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wmpalg

(Not recommended) Matching pursuit

    The wmpalg function no longer supports plotting and is no longer recommended. See Compatibility Considerations.

    Description

    YFIT = wmpalg(MPALG,Y,MPDICT) returns an adaptive greedy approximation, YFIT, of the input signal, Y, in the dictionary, MPDICT. The adaptive greedy approximation uses the matching pursuit algorithm, MPALG. The dictionary, MPDICT, is typically an overcomplete set of vectors.

    [YFIT,R] = wmpalg(___) returns the residual, R, which is the difference vector between Y and YFIT at the termination of the matching pursuit.

    [YFIT,R,COEFF] = wmpalg(___) returns the expansion coefficients, COEFF. The number of expansion coefficients depends on the number of iterations in the matching pursuit.

    [YFIT,R,COEFF,IOPT] = wmpalg(___) returns the column indices of the retained atoms, IOPT. The length of IOPT equals the length of COEFF and is determined by the number of iterations in the matching pursuit.

    [YFIT,R,COEFF,IOPT,QUAL] = wmpalg(___) returns the proportion of retained signal energy, QUAL, for each iteration of the matching pursuit. QUAL is the ratio of the ℓ2 squared norm of the expansion coefficient vector, COEFF, to the ℓ2 squared norm of the input signal, Y.

    [YFIT,R,COEFF,IOPT,QUAL,X] = wmpalg(___) returns the normalized dictionary, X. X contains the unit vectors in the ℓ2 norm corresponding to the columns of MPDICT.

    example

    [YFIT,R,COEFF,IOPT,QUAL,X] = wmpalg(___,Name=Value) returns an adaptive greedy approximation with additional options specified by one or more Name=Value arguments.

    Examples

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    Approximate the cuspamax signal with the dictionary using orthogonal matching pursuit.

    Use a dictionary consisting of sym4 wavelet packets and the DCT-II basis.

    load cuspamax
    yfit = wmpalg('OMP',cuspamax,'lstcpt',{{'wpsym4',2},'dct'});
    plot(cuspamax,'k')
    hold on
    plot(yfit,'linewidth',2)
    hold off
    legend('Original Signal','Matching Pursuit')

    Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Original Signal, Matching Pursuit.

    Obtain the expansion coefficients in the dictionary, the column indices of the selected dictionary atoms, and the proportion of retained signal energy.

    Specify a dictionary consisting of sym4 wavelet packets and the DCT-II basis. Approximate the cuspamax signal with the dictionary using orthogonal matching pursuit.

    load cuspamax;
    [yfit,r,coeff,iopt,qual] = wmpalg('OMP',cuspamax,...
        'lstcpt',{{'wpsym4',2},'dct'});

    This example shows how to set the maximum number of iterations of the orthogonal matching pursuit to 50.

    load cuspamax
    [yfit,r,coeff,iopt,qual] = wmpalg('OMP',cuspamax,...
        'lstcpt',{{'wpsym4',1},{'wpsym4',2},'dct'},...
        'itermax',50);

    This example shows how to allow for a suboptimal choice in the update of the orthogonal matching pursuit.

    Load a signal.

    load cuspamax

    Approximate the signal using weak orthogonal matching pursuit. Relax the requirement to be 0.8 times the optimal assignment.

    [yfit,r,coeff,iopt,qual] = wmpalg('WMP',cuspamax,...
        'lstcpt',{{'wpsym4',1},{'wpsym4',2},'dct'},...
        'wmpcfs',0.8);

    Plot the signal, approximation, residual, and the proportion of retained signal energy for each iteration in the matching pursuit result.

    subplot(3,1,1)
    plot(cuspamax)
    hold on
    plot(yfit)
    hold off
    legend('Signal','Approx.')
    title('Signal and Approximation')
    axis tight
    subplot(3,1,2)
    plot(r)
    title('Residual')
    axis tight
    subplot(3,1,3)
    plot(qual,'s-')
    title('Quality / Iteration')
    ylabel('Quality')
    xlabel('Iteration')

    Figure contains 3 axes objects. Axes object 1 with title Signal and Approximation contains 2 objects of type line. These objects represent Signal, Approx.. Axes object 2 with title Residual contains an object of type line. Axes object 3 with title Quality / Iteration contains an object of type line.

