Discrete Wavelet Transforms: The Relationship of the a Trous and Mallat Algorithms
NAVAL OCEAN SYSTEMS CENTER SAN DIEGO CA
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In a general sense this paper represents an effort to clarify the relationship of discrete and continuous wavelet transforms. More narrowly, it focuses on bringing together two separately motivated implementations of the wavelet transform, the algorithm a trous and Mallats multiresolution decomposition. It is observed that these algorithms are both special cases of a single filter bank structure, the discrete wavelet transform, the behavior of which is governed by ones choice of filters. In fact, the a trous algorithm, originally devised as a computationally efficient implementation, is more properly viewed as a nonorthogonal multiresolution algorithm for which the discrete wavelet transform is exact. A systemative framework for the discrete wavelet transform is provided, and conditions are derived under which it computes the continuous wavelet transform exactly.
- Numerical Mathematics