图像恢复及滤波处理

以你之姓@ 2022-08-18 15:19 310阅读 0赞

**图像恢复及滤波处理**

# **1.基本概念：** #

a.**图像恢复**是通过计算机处理，对质量下降的图像加以重建或恢复的处理过程。因摄像机与物体相对运动、系统误差、畸变、噪声等因素的影响，使图像往往不是真实景物的完善映像。在图像恢复中，需建立造成图像质量下降的退化模型，然后运用相反过程来恢复原来图像，并运用一定准则来判定是否得到图像的最佳恢复。在遥感图像处理中，为消除遥感图像的失真、畸变，恢复目标的反射波谱特性和正确的几何位置，通常需要对图像进行恢复处理，包括辐射校正、大气校正、条带噪声消除、几何校正等内容。

b.图像恢复采用的方法：

1）空间域滤波：均值滤波，顺序统计滤波；

2）频率域滤波：带通、带阻滤波，陷波滤波。

滤波器被广泛地用于图象的预处理，抑制图象噪声,增强对比度，以及强化图象的边沿特征。运用较为广泛的线性滤波器如平均值滤波器，能较好地抑制图象中的加性噪声。 但是，线性滤波器会引起图象的钝化或模糊，使得图象中物体边界产生位移。特别是，在图象受到乘性噪声或脉冲噪声的干扰，如超声波及雷达成像中普遍存在的斑点噪声，线性滤波器就不能取得预期的效果。中值滤波器，就像其名字一样，是用该像素的相邻像素的灰度中值来代替该像素的值，是一种非线性滤波器。

c.**椒盐噪声**：噪声脉冲可正可负，正脉冲是以白点（盐点）出现，负脉冲是以黑点（胡椒点）出现。

d.算术均值滤波器和几何均值滤波器（尤其是后者）更适合处理高斯或均匀等随机噪声。

谐波均值滤波器：对于“盐”噪声效果好，不适用“胡椒”噪声，善于处理高斯噪声那样的脉冲噪声。

缺点是：必须知道噪声是暗噪声还是亮噪声，以便选择合适的Q号。

逆谐波均值滤波器：Q为其阶数，当Q为正数，用于消除“胡椒”噪声；当Q为负数，用于消除“盐”噪声；

当Q=0退变为算术均值滤波器，当Q=-1时，退变为谐波均值滤波器。

e.顺序统计滤波器：

中值滤波器:处理脉冲型加性噪声，重复使用中值滤波器处理图像会使图像模糊，尽可能保持希望的处理次数。

最大值：除去图像中的“胡椒”噪声，但是会从黑色物体的边缘移走一些黑色像素。

最大值：除去图像中的“盐”噪声，但是会从白色物体的边缘移走一些白色像素。

修正的阿尔法均值滤波器：d值可以取0到mn-1之间的任意数，当d=0时，退变为算术均值滤波器,

当d=（mn-1）/2退变为中值滤波器，d取其他值时，在包含多种噪声的情况下非常适用。

f.巴特沃斯IIR滤波器的设计:

在MATLAB下，设计巴特沃斯IIR滤波器可使用butter函数。

Butter函数可设计低通、高通、带通和带阻的数字和模拟IIR滤波器，其特性为使通带内的幅度响应最大限度地平坦，但同时损失截止频率处的下降斜度。在期望通带平滑的情况下，可使用butter函数。

butter函数的用法为：\[b,a\]=butter(n,Wn,/ftype/)

其中n代表滤波器阶数，Wn代表滤波器的截止频率，这两个参数可使用buttord函数来确定。buttord函数可在给定滤波器性能的情况下，求出巴特沃斯滤波器的最小阶数n，同时给出对应的截止频率Wn。

buttord函数的用法为：\[n,Wn\]= buttord(Wp,Ws,Rp,Rs)

