Symmetrical wavelet transforms for edge localization

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Abstract
The performance of a wavelet-based edge detector is characterized by a set of digital filters that implement the wavelet transform. We clarify the issue of whether an implementation filter for the wavelet transform can be centered at the origin while still maintaining the desired spatial domain localization. It is shown that this is possible only when the filter (or the wavelet) possesses an even-symmetry* with respect to the origin. When the filter (or the wavelet) is antisymmetric with respect to the origin, however, the filter coefficients converge in the order of 1/n, producing a poor spatial domain localization. We show that the optimal axis of antisymmetry for the filter is located at the half-sample point to either side of the origin. We also present a scheme to adjust the degree of spatial filtering to balance between two conflicting factors of suppressing noise and preserving edges.

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