图像频域变换及应用

曾经终败给现在 2022-08-18 15:17 170阅读 0赞

图像频域变换及应用

# 1.名词解释： #

## （1）空间域：英文：spatial domain。释文：又称图像空间(imagespace)。由图像像元组成的空间。在图像空间中以长度(距离)为自变量直接对像元值进行处理称为空间域处理。 ##

##  （2）频率域：英文：spatial frequency domain。释文：以空间频率(即波数)为自变量描述图像的特征,可以将一幅图像像元值在空间上的变化分解为具有不同振幅、空间频率和相位的简振函数的线性叠加,图像中各种空间频率成分的组成和分布称为空间频谱。这种对图像的空间频率特征进行分解、处理和分析称为空间频率域处理或波数域处理。  ##

##  （3）二者关系： ##

空间域与频率域可互相转换。在空间频率域中可以引用已经很成熟的频率域技术,处理的一般步骤为：

①对图像施行二维离散傅立叶变换或小波变换,将图像由图像空间转换到频域空间。

②在频率域中对图像的频谱作分析处理,以改变图像的频率特征。即设计不同的数字滤波器,对图像的频谱进行滤波。频率域处理主要用于与图像空间频率有关的处理中。如图像恢复、图像重建、辐射变换、边缘增强、图像锐化、图像平滑、噪声压制、频谱分析、纹理分析等处理和分析中。须注意,空间频率(波数)的单位为米 \-l或(毫米)-1等。

# 2.应用： #

#  （1）对test 目录下的图像：  查看不同图像的傅立叶变换的图像；    查看不同图像的DCT（离散余弦）变换；  （2）对变换后的图像使用空间域图像增强的方法增强效果；  比较5.bmp和10.bmp、6.bmp和9.bmp频率域图像的不同；   （3）使用低通滤波器，查看结果；   （4）使用高通滤波器，查看结果。  3.常用函数：  （1）频率域处理函数：  ① 傅里叶变换fft2()；   ② 傅里叶逆变换ifft2()；  ③ 离散余弦变换dct2()；  ④ 频域滤波函数：dftfilt(f,H)。 #

##  （2）空间域图像增强 ##

① 线性变换 transfor()；

② 指数变换 exp()；

③ 对数变换 log()；

④ 幂次变换 power()；

⑤ 查看直方图 imhist()；

⑥ 使用直方图均衡Histeq()；

⑦ 使用平滑滤波器（imfilter实现函数滤波）；

⑧ 使用锐化滤波器。

## （3）傅里叶变换过程： ##

F = fft2(f)
         S = abs(F); 傅里叶谱
     Fc = fftshift(F); 象限变换
     S2 = log(1+abs(Fc))； 对数变换
     Im2uint8(mat2gray(g))

##  ##

# 4.matlab源码： #

全部文件截图如下：

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(1) adjusttoday.m

function adjusttoday(strFileName);
    
    I = imread(strFileName);
    J = imadjust(I,[0.3 0.7],[0.1 0.9],[]);
    
    J = uint8(I);
    
    imshow(J);

(2) dcttoday.m

function dcttoday(strFileName);
    
    RGB = imread(strFileName);
    I = rgb2gray(RGB);
    J = dct2(I);
    figure, imshow(log(abs(J)), []);
    
    colormap(jet(64));
    colorbar;

(3) dftfilt.m

function g = dftfilt(f, H)
    %DFTFILT Performs frequency domain filtering.
    %   G = DFTFILT(F, H) filters F in the frequency domain using the
    %   filter transfer function H. The output, G, is the filtered
    %   image, which has the same size as F.  DFTFILT automatically pads
    %   F to be the same size as H.  Function PADDEDSIZE can be used to
    %   determine an appropriate size for H.
    %
    %   DFTFILT assumes that F is real and that H is a real, uncentered
    %   circularly-symmetric filter function. 
    
    %   Copyright 2002-2004 R. C. Gonzalez, R. E. Woods, & S. L. Eddins
    %   Digital Image Processing Using MATLAB, Prentice-Hall, 2004
    %   $Revision: 1.5 $  $Date: 2003/08/25 14:28:22 $
    
    % Obtain the FFT of the padded input.
    F = fft2(f, size(H, 1), size(H, 2));
    
    % Perform filtering. 
    g = real(ifft2(H.*F));
    
    % Crop to original size.
    g = g(1:size(f, 1), 1:size(f, 2));

(4) FFTShow.m

function FFTShow(strFilename);
    
    I=imread(strFilename);
    I=rgb2gray(I);
    figure,imshow(I);
    
    str=strcat(strFilename, 'FFT');
    F = fft2(I);
     F2 = log(abs(F));
    figure,imshow(F,[-1 5],'notruesize'),Title(str); colormap(jet); colorbar
    
    D=dct2(I);
     D2 = log(abs(D));
    figure,imshow(D2,[-1 5],'notruesize');  %colormap(jet); colorbar

(5) ffttoday.m

function ffttoday(strFileName);
    
    I = imread(strFileName);
    F = fft2(I);
    S = abs(F);
    Fc = fftshift(F);
    S2 = log(1+abs(Fc));
    outcome = Im2uint8(mat2gray(S2));
    
    subplot(221); imshow(I);
    subplot(222); imshow(F);
    subplot(223); imshow(S);
    subplot(224); imshow(Fc);
    
    figure, imshow(S2);
    figure, imshow(outcome);

(6) Freq.m

function FFTShow(strFilename);
    
    I=imread(strFilename);
    %   I=rgb2gray(I);
    figure,imshow(I);
    
