Abstract: In the calculation of scale invariants of geometric moment, image scale changes will bring errors and cause the fluctuation of scale invariants because of the discrete digital image. The error expansion phenomenon is found through analyzing the anomalistic fluctuation of geometric moment scale invariants with low resolution. It is an unreasonable fluctuation enlarged from the small quantification error. A theory explains the origin of error expansion, and the formula of error calculation is given. Combined with the actual data, it is concluded that the error expansion will cause large fluctuation of geometric moment scale invariants and bring severe interference to its invariance with low resolution. Aiming at the cause of error expansion, a new definition of geometric absolute moment and its invariants is given, which will deny the error expansion efficiently. It is proven that the scale invariants of geometric absolute moment perform better compared with traditional Hu’s invariants on low resolution through comparative experiments.

Key words: geometric moment, scale invariants, low resolution, fluctuation correction

摘要： 几何矩函数不变量的计算过程中，由于数字图像的离散化，图像本身尺度的缩放会带来计算误差，从而引起尺度不变量值的波动。通过分析较低分辨率下几何矩尺度不变量值的不规则波动，推断计算过程中常常会发生误差扩大现象，即离散化带来的较小误差被不合理地放大。从理论上分析了误差扩大现象产生的原因，并给出了具体的误差计算公式。结合几何矩函数的实际数据可知，在较低分辨率下，误差扩大将会引起不变量数值的大幅度波动，严重干扰其不变性；而高分辨率下，这种现象很不明显。之后，针对产生误差扩大的原因给出了一种几何绝对矩及其尺度不变量的定义，可以有效地克服误差扩大现象。对比的实验结果也表明，相对于传统的Hu不变矩，几何绝对矩不变量在低分辨率下可以保持更好的不变性。

关键词: 几何矩, 尺度不变量, 低分辨率, 波动修正