Recently, it has been shown that gradient-based meth- ods are the most powerful approaches for describing the lo- cal content of digital images in the neighborhood of salient points. In practice, salient points are always located on image singularities whatever the detector used. In this pa- per, we show that a more efficient mathematical notion can be used to describe singularities: the H¨older exponent. We propose here to conjointly use the H¨older exponents and the direction of minimal regularity of the bidimensionnal signal singularities to compute a signature describing precisely a region of interest centered on an interest point. H¨older ex- ponents are estimated thanks to the foveal wavelets theory and the resulting descriptor is shown to be more efficient than classical SIFT and PCA-SIFT descriptors in the case of an image registration application.