Dyadic wavelet construction and dyadic wavelet transforms
Abstract:
Orthogonal and bi-orthogonal wavelet transforms have been successfully used in many applications, yet their lack of translation invariance would lead to pseudo-Gibbs phenomena in the neighborhood of discontinuities when signal is reconstructed with processed wavelet coefficients. The dyadic wavelet transform pioneered by Mallat and Zhong was precisely introduced to cope with this , which has been widely used in image processing such as multiscale edge detection because of its translation invariance. This type of dyadic wavelet has a property that the dyadic wavelet does not equal to its reconstruction wavelet, and the two dimensional dyadic wavelet transform is defined by an approximation component, two detail components in horizontal and vertical directions. Turuki et al. proposed dyadic lifting scheme to construct dyadic wavelet with higher number of vanishing moments, which generalized the Sweldens lifting schemes for orthogonal and bi-orthogonal wavelets. In recent years, we investigated and enriched dyadic wavelet theory and applications. Our contributions focus on the following three aspects.
1. Two-dimensional dyadic wavelet transform (2D-DWT)

Currently, two-dimensional dyadic wavelet transform (2D-DWT) is habitually considered as the one presented by Mallat, which is defined by an approximation component,two detail components in horizontal and vertical directions. The reference [1] introduces a new type of two-dimensional dyadic wavelet transform and its application so that dyadic wavelet can be studied and used widely furthermore. (1) Two-dimensional stationary dyadic wavelet transform (2D-SDWT) is proposed, it is defined by approximation coefficients, detail coefficients in horizontal, vertical and diagonal directions,which is essentially the extension of two-dimensional stationary wavelet transform for orthogonal/ biorthogonal wavelet filters. (2) ε-decimated dyadic discrete wavelet transform (DDWT) is introduced and its relation with 2D-SDWT is given, where ε is a sequence of 0's and 1's. (3) Mallat decomposition algorithm based on dyadic wavelet is introduced as a special case of ε-decimated DDWT, and so a face recognition algorithm based on dyadic wavelet is proposed, and experimental results are given to show its effectiveness.

2. Lifting construction of spline dyadic wavelet filters

The dyadic lifting schemes, which generalize Sweldens lifting schemes, have been applied to design dyadic wavelet with higher number of vanishing moments. But the existing dyadic lifting methods cannot give the free parameters (i.e. lifting factors) explicitly under vanishing moment constraints, and the exact vanishing moments of the lifted wavelet is unknown a priori. We provided a solution of these problems for spline dyadic wavelets [2]. The new lifting construction scheme can be applied to design spline dyadic wavelet filters with any number of vanishing moments starting from one single dyadic wavelet with 1 or 2 vanishing moments. Its computational advantage is that the lifting factor parameters can be chosen and given explicitly under vanishing moment constraints. Some spline dyadic wavelet filters are designed by using our method.

3. Construction and properties of spline dyadic wavelet filters

The reference [3] focuses on construction and properties of spline dyadic wavelet that equals to its reconstruction one. A general construction method of finite spline dyadic low pass and high pass filters are given. It proves that finite spline dyadic low pass filters are symmetric about 0 or 1/2, but there are no finite spline high pass filters possessing symmetry with respect to 0 or 1/2. It further shows that there exist infinite spline high pass filters possessing symmetry with respect to 0 or 1/2, which can be constructed and their energy are concentrated and so finite symmetric spline dyadic wavelet filter that equals to its reconstruction one can be obtained approximately. Construction examples for quadratic and cubic spline dyadic wavelet filters are given.

References:

  1. Y.K. Sun, Yong Chen, Hao Feng, Two-dimensional stationary dyadic wavelet transform, decimated dyadic discrete wavelet transform and the face recognition application, Int. J. Wavelets Multiresolut. Inf. Process . 9(3)(2011), pp. 397-416

  2. Y.K. Sun, Lifting Construction of Spline Dyadic Wavelet filters with any Number of Vanishing Moments. Int. J. Wavelets Multiresolut. Inf. Process .7(5)(2009), pp. 693-710.

  3. Y.K. Sun, Chen Ding, Construction and properties of spline dyadic wavelet filters, International Journal of Computer Mathematics, 89(2),254-263, 2012
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