How to calculate the inverse maximal overlap discrete wavelet transform to reconstruct signal from (MODWT) decomposition coefficient?

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Syed Mohammed Shafi

The inverse maximal overlap discrete wavelet transform (MODWT) is a technique to reconstruct a signal from its wavelet decomposition coefficients. The MODWT is a version of the discrete wavelet transform (DWT) that applies wavelet filters in a way that preserves the time domain properties of the signal.

To reconstruct a signal from its MODWT decomposition coefficients, you can use the following steps:

Could you determine the number of levels in the MODWT decomposition? This is equal to the number of times the signal was downsampled during the MODWT decomposition process.

Initialize an array of zeros with the same length as the original signal.

From the highest decomposition level (i.e., the coarsest scale), compute the inverse MODWT coefficients for each group. The inverse MODWT coefficients can be calculated using the following formula:x_n = W_J-1(n) + V_J-1(n) for n = 0, 1, ..., N-1where W_J-1(n) and V_J-1(n) are the MODWT coefficients at level J-1 (the following coarsest scale), and N is the length of the signal. The inverse MODWT coefficients can be computed using the MODWT reconstruction filter bank.

Could you add the inverse MODWT coefficients for each level to the array initialized in step 2?

Once all levels have been reconstructed, the final signal can be obtained by taking the inverse DWT of the array of reconstructed coefficients.

Note that the MODWT reconstruction filter bank differs from the DWT filter bank. The MODWT reconstruction filter bank works with the overlap-add technique used in the MODWT decomposition. In contrast, the DWT filter bank is designed to work with non-overlapping subband decomposition.

The implementation details of the MODWT reconstruction filter bank can vary depending on the specific wavelet used in the decomposition. Therefore, it's essential to consult the literature or documentation for the particular wavelet used in the MODWT decomposition for the correct filter coefficients and implementation details.

Mohammed Nessan

Syed Mohammed Shafi Thank you for your response. I have carefully reviewed the steps you provided. It refers to the sum of the wavelet and scaling coefficients for the next level to obtain the signal of the previous level and so up to the original signal. While these steps are helpful, I have noticed that they do not include any mathematical equations for using the "imodwt" function to reconstruct a signal from any specific level.

Furthermore, I am specifically interested in reconstructing a signal from two levels - detail (D5) and detail (D6) - using only the wavelet coefficient, without the scaling coefficient. To do this, I need to understand the mathematical equations upon which the "imodwt" function in Matlab is based. Because, I have compared the results obtained using the summation process you described and those produced by the "imodwt" function in Matlab and the results are significantly different. I believe a better understanding of the underlying equations will help me reconcile these differences.

Thanks for your assistance.