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The Theory and Use of the Quaternion Wavelet Transform

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Abstract

This paper presents the theory and practicalities of the quaternion wavelet transform (QWT). The major contribution of this work is that it generalizes the real and complex wavelet transforms and derives a quaternionic wavelet pyramid for multi-resolution analysis using the quaternionic phase concept. As a illustration we present an application of the discrete QWT for optical flow estimation. For the estimation of motion through different resolution levels we use a similarity distance evaluated by means of the quaternionic phase concept and a confidence mask. We show that this linear approach is amenable to be extended to a kind of quadratic interpolation.

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Authors and Affiliations

  1. Computer Science Department, GEOVIS Laboratory, Centro de Investigación y de Estudios Avanzados, CINVESTAV, Guadalajara, Jalisco, 44550, Mexico

    Eduardo Bayro-Corrochano

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  1. Eduardo Bayro-Corrochano
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Eduardo Jose Bayro-Corrochano gained his Ph.D. in Cognitive Computer Science in 1993 from the University of Wales at Cardiff. From 1995 to 1999 he has been Researcher and Lecturer at the Institute for Computer Science, Christian Albrechts University, Kiel, Germany, working on applications of geometric Clifford algebra to cognitive systems.

His current research interest focuses on geometric methods for artificial perception and action systems. It includes geometric neural networks, visually guidevsd robotics, color image processing, Lie bivector algebras for early vision and robot maneuvering. He is editor and author of the following books: Geometric Computing for Perception Action Systems, E. Bayro-Corrochano, Springer Verlag, 2001; Geometric Algebra with Applications in Science and Engineering, E. Bayro-Corrochano and G. Sobczyk (Eds.), Birkahauser 2001; Handbook of Computational Geometry for Pattern Recognition, Computer Vision, Neurocomputing and Robotics, E. Bayro-Corrochano, Springer Verlag, 2005.

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Bayro-Corrochano, E. The Theory and Use of the Quaternion Wavelet Transform. J Math Imaging Vis 24, 19–35 (2006). https://doi.org/10.1007/s10851-005-3605-3

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