The Discrete Wavelet Transform: Wedding the .

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 trans- form, the algorithme U trous and Mallat's multiresolution de- composition. It is observed that these algorithms are both spe- cial cases of a single filter bank structure, the discrete wavelet transform, the behavior of which is governed by one's choice of filters. In fact, the h trow algorithm, originally devised as a computationally efficient implementation, is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact. Moreover, it is shown that the commonly used Lagrange i~ trous filters are in one-to-one correspondence with the convolutional squares of the Daubechies filters for orthonormal wavelets of compact support. A systematic framework for the discrete wavelet transform is provided, and conditions are derived under which it com- putes the continuous wavelet transform exactly. Suitable filter constraints for finite energy and boundedness of the discrete transform are also derived. Finally, relevant signal processing parameters are examined, and it is remarked that orthonor- mality is balanced by restrictions on resolution.