Unique reconstruction of band-limited signals by a Mallat-Zhong wavelet transform algorithm |
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Authors: | C J Kicey C J Lennard |
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Institution: | (1) Department of Mathematics and Computer Science, Valdosta State University, Valdosta, Georgia 31698, USA;(2) Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260, USA |
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Abstract: | We show that uniqueness and existence for signal reconstruction from multiscale edges in the Mallat and Zhong algorithm become
possible if we restrict our signals to Paley-Wiener space, band-limit our wavelets, and irregularly sample at the wavelet
transform (absolute) maxima—the edges—while possibly including (enough) extra points at each level. We do this in a setting
that closely resembles the numerical analysis setting of Mallat and Zhong and that seems to capture something of the essence
of their (practical) reconstruction method. Our work builds on a uniqueness result for reconstructing an L2 signal from irregular sampling of its wavelet transform of Grochenig and the related work of Benedetto, Heller, Mallat, and
Zhong. We show that the rate of convergence for this reconstruction algorithm is geometric and computable in advance. Finally,
we consider the effect on the rate of convergence of not sampling enough local maxima. |
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Keywords: | Primary 42C Secondary 46C |
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