陪集和：小波构造中张量积的替代方法
Coset Sum: an alternative to the tensor product in wavelet construction
AI总结：
提出陪集和作为张量积的替代方法，用于从单变量细化掩模构造多变量细化掩模，保持双正交性、精度数并支持快速算法。
摘要：
在多分辨率分析框架下，可以从一对多变量双正交细化掩模获得多变量双正交小波系统。一些多变量细化掩模可以分解为低维细化掩模。张量积是从低维细化掩模构造可分解多变量细化掩模的流行方法。我们提出一种替代方法，称为陪集和，用于从单变量细化掩模构造多变量细化掩模。陪集和具有张量积的许多基本特征，使其在实践中具有吸引力：（1）它保持单变量细化掩模的双正交性，（2）它保持单变量细化掩模的精度数，（3）与其关联的小波系统具有用于计算和反演小波系数的快速算法。在某些情况下，陪集和甚至可以提供比张量积更快的算法。陪集和的这些特征表明，开发和实践替代张量积的方法来构造多变量小波系统是值得的。给出了一些使用二维图像的实验结果来说明我们的发现。
英文摘要：
A multivariate biorthogonal wavelet system can be obtained from a pair of multivariate biorthogonal refinement masks in Multiresolution Analysis setup. Some multivariate refinement masks may be decomposed into lower dimensional refinement masks. Tensor product is a popular way to construct a decomposable multivariate refinement mask from lower dimensional refinement masks. We present an alternative method, which we call coset sum, for constructing multivariate refinement masks from univariate refinement masks. The coset sum shares many essential features of the tensor product that make it attractive in practice: (1) it preserves the biorthogonality of univariate refinement masks, (2) it preserves the accuracy number of the univariate refinement mask, and (3) the wavelet system associated with it has fast algorithms for computing and inverting the wavelet coefficients. The coset sum can even provide a wavelet system with faster algorithms in certain cases than the tensor product. These features of the coset sum suggest that it is worthwhile to develop and practice alternative methods to the tensor product for constructing multivariate wavelet systems. Some experimental results using 2-D images are presented to illustrate our findings.
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