    F = fft2(I);
     F2 = log(abs(F));
    figure,imshow(F2,[-1 5],'notruesize'),Title(strFilename+ 'FFT');% colormap(jet); colorbar
    
    D=dct2(I);
    % D2 = log(abs(D));
    figure,imshow(D,[-1 5],'notruesize'); % colormap(jet); colorbar

(7) hpfilter.m

function H = hpfilter(type, M, N, D0, n)
    %HPFILTER Computes frequency domain highpass filters.
    %   H = HPFILTER(TYPE, M, N, D0, n) creates the transfer function of
    %   a highpass filter, H, of the specified TYPE and size (M-by-N).
    %   Valid values for TYPE, D0, and n are: 
    %
    %   'ideal'    Ideal highpass filter with cutoff frequency D0.  n
    %              need not be supplied. D0 must be positive.
    %
    %   'btw'      Butterworth highpass filter of order n, and cutoff
    %              D0.  The default value for n is 1.0. D0 must be
    %              positive.
    %
    %   'gaussian' Gaussian highpass filter with cutoff (standard
    %              deviation) D0. n need not be supplied. D0 must be
    %              positive.
    
    %   Copyright 2002-2004 R. C. Gonzalez, R. E. Woods, & S. L. Eddins
    %   Digital Image Processing Using MATLAB, Prentice-Hall, 2004
    %   $Revision: 1.4 $  $Date: 2003/08/25 14:28:22 $
    
    % The transfer function Hhp of a highpass filter is 1 - Hlp, 
    % where Hlp is the transfer function of the corresponding lowpass 
    % filter.  Thus, we can use function lpfilter to generate highpass 
    % filters.
    
    if nargin == 4
       n = 1; % Default value of n.
    end
    
    % Generate highpass filter.
    Hlp = lpfilter(type, M, N, D0, n);
    H = 1 - Hlp;

(8) lpfilter.m

function H = lpfilter(type, M, N, D0, n) 
    %LPFILTER Computes frequency domain lowpass filters. 
    %   H = LPFILTER(TYPE, M, N, D0, n) creates the transfer function of 
    %   a lowpass filter, H, of the specified TYPE and size (M-by-N). To 
    %   view the filter as an image or mesh plot, it should be centered 
    %   using H = fftshift(H).  
    % 
    %   Valid values for TYPE, D0, and n are: 
    % 
    %   'ideal'    Ideal lowpass filter with cutoff frequency D0. n need 
    %              not be supplied.  D0 must be positive. 
    % 
    %   'btw'      Butterworth lowpass filter of order n, and cutoff 
    %              D0.  The default value for n is 1.0.  D0 must be 
    %              positive. 
    % 
    %   'gaussian' Gaussian lowpass filter with cutoff (standard 
    %              deviation) D0.  n need not be supplied.  D0 must be 
    %              positive.  
     
    %   Copyright 2002-2004 R. C. Gonzalez, R. E. Woods, & S. L. Eddins 
    %   Digital Image Processing Using MATLAB, Prentice-Hall, 2004 
    %   $Revision: 1.7 $  $Date: 2003/10/13 00:46:25 $ 
     
    % Use function dftuv to set up the meshgrid arrays needed for 
    % computing the required distances.  
    %[U, V] = dftuv(M, N); 
    U = M;V= N; 
    % Compute the distances D(U, V). 
    D = sqrt(U.^2 + V.^2); 
     
    % Begin filter computations. 
    switch type 
    case 'ideal' 
       H = double(D = D0); 
    case 'btw' 
       if nargin == 4 
          n = 1;	 
       end 
       H = 1./(1 + (D./D0).^(2*n)); 
    case 'gaussian' 
       H = exp(-(D.^2)./(2*(D0^2))); 
    otherwise 
       error('Unknown filter type.') 
    end

(9) paddedsize.m

function PQ = paddedsize(AB, CD, PARAM)
    %PADDEDSIZE Computes padded sizes useful for FFT-based filtering.
    %   PQ = PADDEDSIZE(AB), where AB is a two-element size vector,
    %   computes the two-element size vector PQ = 2*AB.
    %
    %   PQ = PADDEDSIZE(AB, 'PWR2') computes the vector PQ such that
    %   PQ(1) = PQ(2) = 2^nextpow2(2*m), where m is MAX(AB).
    %
    %   PQ = PADDEDSIZE(AB, CD), where AB and CD are two-element size
    %   vectors, computes the two-element size vector PQ.  The elements
    %   of PQ are the smallest even integers greater than or equal to 
    %   AB + CD -1.
    %   
    %   PQ = PADDEDSIZE(AB, CD, 'PWR2') computes the vector PQ such that
    %   PQ(1) = PQ(2) = 2^nextpow2(2*m), where m is MAX([AB CD]).
    
    if nargin == 1
       PQ = 2*AB;
    elseif nargin == 2 & ~ischar(CD)
       PQ = AB + CD - 1;
       PQ = 2 * ceil(PQ / 2);
    elseif nargin == 2
       m = max(AB); % Maximum dimension.
    
       % Find power-of-2 at least twice m.
       P = 2^nextpow2(2*m);
       PQ = [P, P];
    elseif nargin == 3
       m = max([AB CD]); %Maximum dimension.
       P = 2^nextpow2(2*m);
       PQ = [P, P];
    else
       error('Wrong number of inputs.')
    end

(10) 测试用的图片：test

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![Center 2][]

[Center]: /images/20220731/36b3aa0f8b98421483f9021ed9fdc0a1.png
[Center 1]: /images/20220731/f374645bc1fa4710b500a0eb7b838cd3.png
[Center 2]: /images/20220731/05c2df87049e4020af6092a1bec03104.png