论文标题
基质值信号和基质值晶格的基质值内部产品
A Matrix-Valued Inner Product for Matrix-Valued Signals and Matrix-Valued Lattices
论文作者
论文摘要
在构造基质值信号之前，提出了矩阵值的内部产物。与矩阵中所有组件的正交性相比，它引入了矩阵值信号较弱的正交性，该基质在正交多波伦构建体中通常使用。由于正交性较弱，构建正交基质值的小波会更容易。在本文中，我们从内部产品的角度重新研究了矩阵值的内部产品，该产品的观点更基本，并提出了一个新的但等效的矩阵值信号标准。我们表明，尽管它没有标量值，但它仍保持大多数标量值的内部产品属性。我们引入了一个新的线性独立性概念，用于矩阵值信号，并介绍一些相关的属性。然后，我们为一组线性独立的矩阵值信号提供了革兰氏 - schmidt正统计学过程。最终，我们定义了矩阵值的晶格。
A matrix-valued inner product was proposed before to construct orthonormal matrix-valued wavelets for matrix-valued signals. It introduces a weaker orthogonality for matrix-valued signals than the orthogonality of all components in a matrix that is commonly used in orthogonal multiwavelet constructions. With the weaker orthogonality, it is easier to construct orthonormal matrix-valued wavelets. In this paper, we re-study the matrix-valued inner product more from the inner product viewpoint that is more fundamental and propose a new but equivalent norm for matrix-valued signals. We show that although it is not scalar-valued, it maintains most of the scalar-valued inner product properties. We introduce a new linear independence concept for matrix-valued signals and present some related properties. We then present the Gram-Schmidt orthonormalization procedure for a set of linearly independent matrix-valued signals. Finally we define matrix-valued lattices.
