<p>In this paper, we introduce the quadratic-phase ridgelet transform (QRLT) by intertwining the key advantages of the quadratic-phase Fourier transform (QPFT) and the classical ridgelet transform. This construction is achieved through the combined use of the Radon transform and the QPFT. As a first step, we develop a family of quadratic-phase ridgelet waveforms by appropriately chirping a one-dimensional wavelet along specific orientations. The formulation is supported by an illustrative example that highlights the distinctive structure of the proposed ridgelets. Subsequently, we define the QRLT and investigate its fundamental properties, including Parseval’s identity and the inversion formula. To demonstrate the efficacy of the transform, we provide an example illustrating the implementation of the QRLT on a bivariate function. Finally, we establish a Heisenberg-type uncertainty inequality associated with the proposed QRLT.</p>

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Quadratic-phase ridgelet transform: a mathematical perspective

  • Shraddha Sachan,
  • Waseem Z. Lone,
  • Amit K. Verma

摘要

In this paper, we introduce the quadratic-phase ridgelet transform (QRLT) by intertwining the key advantages of the quadratic-phase Fourier transform (QPFT) and the classical ridgelet transform. This construction is achieved through the combined use of the Radon transform and the QPFT. As a first step, we develop a family of quadratic-phase ridgelet waveforms by appropriately chirping a one-dimensional wavelet along specific orientations. The formulation is supported by an illustrative example that highlights the distinctive structure of the proposed ridgelets. Subsequently, we define the QRLT and investigate its fundamental properties, including Parseval’s identity and the inversion formula. To demonstrate the efficacy of the transform, we provide an example illustrating the implementation of the QRLT on a bivariate function. Finally, we establish a Heisenberg-type uncertainty inequality associated with the proposed QRLT.