Abstract-A multiscale Canny edge detection is equivalent to finding the local maxima of a wavelet transform. We study the properties of multiscale edges through the wavelet theory. For pattern recognition, one often needs to discriminate different types of edges. We show that the evolution of wavelet local maxima across scales characterize the local shape of irregular structures. Numerical descriptors of edge types are derived. The completeness of a multiscale edge representation is also studied. We describe an algorithm that reconstructs a close approximation of 1-D and 2-D signals from their multiscale edges. For images, the reconstruction errors are below our visual sensitivity. As an application, we implement a compact image coding algorithm that selects important edges and compresses the image data by factors over 30.Index Terms-Edge detection, feature extraction, level crossings, multiscale wavelets.I. INTROJXJCTION P OINTS OF SHARP variations are often among the most important features for analyzing the properties of transient signals or images. In images, they are generally located at the boundaries of important image structures. In order to detect the contours of small structures as well as the boundaries of larger objects, several researchers in computer vision have introduced the concept of multiscale edge detection [18], [23], [25]. The scale defines the size of the neighborhood where the signal changes are computed. The wavelet transform is closely related to multiscale edge detection and can provide a deeper understanding of these algorithms. We concentrate on the Canny edge detector [2], which is equivalent to finding the local maxima of a wavelet transform modulus.There are many different types of sharp variation points in images. Edges created by occlusions, shadows, highlights, roofs, textures, etc. have very different local intensity profiles. To label more precisely an edge that has been detected, it is necessary to analyze its local properties. In mathematics, singularities are generally characterized by their Lipschitz exponents. The wavelet theory proves that these Lipschitz exponents can be computed from the evolution across scales of the wavelet transform modulus maxima. We derive a numerical procedure to measure these exponents. If an edge is smooth, we can also estimate how smooth it is from the decay of the [21] has found counterexamples to these conjectures. In spite of these counterexamples, we show that one can reconstruct a close approximation of the original signal from multiscale edges. The reconstruction algorithm is based on alternate projections. We prove its convergence and derive a lower bound for the convergence rate. Numerical results are given both for 1-D and 2-D signals. The differences between the original and reconstructed images are not visible on a high-quality video monitor.The ability to reconstruct images from multiscale edges has many applications in signal processing. It allows us to process the image information with edge-based algorithms. We describe a com...

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