Specifications and Standards for Reproducibility of Wavelet Transforms

As the number of applications and use of wavelet transforms continues to grow, so does the number of classes and variations of wavelet transform algorithms. All of these algorithms incorporate a filter convolution in some implementation, typically, as part of an iterated filter bank. In contrast to implementations of the classical Fourier transform where there is at most a choice of sign and normalization constant in the complex exponential kernel, for wavelet transform algorithms there are multiple choices including both the signs and normalization constants of the wavelet kernels as well as the phase delays or advances of each of the filters in the wavelet filter bank. These algorithmic details, however, are usually not reported in the literature albeit with certain exceptions such as the FBI fingerprint image compression standard. Nevertheless, it is necessary to specify such details in order to insure the reproducibility of results output by each algorithm regardless of its implementation by any programmer working in any language or any engineer designing any DSP chip. This report itemizes a list of choices that must be specified clearly in order to insure the reproducibility of a sequence of transform coefficients generated by a specific wavelet transform algorithm. Moreover, this report proposes a simple but novel solution to the phase alignment problem for wavelet transforms. The general principles of this solution apply in various specific forms to both non-subsampled and critically subsampled wavelet transforms and to both symmetric and asymmetric wavelet filters.