The quadratic-phase curvelet transform and associated uncertainty relations

The classical curvelet transform’s nonadaptive nature prevents it from locating the specific features of the data required in many applications. To overcome this drawback, this paper introduces a new quadratic-phase curvelet transform (QPCT) by plugging the kernel with the curvelet as the quadratic function. We derive some fundamental properties of the QPCT, including translation, scaling, parity, and linearity. In the sequel, key results such as the inversion formula, Rayleigh’s energy formula, and characterization of the range of the QPCT are obtained. Further, some quantum mechanics results, like classical Heisenberg and logarithmic uncertainty-type inequalities, are extended for the QPCT. The local type and Nazarov’s uncertainty principles associated with the proposed QPCT are also derived. The efficacy of the QPCT, depending on adaptive curvelets, underscores its potential as a robust technique.