Seislet transform and seislet frame

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# From transform to frame

The 1-D and 2-D transforms, defined in the previous sections, are appropriate for analyzing signals, which have a single dominant sinusoid or plane-wave component. In practice, it is common to analyze signals composed of multiple sinusoids (in 1-D) or plane waves (in 2-D). If a range of frequencies or plane-wave slopes is chosen, and the appropriate transform is constructed for each of them, all the transform domains taken together will constitute an overcomplete representation or a frame (Mallat, 2009).

Mathematically, if is the orthonormal seislet transform for -th frequency or plane wave, then, for any data vector ,

 (14)

which means that all transforms taken together constitute a tight frame with constant .

For example, in the 1-D case, one can find appropriate frequencies by autoregressive spectral analysis (Burg, 1975; Marple, 1987). We define the algorithm for the 1-D seislet frame as follows:

1. Select a range of coefficients . When using autoregressive spectral analysis, these coefficients are simply the roots of the prediction-error filter. Alternatively, they can be defined from an appropriate range of frequencies .
2. For each of the coefficients, perform the 1-D seislet transform.

Because of its over-completeness, a frame representation for a given signal is not unique. In order to assure that different frequency components do not leak into other parts of the frame, it is advantageous to employ sparseness-promoting inversion. We adopt a nonlinear shaping regularization scheme (Fomel, 2008), analogous to the sparse inversion method of Daubechies et al. (2004), and define sparse decomposition as an iterative process

 (15) (16)

where is the seislet frame at -th iteration, is an auxiliary quantity, is input data, is the identity operator, and are frame construction and deconstruction operators

where is the seislet transform for an individual frequency, and is a nonlinear shaping operator, such as soft thresholding (Donoho, 1995). The iteration 15-16 starts with and and is related to the linearized Bregman iteration (Osher et al., 2005; Yin et al., 2008). We find that a small number of iterations is usually sufficient for convergence and achieving both model sparseness and data recovery.

Subsections
 Seislet transform and seislet frame

Next: 1-D data analysis with Up: Fomel and Liu: Seislet Previous: Seislet stack

2013-03-02