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Characterization of Signals from Multiscale Edges - CMAP
710 IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE VOL 14 NO 7 JULY 1992 Characterization of Signals from Multiscale Edges Stephane Mallat and Sifen Zhong 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 cross ings multiscale wavelets I INTRODUCTION 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 Manuscript received January 10 1990 revised January 29 1992 This work was supported by the NSF grant IRI 890331 AFOSR grant AFOSR 90 0040 and ONR grant N00014 91 J 1967 Recommended for acceptance by Associate Editor R Woodham S Mallat is with the Courant Institute New York University New York NY 10012 S Zhong was with the Courant Institute New York |
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