The paper discusses the relationship between discrete and continuous wavelet transforms, focusing on two algorithms: the à trous algorithm and Mallat's multiresolution algorithm. Both are shown to be special cases of a single filter bank structure, the discrete wavelet transform (DWT). The à trous algorithm, originally designed for computational efficiency, is viewed as a nonorthonormal multiresolution algorithm, while Mallat's algorithm uses orthonormal wavelets. The paper demonstrates that the Lagrange à trous filters correspond to the convolutional squares of Daubechies filters for orthonormal wavelets. A systematic framework for the DWT is provided, along with conditions under which it computes the continuous wavelet transform exactly. The paper also examines signal processing parameters and the trade-offs involved in choosing bandpass filters, emphasizing differences between orthonormal and nonorthonormal cases. The paper concludes that the DWT can be an exact wavelet transform under certain conditions, with the à trous condition ensuring this exactness. The paper provides a detailed derivation of the algorithms and their relationship to the continuous wavelet transform.The paper discusses the relationship between discrete and continuous wavelet transforms, focusing on two algorithms: the à trous algorithm and Mallat's multiresolution algorithm. Both are shown to be special cases of a single filter bank structure, the discrete wavelet transform (DWT). The à trous algorithm, originally designed for computational efficiency, is viewed as a nonorthonormal multiresolution algorithm, while Mallat's algorithm uses orthonormal wavelets. The paper demonstrates that the Lagrange à trous filters correspond to the convolutional squares of Daubechies filters for orthonormal wavelets. A systematic framework for the DWT is provided, along with conditions under which it computes the continuous wavelet transform exactly. The paper also examines signal processing parameters and the trade-offs involved in choosing bandpass filters, emphasizing differences between orthonormal and nonorthonormal cases. The paper concludes that the DWT can be an exact wavelet transform under certain conditions, with the à trous condition ensuring this exactness. The paper provides a detailed derivation of the algorithms and their relationship to the continuous wavelet transform.