Discrete Gabor dictionary

Gabor functions (sine-modulated Gaussian functions) provide optimal joint time-frequency localization. A real Gabor function can be expressed as:

$$g_\gamma (t) = K(\gamma)e^{-\pi{ \left( {t-u} \over {s} \right) }^2} \sin\left(2 \pi \frac \omega N (t-u)+\phi\right)$$ (6)

N is the size of the signal for which the dictionary is constructed, $K(\gamma)$ is such that $\vert\vert g_{\gamma}\vert\vert=1$. $\gamma=\{ u, \omega, s, \phi \}$ denotes parameters of the dictionary's functions (time-frequency atoms). The length of signal ($N$ points) suggests ranges for the parameters of Gabor functions which can be reasonably fitted to such an epoch. However, within these ranges no particular sampling is a priori defined, and we are confronted with a three-dimensionalcontinuous space; this results in an infinite dictionary size. Therefore, in practical implementations, we use subsets of the possible dictionary functions.

In the dictionary implemented originally by Mallat and Zhang in [1], the parameters of the atoms are chosen from dyadic sequences of integers. Their sampling is governed by an extra parameter--octave $j$ (integer). Scale $s$, which corresponds to an atom's width in time, is derived from the dyadic sequence $s=2^j, 0 \leq j \leq L$ (signal size $N=2^L$). Parameters $u$ and $\omega$, which correspond to an atom's position in time and frequency, are sampled for each octave $j$ with interval $s = 2^j$, or, if oversampling by $l$ is introduced, with interval $2^{j-l}$.

Footnotes

... three-dimensional

The phase $\phi$ is usually optimized in each step of the MP procedure separately for each fitted atom.