In this paper, we focus on the derivation of blur moment invariants. Blur moment invariants are image moment-based features, which preserve their values when the image is convolved by a point-spread function (PSF). Suppose a PSF has $N$-fold rotational symmetry, we prove its geometric moments of the same order are linearly dependent. Depending on this property, a new approach is proposed to determine whether an existing similarity or affine moment invariant also has invariance to $N$-fold symmetric blur. Unlike earlier work, this method is not based on complicated operators and construction formulas. We use it to analyse classical moment-based features, and surprisingly find that five of Hu moment invariants are naturally invariant to $N$-fold symmetric blur. Meanwhile, we first prove the existence of moment invariants to both affine transform and $N$-fold symmetric blur. The experiments using synthetic and real blur image datasets are carried out to test these expectations. And the results show that five Hu moment invariants outperform some widely used blur moment invariants and non-moment image features in image retrieval, classification and template matching.

在本文中，我们专注于模糊矩不变量的推导。模糊矩不变式是基于图像矩的特征，当通过点扩展函数（PSF）对图像进行卷积时，它们将保留其值。假设PSF具有$ñ$倍旋转对称性，我们证明其相同阶数的几何矩是线性相关的。根据这一特性，提出了一种新的方法来确定现有的相似性或仿射矩不变性是否也具有不变性。$ñ$折对称模糊。与早期的工作不同，此方法不是基于复杂的运算符和构造公式。我们用它来分析经典的基于矩的特征，令人惊讶地发现，五个Hu矩不变量自然是$ñ$折对称模糊。同时，我们首先证明仿射变换和$ñ$折对称模糊。使用合成的和真实的模糊图像数据集进行实验以测试这些期望。结果表明，在图像检索，分类和模板匹配中，五个Hu矩不变量优于某些广泛使用的模糊矩不变量和非矩图像特征。