Unique reconstruction of band-limited signals by a Mallat-Zhong wavelet transform algorithm

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 Gröchenig 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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