数字图像处理MATLAB学习笔记（二）

数字图像处理MATLAB学习笔记（二）

Filtering in the Frequency Domain频域处理

主要通过Matlab语言学习傅立叶变换下的频域处理和实践

1. The 2-D Discrete Fourier Transform(2-D DFT)二维离散傅里叶变换

1.1 DFT和IDFT

其中满足的条件：

• $x=0,1,2,\dots ,M-1$
• $y=0,1,2,\dots ,N-1$
• $u=0,1,2,\dots ,M-1$
• $v=0,1,2,\dots ,N-1$

令$f(x,y)$表示一副$M\times N$大小的图像，**二维离散傅里叶变换(2-D DFT)**记为$F(u,v)$，则有
$$F(u,v)=\sum _{x=0}^{M-1}\sum _{y=0}^{N-1}f(x,y)exp(-j2\pi (\frac{ux}{M}+\frac{vy}{N}))$$
由式得，$F(0,0)=mean(sum(f(x,y)))$

**离散傅立叶反变换(Inverse DFT=IDFT)**表达式为：
$$f(x,y)=\frac{1}{MN}\sum _{u=0}^{M-1}\sum _{v=0}^{N-1}F(u,v)exp(j2\pi (\frac{ux}{M}+\frac{vy}{N}))$$
因此，当得知$F(u,v)$时，我们可以通过计算IDFT的平均值得到$f(x,y)$。

1.2 DFT性质

尽管$f(x,y)$是实函数，DFT通常是一个复数变换。

先设两个函数：

• $R(u,v)$：the real components of $F(u,v)$；实部
• $I(u,v)$：the imaginary components of $F(u,v)$；虚部

**傅立叶频谱(Fourier spectrum)**定义为：
$$|F(u,v)|=[R^2(u,v)+I^2(u,v){]}^{\frac{1}{2}}$$
**变换相角(phase angle of transform)**定义为：
$$\varphi (u,v)=acrtan[\frac{I(u,v)}{R(u,v)}]$$
极坐标(polar form)下表示复数(complex function)$F(u,v)$为
$$F(u,v)=|F(u,v)|exp(j\varphi (u,v))$$
**功率谱(Power spectrum)可定义为幅度(magnitude)的平方：
$$P(u,v)=|F(u,v){|}^2=R^2(u,v)+I^2(u,v)$$
如果$f(x,y)$是real function，傅里叶变换是关于远点共轭对称的，即
$$F(u,v)=F^{\ast }(-u,-v)$$
即傅里叶谱频谱关于原点对称，即
$$|F(u,v)|=|F(-u,-v)|$$
同时满足下列equation（$k_1$和$k_2$均为Integer）：
$$F(u,v)=F(u+k_{1}M,v)+F(u,v+K_{2}N)=F(u+k_{1}M,v+k_{2}N)$$
也就是说，DFT在u、v方向上是无穷周期(infinitely periodic)**的，周期由M、N决定。DFT的周期性表达为
$$f(x,y)=f(x+k_{1}M,y)+f(x,y+K_{2}N)=f(x+k_{1}M,y+k_{2}N)$$
即，IDFT获取到的图像也是具有无穷周期的。所以，我们计算时只计算一个周期内的，即$M\times N$大小的图片。

2. Computing and Visualizing the 2-D DFT in MATLAB

DFT和IDFT实现：这里有一个fast Fourier transform(FFT) algorithm，在matlab中使用函数fft2。逆变换则使用函数ifft2。

F = fft2(f)

如果需要对图像进行0填充，则语法变为

F = fft2(f, P, Q)

输出F的大小变为PxQ.

Fourier spectrum实现：使用函数abs

S = abs(F)

abs会计算F中每一个元素的magnitude（real和imaginary components的平方和的平方根）

>> f = imread('Fig0303(a).tif');
>> imshow(f);xlabel('Image');
>> F = fft2(f);
>> subplot(1,2,1);
>> imshow(f);xlabel('Image');
>> subplot(1,2,2);
>> imshow(S, []);xlabel('Fourier spectrum');

周期性处理函数：fftshift。也就是将变换的原点移动到频域矩阵的中心，语法为

Fc = fftshift(F)

Fc就是中心变换后的结果

• 如果F是一个vector，Fc就是F中前一半与后一半向量的互换
• 如果F是一个Matrix，Fc就是F横竖分四份(奇数时前半部分向上取整)，对角互换的结果
• 如果F是一个multidimensional array，Fc就是F

