Seislet transform and seislet frame |

In order to define the new transform, we follow the general recipe for digital wavelet transforms provided by Sweldens and Schröder (1996). In the most general terms, the lifting scheme (Sweldens, 1995) can be defined as follows:

- Organize the input data as a sequence of records.
- Break the data into even and odd components and .
- Find a residual difference between the odd
component and its prediction from the even component:

where is a*prediction*operator. - Find a coarse approximation of the data by updating
the even component

where is an*update*operator. - The coarse approximation becomes the new data, and the sequence of steps is repeated at the next scale level.

with the update operator designed to preserve the DC (zero frequency) component of the signal. Alternatively, the Cohen-Daubechies-Feauveau biorthogonal wavelets (Cohen et al., 1992) are constructed by making the prediction operator a linear interpolation between two neighboring samples,

and by constructing the update operator to preserve the running average of the signal (Sweldens and Schröder, 1996), as follows:

The digital wavelet transform is an efficient operation. Assuming that the prediction and update operation take a constant cost per record, the number of operations at the finest scale is proportional to the total number of records , the next scale computation takes , etc. so that the total number of operations is proportional to , which is smaller than the cost of the Fast Fourier Transform.

The digital wavelet transform is also easily invertible. Reversing the lifting scheme operations provides the inverse transform algorithm, as follows:

- Start with the coarsest scale data representation and the coarsest scale residual .
- Reconstruct the even component by reversing the
operation in equation 2, as follows:

- Reconstruct the odd component
by reversing the operation in equation 1, as follows:

- Combine the odd and even components to generate the data at the previous scale level and repeat the sequence of steps.

Figure 1 shows a classic benchmark image from the image analysis literature and its digital wavelet transform using 2-D biorthogonal wavelets. Thanks to the general smoothness of the ``Lena'' image, the residual differences from equation 2 (stored as wavelet coefficients at different scales) have a small dynamic range, which enables an effective compression of the image in the transform domain. Wavelet compression algorithms are widely used in practice for compression of natural images. As for compression of seismic data, the classic DWT may not be optimal, because it does not take into account the specific nature of seismic data patterns. In the next section, we turn the wavelet transform into the seislet transform, which is tailored for representing seismic data.

lena,linear2
Benchmark ``Lena''
image from image analysis literature (a) and its 2-D digital wavelet
transform using bi-orthogonal wavelets (b).
Figure 1. |
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Seislet transform and seislet frame |

2013-07-26