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Pushing ahead, East, North, West development, facilitating shanty, complete xincheng West Road, stability and reconstruction of the street, Star Ding garden built century Garden District, subject public housing construction, heating the gas expansion accelerated implementation, Jing Rui River and bridge style line solid shaft, further expanded the town development. Beautiful country adhere to the advance and typical for a group to lead. According to \ogical lay based, and industry increased income, and live upgrade civilization, and culture rich connotation\caught built beautiful village, and whole village advance, and easily relocation, and \built\ct near 30 a; depth into ecological, and industry, and culture, and tourism, elements, high standards caught party original city Liu, and jade are Xia AO, model village, in province City set has benchmark. Four is equal benefits to promote sharing, coordinated development of various social undertakings. In the case of rigid spending slowing revenue growth, increase, financial 75% to the people's livelihood, the masses have more access in the sharing. Cuoban something good and benefit. Adhere to the affordable, best, carefully cuoban village oil, water and heat, cultural, city and County columns practical, developments benefit families, bring more welfare to the masses. Development of social convenience. Acceptance of balanced development of compulsory education through the national assessment, the County vocational education center was named \onal education in China\batch of national reform and development model through provincial ... Five is to deepen reform and innovation to break bottlenecks and endogenous development of effective fire. As an engine of dynamism, adding reform key to adopting reform measures stress energy and break bottlenecks and promoting development. Pay close attention to the key areas of reform. Conscientiously implement the provincial deployment and cancel, adjustment, to undertake the administrative approval items according to Regulation 347 to further improve administrative efficiency continuing deepening reform for poverty alleviation, speeding up rural land right registration certification pilot project, encouraging the orderly transfer of land management, utilize the land resources, in order to promote scale management laid the Foundation. Actively cultivate the market. Firmly establish a market-oriented, nurturing family farms, industrial users, professional cooperation organization new market management main body 456, effectively improve the industry, intensive and marketization of encouraging support social capital into the small micro-credit, urban development, culture and tourism, public services and other areas, easing development shortfall. To promote mass innovation. o support the rural sector to use new thinking and new ideas, creative work incentives such as actively promoting financial intermediation, bring added 159 in non public enterprises, individually-1540, created a傅里叶变换及其在数字图像处理中的应用

王家硕 学号：1252015

一、 Fourier变换

1. 一维连续傅里叶变换

设 f (x)为x的实变函数，如果f (x)满足下面的狄里赫莱条件： （1）具有有限个间隔点。 （2）具有有限个极点。 （3）绝对可积。

则 f (x)的傅里叶变换（Fourier Transformation，FT）定义为: Fourier正变换：F(?)? Fourier逆变换：f(t)?f式中：j?f[f(t)]??1?????f(t)e?j?tdt；

f(?)e?j?td?，

[F(t)]?12???????1，ω 为频域变量。

f (x)与F(w)构成傅里叶变换对，可以证明傅里叶变换对总是存在的。由于f (x)为实函数，则它的傅里叶变换F(w)通常是复函数，于是F(w)可写成

F(w) ??R(w) ??jI (w) (1)

式中：R(w)和I (w)分别是F(w)的实部和虚部。公式1可表示为指数形式：

式中：

F(w) 为f (x)的傅里叶幅度谱，f(w)为f (x)的相位谱。

2. 二维连续傅里叶变换

如果二维函数f (x, y)是连续可积的，即则二维连续傅里叶变换对可表示为：

??????????f(x,y)|dxdy??，且F(u, v)是可积的，

??F(u,v)?F(x,y)?????f(x,y)e?j?tdt

F(u,v)ej?tdt

??????????对于图像 f (x, y)，F(u, v)是它的频谱。变量u 是对应于x 轴的空间频率，变量v 是

对应于y轴的空间频率，与在一维的情况类似，可定义二维傅里叶变换的幅度谱和相位谱为：

Pushing ahead, East, North, West development, facilitating shanty, complete xincheng West Road, stability and reconstruction of the street, Star Ding garden built century Garden District, subject public housing construction, heating the gas expansion accelerated implementation, Jing Rui River and bridge style line solid shaft, further expanded the town development. Beautiful country adhere to the advance and typical for a group to lead. According to \ogical lay based, and industry increased income, and live upgrade civilization, and culture rich connotation\caught built beautiful village, and whole village advance, and easily relocation, and \built\ct near 30 a; depth into ecological, and industry, and culture, and tourism, elements, high standards caught party original city Liu, and jade are Xia AO, model village, in province City set has benchmark. Four is equal benefits to promote sharing, coordinated development of various social undertakings. In the case of rigid spending slowing revenue growth, increase, financial 75% to the people's livelihood, the masses have more access in the sharing. Cuoban something good and benefit. Adhere to the affordable, best, carefully cuoban village oil, water and heat, cultural, city and County columns practical, developments benefit families, bring more welfare to the masses. Development of social convenience. Acceptance of balanced development of compulsory education through the national assessment, the County vocational education center was named \onal education in China\batch of national reform and development model through provincial ... Five is to deepen reform and innovation to break bottlenecks and endogenous development of effective fire. As an engine of dynamism, adding reform key to adopting reform measures stress energy and break bottlenecks and promoting development. Pay close attention to the key areas of reform. Conscientiously implement the provincial deployment and cancel, adjustment, to undertake the administrative approval items according to Regulation 347 to further improve administrative efficiency continuing deepening reform for poverty alleviation, speeding up rural land right registration certification pilot project, encouraging the orderly transfer of land management, utilize the land resources, in order to promote scale management laid the Foundation. Actively cultivate the market. Firmly establish a market-oriented, nurturing family farms, industrial users, professional cooperation organization new market management main body 456, effectively improve the industry, intensive and marketization of encouraging support social capital into the small micro-credit, urban development, culture and tourism, public services and other areas, easing development shortfall. To promote mass innovation. o support the rural sector to use new thinking and new ideas, creative work incentives such as actively promoting financial intermediation, bring added 159 in non public enterprises, individually-1540, created a