其中Wp和Ws分别是通带和阻带的拐角频率（截止频率），其取值范围为0至1之间。当其值为1时代表采样频率的一半。Rp和Rs分别是通带和阻带区的波纹系数。

g.不同类型（高通、低通、带通和带阻）滤波器对应的Wp和Ws值遵循以下规则：

(1) 高通滤波器：Wp和Ws为一元矢量且Wp>Ws；

(2) 低通滤波器：Wp和Ws为一元矢量且Wp<Ws；

(3) 带通滤波器：Wp和Ws为二元矢量且Wp<Ws，如Wp=\[0.2,0.7\],Ws=\[0.1,0.8\];

(4) 带阻滤波器：Wp和Ws为二元矢量且Wp>Ws，如Wp=\[0.1,0.8\],Ws=\[0.2,0.7\]。

h.契比雪夫I型IIR滤波器的设计

在期望通带下降斜率大的场合，应使用椭圆滤波器或契比雪夫滤波器。在MATLAB下可使用cheby1函数设计出契比雪夫I型IIR滤波器。

cheby1函数可设计低通、高通、带通和带阻契比雪夫I型滤IIR波器，其通带内为等波纹，阻带内为单调。契比雪夫I型的下降斜度比II型大，但其代价是通带内波纹较大。

cheby1函数的用法为：\[b,a\]=cheby1(n,Rp,Wn,/ftype/)

在使用cheby1函数设计IIR滤波器之前，可使用cheblord函数求出滤波器阶数n和截止频率Wn。cheblord函数可在给定滤波器性能的情况下，选择契比雪夫I型滤波器的最小阶和截止频率Wn。

cheblord函数的用法为：\[n,Wn\]=cheblord(Wp,Ws,Rp,Rs)

# 2.常用函数： #

Spfilt：空间域滤波；

Lpfilter：低通滤波；

Imnoise2：生成指定分布模型的噪声；

Imnoise3：生成周期噪声 。

# 3.应用： #

## （1）处理过程： ##

• 无padding过程

直接在频率域产生于图像大小一致的滤波器。

• 有padding过程

频域滤波函数乘以![Center][]的反变换，用0延拓，然后再进行傅里叶正变换。

a.无padding过程:

显示原图像：
    f=imread(windmill_noise.png'); 
    figure,subplot(2,2,1),imshow(f); 
    title(‘原图像');
    F=fft2(f);
    [M N]=size(f);
    f=double(f);
    fc=fftshift(F); 
    s=log(1+abs(fc)); 
    subplot(2,2,2),imshow(s,[]);
    title(‘原图像频谱');<span style="color:#cc66cc;">

理想带阻滤波器滤波效果：
    [u1,v1]=dftuv(M,N); 
    d1=sqrt(u1.^2+v1.^2); 
    w=5;
     D0=15;
    ba= d1<(D0-w/2) | d1>(D0+w/2);
    H=double(ba);
    g=real(ifft2(H.*fft2(f)));
    subplot(2,2,3),imshow(g,[]); 
    title(‘理想带阻');
    F=fft2(g); 
    fc=fftshift(F); 
    s=log(1+abs(fc));      
    subplot(2,2,4),imshow(s,[]); 
    title(理想带阻频谱');

巴特沃斯带阻滤波：
    n=3；
    H = 1./(1 + (w*d1./(d1.^2-D0^2)).^(2*n));
    g=real(ifft2(H.*fft2(f)));
    figure,subplot(2,2,1),imshow(g,[]);
     title(‘btr');
    F=fft2(g); 
    fc=fftshift(F); 
    s=log(1+abs(fc));      
    subplot(2,2,2),imshow(s,[]); 
     title(‘btr频谱图');

高斯带阻滤波：
    H = 1-exp(-1/2*(((d1.^2)-D0^2)./(d1*w)).^2);
    g=real(ifft2(H.*fft2(f)));
    subplot(2,2,3),imshow(g,[]); 
    title(‘高斯带阻');
    F=fft2(g); 
    fc=fftshift(F); 
    s=log(1+abs(fc));      
    subplot(2,2,4),imshow(s,[]); 
    title(‘高斯带阻频谱');

## （2）主要MATLAB源码： ##

① dftfilt.m

function g=dftfilt(f,H) 
    F=fft2(f,size(H,1),size(H,2)); 
    g=real(ifft2(H.*F)); 
    g=g(1:size(f,1),1:size(f,2));

