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The lifting scheme for the seislet transform

The lifting scheme (Sweldens, 1995) provides a convenient approach for designing digital wavelet transforms. The general recipe is as follows:

  1. Organize the input data as a sequence of records. For OC-seislet transform of 2-D seismic reflection data, the input is in the `frequency'-`midpoint wavenumber'-`offset' domain after the log-stretched NMO correction (Bolondi et al., 1982), and the transform direction is offset.
  2. Divide the data records (along the offset axis in the case of the OC-seislet transform) into even and odd components $ \mathbf{e}$ and $ \mathbf{o}$ . This step works at one scale level.
  3. Find the residual difference $ \mathbf{r}$ between the odd component and its prediction from the even component:

    $\displaystyle \mathbf{{r}} = \mathbf{{o}} - \mathbf{P[{e}]}\;,$ (1)

    where $ \mathbf{P}$ is a prediction operator. For example, one can obtain Cohen-Daubechies-Feauveau (CDF) 5/3 biorthogonal wavelets (Cohen et al., 1992) by defining the prediction operator as a linear interpolation between two neighboring samples,

    $\displaystyle \mathbf{P[e]}_k = \left(\mathbf{e}_{k-1} + \mathbf{e}_{k}\right)/2\;,$ (2)

    where $ k$ is an index number at the current scale level.
  4. Find an approximation $ \mathbf{c}$ of the data by updating the even component:

    $\displaystyle \mathbf{{c}} = \mathbf{{e}} + \mathbf{U[{r}]}\;,$ (3)

    where $ \mathbf{U}$ is an update operator. Constructing the update operator for CDF 5/3 biorthogonal wavelets aims at preserving the running average of the signal (Sweldens and Schröder, 1996):

    $\displaystyle \mathbf{U[r]}_k = \left(\mathbf{r}_{k-1} + \mathbf{r}_{k}\right)/4\;.$ (4)

  5. The coarse approximation $ \mathbf{{c}}$ becomes the new data, and the sequence of steps is repeated on the new data to calculate the transform coefficients at a coarser scale level.

Next, we define new prediction and update operators using offset-continuation operators.


next up previous [pdf]

Next: OC-seislet structure Up: Theoretical basis Previous: Theoretical basis

2013-07-26