Exponential moments (EMs) are important radial orthogonal moments, which have good image description ability and have less information redundancy compared with other orthogonal moments. Therefore, it has been used in various fields of image processing in recent years. However, EMs can only take integer order, which limits their reconstruction and antinoising attack performances. The promotion of fractional-order exponential moments (FrEMs) effectively alleviates the numerical instability problem of EMs; however, the numerical integration errors generated by the traditional calculation methods of FrEMs still affect the accuracy of FrEMs. Therefore, the Gaussian numerical integration (GNI) is used in this paper to propose an accurate calculation method of FrEMs, which effectively alleviates the numerical integration error. Extensive experiments are carried out in this paper to prove that the GNI method can significantly improve the performance of FrEMs in many aspects.

中文翻译：

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分数阶指数矩的精确计算

指数矩（EMs）是重要的径向正交矩，与其他正交矩相比，它们具有良好的图像描述能力和较少的信息冗余。因此，近年来，它已被用于图像处理的各个领域。但是，EM只能采用整数顺序，这限制了它们的重建和抗噪攻击性能。分数阶指数矩（FrEMs）的推广有效地缓解了EMs的数值不稳定性问题。但是，传统的FrEMs计算方法所产生的数值积分误差仍然会影响FrEMs的精度。因此，本文采用高斯数值积分（GNI）提出了一种精确的FrEMs计算方法，有效地减轻了数值积分误差。