② dftuv.m

function [u,v]=dftuv(m,n) 
    u=0:(m-1); 
    v=0:(n-1); 
    idx=find(u>m/2); 
    u(idx)=u(idx)-m; 
    idy=find(v>n/2); 
    v(idy)=v(idy)-n; 
    [v,u]=meshgrid(v,u);

③ pad1.m

④ pad2.m

%本文件包含paddedsize.m和dftuv.m及dftfilt.m三个文件 
    f=imread('D:\test\windmill_noise.png'); 
    figure,subplot(2,2,1),imshow(f); title('原图像');
    F=fft2(f);
    [M N]=size(f);
    f=double(f);
    fc=fftshift(F); 
    s=log(1+abs(fc)); 
    subplot(2,2,2),imshow(s,[]);  title('原图像频谱');                  %原图像频谱 
    %##########################
    % PQ=paddedsize(size(f));  
    % [u,v]=dftuv(PQ(1),PQ(2)); 
    [u1,v1]=dftuv(M,N); %没有padding
    d1=sqrt(u1.^2+v1.^2); 
    %d0=0.05*PQ(2); 
    %d0=0.1*N; 
    d0=15;
    % F=fft2(f,PQ(1),PQ(2)); 
    % H=1-(1-exp(-(u1.^2+v1.^2)/(2*(d0^2))));       %高斯高通滤波 
    % %g=dftfilt(f,H);
    % g=real(ifft2(H.*fft2(f)));
    % subplot(2,2,3),imshow(g,[]); title('高斯high通滤波');
    % F=fft2(g); 
    % fc=fftshift(F); 
    % s=log(1+abs(fc));
    % subplot(2,2,4),imshow(s,[]); title('高斯滤波后的图像频谱');      %高斯滤波后的图像频谱 
    % %###########################
    % d=sqrt(u.^2+v.^2); 
    % H=1-1./(1+(d1/d0).^2);                  %巴特沃兹高通滤波 
    % %g=dftfilt(f,H); 
    % g=real(ifft2(H.*fft2(f)));
    % figure,subplot(2,2,1),imshow(g,[]); title('巴特沃兹high通滤波');
    % F=fft2(g); 
    % fc=fftshift(F); 
    % s=log(1+abs(fc));      
    % subplot(2,2,2),imshow(s,[]); title('巴特沃兹high通滤波后的图像频谱');     %巴特沃兹低通滤波后的图像频谱 
    % %##########################
    % h=double(d1<=d0);                    %理想高通滤波               
    % %g=dftfilt(f,H);
    % g=real(ifft2(H.*fft2(f)));
    % H=1-double(d1<=d0); 
    % subplot(2,2,3),imshow(g,[]); title('理想high通滤波');
    % F=fft2(g); 
    % fc=fftshift(F); 
    % s=log(1+abs(fc)); 
    % subplot(2,2,4),imshow(s,[]); title('理想high通滤波后的图像频谱');        %理想低通滤波后的图像频谱
    %#####################
     w=5;%带宽                              理想带阻滤波
    % % D0=sqrt(209*209+298*298);
     D0=15;
    ba= d1<(D0-w/2) | d1>(D0+w/2);
    H=double(ba);
    %g=dftfilt(f,H); 
    g=real(ifft2(H.*fft2(f)));
    subplot(2,2,3),imshow(g,[]); title('理想带阻滤波');
    F=fft2(g); 
    fc=fftshift(F); 
    s=log(1+abs(fc));      
    subplot(2,2,4),imshow(s,[]); title('理想带阻滤波后的图像频谱');     %理想带阻滤波后的图像频谱 
    %###########################
    n=3;                                %3阶巴特沃斯滤波器
    H = 1./(1 + (w*d1./(d1.^2-D0^2)).^(2*n));
    %g=dftfilt(f,H); 
    g=real(ifft2(H.*fft2(f)));
    figure,subplot(2,2,1),imshow(g,[]); title('巴特沃斯带阻滤波');
    F=fft2(g); 
    fc=fftshift(F); 
    s=log(1+abs(fc));      
    subplot(2,2,2),imshow(s,[]);  title('巴特沃斯带阻滤波图像频谱');    %巴特沃斯带阻滤波后的图像频谱 
    %#########################
    H = 1-exp(-1/2*(((d1.^2)-D0^2)./(d1*w)).^2);%高斯带阻滤波器
    %g=dftfilt(f,H); 
    g=real(ifft2(H.*fft2(f)));
    subplot(2,2,3),imshow(g,[]); title('高斯带阻滤波器');
    F=fft2(g); 
    fc=fftshift(F); 
    s=log(1+abs(fc));      
    subplot(2,2,4),imshow(s,[]); title('高斯带阻滤波后的图像频谱');     %高斯带阻滤波后的图像频谱 
    
