cwt





EXAMPLE: :










N = 2048 ; H = 0.3 ; Q = linspace(-4,4,11) ;
[x] = fbmlevinson(N,H) ;
[wt,scale] = cwt(x,2^(-6),2^(-1),36,0) ;
[maxmap] = findWTLM(wt,scale)


 Vizualisation in Matlab: 






viewWTLM(maxmap,scale,wt) ,
axis([1024 - 64 1024 + 64 0 log2(max(scale))]) ,


 Vizualisation in Scilab: Not implemeted yet!... 




 
 




 	

19.83   viewmat   Vizualisation of a matrix


Author: Paulo Goncalves - Bertrand Guiheneuf





Two dimensional display of a matrix















USAGE :





viewmat(Z [,X,Y])






INPUT PARAMETERS :



1








REMARK :




 viewmat changes the color map





SEE ALSO: :




plot3d, grayplot





EXAMPLE: :





//Signal synthesis:
x = oscillsing(1,1,0,128) ; 
X = x(:)*x(:)' ; 
//Matrix vizualisation: 

viewmat(abs(X))


 
 
 	

19.84   wave2gifs   Computation of IFS coef. with Discrete Wavelet coefficients


Author: Khalid Daoudi





Computes the GIFS cefficients of a
1-D real signal as the ratio between (synchrounous) wavelets coefficients
at successive scales. You have to compute the wavelet
coefficients of the given signal (using FWT) before using wave2gifs.















USAGE :




[Ci, Ci_idx, Ci_lg, pc0, pc_ab]=wave2gifs(wt, wt_idx, wt_lg, [M0, a, b])





INPUT PARAMETERS :



1








OUTPUT PARAMETERS :



1








DESCRIPTION :









PARAMETERS :



1








ALGORITHM DETAILS :









SEE ALSO: :




 
FWT and MakeQMF.





EXAMPLE: :






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