Process of Discrete Wavelet Transform (II): Wavelet Reconstruction

Another part of the wavelet transform story is wavelet reconstruction, which assembles the components back to the signal without loss of information

Part I: Wavelet Decomposition

Part II: Wavelet Reconstruction

Part III: Wavelet Partial Reconstruction

In the previous part, we have talked about wavelet decomposition process. That is only part of the story of wavelet transform. In this post, I will illustrate the general process of Discrete Wavelet Reconstruction.

2. Wavelet Reconstruction

The another part of the story is how to assemble those components back into the original signal without loss of information. This process is called wavelet reconstruction, or synthesis.

The mathematical manipulation or function that effects synthesis is called the inverse discrete wavelet transform (IDWT).

Relating to the decomposition, wavelet reconstruction can be divided into single-level reconstruction and multi-level wavelet reconstruction.

3. Single-level reconstruction

Opposite to wavelet decomposition Where wavelet analysis involves filtering and downsampling, the wavelet reconstruction process consists of upsampling and filtering.

Upsampling is the process of lengthening a signal component by inserting zeros between samples.

For the single-level (or one-stage) wavelet reconstruction, the first-level approximation coefficient (cA1) and detailed coefficient (cD1) are upsampled first, and then pass two reconstruction lowpass (L’) and hightpass (H’) to obtain the synthesized original signal (S’). The reconstructed signal is normally not equal to the original signal (S) because errors might be caused by the synthesis process, but the error is usually very small.

3. Multi-level Reconstruction

It is not hard to understand the process of multi-level wavelet reconstruction after knowing the process of one-stage wavelet reconstruction.

Let’s take a three-level wavelet reconstruction for example, cA3 and cD3, the approximation and detail coefficients at the level 3 are upsampled and then passed reconstruction lowpass (L’) and hightpass (H’) to get the approximation coefficient (cA2) at the level 2. The process is repeated for cA2 and cD2, the approximation and detail coefficients at the level 2 to get cA1, the approximation at the level 1. Then we continue the process for cA1 and cD1, the approximation and detail coefficients at the level 1, and finally we get the reconstructed signal.

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