    
    % %###########################
    % v0=15;
    % u0=0;
    % d1=((u-M/2-u0).^2 + (v-N/2-v0).^2).^(1/2);
    % d2=((u-M/2-u0).^2 + (v-N/2+v0).^2).^(1/2);
    % H=double(d0<d1 & d0<d2);%理想陷波带阻滤波器
    % g=dftfilt(f,H); 
    % subplot(2,2,3),imshow(g,[]); 
    % F=fft2(g); 
    % fc=fftshift(F); 
    % s=log(1+abs(fc));      
    % subplot(2,2,4),imshow(s,[]);      %理想陷波带阻滤波后的图像频谱 
    % %H=1/(1+(D0^2/(d1.*d2)).^2);

⑤ pad3.m

⑥ 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]). 
        
    %   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 $
    
    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

以上代码的处理结果：

![Center 1][] ![Center 2][]

![Center 3][]

![Center 4][]

![Center 5][]

![Center 6][]

# 4.附录： #

可能用到的其他函数代码：

① 生成指定分布模型的噪声: imnoise2.m

function R = imnoise2(type, M, N, a, b)
    %IMNOISE2 Generates an array of random numbers with specified PDF.
    %   R = IMNOISE2(TYPE, M, N, A, B) generates an array, R, of size
    %   M-by-N, whose elements are random numbers of the specified TYPE
    %   with parameters A and B. If only TYPE is included in the
    %   input argument list, a  single random number of the specified
    %   TYPE and default parameters shown below is generated. If only
    %   TYPE, M, and N are provided, the default parameters shown below
    %   are used.  If M = N = 1, IMNOISE2 generates a single random
    %   number of the specified TYPE and parameters A and B.
    %
    %   Valid values for TYPE and parameters A and B are:
    % 
    %   'uniform'       Uniform random numbers in the interval (A, B).
    %                   The default values are (0, 1).  
    %   'gaussian'      Gaussian random numbers with mean A and standard
    %                   deviation B. The default values are A = 0, B = 1.
    %   'salt & pepper' Salt and pepper numbers of amplitude 0 with
    %                   probability Pa = A, and amplitude 1 with
    %                   probability Pb = B. The default values are Pa =
    %                   Pb = A = B = 0.05.  Note that the noise has
    %                   values 0 (with probability Pa = A) and 1 (with
    %                   probability Pb = B), so scaling is necessary if
    %                   values other than 0 and 1 are required. The noise
    %                   matrix R is assigned three values. If R(x, y) =
    %                   0, the noise at (x, y) is pepper (black).  If
    %                   R(x, y) = 1, the noise at (x, y) is salt
    %                   (white). If R(x, y) = 0.5, there is no noise
    %                   assigned to coordinates (x, y). 
    %   'lognormal'     Lognormal numbers with offset A and shape
    %                   parameter B. The defaults are A = 1 and B =
    %                   0.25.
    %   'rayleigh'      Rayleigh noise with parameters A and B. The
    %                   default values are A = 0 and B = 1. 
    %   'exponential'   Exponential random numbers with parameter A.  The
    %                   default is A = 1.
    %   'erlang'        Erlang (gamma) random numbers with parameters A
    %                   and B.  B must be a positive integer. The
    %                   defaults are A = 2 and B = 5.  Erlang random
    %                   numbers are approximated as the sum of B
    %                   exponential random numbers.
    
