The Hyperanalytic Wavelet Transform
University College London
given at strobl07 (19.06.07 10:00)
The extension of the analytic wavelet transform to higher dimensions is discussed. Such extensions require an appropriate definition of the extension of the analytic signal, and a careful combination of the calculation of an analytic signal with the operation of localization. The hyperanalytic wavelet transform is defined to ensure suitable properties of the transform coefficients.
The Analytic Wavelet Transform (AWT) is an important analysis tool, that has been used to characterise 1-D singularities and oscillations. To name but a few of the properties of the transform, the magnitude of the AWT will not oscillate around singularities, it is locally shift invariant, and using suitable tensor products, directional 2-D decompositions may be constructed, see for example Selesnick et al. (2005).
Extending the AWT to higher dimensions is not trivial. This task is complicated by the fact that more than one extension of the analytic signal exists, and using a hyperanalytic decomposition function, does not necessitate that hyperanalytic decomposition coefficients are constructed (see Olhede & Metikas (2006)). The extension of the AWT to 2-D thus necessitates making several choices when defining the transform. These choices are made to define wavelet coefficients with desirable properties, rather than wavelet functions.
The hyperanalytic wavelet transform is defined in a series of steps. First to extend the analytic signal, the hyperanalytic signal is used, i.e. a limit of a set of solutions to a set of generalized Cauchy-Riemann equations. We single out the hypercomplex, and monogenic signals (Olhede & Metikas (2006)), as particularly useful constructions. These objects must be combined with a localisation operation, where we propose to use a family of generalized or multidimensional Morse wavelets (see Metikas & Olhede (2007)), to this purpose.
This set of actions constructs families of wavelets that may be used to calculate hyperanalytic wavelet coefficients. The properties of the coefficients in terms of their deterministic structure, i.e. the stability of the coefficients to small local perturbations in spatial locations such as translations and rotations, are given.
The stochastic properties of the hyperanalytic decomposition of random field observations, are also outlined. We discuss how these properties facilitate the usage of the hyperanalytic wavelet transform coefficients for data analysis and estimation, see also Metikas and Olhede (2007), and Olhede (2007). We outline how existing complex-valued decompositions fit into the general framework of the hyperanalytic wavelet transform, see for example (Chan et al (2005), Selesnick et al (2007)).
W. L. Chan, H. Choi and R. Baraniuk,
Coherent image processing using quaternion wavelets, in Proceedings of SPIE, Volume 5914
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G. Metikas & S. C. Olhede, Multiple Multidimensional Morse Wavelets. IEEE Trans. Signal Proc. vol. 55, pp. 921-936, 2007.
S. C. Olhede, Hyperanalytic Denoising. IEEE Trans. Image Proc, vol 16, pp. 1522—1537, 2007.
S. C. Olhede and G. Metikas, The Hyperanalytic Wavelet Transform, TR-06-02, available at: http://arxiv.org/abs/math.ST/0605623, 2006 .
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