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    框架正交性的证明和重构公式分析

    SIMPLE PROOF OF ORTHOGONAL FRAME AND ANALYSIS OF ITS SPECIAL EXAMPLES

    • 摘要: 首先分析了小波变换中“框架”的一般定义 ,介绍了一种物理意义直观的定义方式 .其次给出了正交框架充要性的一个简单但是严格的证明 ,其几何意义和物理意义非常明显 .然后 ,利用对偶基概念讨论了一般框架下信号的重构公式 ,并给出了对偶框架的具体计算公式 ,该公式不包含任何算符 ,可以明显看出基矢的相关性与框架类型之间的关系 .由于这些公式没有包含各种算符 ,对于实际的信号分析工作者而言具有重要的实用价值 ,同时对于深入理解框架也十分有益 .最后通过几个简单直观的实例指出框架分析中应该注意的一些问题

       

      Abstract: Frame is an important concept in wavelet transform. First of all, the concept of frame is analyzed, and a reasonable definition is introduced with definite physical meaning. Then a simple proof of sufficient and necessary criterion for orthogonal frame is presented. After that, by making use of dual basis, the general reconstruction formula about frame is discussed, and the computation formula for the dual frame is presented, in which no operators are contained. The fact that there are no operators in the formula of dual frame is of great practical importance for signal processing, and of great help in understanding the concept “frame”. Finally, some important problems about frame are pointed out by analyzing some simple examples.

       

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