wmpdictionary

R2026b

(To be removed) Dictionary for matching pursuit

wmpdictionary will be removed in a future release. Use sensingDictionary instead. For more information, see Version History.

Syntax

MPDICT = wmpdictionary(N)

[MPDICT,NBVECT] = wmpdictionary(N)

[MPDICT,NBVECT]= wmpdictionary(N,Name,Value)

[MPDICT,NBVECT,LST] = wmpdictionary(N,Name,Value)

[MPDICT,NBVECT,LST,LONGS] = wmpdictionary(N,Name,Value)

Description

MPDICT = wmpdictionary(N) returns the N-by-P dictionary, MPDICT, for the default subdictionaries {{'sym4',5},{'wpsym4',5},'dct','sin'}. The column dimension of MPDICT depends on N.

[MPDICT,NBVECT] = wmpdictionary(N) returns the row vector, NBVECT, which contains the number of vectors in each subdictionary. The order of the elements in NBVECT corresponds to the order of the subdictionaries and any prepended or appended subdictionaries. The sum of the elements in NBVECT is the column dimension of MPDICT.

[MPDICT,NBVECT]= wmpdictionary(N,Name,Value) returns the dictionary, MPDICT, using additional options specified by one or more Name,Value pair arguments.

[MPDICT,NBVECT,LST] = wmpdictionary(N,Name,Value) returns the cell array, LST, with descriptions of the subdictionaries.

[MPDICT,NBVECT,LST,LONGS] = wmpdictionary(N,Name,Value) returns the cell array, LONGS, containing the number of vectors in each subdictionary. LONGS is only useful for wavelet subdictionaries. In wavelet subdictionaries, the corresponding element in LONGS gives the number of scaling functions at the coarsest level and wavelet functions by level.

Examples

Discrete Cosine Transform and Kronecker Delta Dictionary

Create a DCT and shifted Kronecker delta dictionary to represent a signal of length 100.

mpdict = wmpdictionary(100,'lstcpt',{'dct','RnIdent'});

Input Arguments

N — Input signal length

positive integer

Length of your input signal, specified as a positive integer. The dictionary atoms are constructed to have N elements. N equals the row dimension of the dictionary, MPDICT.

Data Types: double

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: MPDICT = wmpdictionary(100,lstcpt={'dct','RnIdent'})

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

Example: MPDICT = wmpdictionary(100,'lstcpt',{'dct','RnIdent'})

addbeg — Prepended subdictionary

matrix

Prepended subdictionary, specified as an N-by-M matrix, where N is the length of the input signal. wmpdictionary does not check that the M column vectors of the prepended dictionary form a basis. If you do not specify a value for lstcpt, the subdictionary is prepended to the default dictionary. The column vectors in the prepended subdictionary do not have to be unit-norm.

Data Types: double

addend — Appended subdictionary

matrix

Appended subdictionary, specified as an N-by-M matrix, where N is the length of the input signal. wmpdictionary does not check that the M column vectors of the prepended dictionary form a basis. If you do not specify a value for lstcpt, the subdictionary is appended to the default dictionary. The column vectors in the appended subdictionary do not have to be unit-norm.

Data Types: double

lstcpt — Valid subdictionaries

cell array

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:

  $${\varphi }_k(n)=\{\begin{array}{ll}\frac{1}{\sqrt{N}}\hfill & k=0\hfill \\ \sqrt{\frac{2}{N}}\cos \left(\frac{\pi }{N}\left(n+\frac{1}{2}\right)k\right)\hfill & k=1,2,\dots ,N-1.\hfill \end{array}$$

• 'sin' Sine subdictionary. The sine subdictionary is

  $${\varphi }_k(t)=\sin (2\pi kt)k=1,2,\dots \lceil \frac{N}{2}\rceil 0\le t\le 1$$

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

• 'cos' Cosine subdictionary. The cosine subdictionary is

  $${\varphi }_k(t)=\cos (2\pi kt)k=1,2,\dots \lceil \frac{N}{2}\rceil 0\le t\le 1$$

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

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

  $$p_n(t)=t^{n-1}n=1,2,\dots 200\le t\le 1$$

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

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

  $${\varphi }_k(n)=\delta (n-k)k=0,1,\dots N$$

Data Types: double

Output Arguments

MPDICT — Matching pursuit dictionary

matrix

Matching pursuit dictionary, returned as a matrix. MPDICT is an N-by-P matrix with the row dimension, N, equal to the length of the input signal. The column dimension of the matrix depends on the size of the concatenated subdictionaries.

