Wavelets behind the scenes: Practical aspects, insights, and perspectives

Over the years, wavelet-based analyses have been responsible for remarkable achievements in physics and related sciences. Nevertheless, a deep inspection on wavelet-based strategies described in recent scientific papers, dissertations, and theses reveals that a significant number of authors, i.e., students and even researchers with a modest background on signal analysis, still misunderstand the fundamentals of wavelets. One classical source of confusion, for instance, involves two different but related approaches used to perform discrete-time wavelet transformations and their inverses: algebra- and filter-based. Although the latter is usually adopted in practice, the former reveals the beauty of multiresolution analysis over Heisenberg's uncertainty principle, showing what really happens behind the scenes. Thus, based on the solid and easy-to-follow explanations provided in this smoothly written tutorial-review article, interested readers will definitively comprehend the different types of wavelet transforms and their specific applications, getting hands-on experience and insights on how to extract the most of their research by using that powerful tool. Because its mission is clarification, example wavelet-related applications are provided in this document to stimulate state-of-the-art research in a diversity of branches in physics.