    %   Copyright 2002-2004 R. C. Gonzalez, R. E. Woods, & S. L. Eddins
    %   Digital Image Processing Using MATLAB, Prentice-Hall, 2004
    %   $Revision: 1.5 $  $Date: 2003/10/12 23:37:29 $
    
    % Set default values.
    if nargin == 1
       a = 0; b = 1;
       M = 1; N = 1;
    elseif nargin == 3
       a = 0; b = 1;
    end
    
    % Begin processing. Use lower(type) to protect against input being
    % capitalized. 
    switch lower(type)
    case 'uniform'
       R = a + (b - a)*rand(M, N);
    case 'gaussian'
       R = a + b*randn(M, N);
    case 'salt & pepper'
       if nargin <= 3
          a = 0.05; b = 0.05;
       end
       % Check to make sure that Pa + Pb is not > 1.
       if (a + b) > 1
          error('The sum Pa + Pb must not exceed 1.')
       end
       R(1:M, 1:N) = 0.5;
       % Generate an M-by-N array of uniformly-distributed random numbers
       % in the range (0, 1). Then, Pa*(M*N) of them will have values <=
       % a. The coordinates of these points we call 0 (pepper
       % noise). Similarly, Pb*(M*N) points will have values in the range
       % > a & <= (a + b).  These we call 1 (salt noise). 
       X = rand(M, N);
       c = find(X <= a);
       R(c) = 0;
       u = a + b;
       c = find(X > a & X <= u);
       R(c) = 1;
    case 'lognormal'
       if nargin <= 3
          a = 1; b = 0.25;
       end
       R = a*exp(b*randn(M, N));
    case 'rayleigh'
       R = a + (-b*log(1 - rand(M, N))).^0.5;
    case 'exponential'
       if nargin <= 3
          a = 1;
       end
       if a <= 0
          error('Parameter a must be positive for exponential type.')
       end
       k = -1/a;
       R = k*log(1 - rand(M, N));
    case 'erlang'
       if nargin <= 3
          a = 2; b = 5;
       end
       if (b ~= round(b) | b <= 0)
          error('Param b must be a positive integer for Erlang.')
       end
       k = -1/a;
       R = zeros(M, N);
       for j = 1:b         
          R = R + k*log(1 - rand(M, N));
       end
    otherwise
       error('Unknown distribution type.')
    end<span style="color:#3333ff;">
    </span>

② 生成周期噪声:  imnoise3.m

function [r, R, S] = imnoise3(M, N, C, A, B)
    %IMNOISE3 Generates periodic noise.
    %    [r, R, S] = IMNOISE3(M, N, C, A, B), generates a spatial
    %    sinusoidal noise pattern, r, of size M-by-N, its Fourier
    %    transform, R, and spectrum, S.  The remaining parameters are:
    %
    %    C is a K-by-2 matrix with K pairs of freq. domain coordinates (u,
    %    v) that define the locations of impulses in the freq. domain. The
    %    locations are with respect to the frequency rectangle center at
    %    (floor(M/2) + 1, floor(N/2) + 1).  The impulse locations are spe-
    %    cified as increments with respect to the center. For ex, if M =
    %    N = 512, then the center is at (257, 257). To specify an impulse
    %    at (280, 300) we specify the pair (23, 43); i.e., 257 + 23 = 280,
    %    and 257 + 43 = 300. Only one pair of coordinates is required for
    %    each impulse.  The conjugate pairs are generated automatically. 
    %
    %    A is a 1-by-K vector that contains the amplitude of each of the
    %    K impulse pairs. If A is not included in the argument, the
    %    default used is A = ONES(1, K).  B is then automatically set to
    %    its default values (see next paragraph).  The value specified
    %    for A(j) is associated with the coordinates in C(j, 1:2). 
    %
    %    B is a K-by-2 matrix containing the Bx and By phase components
    %    for each impulse pair.  The default values for B are B(1:K, 1:2)
    %    = 0.
      
