The goal of the book is to develop a deeper understanding of the roles of time-frequency or Fourier and Gabor analysis, and time-scale or wavelet analysis, when the various tools are properly assembled in a larger context. While researchers at the forefront of developments in time-frequency scale (TFS) analysis are well aware of the benefits of such a unified approach, there remains a gap in the larger community of practitioners concerning precisely the strengths and limitations of Gabor analysis versus wavelets. The book fills this gap by presenting the interface between time-frequency and time-scale methods as a rich area of work. Topics and features: • Addresses new TFS methods and algorithms • Presents an integrated approach to using Fourier/Gabor methods and wavelet methods • Real world applications of Gabor theory, wavelets to PDEs and wavelets to compression • Develops the ideas of phase space and the uncertainty principle focusing on the role of the metaplectic group and the flexibility in constructing analysis tools • Explanations at the end of each chapter address historical context, major developments and new directions.