A basic study on signal processing and restoration using Clifford algebra

数字信号处理是与酉矩阵相关的矩阵理论。本研究的目的是探讨将现有的许多信号处理方法所基于的复数域扩展到Clifford代数、四元数和超复数域的可能性。我们从对称群的角度研究了精确离散傅立叶变换（DFT），发现了二维DFT与置换之间的关系。从这个角度出发，我们研究了将结果推广到具有2阶以上本质对称的一般酉矩阵。将DFT推广到四元数和Clifford代数是通过用四元数的三个单位向量代替虚单位来实现的，但这种推广本质上与2阶置换有关。因此，沿着这个思想的推广不能应用于具有大于2阶对称性的一般酉矩阵。在本研究中，我们从基本和理论上探讨了将复数推广到四元数和Clifford代数中特征值为有限阶的酉矩阵的问题。我们发现了理论延伸的可能性。当酉矩阵对一维信号具有e阶对称性时，将这些酉矩阵应用于二维信号，我们就有了e^2阶对称性。但在不降低e^2阶对称性的情况下，利用Clifford代数Cl_n的e^2体积元，我们可能无法将酉矩阵嵌入到c4矩阵中，从而保持e^2阶的对称性。利用元素互为不相交双向量的子集B和交换子代数R_<2n, 0 >^+，得到了e^2阶对称性的约简，但这种嵌入到Cl_n，-矩阵中仍然保持了大约一半的e^2对称性。当二维信号的对称性大于2阶时，我们可以用Kronecker积来研究cl_n矩阵的对称性，但对于大于三维的信号，我们必须用张量积来将酉矩阵嵌入到c4矩阵中。这个问题仍然悬而未决。我们将研究这些问题。少

Digital signal processing is the theory of matrices related to unitary matrices. The goal of this research is a basic study to investigate the possibility for extending the field of complex numbers on which the existing many methods of signal processings are based to the Clifford algebra, the fields of quaternion and hyper complex numbers. We investigated precisely discrete Fourier Transform (DFT) from the point of view for symmetry groups, we found a relation between 2-dimesional DFT and permutations. From the point of view we investigated to extend the results to general unitary matrices which have intrinsically symmetries with more than 2 order. The extension of DFT to quaternions and Clifford algebra is done by replaced the imaginary units by three units vectors of quaternions, but this extension is intrinsically related to a permutation with order 2. Therefore, the extension along this idea cannot be applied to general unitary matrices which have symmetries with more than 2 order. … More In this research, we investigated basically and theoretically to extend complex numbers to quaternions and Clifford algebra for unitary matrices of which eigenvalues are of finite orders., and we found the possibility of theoretical extension. When the unitary matrices have the symmetry of e order for one-dimensional signals, applying these unitary matrices to two-dimensional signals, we have a symmetry of e^2 order. But without reducing the symmetry of e^2 order, using e^2 volume elements of Clifford algebra Cl_n, it is indicated that we may not be able to embed unitary matrices into C4-matrices so that we can keep the symmetry of e^2 order. Using a subset B of which elements are disjoint bivectors each other, and the commutative subalgebra R_<2n,o>^+, it yields the reduction of the symmetry of e^2 order, but this embedding to Cl_n,-matrices still keeps about a half of e^2 symmetries. When the symmetries of two-dimensional signals are of more than 2 order, we can investigate the symmetries by Kronecker product for Cl_n-matrices, and but, for more than three-dimensional signals, we have to use tensor product for embedding unitary matrices to C4-matrices. This problem is still open. We shall study the problems. Less