    %   Copyright 2002-2004 R. C. Gonzalez, R. E. Woods, & S. L. Eddins
    %   Digital Image Processing Using MATLAB, Prentice-Hall, 2004
    %   $Revision: 1.5 $  $Date: 2004/11/04 22:32:42 $
    
    % Process input parameters.
    [K, n] = size(C);
    if nargin == 3
       A(1:K) = 1.0;
       B(1:K, 1:2) = 0;
    elseif nargin == 4
       B(1:K, 1:2) = 0;
    end
    
    % Generate R.
    R = zeros(M, N);
    for j = 1:K
       u1 = M/2 + 1 + C(j, 1); v1 = N/2 + 1 + C(j, 2);
       R(u1, v1) = i * (A(j)/2) * exp(i*2*pi*C(j, 1) * B(j, 1)/M);
       % Complex conjugate.
       u2 = M/2 + 1 - C(j, 1); v2 = N/2 + 1 - C(j, 2);
       R(u2, v2) = -i * (A(j)/2) * exp(i*2*pi*C(j, 2) * B(j, 2)/N);
    end
    
    % Compute spectrum and spatial sinusoidal pattern.
    S = abs(R);
    r = real(ifft2(ifftshift(R)));

③ 低通滤波: 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: 1.8 $  $Date: 2004/11/04 22:33:16 $
    
    % Use function dftuv to set up the meshgrid arrays needed for
    % computing the required distances. 
    [U, V] = dftuv(M, 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

④ 带通滤波：bpfilter.m

function H=bpfilter(type, M, N, D0,n,W)
    if nargin == 4
       n = 1; % Default value of n.
    end
    
    % Generate highpass filter.
    Hlp = bsfilter(type, M, N, D0, n,W);
    H = 1 - Hlp;

⑤ 带阻滤波：bsfilter.m

function H=bsfilter(type, M, N, D0,n,W)
    [U, V] = dftuv(M, N);
    
    % Compute the distances D(U, V).
    D = sqrt(U.^2 + V.^2);
    
    % Begin filter computations.
    switch type
    case 'ideal'
       if D < (D0-W/2)|D > (D0+W/2)
           H=1;
       else
           H=0;
       end
    case 'btw'
       if nargin == 4
          n = 1;	
       end
       H = 1./(1 + (D./(D.^2-D0^2)).^(2*n));
    case 'gaussian'
       H = 1-exp-1/2*(((D.^2)-D0^2)./(D*W)).^2;
    otherwise
       error('Unknown filter type.')
    end

⑥ 高通滤波：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;

⑦ adpmedian.m

function f=adpmedian(g,smax)
    if(smax<=1)||(smax/2==round(smax/2))||(smax~=round(smax))
        error('smax must be an odd inger>1')
    end
    [M,N]=size(g);
    f=g;
    f(:)=0;
    alreadyprocessed=false(size(g));
    for k=3:2:smax
        zmin=ordfilt2(g,1,ones(k,k),'symmetric');
        zmax=ordfilt2(g,k*k,ones(k,k),'symmetric');
        zmed=medfilt2(g,[k,k],'symmetric');
        processusinglevelB=(zmed>zmin)&(zmax>zmed)&~alreadyprocessed;
        zB=(g>zmin)&(zmax>g);
        outputzxy=processusinglevelB &zB;
        outputzmed=processusinglevelB &~zB;
        f(outputzxy)=g(outputzxy);
        f(outputzmed)=g(outputzmed);
        alreadyprocessed=alreadyprocessed|processusinglevelB;
        if all( alreadyprocessed(:))
            break;
        end
    end

⑧ 改变图像的存储类：changeclass.m

function image = changeclass(class, varargin)
    %CHANGECLASS changes the storage class of an image.
    %  I2 = CHANGECLASS(CLASS, I);
    %  RGB2 = CHANGECLASS(CLASS, RGB);
    %  BW2 = CHANGECLASS(CLASS, BW);
    %  X2 = CHANGECLASS(CLASS, X, 'indexed');
    
    %  Copyright 1993-2002 The MathWorks, Inc.  Used with permission.
    %  $Revision: 1.2 $  $Date: 2003/02/19 22:09:58 $
    
    switch class
    case 'uint8'
       image = im2uint8(varargin{:});
    case 'uint16'
       image = im2uint16(varargin{:});
    case 'double'
       image = im2double(varargin{:});
    otherwise
       error('Unsupported IPT data class.');
    end