这里还可以进行逆操作，函数为ifftshift

根据上一节总结的，$F(u,v)$的real和imaginary components使用如下函数表示

• Real：function real(F)
• Imaginary：function imag(F)

phase angle of transform实现，使用atan2函数，传入real components和imaginary components

phi = atan2(I, R) = atan2(imag(F),real(F))

或者使用angle函数，直接传入F，不用再转换为real和imaginary

phi = angle(F)

此时，得到DFT有：

F = S.*exp(i*phi)

Note：这里的i指的是复数，即 $i=0+1\times i$

3. Filtering in the Frequency Domain频域滤波

3.1 Fundamentals基本知识

空间和频域上的线性滤波都是基于Convolution操作的。可以用公式表达为
$$\begin{gathered} f(x,y)\odot h(x,y)\iff H(u,v)F(u,v) \\ f(x,y)h(x,y)\iff H(u,v)\odot F(u,v) \end{gathered}$$
频域滤波的main idea就是选择一个filter transfer function用特定的规则修改F(u,v)。

函数解释：

• h(x,y)：filtering mask卷积核
• H(u,v)：filter transfer function滤波传递函数。

在处理discrete quantities时，F和H都是周期循环的，所以离散频域上的卷积也是周期性的，即DFT的卷积操作也称为circular convolution循环卷积。为了得到相同大小结果，我们需要进行0-padding进行填充。

• 原因：原因比较复杂，这个看了两遍才看懂。简单来讲，就是因为在进行周期性卷积操作时，如果函数非零部分的延续周期很靠近，周期性卷积就会造成相邻函数的重叠。如果进行0-padding，则可以避开这些重叠的部分，因为值是0，对结果无影响。

解决办法：构造两个padded function，每个输出大小都是$P\times Q$。假设$f(x,y)$和$h(x,y)$的大小分别为$A\times B$和$C\times D$。

则，输出大小需满足的条件为：
$$\begin{gathered} P\ge A+C-1 \\ Q\ge B+D-1 \end{gathered}$$
通过函数实现：function paddedsize

function PQ = paddedsize(AB, CD, PARAM)
%PADDEDSIZE Computes padded size 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) && strcmpi(PARAM, 'pwr2')
    m = max([AB CD]); % Maximum dimension
    P = 2^nextpow2(2*m);
    PQ = [P, P];
else
    error('Wrong number of inputs.');
end

这里还要自定义两个函数：lpfilter和dfuv

• function lpfilter：得到指定类型的低通滤波器

• function dftuv：计算网格频率矩阵U和V

我们通过调用上述函数，获得对比图为：

>> f = imread('Fig0305(a).tif');
>> imshow(f);
>> [M, N] = size(f);
>> [f, revertClass] = tofloat(f);
>> F = fft2(f);
>> sig = 10;
>> H = lpfilter('gaussian', M, N, sig);
>> G = H.*F;
>> g = ifft2(G);
>> g = revertClass(g);
>> imshow(g);
>> PQ = paddedsize(size(f)); % f is floating point.
>> Fp = fft2(f, PQ(1), PQ(2)); % computing the FFT with padding
>> Hp = lpfilter('gaussian', PQ(1), PQ(2), 2*sig);
>> Gp = Hp.*Fp;
>> gp = ifft2(Gp);
>> gpc = gp(1:size(f,1), 1:size(f,2));
>> gpc = revertClass(gpc);
>> imshow(gpc);
>> title('Image lowpass-filtered with padding');

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3.2 Basic Steps in DFT Filtering DFT滤波基本操作

1. image数据类型转换：使用tofloat函数转换为浮点型数据。

2. 获得图片padding扩充大小：使用paddedsie函数获得，传入参数为f的size。

3. 傅里叶变换：需要加上padding值，得到F。

4. 生成一个大小为PQ(2)xPQ(2)的H滤波函数。

5. 使用滤波器H乘FFT交换得到G。

6. 获得G的逆FFT交换得到g。

7. 裁剪左上角位置图像，也就是原图大小。

8. 滤波图像数据类型转换为原类型即可。

3.3 An M-function for Filtering in the Frequency Domain

设计一个函数，使得可以自动地进行频域滤波操作。函数名为dftfilt。

function g = dftfilt(f, H, classout)
%DFTFILT Performs frequency domain filtering
%   g = DFTFILT(f, H, CLASSOUT) 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.
% 
%   Valid values of CLASSOUT are
% 
%   'original' The output is of the same class as the input.
%              This is the default if CLASSOUT is not included
%              in the call
%   'fltpoint' The output is floating point of class single, unless
%              both f and H are of class double, in which case the
%              output also is of class double.
% 
%   DFTFILE automatically pads f to be the same size as H. Both f
%   and H must be real. In addition, H must be an uncentered,
%   circularly-symmetric filter function.

