ABSTRACT
The detection of singularities with multiscale transforms has been widely stud ied not only in mathematics but also in signal processing, computer vision, etc. For contour detection, the signal can be smoothed at different scales and the edges are found for example as local extrema of the gradient modulus of the smoothed signal (Canny) or as zero-crossings of the Laplacian of the smoothed signal (Marr and Hilbreth). Mallat and al. [3] showed the equiv alence between the Canny edge detector and the local maxima of a wavelet transform modulus detection. The signal regularity at each point x is mea sured by the decay across scales of its wavelet coefficients centered at x. For this analysis, wavelet transforms with dyadic sampling, particularly orthonor mal bases, may introduce distortion since they are not translation invariant. A particular class of wavelet, derivatives of B-splines, leads to fast and effi cient algorithms for edge detection of ID and 2D signals by a discrete dyadic finite scale analysis (Mallat and Zhong). For this class, the scaling function φ is a quadratic spline if the wavelet ψ is the 1st derivative of a cubic spline. More generally, a wavelet which is the pth derivative of a smoothing function is well suited to characterize singularities of a (p-l ) th derivative through mul tiscale analysis using B-spline functions. The two scale difference equation is a subdivision equation and the wavelet coefficients are obtained at each scale from the differentiation formulas. This formulation allows us to generalize the algorithm to nonuniformly sampled data to locate singularities.