⑨ 空间域滤波: spfilt.m

function f = spfilt(g, type, m, n, parameter)
    %SPFILT Performs linear and nonlinear spatial filtering.
    %   F = SPFILT(G, TYPE, M, N, PARAMETER) performs spatial filtering
    %   of image G using a TYPE filter of size M-by-N. Valid calls to
    %   SPFILT are as follows: 
    %
    %     F = SPFILT(G, 'amean', M, N)       Arithmetic mean filtering.
    %     F = SPFILT(G, 'gmean', M, N)       Geometric mean filtering.
    %     F = SPFILT(G, 'hmean', M, N)       Harmonic mean filtering.
    %     F = SPFILT(G, 'chmean', M, N, Q)   Contraharmonic mean
    %                                        filtering of order Q. The
    %                                        default is Q = 1.5.
    %     F = SPFILT(G, 'median', M, N)      Median filtering.
    %     F = SPFILT(G, 'max', M, N)         Max filtering.
    %     F = SPFILT(G, 'min', M, N)         Min filtering.
    %     F = SPFILT(G, 'midpoint', M, N)    Midpoint filtering.
    %     F = SPFILT(G, 'atrimmed', M, N, D) Alpha-trimmed mean filtering.
    %                                        Parameter D must be a
    %                                        nonnegative even integer;
    %                                        its default value is D = 2.
    %
    %   The default values when only G and TYPE are input are M = N = 3,
    %   Q = 1.5, and D = 2. 
    
    %   Copyright 2002-2004 R. C. Gonzalez, R. E. Woods, & S. L. Eddins
    %   Digital Image Processing Using MATLAB, Prentice-Hall, 2004
    %   $Revision: 1.6 $  $Date: 2003/10/27 20:07:00 $
    
    % Process inputs.
    if nargin == 2
       m = 3; n = 3; Q = 1.5; d = 2;
    elseif nargin == 5
       Q = parameter; d = parameter;
    elseif nargin == 4
       Q = 1.5; d = 2;
    else 
       error('Wrong number of inputs.');
    end
    
    % Do the filtering.
    switch type
    case 'amean'
       w = fspecial('average', [m n]);
       f = imfilter(g, w, 'replicate');
    case 'gmean'
       f = gmean(g, m, n);
    case 'hmean'
       f = harmean(g, m, n);
    case 'chmean'
       f = charmean(g, m, n, Q);
    case 'median'
       f = medfilt2(g, [m n], 'symmetric');
    case 'max'
       f = ordfilt2(g, m*n, ones(m, n), 'symmetric');
    case 'min'
       f = ordfilt2(g, 1, ones(m, n), 'symmetric');
    case 'midpoint'
       f1 = ordfilt2(g, 1, ones(m, n), 'symmetric');
       f2 = ordfilt2(g, m*n, ones(m, n), 'symmetric');
       f = imlincomb(0.5, f1, 0.5, f2);
    case 'atrimmed'
       if (d <= 0) | (d/2 ~= round(d/2))
          error('d must be a positive, even integer.')
       end
       f = alphatrim(g, m, n, d);
    otherwise
       error('Unknown filter type.')
    end
    
    %-------------------------------------------------------------------%
    function f = gmean(g, m, n)
    %  Implements a geometric mean filter.
    inclass = class(g);
    g = im2double(g);
    % Disable log(0) warning.
    warning off;
    f = exp(imfilter(log(g), ones(m, n), 'replicate')).^(1 / m / n);
    warning on;
    f = changeclass(inclass, f);
    
    %-------------------------------------------------------------------%
    function f = harmean(g, m, n)
    %  Implements a harmonic mean filter.
    inclass = class(g);
    g = im2double(g);
    f = m * n ./ imfilter(1./(g + eps),ones(m, n), 'replicate');
    f = changeclass(inclass, f);
    