    Obtain a matching pursuit of electricity consumption measured every minute over a 24-hour period.

    Load and plot data. The data shows electricity consumption sampled every minute over a 24-hour period. Because the data is centered, the actual usage values are not interpretable.

    load elec35_nor;
    y = signals(32,:);
    plot(y)
    xlabel('Minutes')
    ylabel('Usage')
    set(gca,'xlim',[1 1440])

    Figure contains an axes object. The axes object contains an object of type line.

    Specify a dictionary for matching pursuit consisting of the Daubechies' extremal-phase wavelet with 2 vanishing moments at level 2, the Daubechies' least-asymmetric wavelet with 4 vanishing moments at levels 1 and 4, the discrete cosine transform-II basis, and the sine basis.

    dictionary = {{'db4',2},'dct','sin',{'sym4',1},{'sym4',4}};

    Implement orthogonal matching pursuit to obtain a signal approximation in the dictionary. Use 35 iterations. Plot the result.

    [yfit,r,coef,iopt,qual] = wmpalg('OMP',y,...
        'lstcpt',dictionary,'itermax',35);
    plot(y)
    hold on
    plot(yfit,'r')
    xlabel('Minutes')
    ylabel('Usage');
    legend('Original Signal','OMP','Location','NorthEast')
    set(gca,'xlim',[1 1440])

    Figure contains an axes object. The axes object contains 2 objects of type line. These objects represent Original Signal, OMP.

    Input Arguments

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    Matching pursuit algorithm, specified as one of the following:

    • 'BMP' — Basic matching pursuit

    • 'OMP' — Orthogonal matching pursuit

    • 'WMP' — Weak orthogonal matching pursuit

    See Matching Pursuit Algorithms.

    Matching pursuit dictionary, specified as a matrix. MPDICT is a N-by-P matrix, where N is equal to the length of the input signal, Y. In matching pursuit, MPDICT is commonly a frame, or overcomplete set of vectors. You may use the name-value argument 'lstcpt' to specify a dictionary instead of using MPDICT.

    Data Types: double

    Signal for matching pursuit, specified as a vector. The row dimension of MPDICT must match the length of Y.

    Name-Value Arguments

    Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

    Example: yfit = wmpalg('OMP',y,lstcpt={'dct'}) specifies the DCT-II dictionary.

    Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

    Example: yfit = wmpalg('OMP',y,'lstcpt',{'dct'})

    Positive integer fixing the maximum number of iterations of the matching pursuit algorithm. If you do not specify a 'maxerr' value, the number of expansion coefficients, COEFF, the number of dictionary vector indices, IOPT, and the length of the QUAL vector equal the value of 'itermax'.

    Data Types: double

    A cell array of cell arrays with valid subdictionaries. Each cell array describes one subdictionary. Valid subdictionaries are:

    • A valid Wavelet Toolbox™ orthogonal or biorthogonal wavelet family short name with the number of vanishing moments and an optional decomposition level and extension mode. For example, {'sym4',5} denotes the Daubechies least-asymmetric wavelet with 4 vanishing moments at level 5 and the default extension mode 'per'. If you do not specify the optional level and extension mode, the decomposition level defaults to 5 and the extension mode to 'per'.

    • A valid Wavelet Toolbox orthogonal or biorthogonal wavelet family short name preceded by wp with the number of vanishing moments and an optional decomposition level and extension mode. For example, {'wpsym4',5} denotes the Daubechies least-asymmetric wavelet packet with 4 vanishing moments at level 5. If you do not specify the optional level and extension mode, the decomposition level defaults to 5 and the extension mode to 'per'.

