Inspired by quaternion algebra and the idea of fractional-order transformation, we propose a new set of quaternion fractional-order generalized Laguerre orthogonal moments (QFr-GLMs) based on fractional-order generalized Laguerre polynomials. Firstly, the proposed QFr-GLMs are directly constructed in Cartesian coordinate space, avoiding the need for conversion between Cartesian and polar coordinates; therefore, they are better image descriptors than circularly orthogonal moments constructed in polar coordinates. Moreover, unlike the latest Zernike moments based on quaternion and fractional-order transformations, which extract only the global features from color images, our proposed QFr-GLMs can extract both the global and local color features. This paper also derives a new set of invariant color-image descriptors by QFr-GLMs, enabling geometric-invariant pattern recognition in color images. Finally, the performances of our proposed QFr-GLMs and moment invariants were evaluated in simulation experiments of correlated color images. Both theoretical analysis and experimental results demonstrate the value of the proposed QFr-GLMs and their geometric invariants in the representation and recognition of color images.

受四元数代数和分数阶变换的思想的启发，我们提出了基于分数阶广义Laguerre多项式的一组新的四元数分数阶广义Laguerre正交矩（QFr-GLM）。首先，提出的QFr-GLM直接在笛卡尔坐标空间中构建，从而避免了在笛卡尔坐标和极坐标之间进行转换的需求。因此，它们比极坐标中构造的圆形正交矩更好。此外，与基于四元数和分数阶变换的最新Zernike矩不同，后者仅从彩色图像中提取全局特征，而我们提出的QFr-GLM可以提取全局和局部颜色特征。本文还通过QFr-GLM推导了一组新的不变彩色图像描述符，可以在彩色图像中识别几何不变的图案。最后，在相关彩色图像的仿真实验中评估了我们提出的QFr-GLM和不变矩的性能。理论分析和实验结果均证明了所提出的QFr-GLM及其几何不变式在彩色图像表示和识别中的价值。