Existence of uncertainty minimizers for the continuous wavelet transform |
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Authors: | Simon Halvdansson Jan-Fredrik Olsen Nir Sochen Ron Levie |
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Affiliation: | 1. Department of Mathematics, NTNU Norwegian University of Science and Technology, Trondheim, Norway;2. Centre for Mathematical Sciences, Lund University, Lund, Sweden;3. School of Mathematical Sciences, Tel Aviv University, Tel Aviv, Israel;4. Faculty of Mathematics, Technion - Israel Institute of Technology, Haifa, Israel |
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Abstract: | Continuous wavelet design is the endeavor to construct mother wavelets with desirable properties for the continuous wavelet transform (CWT). One class of methods for choosing a mother wavelet involves minimizing a functional, called the wavelet uncertainty functional. Recently, two new wavelet uncertainty functionals were derived from theoretical foundations. In both approaches, the uncertainty of a mother wavelet describes its concentration, or accuracy, as a time-scale probe. While an uncertainty minimizing mother wavelet can be proven to have desirable localization properties, the existence of such a minimizer was never studied. In this paper, we prove the existence of minimizers for the two uncertainty functionals. |
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Keywords: | continuous wavelet uncertainty minimizer uncertainty principle wavelet design |
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