# imodwt

Inverse maximal overlap discrete wavelet transform

## Description

example

xrec = imodwt(w) reconstructs the signal based on the maximal overlap discrete wavelet transform (MODWT) coefficients in w. By default, imodwt assumes that you obtained w using the 'sym4' wavelet with periodic boundary handling. If you do not modify the coefficients, xrec is a perfect reconstruction of the signal.

example

xrec = imodwt(w,wname) reconstructs the signal using the orthogonal wavelet wname. wname must be the same wavelet used to analyze the signal input to modwt.

example

xrec = imodwt(w,Lo,Hi) reconstructs the signal using the orthogonal scaling filter Lo and the wavelet filter Hi. The Lo and Hi filters must be the same filters used to analyze the signal input to modwt.

example

xrec = imodwt(___,lev) reconstructs the signal up to level lev. xrec is a projection onto the scaling space at level lev. The default level is 0, which results in perfect reconstruction if you do not modify the coefficients.

example

xrec = imodwt(___,'reflection') uses the reflection boundary condition in the reconstruction. If you specify 'reflection', imodwt assumes that the length of the original signal length is one half the number of columns in the input coefficient matrix. By default, imodwt assumes periodic signal extension at the boundary.

You must enter the entire character vector 'reflection'. If you added a wavelet named 'reflection' using the wavelet manager, you must rename that wavelet prior to using this option. 'reflection' may be placed in any position in the input argument list after x.

## Examples

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Obtain the MODWT of an ECG signal and demonstrate perfect reconstruction.

Load the ECG signal data and obtain the MODWT.

Obtain the MODWT and the Inverse MODWT.

w = modwt(wecg);
xrec = imodwt(w);

Use the L-infinity norm to show that the difference between the original signal and the reconstruction is extremely small. The largest absolute difference between the original signal and the reconstruction is on the order of $1{0}^{-12}$, which demonstrates perfect reconstruction.

norm(abs(xrec'-wecg),Inf)
ans = 2.3255e-12

Obtain the MODWT of Deutsche Mark-U.S. Dollar exchange rate data and demonstrate perfect reconstruction.

Load the Deutsche Mark-U.S. Dollar exchange rate data.

Obtain the MODWT and the Inverse MODWT using the 'db2' wavelet.

wdm = modwt(DM_USD,'db2');
xrec = imodwt(wdm,'db2');

Use the L-infinity norm to show that the difference between the original signal and the reconstruction is extremely small. The largest absolute difference between the original signal and the reconstruction is on the order of $1{0}^{-13}$, which demonstrates perfect reconstruction.

norm(abs(xrec'-DM_USD),Inf)
ans = 1.6370e-13

Obtain the MODWT of an ECG signal using the Fejér-Korovkin filters.

Create the 8-coefficient Fejér-Korovkin filters. Use the filters to obtain the MODWT of the ECG data.

[~,~,Lo,Hi] = wfilters("fk8");
wtecg = modwt(wecg,Lo,Hi);

Obtain the inverse MODWT using the filters.

xrec = imodwt(wtecg,Lo,Hi);

Obtain a second inverse MODWT using the wavelet name. Confirm both inverse transforms are equal.

xrec2 = imodwt(wtecg,"fk8");
max(abs(xrec-xrec2))
ans = 0

Plot the original data and one of the reconstructions.

subplot(2,1,1)
plot(wecg)
title("ECG Signal")
subplot(2,1,2)
plot(xrec)
title("Reconstruction")

Obtain the MODWT of an ECG signal down to the maximum level and obtain the projection of the ECG signal onto the scaling space at level 3.

Obtain the MODWT.

wtecg = modwt(wecg);

Obtain the projection of the ECG signal onto ${V}_{3}$, the scaling space at level three by using the imodwt function.

v3proj = imodwt(wtecg,3);

Plot the original signal and the projection.

subplot(2,1,1)
plot(wecg)
title('Original Signal')
subplot(2,1,2)
plot(v3proj)
title('Projection onto V3')

Note that the spikes characteristic of the R waves in the ECG are missing in the ${V}_{3}$ approximation. You can see the missing details by examining the wavelet coefficients at level three.

