摘要：1873年由德国学者恩斯特·卡尔·阿贝（Ernst Karl Abbe，1840-1905）提出的二次成像理论及其相应的空间滤波实验，是空间滤波技术先导。随着后来对空间滤波技术理论的完善，使其广泛应用到科学技术的各个领域。65785

文章通过二维傅里叶分析将平面光场的复振幅分布分解为不同空间频率（即不同方向）的平面光的叠加，结合光的标量衍射理论得到透镜的傅里叶变换性质。接着提出阿贝成像和空间滤波技术，并对其进行二维傅里叶分析。最后对空间滤波进行MATLAB模拟加深对空间滤波的理解，并了解空间滤波在图像处理方面的应用。

阿贝成像及其空间滤波实验启示人们可以通过改变光波的频谱结构，使其像按照人们的要求得到预期的改善。在此基础上发展的光学信息处理技术，可用光学的方法对信息实施某种运算或变换以达到对感兴趣的信息进行提取、编码、存储、识别和恢复的目的。这种处理方法具有二维、并行和实时处理的优越性。

毕业论文关键字：二维傅里叶变换，标量衍射，透镜的傅里叶变换，空间滤波

Abstract: Principle of second image formation and its corresponding spatial filtering experiment, put forward by German scholar Ernst Karl Abbe in 1873 (Ernst Karl Abbe，1840-1905), was the forerunner of spatial filtering techniques. Subsequently, it is widely applied in all fields of science and technology with the development of spatial filtering technique theory.

    In this paper, complex amplitude distribution is decomposed into superposition of plane light in different spatial frequency (that is different direction) through Fourier analysis in two dimensions, so as to obtain Fourier transform properties of lens combing scalar diffraction theory of light. Then Abbe’s imagery and spatial filtering techniques are proposed, meanwhile, two-dimensional Fourier is analyzed. And finally, MATLAB simulation of spatial filtering is conducted in order to deepen the understanding of it, besides, the application of spatial filtering in the aspect of image processing is also described.

Abbe’s imagery and its spatial filtering experiment can enlighten us that the images can be improved as our expectation and requirement through changing the frequency-spectrum structure of light wave. Therefore, the information which is interested by us can be extracted, coded, stored, identified, and restored by utilizing certain operation and conversion with the method of optics, which is optical information processing technique developed on the basis of this results. Furthermore, this processing method owns the advantages of two-dimensional, parallel, and real-time processing.

Keywords: Two-Dimensional Fourier Transform, Scalar Diffraction Theory, Fourier Transform Properties of Lens, Spatial Filtering

目 录

1 绪论 5

1.1 课题背景 5

1.2 阿贝成像简介 6

1.3 本文的主要内容 6

2 二维傅里叶分析 7

2.1 平面波的空间分布 7

2.2 数学上的二维傅里叶变换 8

2.3 单色平面光场的二维傅里叶分析 8

3 光的标量衍射理论 9

3.1 菲涅尔——基尔霍夫衍射公式 9

3.2 菲涅尔衍射 10

3.3弗琅禾费衍射 11

4  透镜的傅里叶变换性质