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wdcbm2

Thresholds for wavelet 2-D using Birgé-Massart strategy

    Description

    example

    [thr,nkeep] = wdcbm2(C,S,alpha,M) returns level-dependent thresholds thr and numbers of coefficients to be kept nkeep, for denoising or compressing an image. wdcbm2 uses a wavelet coefficients selection rule based on the Birgé-Massart strategy to obtain the thresholds.

    [C,S] is the wavelet decomposition structure of the image to be denoised or compressed, at level N = size(S,1)-2. alpha and M are real numbers greater than 1.

    wdcbm2(C,S,alpha) is equivalent to wdcbm2(C,S,alpha,prod(S(1,:))).

    Examples

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    Load an image.

    load detfingr
    nbc = size(map,1);

    Obtain the wavelet decomposition of the image at level 3 using the sym4 wavelet.

    wname = "sym4";
    lev = 3;
    [c,s] = wavedec2(X,lev,wname);

    Use wdcbm2 to select level-dependent thresholds for image compression. Use the suggested parameters.

    alpha = 1.5;
    m = 2.7*prod(s(1,:));
    [thr,nkeep] = wdcbm2(c,s,alpha,m)
    thr = 3×3
    
       21.4814   46.8354   40.7907
       21.4814   46.8354   40.7907
       21.4814   46.8354   40.7907
    
    
    nkeep = 1×3
    
             624         961        1765
    
    

    Use wdencmp2 for compressing the image using the thresholds. Use hard thresholding.

    [xd,cxd,sxd,perf0,perfl2] = ...
        wdencmp("lvd",c,s,wname,lev,thr,"h");

    Plot the original and compressed images.

    colormap(pink(nbc))
    subplot(2,2,1)
    image(wcodemat(X,nbc))
    title("Original Image")
    subplot(2,2,2)
    image(wcodemat(xd,nbc))
    title("Compressed Image")
    xlab1 = ['2-norm rec.: ',num2str(perfl2)];
    xlab2 = [' %  -- zero cfs: ',num2str(perf0), ' %'];
    xlabel([xlab1 xlab2]);

    Figure contains 2 axes objects. Axes object 1 with title Original Image contains an object of type image. Axes object 2 with title Compressed Image contains an object of type image.

    Input Arguments

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    Wavelet decomposition of the image to be denoised or compressed, specified as a real-valued vector. The vector C contains the approximation and detail coefficients organized by level. The bookkeeping matrix S is used to parse C. See wavedec2.

    Data Types: double

    Bookkeeping matrix, specified as an integer-valued matrix. The matrix S contains the dimensions of the wavelet coefficients by level and is used to parse the wavelet decomposition vector C. See wavedec2.

    Data Types: double

    Sparsity parameter to use in the Birgé-Massart strategy, specified as a real-valued scalar greater than 1. Typically, alpha = 1.5 for compression and alpha = 3 for denoising. For more information, see Wavelet Coefficients Selection.

    Data Types: double

    Factor to use in the Birgé-Massart strategy, specified as a real-valued scalar greater than 1. The default value is prod(S(1,:)), the length of the coarsest approximation coefficients. Recommended values for M are from prod(S(1,:)) to 6*prod(S(1,:)). For more information, see Wavelet Coefficients Selection.

    Data Types: double

    Output Arguments

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    Level-dependent thresholds, returned as a 3-by-N matrix, where N = size(S,1)-2. thr(:,i) contains the thresholds in the three orientations: horizontal, diagonal, and vertical, for level i. The thresholds are obtained using a wavelet coefficients selection rule based on the Birgé-Massart strategy. For more information, see Wavelet Coefficients Selection.

    Number of coefficients to be kept at each level, returned as a vector of length N, where N = size(S,1)-2. nkeep(i) contains the number of coefficients to be kept at level i.

    Data Types: double

    More About

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    Wavelet Coefficients Selection

    Thresholds are obtained using a wavelet coefficients selection rule based on the Birgé-Massart strategy. The values N = prod(S(1,:)), M and alpha define the strategy.

    • At level N+1 (and coarser levels), everything is kept.

    • For level i from 1 to N, the ni largest coefficients are kept, where ni = M / (N+2-i)alpha.

    The default value of M = prod(S(1,:)) corresponds to the formula nN+1 = M / (N+2-(N+1))alpha = M.

    References

    [1] Birgé, Lucien, and Pascal Massart. “From Model Selection to Adaptive Estimation.” In Festschrift for Lucien Le Cam: Research Papers in Probability and Statistics, edited by David Pollard, Erik Torgersen, and Grace L. Yang, 55–87. New York, NY: Springer, 1997. https://doi.org/10.1007/978-1-4612-1880-7_4.

    Version History

    Introduced before R2006a