3 Tests of non-gaussianity

The most direct and obvious way of analysing the statistical properties of an image is to use the distribution of the pixel brightnesses, or temperatures, together with the skewness and kurtosis. If the two quantities are different from zero, they indicate that the signal is non-Gaussian. However, a weak non-Gaussian signal will hardly be detected through the moments of the temperature distribution. Another way of addressing the problem is to use the coefficients in the wavelet decomposition and to study their statistical properties which in turn characterise the signal. In fact, the wavelet coefficients are quite sensitive to variations (even weak ones) in the signal, temperature or brightness, and hence to the statistical properties of the underlying process. We have developed two tests which exhibit the non-Gaussian characteristics of a signal using the wavelet coefficients. Since our test maps are not skewed in the following we focus only on the results obtained using the fourth moment.

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, $\partial/\partial x$ and $\partial/\partial y$, 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 ${\mathcal{G}}_L=\left(\partial/\partial x\right)_L^2 +\left(\partial/\partial y\right)_L^2$,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 $\partial/\partial y$ and $\partial/\partial x$, as in the multi-scale gradient method, as well as with the cross derivative $\partial^2/\partial x\partial y$. 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.