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tokens: of freedom is 1. The errors on the elements of the Hessian matrix are propagated to . Since Eq. ( 9 ) is non-linear, we compute the error map on from Gaussian realizations of the errors on the coefficients of the matrix. The uncertainty in , computed within bins of values of the map, is , independent of the intensity at 353 GHz (Fig. A.1 ). Fig. A.1 Two independent estimates of the uncertainties in the determination of angles from the Hessian analysis, computed within bins of values of the map. The red squares represent the noise obtained by propagation of the errors on the coefficients of the Hessian matrix. The noise computed with the toy model (Sect. A.2 ) is plotted with black squares. Fig. A.2 Distribution functions of orientation angles for the selected ridges derived from the Hessian analysis on . They are computed within three bins of Galactic latitudes with equal numbers of data points. The mean absolute latitudes are , , and for the DFs plotted in black, orange and red, respectively. The image shows that the method does not introduce any bias as a function of Galactic latitude in estimating . Appendix A.2: Test on a toy model of the sky We have tested the Hessian algorithm on a toy model of the full sky built from a realization of a Gaussian map, , with an angular power spectrum, equal to a power law of index , computed with the procedure SYNFAST of HEALPix . The spectral index used is within the range of values found for power spectra of dust maps ( Miville-Deschênes et al. 2007 ). The mean value of is 0. We form, where , , , and are four factors chosen to fit the latitude profile, and the longitude profile at Galactic latitude , measured on the map. The values of these factors, including their sign, depend on the realization of . The toy model matches the large-scale structure of the Galactic dust emission, but it does not have its filamentary structure because it is computed from an isotropic Gaussian map. It assumes that the amplitude of the brightness fluctuations at a given scale is proportional to the brightness. This is in agreement with what has been reported for the emission at 100 m of the diffuse ISM ( Miville-Deschênes et al. 2007 ). We use the toy model to quantify the impact of the structure of the diffuse Galactic emission and data noise on the Hessian analysis. We run the Hessian analysis on the map with and without data noise computed as a Gaussian realization of the noise map on . The DFs of orientation angles, computed for the model with added noise, are flat at all Galactic latitudes, as shown in Fig. A.2 . We find no bias on the angle introduced by the large-scale gradient of the emission with Galactic latitude. We use the difference between the two angle maps computed on the map, with and without noise, to estimate the contribution of the data noise to the uncertainty in the Hessian angle (black dots in Fig. A.1 ). This uncertainty decreases for increasing values of (i.e. increasing signal-to-noise ratio).