ABSTRACT
This chapter describes the theory of generalized wavelet transforms, including the fractional wavelet transform, fractional Stockwell transform, linear canonical wavelet transform, linear canonical Stockwell transform, linear canonical ridgelet transform, linear canonical curvelet and ripplet transforms and the linear canonical shearlet transform. Besides the formulation of orthogonality relation, inversion formula and range characterization, it aims to study the net time-fractional-frequency resolution and demonstrate the constant Q-property of the fractional wavelet transform. The chapter deals with ramification of the classical Stockwell transform, namely the linear canonical Stockwell transform. The notion of linear canonical Stockwell transform essentially relies on the idea of intertwining the merits of the well-known linear canonical transform with the Stockwell transform, facilitated by an appropriate convolution operation. The extension of the classical wavelet theory to the realm of fractional and linear canonical transforms has paved the way for further insights into the theory of directional representation systems, such as ridgelets, curvelets, ripplets, shearlets and so on.