3. 一维离散傅里叶变换

对一个连续函数f (x)等间隔采样可得到一个离散序列。设共采样N个，则这个离散序列 可表示为{ f (0), f (1), f (2),?, f (N ?1)}。则其离散傅里叶变换F(u)为：

离散傅里叶反变换（IDFT）为：

式中：x ??0, 1, 2,?, N ?1。 令

，则上述公式变成：

4. 二维离散傅里叶变换

二维离散傅里叶变换：

二、 傅里叶变换在图像处理中的应用

傅立叶变换在图像处理中有非常重要的作用，被广泛应用于图像增强与图像去噪、图像分割之边缘检测、图像特征提取（形状、纹理）、图像压缩等方面。

1. 基于傅里叶变换的图像增强

Pushing ahead, East, North, West development, facilitating shanty, complete xincheng West Road, stability and reconstruction of the street, Star Ding garden built century Garden District, subject public housing construction, heating the gas expansion accelerated implementation, Jing Rui River and bridge style line solid shaft, further expanded the town development. Beautiful country adhere to the advance and typical for a group to lead. According to \ogical lay based, and industry increased income, and live upgrade civilization, and culture rich connotation\caught built beautiful village, and whole village advance, and easily relocation, and \built\ct near 30 a; depth into ecological, and industry, and culture, and tourism, elements, high standards caught party original city Liu, and jade are Xia AO, model village, in province City set has benchmark. Four is equal benefits to promote sharing, coordinated development of various social undertakings. In the case of rigid spending slowing revenue growth, increase, financial 75% to the people's livelihood, the masses have more access in the sharing. Cuoban something good and benefit. Adhere to the affordable, best, carefully cuoban village oil, water and heat, cultural, city and County columns practical, developments benefit families, bring more welfare to the masses. Development of social convenience. Acceptance of balanced development of compulsory education through the national assessment, the County vocational education center was named \onal education in China\batch of national reform and development model through provincial ... Five is to deepen reform and innovation to break bottlenecks and endogenous development of effective fire. As an engine of dynamism, adding reform key to adopting reform measures stress energy and break bottlenecks and promoting development. Pay close attention to the key areas of reform. Conscientiously implementhe provincial deployment and cancel, adjustment, to undertake the administrative approval items according to Regulation 347 to further improve administrative efficiency continuing deepening reform for poverty alleviation, speeding up rural land right regstration certification pilot project, encouraging the orderly transfer of land management, utilize the land resources, in order to promote scale management laid the Foundation. Actively cultivate the market. Firmly establish a market-oriented, nurturing family farms, industrial users, professional cooperation organization new market management main body 456, effectively improve the industry, intensive and marketization of encouraging support social capital into the small micro-credit, urban development, culture and tourism, public services and other areas, easing development shortfall. To promote mass innovation. o support the rural sector to use new thinking and new ideas, creative work incentives such as actively promoting financial intermediation, bring added 159 in non public enterprises, individually-1540, created a在图像处理中，图像高频分量：图像突变部分；在某些情况下指图像边缘信息，某些情况下指噪声，更多是两者的混合；低频分量：图像变化平缓的部分，也就是图像轮廓信息。高通滤波器：让图像使低频分量抑制，高频分量通过。低通滤波器：与高通相反，让图像使高频分量抑制，低频分量通过。

图像频域滤波增强技术是在频率域空间对图像进行滤波，因此需要将图像从空间域通过傅里叶变换频率域，具体操作如下：

假定原图像f(x,y)，经傅里叶变换为F(u,v)，频率域增强就是选择合适的滤波器函数H(u,v)对F(u,v)的频谱成分进行调整，然后经逆傅里叶变换得到增强的图像g(x,y)。该过程可以通过下面的流程描述：

（１）对原始图像f(x,y)进行傅里叶变换得到F(u,v)； （２）将F(u,v)与传递函数H(u,v)进行卷积运算得到g(u,v)； （３）将g(u,v)进行傅里叶逆变换得到增强图像g(x,y)。

频域滤波的核心在于如何确定传递函数，即H(u,v)。常用的频率域低通滤波器H(u,v)有4种。理想低通滤波器、巴特沃斯（Butterworth）低通滤波器、指数低通滤波器、梯形低通滤波器。

2. 图像增强MatLab实现

clear all;

I1=imread('cameraman.tif'); %转载图像

fftI1=fft2(I1); %二维离散傅立叶变换

sfftI1=fftshift(fftI1); %直流分量移到频谱中心 RR1=real(sfftI1); %取傅立叶变换的实部 II1=imag(sfftI1); %取傅立叶变换的虚部 A1=sqrt(RR1.^2+II1.^2); %计算频谱幅值 A1=(A1-min(min(A1)))/(max(max(A1))-min(min(A1)))*225;%归一化 subplot(3,2,2);imshow(A1); %显示原图像的频谱 figure,imshow(I1,[]); %把图像显示出来 % I1=imnoise(I1,'salt & pepper'); % figure,imshow(I1);

snoise=0.1*randn(size(I1));