This technical report, authored by Stephane Mallat and Sifen Zhong, explores the characterization of signals from multiscale edges using wavelet transforms. The authors focus on the Canny edge detector, which is equivalent to finding the local maxima of a wavelet transform modulus. They derive numerical descriptors to characterize different types of edges based on the evolution of wavelet transform maxima across scales. The report also discusses the completeness and stability of multiscale edge representations and presents an algorithm for reconstructing signals from their multiscale edges. The reconstruction algorithm is based on alternate projections and is shown to converge with a lower bound on the convergence rate. Numerical results demonstrate that the reconstruction errors are below visual sensitivity for one and two-dimensional signals. Additionally, the report introduces a compact image coding algorithm that selects important edges and compresses image data by factors over 30. The article is organized into sections covering multiscale edge detection, the dyadic wavelet transform, analysis of multiscale information, and signal reconstruction from multiscale edges.This technical report, authored by Stephane Mallat and Sifen Zhong, explores the characterization of signals from multiscale edges using wavelet transforms. The authors focus on the Canny edge detector, which is equivalent to finding the local maxima of a wavelet transform modulus. They derive numerical descriptors to characterize different types of edges based on the evolution of wavelet transform maxima across scales. The report also discusses the completeness and stability of multiscale edge representations and presents an algorithm for reconstructing signals from their multiscale edges. The reconstruction algorithm is based on alternate projections and is shown to converge with a lower bound on the convergence rate. Numerical results demonstrate that the reconstruction errors are below visual sensitivity for one and two-dimensional signals. Additionally, the report introduces a compact image coding algorithm that selects important edges and compresses image data by factors over 30. The article is organized into sections covering multiscale edge detection, the dyadic wavelet transform, analysis of multiscale information, and signal reconstruction from multiscale edges.