摘要本文是在信号处理、模式识别的实际应用研究中提出的基础性问题（代数问题）。即是从实际应用的需求出发，参照矩阵的运算和性质，着重于矩体乘法的定义和算法，逐渐展开对其性质的讨论和后续概念的发散，以期得到应用的实现。

首先定义矩体乘法运算法则。在给出矩体的可部分内积化条件等前置概念后，排除可能产生多义性的要素，最终给出矩体乘法运算法则的定义，并用映射理论进行论证。

随后给出矩体乘法的算法。基于矩体乘法运算法则，给出矩体乘法的两种算法。其中，在矩体乘法算法二中，提出矩体二维化和进化的概念，将复杂、多维度、难以直观想象的矩体退化成容易理解的结构，为后文的诸多论述提供了理论依据。60092

再介绍一些特殊矩体。根据已定义的矩体乘法，定义结构对称矩体等特殊矩体，对矩体乘法的后续概念展开研究和分析。重点定义了矩体的转置及其性质。

最后矩体理论之于视频图像处理的实际应用。用Matlab实验的方式验证了矩体的SVD算法在视频图像处理中的可行性，并与其他算法进行了去噪性能的对比，证明其确具有一定优势。

本文的创新之处即在于给出了矩体乘法新的运算法则，并对之进行了算法的客观描述，其正确性与应用性都还有待深入验证。

关键词  矩体，乘法，算法，特殊矩体，图像处理

毕业设计说明书（论文）外文摘要

Title    Operations and Application of Cuboid            

Abstract

This paper focuses on a basic problem (algebra problem) proposed in practical application of signal processing and pattern recognition, which focuses on defining the multiplication of cuboid and its algorithm, and gradually carries on discussion of its properties and subsequent definitions so as to realize application.

Firstly, a new cuboid multiplication is introduced. After proposing some prepositive definitions such as the condition of partial inner production and eliminating the ambiguity, we present the definition of cuboid multiplication and prove it with mapping theory.

Then, we put forward two algorithms of cuboid multiplication. In the second algorithm, the bidimensionality and evolution of cuboid are raised, which simplify the complicated multi-dimensional cuboid into an understandable structure to offer theoretical bases for the following discussion. 

Afterwards, according to the defined multiplication, we introduce some special cuboids like structurally symmetrical cuboid, and analyse some subsequent definitions of cuboid multiplication with emphasis on the transposition of cuboid and its properties.

Finally, practical application, especially the video and image processing based on cuboid theory is presented. We test the feasibility of the cuboid SVD algorithm by using Matlab software and compared its denoising performance with other methods to testify its advantage.

Keywords cuboid multiplication,algorithm,special cuboid,image processing

1  绪论 1

1．1  矩体课题概述 1

1．2  矩体的基本概念 3

1．3  矩体的简单运算 5

1．4  论文完成的主要工作及创新性成果 6

2  矩体的乘法运算 8

2．1  可部分内积化条件 8

2．2  矩体乘法法则 10

2．3  矩体乘法性质初探