Plot the level-three wavelet coefficients.

figure
plot(wtecg(3,:))
title('Level-Three Wavelet Coefficients')

Obtain the inverse MODWT using reflection boundary handling for Southern Oscillation Index data. The sampling period is one day. imodwt with the 'reflection' option assumes that the input matrix, which is the modwt output, is twice the length of the original signal length. imodwt reflection boundary handling reduces the number of wavelet and scaling coefficients at each scale by half.

wsoi = modwt(soi,4,'reflection');
xrecsoi = imodwt(wsoi,'reflection');

Use the L-infinity norm to show that the difference between the original signal and the reconstruction is extremely small. The largest absolute difference between the original signal and the reconstruction is on the order of $1{0}^{-11}$, which demonstrates perfect reconstruction.

norm(abs(xrecsoi'-soi),Inf)
ans = 1.6421e-11

Load the 23 channel EEG data Espiga3 [2]. The channels are arranged column-wise. The data is sampled at 200 Hz.

Obtain the maximal overlap discrete wavelet transform down to the maximum level.

w = modwt(Espiga3);

Reconstruct the multichannel signal. Plot the original data and reconstruction.

xrec = imodwt(w);
subplot(2,1,1)
plot(Espiga3)
title('Original Data')
subplot(2,1,2)
plot(xrec)
title('Reconstruction')

## Input Arguments

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MODWT transform of a signal or multisignal down to level L, specified as a matrix or 3-D array, respectively. w is an L+1-by-N matrix for the MODWT of an N-point signal, and an L+1-by-N-by-NC array for the MODWT of an N-by-NC multisignal. By default, imodwt assumes that you obtained the MODWT using the 'sym4' wavelet with periodic boundary handling.

Data Types: single | double
Complex Number Support: Yes

Synthesis wavelet, specified as a character vector or string scalar. The wavelet must be orthogonal. Orthogonal wavelets are designated as type 1 wavelets in the wavelet manager, wavemngr.

Valid built-in orthogonal wavelet families are: Best-localized Daubechies ("bl"), Beylkin ("beyl"), Coiflets ("coif"), Daubechies ("db"), Fejér-Korovkin ("fk"), Haar ("haar"), Han linear-phase moments ("han"), Morris minimum-bandwidth ("mb"), Symlets ("sym"), and Vaidyanathan ("vaid").

For a list of wavelets in each family, see wfilters. You can also use waveinfo with the wavelet family short name. For example, waveinfo("db"). Use wavemngr("type",wn) to determine if the wavelet wn is orthogonal (returns 1). For example, wavemngr("type","db6") returns 1.

The synthesis wavelet must be the same wavelet used in the analysis with modwt.

Filters, specified as a pair of even-length real-valued vectors. Lo is the scaling filter, and Hi is the wavelet filter. Lo and Hi must be the same filters used in the analysis with modwt. The filters must satisfy the conditions for an orthogonal wavelet. The lengths of Lo and Hi must be equal. See wfilters for additional information. You cannot specify both wname and a filter pair Lo,Hi.

Note

To agree with the usual convention in the implementation of modwt in numerical packages, the roles of the analysis and synthesis filters returned by wfilters are reversed in imodwt. See Inverse MODWT with Specified Filters.

Data Types: single | double

Reconstruction level, specified as a nonnegative integer between 0 and size(w,1)-2. The level must be less than the level used to obtain w from modwt. If lev is 0 and you do not modify the coefficients, imodwt produces a perfect reconstruction of the signal.

## Output Arguments

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Reconstructed version of the original signal or multisignal based on the MODWT and the level of reconstruction, returned as a vector or matrix.

## References

[1] Percival, Donald B., and Andrew T. Walden. Wavelet Methods for Time Series Analysis. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge ; New York: Cambridge University Press, 2000.

[2] Mesa, Hector. “Adapted Wavelets for Pattern Detection.” In Progress in Pattern Recognition, Image Analysis and Applications, edited by Alberto Sanfeliu and Manuel Lazo Cortés, 3773:933–44. Berlin, Heidelberg: Springer Berlin Heidelberg, 2005. https://doi.org/10.1007/11578079_96.

## Version History

Introduced in R2015b