% Convert the input to floating point.
[f, revertClass] = tofloat(f);

% Obtain the FFT of the padded input.
F = fft2(f, size(H, 1), size(H, 2));

% Perform filtering
g = ifft2(H.*F);

% Crop to original size.
g = g(1:size(f, 1), 1:size(f, 2)); % g is of class single here.

% Convert the output to the same class as the input if so specified.
if nargin == 2 || strcmp(classout, 'original')
    g = revertClass(g);
elseif strcmp(classout, 'fltpoint')
    return
else
    error('Undefined class for the output image.');
end

4 Obtaining Frequency Domain Filters from Spatial Filters

function freqz2：计算FIR(有限脉冲响应)滤波的响应频率。是一个Linear Filter。

H = freqz2(h, R, C)

参数信息：

• h：二维空间滤波器
• H：二维频域滤波器
• R, C：通常传入padding值，即PQ(1)和PQ(2)

Example：空间与频域滤波器对比

图片大小为600x600

>> f = imread('Fig0309(a).tif');
>> f = tofloat(f);
>> F = fft2(f);
>> S = fftshift(log(1 + abs(F)));
>> imshow(S, []);

本例中，我们使用Sobel空间滤波器来增强垂直边缘。

>> h = fspecial('sobel')' % 符号'表示矩阵转置
h =
     1     0    -1
     2     0    -2
     1     0    -1

然后，我们比较sobel滤波器下空间滤波和频域滤波的区别。

>> freqz2(h)
>> PQ = paddedsize(size(f));   % 生成H滤波器
>> H = freqz2(h, PQ(1), PQ(2));
>> H1 = ifftshift(H);
>> imshow(abs(H), []);
>> imshow(abs(H1), []);

接下来生成滤波图像

>> gs = imfilter(f, h); % spatial domain
>> gf = dftfilt(f, H1); % obtained in frequency domain
>> imshow(gs, []);
>> imshow(gf, []);

生成的图片偏灰色调，主要是gs和gf中存在负数。由于sobel模版适用于相应绝对值检测垂直边缘，所以图像的绝对值更具有相关性。

>> imshow(abs(gs), []);
>> imshow(abs(gf), []);

为了边缘显示更清晰，可以采用创建阈值化的二值图像。

>> imshow(abs(gs) > 0.2*abs(max(gs(:)))); % 选用0.2只显示强度比gs最大值的20%还要大的边缘
>> imshow(abs(gf) > 0.2*abs(max(gs(:)))); % 选用0.2只显示强度比gf最大值的20%还要大的边缘

通过计算gs与gf的差距，发现最大的差距仅为1.4063e-06，最小差别为0。差距非常小，可以忽略。

5 Generating Filters Directly in the Frequency Domain

5.1 Creating Meshgrid Arrays for Use in Implementing Filters in the Frequency Domain 建立用于实现频域滤波器的网格数组

这里需要创建的函数为：dftuv(M, N)

因为FFT计算假设变换的原点位置在左上角，距离计算也是根据该点进行。

function [U, V] = dftuv(M, N)
% DFTUV Computes meshgrid frequency matrices.
% [U, V] = DFTUV(M, N) computes meshgrid frequency matrices U and V. U and
% V are useful for computing frequency-domain filter functions that can be
% used with DFTFILT. U and V are both M-by-N.
% more details to see the textbook Page 93
%
% [U，V] = DFTUV（M，N）计算网格频率矩阵U和V。 U和V对于计算可与DFTFILT一起使用的
% 频域滤波器函数很有用。 U和V都是M-by-N。

% Set up range of variables.
% 设置变量范围
u = 0 : (M - 1);
v = 0 : (N - 1);

% Compute the indices for use in meshgrid.
% 计算网格的索引，即将网络的原点转移到左上角，因为FFT计算时变换的原点在左上角。
idx = find(u > M / 2);
u(idx) = u(idx) - M;
idy = find(v > N / 2);
v(idy) = v(idy) - N;

% Compute the meshgrid arrays.
% 计算网格矩阵
[V, U] = meshgrid(v, u);

5.2 Lowpass(Smoothing) Frequency Domain FIlters

一个理想的低通滤波器(ILPF)具有如下的传递方程
$$H(u,v)=\{\begin{array}{l}1,\text{if}D(u,v)\le D_0\\ 0,\text{if}D(u,v)>D_0\end{array}$$
参数信息：