    • 'dct' Discrete cosine transform-II basis. The DCT-II orthonormal basis is:

      ϕk(n)={1Nk=02Ncos(πN(n+12)k)k=1,2,,N1.

    • 'sin' Sine subdictionary. The sine subdictionary is

      ϕk(t)=sin(2πkt)k=1,2,N20t1

      where t is a linearly-spaced N-point vector.

    • 'cos' Cosine subdictionary. The cosine subdictionary is

      ϕk(t)=cos(2πkt)k=1,2,N20t1

      where t is a linearly-spaced N-point vector.

    • 'poly' Polynomial subdictionary. The polynomial subdictionary is:

      pn(t)=tn1n=1,2,200t1

      where t is a linearly-spaced N-point vector.

    • 'RnIdent' The shifted Kronecker delta subdictionary. The shifted Kronecker delta subdictionary is:

      ϕk(n)=δ(nk)k=0,1,N

    Data Types: double

    Cell array containing the name of the norm and the maximum relative error in the norm expressed as a percentage. Valid norms are 'L1', 'L2', and 'Linf'. The relative error expressed as a percentage is

    100RY

    where R is the residual at each iteration and Y is the input signal. For example, {'L1',10} sets maximum acceptable ratio of the L1 norms of the residual to the input signal to 0.10.

    If you specify 'maxerr', the matching pursuit terminates when the first of the following conditions is satisfied:

    • The number of iterations reaches the minimum of the length of the input signal, Y, or 500:
      min(length(Y),500)

    • The relative error falls below the percentage you specify with the 'maxerr' name-value pair.

    Data Types: double

    Optimality factor for weak orthogonal matching pursuit. The optimality factor is a real number in the interval (0,1]. This name-value argument is only valid when MPALG is 'WMP'.

    Data Types: double

    Output Arguments

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    Adaptive greedy approximation of the input signal, Y, in the dictionary

    Data Types: double

    Residual after matching pursuit terminates

    Data Types: double

    Expansion coefficients in the dictionary. The selected dictionary atoms weighted by the expansion coefficients yield the approximated signal, YFIT.

    Data Types: double

    Column indices of the selected dictionary atoms. Using the column indices in IOPT with the expansion coefficients in COEFF, you can form the approximated signal, YFIT.

    Data Types: double

    Proportion of retained signal energy for each iteration in the matching pursuit. QUAL is a vector with each element equal to

    ||αk||22||Y||22

    where αk is the vector of expansion coefficients after the k-th iteration.

    Data Types: double

    The normalized matching pursuit dictionary. X is an N-by-P matrix where N is the length of the input signal, Y. The columns of X have unit norm.

    Data Types: double

    References

    [1] Cai, T. Tony, and Lie Wang. “Orthogonal Matching Pursuit for Sparse Signal Recovery With Noise.” IEEE Transactions on Information Theory 57, no. 7 (July 2011): 4680–88. https://doi.org/10.1109/TIT.2011.2146090.

    [2] Donoho, D.L., M. Elad, and V.N. Temlyakov. “Stable Recovery of Sparse Overcomplete Representations in the Presence of Noise.” IEEE Transactions on Information Theory 52, no. 1 (January 2006): 6–18. https://doi.org/10.1109/TIT.2005.860430.

    [3] Mallat, S.G. and Zhifeng Zhang. “Matching Pursuits with Time-Frequency Dictionaries.” IEEE Transactions on Signal Processing 41, no. 12 (December 1993): 3397–3415. https://doi.org/10.1109/78.258082.

    [4] Tropp, J.A. “Greed Is Good: Algorithmic Results for Sparse Approximation.” IEEE Transactions on Information Theory 50, no. 10 (October 2004): 2231–42. https://doi.org/10.1109/TIT.2004.834793.

    Version History

    Introduced in R2012a

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    Errors starting in R2022a

    Not recommended starting in R2022a