For the first discriminator, we study the statistical properties of the
distribution of the multi-scale gradient coefficients.
This method is appropriate when dealing with a non-Gaussian process
characterised by sharp edges and consequently by strong gradients in the
signal.
Indeed, in any region where the analysed function is smooth the wavelet
coefficients are small. On the contrary, any abrupt change in the behaviour of
the function increases the amplitude of the coefficients around the
singularity.
The detection of non-gaussianity is thus based on the search of these
gradients. In the dyadic wavelet decomposition, one can discriminate between
the coefficients associated with vertical and horizontal gradients and the
other coefficients. In our case, the vertical and horizontal gradients are
analogous to the partial derivatives, and
, of the signal.
Mallat & Zhong (1992) give
a thorough treatment of the characterisation of signals from multi-scale edges.
We compute the quadratic sum of the coefficients, the quantity
,at each decomposition level *L*. This quantity represents the squared
amplitude of the
multi-scale gradient of the image. In the following, we will however refer to
it as the multi-scale gradient coefficient.

The second statistical discriminator is based on the study of the wavelet coefficients related to the horizontal, vertical and diagonal gradients. These coefficients are associated with the partial derivatives and , as in the multi-scale gradient method, as well as with the cross derivative . The coefficients are computed at each decomposition level and their excess of kurtosis with respect to a Gauss distribution exhibits the non-Gaussian signature of the studied signal. In this context, the wavelet coefficients associated with the first derivatives are obviously closely related to the multi-scale gradient.

In the following, we first apply our two tests to purely Gaussian and non-Gaussian maps. We then test the detectability of a non-Gaussian signal added to a Gaussian one with same power spectrum and with increasing mixing ratios.

Copyright The European Southern Observatory (ESO)