Linear canonical ripplet transform: theory and localization operators.

The ripplet transform is a refinement of the ordinary curvelet transform based on the non-parabolic scaling law for resolving two-dimensional singularities. However, the ripplet transform offers flexibility only in the angular window which is not feasible for optimizing the concentration of the ripplet spectrum. Therefore, there arises a fundamental question “whether it is possible to induce flexibility in both the radial and angular windows of the ordinary ripplet transform”. The aim of this study is to address this issue by constructing a new family of ripplet waveforms based on the linear canonical transform having certain extra degrees of freedom for resolving higher dimensional singularities. To begin with, we examine the fundamental aspects of the novel ripplet transform including the formulation of reconstruction and Rayleigh’s energy formulae. In the sequel, we obtain a Heisenberg-type uncertainty inequality associated with the novel ripplet transform. Towards the culmination, we introduce and study a new class of localization operators associated with the novel ripplet transform. [ABSTRACT FROM AUTHOR]