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The Hyperanalytic Wavelet Transform

Olhede, SC; Metikas, G; The Hyperanalytic Wavelet Transform. IEEE Transactions on Signal Processing , 57 (34) pp. 6-3441. 10.1109/TSP.2009.2023397.

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In this paper novel classes of 2-D vector-valued spatial domain wavelets are defined, and their properties given. The wavelets are 2-D generalizations of 1-D analytic wavelets, developed from the Generalized Cauchy-Riemann equations and represented as quaternionic functions. Higher dimensionality complicates the issue of analyticity, more than one `analytic' extension of a real function is possible, and an `analytic' analysis wavelet will not necessarily construct `analytic' decomposition coefficients. The decomposition of locally unidirectional and/or separable variation is investigated in detail, and two distinct families of hyperanalytic wavelet coefficients are introduced, the monogenic and the hypercomplex wavelet coefficients. The recasting of the analysis in a different frame of reference and its effect on the constructed coefficients is investigated, important issues for sampled transform coefficients. The magnitudes of the coefficients are shown to exhibit stability with respect to shifts in phase. Hyperanalytic 2-D wavelet coefficients enable the retrieval of a phase-and-magnitude description of an image in phase space, similarly to the description of a 1-D signal with the use of 1-D analytic wavelets, especially appropriate for oscillatory signals. Existing 2-D directional wavelet decompositions are related to the newly developed framework, and new classes of mother wavelets are introduced.

Type: Article
Title: The Hyperanalytic Wavelet Transform
DOI: 10.1109/TSP.2009.2023397
Publisher version: http://dx.doi.org/10.1109/TSP.2009.2023397
Additional information: 49 pages, 14 Postscript figures
Keywords: math.ST, math.ST, quant-ph, stat.TH, 42C40; 62G08
UCL classification: UCL > School of BEAMS
UCL > School of BEAMS > Faculty of Maths and Physical Sciences
URI: http://discovery.ucl.ac.uk/id/eprint/80762
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