NBVECT — Number of vectors in subdictionaries

vector

Number of vectors in subdictionaries, returned as a vector. NBVECT is a row vector containing the number of elements in each subdictionary. The order of the elements in NBVECT corresponds to the order of the subdictionaries and any prepended or appended subdictionaries.

LST — Dictionary description

cell array

Dictionary description, returned as a cell array. LST is a 1-by-L cell array, where L is the number of subdictionaries. Each element of the cell array contains a description of a subdictionary. If you specify a prepended or appended subdictionary, the first element of LST is 'AddBeg' or 'AddEnd'. If you specify a level for the wavelet or wavelet packet, the corresponding element of LST is a 1-by-2 cell array containing the wavelet or wavelet packet name in the first element and the level in the second element.

LONGS — Number of elements for each subdictionary

cell array

Number of elements for each subdictionary, returned as a cell array. LONGS is useful only for wavelet subdictionaries. If you specify a wavelet subdictionary, the corresponding element of LONGS provides the number of scaling functions at the coarsest level and the number of wavelets at each level.

More About

Matching Pursuit

Matching pursuit refers to a number of greedy or weak-greedy algorithms for computing an adaptive nonlinear expansion of a signal in a dictionary. In the majority of matching pursuit applications, a dictionary is an overcomplete set of vectors. The elements of the dictionary are referred to as atoms and are typically constructed to have certain time/frequency or time/scale properties. Matching pursuit takes the NP-hard problem of finding the best nonlinear expansion in a dictionary and implements it in an energy-preserving formulation that guarantees convergence. See Matching Pursuit Algorithms for more details.

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

R2022a: wmpdictionary will be removed

The wmpdictionary function will be removed in a future release. Use sensingDictionary instead.

Functionality	Result	Use Instead	Compatibility Considerations
MPDICT = wmpdictionary(N)	Still runs	Execute these steps: Create a sensingDictionary using pre-built support for wavelet and DCT frames: A1 = sensingDictionary('Size',N,... 'Type',{'dwt','dct'},... 'Name',{'sym4'},... 'Level',[5]); Create a custom sensingDictionary: T = linspace(0,1,N)'; K = 1:ceil(N/2); T1 = repmat(T,1,numel(K)); K1 = repmat(K,numel(T),1); Amat = sin(2*pi*(K1.*T1)); A2 = sensingDictionary('CustomDictionary',Amat); Concatenate the results: MPDICT = [A1 A2];	sensingDictionary provides pre-built support for a variety of frames, including Fourier, Gaussian and Bernoulli random distributions, and Walsh code. sensingDictionary does not currently support wavelet packet bases.
MPDICT = wmpdictionary(N,'lstcpt',dtypes), where dtypes is a cell array of cell arrays with valid subdictionaries	Still runs	Each cell array in dtypes describes one subdictionary. To specify a subdictionary in sensingDictionary, use the Type, Name, and Level name-value arguments. For example: Replace mpdict = wmpdictionary(100,'lstcpt',{'dct','RnIdent'}); with D = sensingDictionary('Size',100,'Type',{'dct','eye'}) Replace mpdict = wmpdictionary(100,... 'lstcpt',{{'db4',3},'dct'}); with x = sensingDictionary('Size',100,... 'Type',{'dwt','dct'},... 'Name',{'db4'},... 'Level',[3 0])	For the wavelet option, sensingDictionary and wmpdictionary behave differently. wmpdictionary returns the wavelets at all levels and the scaling functions at the final level. sensingDictionary returns the wavelets at only the final level. For example, mpdict = wmpdictionary(100,'lstcpt',{{'db1',2}}); returns the scaling functions for level 2, the wavelets for level 2, and the wavelets for level 1, whereas A = sensingDictionary('Size',100,'Type',{'dwt'},'Name',{'db1'},'Level',2); only returns the wavelets at level 2.
[~,NBVECT] = wmpdictionary(N) or [~,NBVECT] = wmpdictionary(N,'lstcpt')	Still runs	NBVECT is the number of vectors in each subdictionary. The number of vectors in a subdictionary of a sensingDictionary object depends on the associated basis type. For a non-random basis type, the number of vectors is N. For a random basis type, the number of vectors is the column size you specified when you created the sensingDictionary object. For a custom sensingDictionary, the number of vectors is the column size you specified when you created the sensingDictionary object.	You can also use the subdict method of sensingDictionary to extract the vectors.