【正文】 摘要 ： 1873 年由德國(guó)學(xué)者恩斯特卡爾阿貝（ Ernst Karl Abbe， 18401905）提出的二次成像理論及其相應(yīng)的空間濾波實(shí)驗(yàn)，是空間濾波技術(shù)先導(dǎo)。隨著后來(lái)對(duì)空間濾波技術(shù)理論的完善，使其廣泛應(yīng)用到科學(xué)技術(shù)的各個(gè)領(lǐng)域。 文章通過(guò)二維傅里葉分析將平面光場(chǎng)的復(fù)振幅分布分解為不同空間頻率（即不同方向）的平面光的疊加，結(jié)合光的標(biāo)量衍射理論得到透鏡的傅里葉變換性質(zhì)。接著提出阿貝成像和空間濾波技術(shù)，并對(duì)其進(jìn)行二維傅里葉分析。最后對(duì)空間濾波進(jìn)行 MATLAB 模擬加深對(duì)空間濾波的理解，并了解空間濾波在圖像處理方 面的應(yīng)用。 阿貝成像及其空間濾波實(shí)驗(yàn)啟示人們可以通過(guò)改變光波的頻譜結(jié)構(gòu)，使其像按照人們的要求得到預(yù)期的改善。在此基礎(chǔ)上發(fā)展的光學(xué)信息處理技術(shù)，可用光學(xué)的方法對(duì)信息實(shí)施某種運(yùn)算或變換以達(dá)到對(duì)感興趣的信息進(jìn)行提取、編碼、存儲(chǔ)、識(shí)別和恢復(fù)的目的。這種處理方法具有二維、并行和實(shí)時(shí)處理的優(yōu)越性。 關(guān)鍵字： 二維傅里葉變換，標(biāo)量衍射，透鏡的傅里葉變換，空間濾波 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， 18401905), 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, plex amplitude distribution is deposed 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 bing scalar diffraction theory of light. Then Abbe’s imagery and spatial filtering techniques are proposed, meanwhile, twodimensional 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 frequencyspectrum 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 twodimensional, parallel, and realtime processing. Keywords: TwoDimensional Fourier Transform, Scalar Diffraction Theory, Fourier Transform Properties of Lens, Spatial Filtering 目 錄 1 緒論 ..................................................... 6 課題背景 ............................................... 6 阿貝成像簡(jiǎn)介 ........................................... 6 本文的主要內(nèi)容 ......................................... 7 2 二維傅里葉分析 ........................................... 7 平面波的空間分布 ....................................... 7 數(shù)學(xué)上的二維傅里葉變換 ................................. 8 單色平面光場(chǎng)的二維傅里葉分析 ............................ 8 3 光的標(biāo)量衍射理論 ......................................... 9 菲涅爾 —— 基爾霍夫衍射公式 .............................. 9 菲涅爾衍射 ............................................. 11 ............................................ 12 4 透鏡的傅里葉變換性質(zhì) .................................... 12 透鏡的相位調(diào)制作用 .................................... 12 透鏡的傅里葉變換性質(zhì) ................................... 13 5 阿貝成像及空間濾波 ....................................... 15 ............................................ 15 空間濾波 ............................................... 16 6 空間濾波實(shí)驗(yàn)的 MATLAB 仿真 ................................ 21 MATLAB 模擬空間濾波實(shí)驗(yàn)的基本 步驟 ....................... 21 阿貝 —— 波特空間濾波實(shí)驗(yàn)的仿真 .......................... 22 圖像的空間濾波實(shí)驗(yàn) ..................................... 26 參考文獻(xiàn) ...................................................... 30 致謝 .......................................................... 31 1 緒論 課題背景 阿貝在蔡司公司從事顯微鏡的設(shè)計(jì)和研究時(shí)，用傳統(tǒng)的幾何光學(xué)計(jì)算方法對(duì)顯微鏡物鏡頭的像差進(jìn)行修正。當(dāng)時(shí)的光學(xué)設(shè)計(jì)家認(rèn)為設(shè)計(jì)優(yōu)良顯微鏡的關(guān)鍵在于減低像差和提高放大倍率，認(rèn)為顯微鏡的分辨率是無(wú)限的。減小光學(xué)鏡頭像差的一個(gè)簡(jiǎn)單辦法是用減小鏡頭的孔徑，于是蔡司公司出產(chǎn)了一批小孔徑的顯微鏡，想以此對(duì)像差進(jìn)行巨大的改進(jìn)，但結(jié)果反而不如以前生產(chǎn)的孔徑較大、未精心校正像差的顯微鏡。阿貝為了探索其中原因，從理論和試驗(yàn)兩個(gè)方面進(jìn)行研究，于 1873 年在德國(guó)的顯微鏡學(xué)報(bào)上發(fā)表了它的顯微