icwtlin

Inverse continuous wavelet transform (CWT) for linearly spaced scales

Syntax

xrec = icwtlin(cwtstruct)
xrec = icwtlin(wav,meanSIG,cfs,scales,dt)
xrec = icwtin(...,'plot')
xrec = icwtlin (...,'signal',SIG,'plot')
xrec = icwtlin(...,Name,Value)

Description

xrec = icwtlin(cwtstruct) returns the inverse continuous wavelet transform (CWT) of the CWT coefficients obtained at linearly-spaced scales.

    Note:   To use icwtlin you must:

    • Use linearly-spaced scales in the CWT. icwtlin does not verify that the scales are linearly-spaced.

    • Use one of the supported wavelets. See Input Arguments for a list of supported wavelets.

xrec = icwtlin(wav,meanSIG,cfs,scales,dt) returns the inverse CWT of the coefficients in cfs. The inverse CWT is obtained using the wavelet wav, the linearly spaced scales scales, the sampling period dt, and the mean signal value meanSig.

xrec = icwtin(...,'plot') plots the reconstructed signal xrec along with the CWT coefficients and CWT moduli. If the analyzing wavelet is complex-valued, the plot includes the real and imaginary parts of the CWT coefficients.

xrec = icwtlin (...,'signal',SIG,'plot') places a radio button in the bottom-left corner of the plot. Enabling the radio button superimposes the plot of the input signal SIG on the plot of the reconstructed signal. SIG can be a structure array, a cell array, or a vector. If SIG is a structure array, there must be two fields: val and period. The val field contains the signal and the period field contains the sampling period. If SIG is a cell array, SIG{1} contains the signal and SIG{2} is the sampling period.

xrec = icwtlin(...,Name,Value) returns the inverse CWT transform with additional options specified by one or more Name,Value pair arguments.

Input Arguments

cwtstruct

A structure array that is the output of cwtft or constructed from the output of cwt. If you obtain cwtstruct from cwtft, the structure array contains six fields:

  • cfs — CWT coefficient matrix

  • scales — Vector of linearly spaced scales. The scale vector must be linearly-spaced to ensure accurate reconstruction. icwtlin does not check that the spacing of your scale vector is linear.

  • wav — Analyzing wavelet. icwtlin uses this wavelet as the reconstruction wavelet. The supported wavelets are:

    • 'dog' — An m-th order derivative of Gaussian wavelet where m is a positive even integer

    • 'morl' — Analytic Morlet wavelet

    • 'morlex' — Nonanalytic Morlet wavelet

    • 'morl0' — Nonanalytic Morlet wavelet with exact zero mean

    • 'mexh' — Mexican-hat wavelet. This argument represents a special case of the derivative of Gaussian wavelet with m=2.

    • 'paul' — Paul wavelet

  • omega — Angular frequencies used in the Fourier transform in radians/sample

  • MeanSIG — Signal mean

  • dt — Sampling period in seconds

If you create cwtstruct from the output of cwt, cwtstruct contains all of the preceding fields except omega.

Using cwt to obtain the CWT coefficients, the valid analyzing wavelets are:

  • Coiflets — 'coif1','coif2' ,'coif3' ,'coif4', 'coif5'

  • Biorthogonal wavelets — 'bior2.2', 'bior2.4', 'bior2.6', 'bior2.8', 'bior4.4', 'bior5.5', bior6.8

  • Reverse biorthogonal wavelets — 'rbio2.2', 'rbio2.4', 'rbio2.6', 'rbio2.8', 'rbio4.4', 'rbio5.5', 'rbio6.8'

  • Complex Gaussian wavelets — 'cgau2', 'cgau4', 'cgau6', 'cgau8'

Name-Value Pair Arguments

'IdxSc'

Vector of scales to use in the signal reconstruction. Specifying a subset of scales results in a scale-localized approximation of the analyzed signal.

Output Arguments

xrec

Reconstructed signal. Signal approximation based on the input CWT coefficient matrix, analyzing wavelet, selected scales, and sampling period.

The purpose of the CWT inversion algorithm is not to produce a perfect reconstruction of the input signal. The inversion preserves time and scale-localized features in the reconstructed signal. The amplitude scaling in the reconstructed signal, however, can be significantly different. This difference in scaling can occur whether or not you use all the CWT coefficients in the inversion.

Examples

Compute the inverse CWT of a sum of sine waves with disjoint support.

% Define the signal
N = 100;
t = linspace(0,1,N);
Y = sin(8*pi*t).*(t<=0.5) + sin(16*pi*t).*(t>0.5) ;

% Define parameters before analysis
dt = 0.001;
maxsca = 1; s0 = 2*dt; ds = 2*dt;
scales = s0:ds:maxsca;
wname = 'morl';
SIG = {Y,dt};
WAV = {wname,[]};

% Compute the CWT using cwtft with linear scales
cwtS = cwtft(SIG,'scales',scales,'wavelet',WAV);
% Compute inverse CWT using linear scales
Yrec = icwtlin(cwtS,'Signal',Y,'plot');

Reconstruct an approximation to a noisy Doppler signal based on thresholded coefficients. Use the universal threshold. Assume the sampling period is 0.05 seconds.

load noisdopp;
Y = noisdopp;
N = length(Y);

% Define parameters before analysis
% Assume sampling period is 0.05
dt = 0.05;
maxsca = 100; s0 = 2*dt; ds = 4*dt;
scales = s0:ds:maxsca;
wname = 'morl';
SIG = {Y,dt};
WAV = {wname,[]};

% Compute CWT
cwtS = cwtft(SIG,'scales',scales,'wavelet',WAV,'plot');

% Select subset of coefficients
cwtS1 = cwtS;
Hfreq = cwtS.cfs(1:10,:);
% Set threshold
thr = sqrt(2*log(N))*median(abs(Hfreq(:)))/0.6745;
newCFS = cwtS.cfs;
% Set coefficients smaller than threshold in absolute value to 0
newCFS(abs(newCFS)<thr) = 0;
cwtS1.cfs = newCFS;


% Reconstruction from the modified structure
YRDen = icwtlin(cwtS1,'signal',Y,'plot');

Enable the Reconstructed Signal On/Off radio button in the bottom-left corner.

Alternatives

  • icwtft — Computes the inverse for the CWT obtained using cwtft with logarithmically spaced scales. If you use linearly spaced scales in cwtft, or you obtain the CWT with cwt, use icwtlin to compute the inverse.

More About

expand all

Algorithms

See [4] for a description of the inverse CWT algorithm for linearly spaced scales. The icwtlin function uses heuristic scaling factors for the analyzing wavelets. These scaling factors can result in significant differences in the amplitude scaling of the reconstructed signal.

References

[1] Daubechies, I. Ten Lectures on Wavelets, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1992.

[2] Farge, M. "Wavelet Transforms and Their Application to Turbulence", Ann. Rev. Fluid. Mech., 1992, 24, 395–457.

[3] Mallat, S. A Wavelet Tour of Signal Processing, San Diego, CA: Academic Press, 1998.

[4] Sun,W. "Convergence of Morlet's Reconstruction Formula", preprint, 2010.

[5] Torrence, C. and G.P. Compo. "A Practical Guide to Wavelet Analysis", Bull. Am. Meteorol. Soc., 79, 61–78, 1998.

See Also

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