Edges correspond to significant extrema points of a suitable time scale decomposition of the analysed image. We then try to generalize the existing results (Mallat, Jain, Tabbone) to different wavelets and to a larger class of edges. The work represents a first attempt to extend to 2 dimensions the atomic representation for 1D signals. The main aim is to provide the trajectories of significant points in the time scale space that can be associated to singularities in the image by using a continuous, symmetric and compactly supported wavelet. Each atom is characterized by the slope, the location and the decay along scales. Only positive decays are admissible, since the negative one are associated to noise. To strengthen these results we use a particular operator: the wavelet transform is independently evaluated along two orthogonal directions and the results are combined using a simple algebraic operation. In that way the smoothing is anisotropic, since it occurs just along the analysed direction, allowing the emphasis of each single point of the edge and reducing the amount of smoothing. This allows to delay the interference between edges and to recognize the noise from its behaviour along scales reducing ambiguities. The first one allows to accomplish the good localization property, the second one uncorrelates the smoothing along the two directions and allows a good detection property. Scale linking information is exploited by solving the equation describing atoms trajectories, assuming the first scale as the initial condition. In fact, it is important to follow edges before they completely interfere, otherwise the coarse to fine approach is not applicable. Experimental results show the preliminary tests of the proposed approach. They are encouraging since they reveal better results than some classical methods in detecting both important and fine edges, even in its first rough implementation.
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