A novel set of fractional orthogonal polar harmonic transforms for gray-scale and color image analysis are presented in this paper. These transforms are divided into two groups. The first group contains fractional polar complex exponential transforms (FrPCETs), fractional polar cosine transforms (FrPCTs), and fractional polar sine transforms (FrPSTs) for gray-scale images. The second group contains the fractional quaternion polar complex exponential transforms (FrQPCETs), fractional quaternion polar cosine transforms (FrQPCTs), and fractional quaternion polar sine transforms (FrQPSTs) for color images. All mathematical formulae for the basis functions, orthogonality relations and reconstruction forms are derived and their validity are proved. The required mathematical forms for invariance to rotation, scaling and translation (RST) are derived. A series of experiments is performed to test the validity of the proposed fractional polar harmonic transforms (FrPHTs) and the fractional quaternion polar harmonic transforms (FrQPHTs). The performances of the proposed FrPHTs and FrQPHTs are outperformed the classical polar harmonic transforms, the quaternion polar harmonic transforms and the existing fractional orthogonal transforms in terms of accuracy and numerical stability, digital image reconstruction, RST invariances, robustness to noise and computational efficiency.

本文提出了一套新颖的分数阶正交极谐波变换，用于灰度和彩色图像分析。这些转换分为两组。第一组包含灰度图像的分数极复数指数变换（FrPCET），分数极余弦变换（FrPCT）和分数极正弦变换（FrPST）。第二组包含彩色图像的分数四元数极性复数指数变换（FrQPCET），分数四元数极性余弦变换（FrQPCTs）和分数四元数极性正弦变换（FrQPST）。推导了基本函数，正交关系和重构形式的所有数学公式，并证明了其有效性。推导了不变的旋转，缩放和平移（RST）所需的数学形式。进行了一系列实验，以测试所提出的分数极性谐波变换（FrPHTs）和分数四元数极性谐波变换（FrQPHTs）的有效性。提出的FrPHT和FrQPHT的性能在准确性和数值稳定性，数字图像重建，RST不变性，抗噪性和计算效率方面均优于经典的极性谐波变换，四元数极性谐波变换和现有的分数正交变换。