    %-------------------------------------------------------------------%
    function f = charmean(g, m, n, q)
    %  Implements a contraharmonic mean filter.
    inclass = class(g);
    g = im2double(g);
    f = imfilter(g.^(q+1), ones(m, n), 'replicate');
    f = f ./ (imfilter(g.^q, ones(m, n), 'replicate') + eps);
    f = changeclass(inclass, f);
    
    %-------------------------------------------------------------------%
    function f = alphatrim(g, m, n, d)
    %  Implements an alpha-trimmed mean filter.
    inclass = class(g);
    g = im2double(g);
    f = imfilter(g, ones(m, n), 'symmetric');
    for k = 1:d/2
       f = imsubtract(f, ordfilt2(g, k, ones(m, n), 'symmetric'));
    end
    for k = (m*n - (d/2) + 1):m*n
       f = imsubtract(f, ordfilt2(g, k, ones(m, n), 'symmetric'));
    end
    f = f / (m*n - d);
    f = changeclass(inclass, f);

⑩ 入口函数：paddedsize.m

function PQ=paddedsize(AB,CD,PARAM) 
    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); 
          P=2^nextpow2(2*m); 
          PQ=[P,P]; 
      elseif nargin==3 
          m=max([AB,CD]); 
          P=2^nextpow2(2*m); 
          PQ=[P,P]; 
      else 
          error('wrong number of inputs.') 
      end 
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
    function [u,v]=dftuv(m,n) 
    u=0:(m-1); 
    v=0:(n-1); 
    idx=find(u>m/2); 
    u(idx)=u(idx)-m; 
    idy=find(v>n/2); 
    v(idy)=v(idy)-n; 
    [v,u]=meshgrid(v,u); 
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
    function g=dftfilt(f,H) 
    F=fft2(f,size(H,1),size(H,2)); 
    g=real(ifft2(H.*F)); 
    g=g(1:size(f,1),1:size(f,2)); 
    %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% 
    %本文件包含paddedsize.m和dftuv.m及dftfilt.m三个文件 
    f=imread('D:\test\windmill_noise.png'); 
    figure,subplot(2,2,1),imshow(f); 
    F=fft2(f); 
    fc=fftshift(F); 
    s=log(1+abs(fc)); 
    subplot(2,2,2),imshow(s,[]);                    %原图像频谱 
    PQ=paddedsize(size(f));  
    [u,v]=dftuv(PQ(1),PQ(2)); 
    d0=0.05*PQ(2); 
    F=fft2(f,PQ(1),PQ(2)); 
    H=exp(-(u.^2+v.^2)/(2*(d0^2)));       %高斯低通滤波 
    g=dftfilt(f,H); 
    subplot(2,2,3),imshow(g,[]); 
    F=fft2(g); 
    fc=fftshift(F); 
    s=log(1+abs(fc)); 
    subplot(2,2,4),imshow(s,[]);       %高斯滤波后的图像频谱 
    d=sqrt(u.^2+v.^2); 
    H=1./(1+(d/d0).^2);                  %巴特沃兹低通滤波 
    g=dftfilt(f,H); 
    figure,subplot(2,2,1),imshow(g,[]); 
    F=fft2(g); 
    fc=fftshift(F); 
    s=log(1+abs(fc));      
    subplot(2,2,2),imshow(s,[]);      %巴特沃兹低通滤波后的图像频谱 
    h=double(d<=d0);                    %理想低通滤波               
    g=dftfilt(f,H); 
    H=double(d<=d0); 
    subplot(2,2,3),imshow(g,[]); 
    F=fft2(g); 
    fc=fftshift(F); 
    s=log(1+abs(fc)); 
    subplot(2,2,4),imshow(s,[]);         %理想低通滤波后的图像频谱

[Center]: /images/20220731/428fab6dc1aa4353802a8bc8b6b31097.png
[Center 1]: https://img-blog.csdn.net/20151226170820396?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQv/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/Center
[Center 2]: /images/20220731/1288d1feb7a240afa0175944bdfb77ae.png
[Center 3]: /images/20220731/e407de338a12426fa775d329793ede47.png
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[Center 5]: /images/20220731/0c505b138dea4f3db975b2fe4da7cc62.png
[Center 6]: /images/20220731/8d1113c022724e198cabeaf467d8f013.png