Once we arrive at our discrete wavelet coefficients, we need a way to reconstruct them back into the original signal (or a modified original signal if we played around with the coefficients). In order to do this, we utilize the process known as the inverse discrete wavelet transform.

Much like the DWT can be explained by using filter bank theory, so can the reconstruction of the IDWT. The process is simply reversed. The DWT coefficients are first upsampled (the approximation and the detail coefficients are handled separately) by placing zeros in between every coefficient, effectively doubling the lengths of each. These are then convolved with the reconstruction scaling filter for approximation coefficients (the reconstruction scaling filter is simply the original scaling filter that has been flipped left to right) and the reconstruction wavelet filter for the detail coefficients. These results are then added together to arrive at the original signal.

Similar to how we made the signal periodic before doing our DWT calculations on it, we must make our dwt coefficients periodic before convolving to obtain the original signal. This is done by simply taking the first N/2-1 coefficients from the DWT coefficients, and appending them to the end. Remember that N is the length of our scaling filter.

After the convolution and addition, to grab the part of the signal we want away from the convolution ‘junk’, we grab the coefficients from N to the length of the signal + N -1. This will give us our original signal.

*NOTE: THIS MAY NOT BE CORRECT. WE BELIEVE THE THEORY IS SOUND BUT WE ARE ARRIVING AT ANOMOLOUS RESULTS AND BELIEVE THE REASON WHY IS DUE TO THIS PERIODIZATION. FURTHER DETAILS WILL BE ADRESSED FURTHER DOWN.*

If you are looking for a graphical description of this process, simply look at the figure below:

Please keep in mind that the pattern of reconstruction must match the pattern of the DWT’s deconstruction, so in our case for multi level reconstruction we simply reconstruct the approximation coefficients in order from the finest scale to the coarsest scale. In the above figure, the lower paths are the approximation coefficients and the higher paths are the detail coefficients; also the junction of two arrowheads corresponds to an addition.