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<p>This paper introduces a novel generalisation of the Hilbert transform by integrating it with the two-dimensional quadratic phase Fourier transform (2D-QPFT). This integration extends the analytic signal framework into non-stationary and multi-dimensional signal domains. Within the QPFT domain, the paper proposes and thoroughly examines the concepts of “half-planned" and “cross-planned" Hilbert transforms. Theoretical analyses are complemented by simulation studies that demonstrate the superiority of the proposed 2D Hilbert-QPFT over its linear canonical transform counterpart, particularly in terms of localisation and directional selectivity. The paper further demonstrates the utility of the new formulation through its application to Hilbert filtering and the challenging task of wrapped phase map estimation, showcasing improved phase reconstruction accuracy.</p>

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Generalized 2D Hilbert Quadratic-phase Transform with Applications to Wrapped Phase Map Determination

  • 期刊论文

  • 发表日期：

    2026年3月6日

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  • Musadiq Shaheen,
  • Firdous A. Shah

摘要

This paper introduces a novel generalisation of the Hilbert transform by integrating it with the two-dimensional quadratic phase Fourier transform (2D-QPFT). This integration extends the analytic signal framework into non-stationary and multi-dimensional signal domains. Within the QPFT domain, the paper proposes and thoroughly examines the concepts of “half-planned" and “cross-planned" Hilbert transforms. Theoretical analyses are complemented by simulation studies that demonstrate the superiority of the proposed 2D Hilbert-QPFT over its linear canonical transform counterpart, particularly in terms of localisation and directional selectivity. The paper further demonstrates the utility of the new formulation through its application to Hilbert filtering and the challenging task of wrapped phase map estimation, showcasing improved phase reconstruction accuracy.