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 fil- ter 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 mul- tiple choices including both the signs and normalization constants of the wavelet kernels as well as the phase de- lays 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 or- der to insure the reproducibility of results output by each algorithm regardless of its implementation by any pro- grammer working in any language or any engineer design- ing any DSP chip. This report itemizes a list of choices that must be specified clearly in order to insure the re- producibility of a sequence of transform coecients gen- erated by a specific wavelet transform algorithm. More- over, 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 asymmet- ric wavelet filters.