从高阶自相关重建有限阿贝尔群上的有理函数，给出确定性算法与理论极限。
Reconstructing Rational Functions on Finite Abelian Groups with Higher Autocorrelations
- 提出显式重构算法，利用不超过3r+3阶自相关恢复函数
- 证明3r+2阶自相关不足以唯一确定函数，3r+3阶足够
- 适用于晶体学、视觉系统等需模式恢复的领域
有限阿贝尔群上整数值或有理数值函数的高阶自相关自然出现在X射线晶体学中，并在计算机视觉、相关断层成像、相关光谱学和模式识别中有应用。本文研究从高阶自相关重构有限阿贝尔群上的有理值函数问题。我们给出了一个显式的重构算法，并证明了阶数不超过3r+3的自相关始终足以在平移等价意义下唯一确定函数，其中r为群的秩。我们还构造了反例，说明在3r+2阶以下的自相关无法唯一确定某些有理函数。特别地，我们以群的秩为参数，给出了其正则表示分离度的精确上界。
原文摘要 · Abstract (English)
The higher-order autocorrelations of integer-valued or rational-valued functions on finite Abelian groups appear naturally in X-ray crystallography, and have applications in computer vision systems, correlation tomography, correlation spectroscopy, and pattern recognition. In this paper, we consider the problem of reconstructing a rational-valued function on finite Abelian groups from its higher-order autocorrelations. We describe an explicit reconstruction algorithm, and prove that the autocorrelations up to order $3r+3$ are always sufficient to determine the data up to translation, where $r$ is the rank of the group. We also provide examples of rational-valued functions on finite Abelian group which are not determined by their autocorrelations up to order $3r+2$. In particular, we provide a sharp upper bound on the separating degree of the regular representation of a finite Abelian group in terms of its rank.
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