卷积及其应用+文献综述
关键词 卷积 性质 应用 图像滤波
毕业论文设计说明书外文摘要
Title Convolution and Its Application
Abstract
Convolution is a mathematical method of integral transformation which generates the third function of a mathematical operator by two functions f and g, it means the overlapping portions of the area flipping and panning by functions f and g, is an important operation in mathematics. Through data access, we studied the definition, basic properties, geometric meaning and the relevant equations or inequalities. It has been widely applied in Mathematics, Science, and Engineering replaced by ordinary function multiplication because of its good properties. For example, weighted moving average in Statistics; any linear system with the superposition principle in Physics; Image blur, Sharpening and Edge detection and so on in Image processing. It is also widely used in electronic engineering and signal processing. Through the study of convolution, we can understand and the close relationship between the convolution and the science and engineering, and further explore the convolution application in the field of science.
Keywords Convolution Property Application Image Filtering
目 次
1 引言 1
1.1 研究背景与意义1
1.2 国内外研究现状1
2 卷积基本知识 3
2.1 定义 3
2.2 性质 3
2.3 卷积定理5
2.4 卷积等式7
2.5 卷积不等式8
3 卷积的存在性10
4 卷积的计算 12
4.1 利用卷积的定义12
4.2 图解法13
4.3 利用卷积的特性计算14
4.4 利用傅里叶变换计算15
5 卷积与反卷积17
6 卷积应用19
6.1 概率密度19
6.2 图像处理20
6.3 数字信号处理20
6.4 卷积码22
6.5 地震中的应用23
结论 25
致谢 26
参考文献27
1 引言
1.1 研究背景与意义
卷积是在信号与线性系统的背景里诞生的，假若我们脱离这个背景基础来单独谈论卷积，则是没有任何意义的，除了在数学离散求和的情况下。
信号经过一个线性系统之后所发生的变化，称为信号与线性系统。这个所谓的系统，所带来的输入信号和输出信号的数学关系式之间的线性的运算关系，就是线性系统的含义[7]。
卷积不仅是分析数学中的一种重要的运算，在科学和工程方面也都有着广泛的应用。本文从卷积的定义及性质出发，研究了卷积定理，相关的等式或不等式，以及卷积在各个相关领域的应用。卷积是为了简化研究问题而人为定义的一种运算。数学和工程里很多地方都用到卷积，但是不同的地方卷积有着不同的意义。