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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 $\mathbf{F}_n$ is the orthonormal seislet transform for $n$-th frequency or plane wave, then, for any data vector $\mathbf{d}$,

\sum\limits_{n=1}^N \Vert\mathbf{F}_n\,\mathbf{d}\Vert^2 = \...
..._{n=1}^N \Vert\mathbf{d}\Vert^2 = N\,\Vert\mathbf{d}\Vert^2\;,
\end{displaymath} (14)

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

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 $Z_1, Z_2, \ldots, Z_k$. 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 $\omega_1, \omega_2, \ldots, \omega_k$.
  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

$\displaystyle \widehat{\mathbf{f}}_{k+1}$ $\textstyle =$ $\displaystyle \mathbf{S}[\mathbf{F}\,\mathbf{d}+(\mathbf{I}-\mathbf{F}\,
\mathbf{F}')\,\widehat{\mathbf{f}}_{k}]\;,$ (15)
$\displaystyle \mathbf{f}_{k+1}$ $\textstyle =$ $\displaystyle \mathbf{f}_k + \mathbf{F}\,\mathbf{d}- \mathbf{F}\,\mathbf{F}' \widehat{\mathbf{f}}_{k+1}\;,$ (16)

where $\mathbf{f}_k$ is the seislet frame at $k$-th iteration, $\widehat{\mathbf{f}}_k$ is an auxiliary quantity, $\mathbf{d}$ is input data, $\mathbf{I}$ is the identity operator, $\mathbf{F}$ and $\mathbf{F}'$ are frame construction and deconstruction operators

\mathbf{F} & \equiv &
\left[\begin{array}{cccc}\mathbf{F}_1 ...
...hbf{F}_2^{-1} &
\cdots &\mathbf{F}_k^{-1}\end{array}\right]\;,

where $\mathbf{F}_j$ is the seislet transform for an individual frequency, and $\mathbf{S}$ is a nonlinear shaping operator, such as soft thresholding (Donoho, 1995). The iteration 15-16 starts with $\mathbf{f}_0=\mathbf{0}$ and $\widehat{\mathbf{f}}_0=\mathbf{F}\,\mathbf{d}$ 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.

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