Abstract Orthogonal moments were successfully used to extract features from gray-scale and color images. Recently, scientists show that orthogonal moments of fractional-orders have better capabilities to extract the fine features. In this work, novel orthogonal generic fractional-order Jacobi-Fourier moments are proposed for image processing, pattern recognition and computer vision applications. Novel orthogonal Jacobi-Fourier polynomials of fractional-order were derived and defined in polar coordinates. The mathematical equation for orthogonality was formulated and a three-term recurrence relation was derived for easier computation of these polynomials. The proposed orthogonal fractional-order Jacobi-Fourier moments are generic where other orthogonal fractional-order moments are derived as special cases by choosing different values of the controlling parameters. The invariance to geometric transformations, rotation, scaling and translations, is proved where the required mathematical formulae for these invariances are presented. The proposed new fractional-order moments were tested using different datasets of gray-scale and color images in terms of image reconstruction, invariance to geometric transformations, robustness to noise, image recognition, and computational times where their performance were compared with the recent existing orthogonal integer- and fractional-order moments. The proposed generic fractional-order Jacobi-Fourier moments outperformed all existing orthogonal moments.

中文翻译：

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用于图像分析的新型分数阶通用 Jacobi-Fourier 矩

摘要 正交矩成功地用于从灰度和彩色图像中提取特征。最近，科学家们表明分数阶的正交矩具有更好的提取精细特征的能力。在这项工作中，为图像处理、模式识别和计算机视觉应用提出了新颖的正交通用分数阶 Jacobi-Fourier 矩。在极坐标中导出和定义了新的分数阶正交雅可比-傅立叶多项式。公式化了正交性的数学方程，并导出了三项递推关系，以便于计算这些多项式。所提出的正交分数阶 Jacobi-Fourier 矩是通用的，其中其他正交分数阶矩是通过选择不同的控制参数值作为特殊情况导出的。几何变换、旋转、缩放和平移的不变性在提出这些不变性所需的数学公式的地方得到证明。在图像重建、几何变换不变性、噪声鲁棒性、图像识别和计算时间方面，使用不同的灰度和彩色图像数据集对提出的新分数阶矩进行了测试，并将它们的性能与最近现有的正交矩进行了比较。整数阶矩和分数阶矩。提出的通用分数阶 Jacobi-Fourier 矩优于所有现有的正交矩。