Robotics Research Technical Report (Classic Reprint): Characterization of Signals From Multiscale Edges: Characterization of Signals from Multiscale Edges (Classic Reprint) - Softcover

Mallat, Stephane

9781332111657: Robotics Research Technical Report (Classic Reprint): Characterization of Signals From Multiscale Edges: Characterization of Signals from Multiscale Edges (Classic Reprint)

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ISBN 10: 1332111653 ISBN 13: 9781332111657

Verlag: Forgotten Books, 2018

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This book delves into the fascinating world of multiscale edge detection, a cornerstone of image and signal analysis. The author, drawing on the powerful framework of wavelet theory, explores the nuanced relationship between multiscale edges and the underlying structure of signals. This book goes beyond simple edge detection, investigating the different types of edges, their characteristics, and the information they hold. The author develops a method for numerically classifying edges based on their local properties, providing a richer understanding of their significance. The author then tackles the question of signal reconstruction from multiscale edges, revealing the potential and limitations of using these features to represent signals faithfully. A key contribution of the book is the development of a novel algorithm for reconstructing signals from their multiscale edges, utilizing an elegant process of alternating projections. This book offers a comprehensive and insightful exploration of multiscale edge detection, showcasing its potential in various fields, including pattern recognition, image compression, and noise reduction.

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Excerpt from Robotics Research Technical Report: Characterization of Signals From Multiscale Edges

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This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

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Excerpt from Robotics Research Technical Report: Characterization of Signals From Multiscale Edges

Points 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 [24, 29, 31]. 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 [3[, 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. 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 wavelet transform maxima across scales. Lipschitz exponents and smoothing factors arc numerical descriptors that allow us to discriminate the intensity profiles of different types of edges.

An important open problem in computer vision is to understand how much information is carried by multiscale edges and how stable is a multiscale edge representation. This issue is important in pattern recognition where one needs to know whether some interesting information is lost when representing a pattern with edges. We study the reconstruction of one and two-dimensional signals from multiscale edges detected by the wavelet transform modulus maxima. It has been conjectured [22, 24] that multiscale edges characterize uniquely one and two-dimensional signals, but recently Meyer [27] has found counter-examples to these conjectures. In spite of these counter-examples, 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 one and two-dimensional 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 compact image coding algorithm that keeps only the "important" edges. The image that is recovered from these main features has lost some small details but is visually of good quality. Examples with compression ratio over 30 are shown.

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