Multiresolution representations using the auto-correlation functions of compactly supported wavelets, (with G.Beylkin), Research Note, Schlumberger-Doll Research, Aug. 1991.

Abstract

In this paper we propose a "hybrid" shift-invariant multiresolution representation which utilizes dilations and translations of the auto-correlation functions of compactly supported wavelets. In this representation, the exact filters for the decomposition are the auto-correlation of the quadrature mirror filter coefficients of the compactly supported wavelets. The decomposition filters are, therefore, exactly symmetric. Moreover, the auto-correlation functions of the compactly supported wavelets may be viewed as pseudo-differential operators of the even order and behave, essentially, as the derivative operators of the same order. This allows us to relate the zero-crossings in this representation to the locations of edges at different scales in the signal. The recursive definition of the compactly supported wavelets and, therefore, their auto-correlation functions, allows us to construct fast recursive algorithms to generate the multiresolution representations. Though it is not an orthogonal representation, there is a simple relation with the wavelet-based orthogonal representations on each scale. We describe a simple reconstruction algorithm to recover functions from such expansions. A remarkable feature of the representation using the auto-correlation function of compactly supported wavelets is a natural interpolation algorithm associated with it. This interpolation algorithm, the so-called symmetric iterative interpolation, is due to Dubuc [1] and Deslauriers and Dubuc [2]. The coefficients of the interpolation scheme of [1] and [2] generated from the Lagrange polynomials are the auto-correlation coefficients of quadrature mirror filters associated with the compactly supported wavelets of Daubechies. Finally, we consider the reconstruction of signals from zero-crossings (and slopes at zero-crossings) of their multiresolution representations. Our approach permits a non-iterative reconstruction from zero-crossings (and slopes at zero-crossings). Using the interpolation algorithm mentioned above, we locate the zero-crossings and compute slopes at these points within the prescribed numerical accuracy. We then set up a linear system, where the entries of the matrix are computed from the values of the auto-correlation function and its derivative at the integer translates of zero-crossings. The original signal is reconstructed within the prescribed accuracy by solving this linear system.

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