• $D(u,v)$：点(u,v)到滤波器中心的距离
• $D_0$：正数
• 满足$D(u,v)=D_0$的轨迹为圆

因为滤波器H要乘一幅图像的Fourier transform，我们看到理想滤波器切断(0)了园外的F(u,v)的分量，保留圆上和圆内的点(1)。ILPF通常用于解释振铃(ringing)和重叠误差(wraparound error)等现象。

Butterworth lowpass filter(BLFT)：布特沃斯低通滤波器，具有从滤波中心到$D_0$的截止频率，表达式为
$$H(u,v)=\frac{1}{1+[D(u,v)/D_0{]}^{2n}}$$
这里默认n=1。当$D(u,v)=D_0$，滤波器降到最大值1的0.5，即$H(u,v)=0.5$。

Gaussian Lowpass Filter(GLPF)：高斯低通滤波器，用的最多的一个
$$H(u,v)=exp(\frac{-D^2(u,v)}{2{\sigma }^2})$$
参数：$\sigma =D_0$，即标准差

根据截止参数，我们可以得到
$$H(u,v)=exp(\frac{-D^2(u,v)}{2D_0^2})$$
当$D(u,v)=D_0$，滤波器降到最大值1的0.607

Example：Lowpass FIlter Test

>> f = imread('Fig0313(a).tif');
>> [f, revertClass] = tofloat(f);
>> PQ = paddedsize(size(f));
>> [U, V] = dftuv(PQ(1), PQ(2));
>> D = hypot(U, V); % Compute faster than D=sqrt(U.^2+V.^2)
>> D0 = 0.05*PQ(2);
>> F = fft2(f, PQ(1), PQ(2)); % Needed for the spectrum.
>> H = exp(-(D.^2)/(2*(D0.^2))); % GLPF
>> subplot(2,2,1);
>> imshow(f);title('image');
>> subplot(2,2,2);
>> imshow(fftshift(H));
>> title('GLPF shows as an Image');
>> subplot(2,2,3);
>> imshow(log(1+abs(fftshift(F))), []);
>> title("Spectrum of original image");
>> subplot(2,2,4);
>> imshow(g);title('Filtered image');

我们也可以将上述三种低通滤波器合并至一个M-Function文件中，函数为lpfilter

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 value 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.
%
% 得到指定类型的低通滤波器

% 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('Unkown filter type.')
end

5.3 Wireframe and Surface Plotting线框图和表面图的绘制

主要用到的函数：

function mesh：绘制一个MxN大小的二维函数H的线框图

mesh(H)
mesh(H(1:k:end, 1:k:end))

function colormap：上色用的，默认为黑色[0 0 0]

colormap([0 0 0])

function view：观察点，可以控制观察点视角位置

view(az, el) % az表示方位角，el表示仰角

Example Wireframe plotting：

>> H = fftshift(lpfilter('gaussian', 500, 500, 50)); % 绘制高斯低通滤波器
>> mesh(double(H(1:10:500, 1:10:500)))  % 为了防止数据稠密，选取50个点做线框图
>> axis tight
>> colormap([0 0 0])
>> axis off
>> grid off
>> view(-25, 30)
>> view(-25, 0)

function surf：绘制表面图

surf(H)

function colormap：颜色转换

colormap(gray) % 转换为灰度

Example Surface Plotting：

>> H = fftshift(lpfilter('gaussian', 500, 500, 50)); % 绘制高斯低通滤波器
>> surf(double(H(1:10:500, 1:10:500)))
>> subplot(1,2,1)
>> surf(double(H(1:10:500, 1:10:500)))
>> axis tight
>> subplot(1,2,2)
>> surf(double(H(1:10:500, 1:10:500)))
>> axis tight
>> colormap(gray)
>> axis off
>> shading interp % 平滑曲线

如果是绘制含有两个变量的解析函数，我们用function meshgrid产生坐标值，并从这些坐标值中产生我们用于传入mesh和surf的离散矩阵。

6 Highpass (Sharpening) Frequency Domain Filters高通(锐化)频域滤波器

高通(锐化)是低通(平滑)的反处理，即通过消弱Fourier Transform的低频并保持高频相对不变来sharpen image。

相对于Lowpass滤波器的transfer function，highpass滤波器的transfer function表达式为：
$$H_{HP}(u,v)=1-H_{LP}(u,v)$$
则，对于上述提到的三个Lowpass filter transfer function，我们可以得到Highpass filter transfer function：

• Ideal Highpass filter：

$$H(u,v)=\{\begin{array}{l}0,\text{if}D(u,v)\le D_0\\ 1,\text{if}D(u,v)>D_0\end{array}$$

• Butterworth Filter：

$$H(u,v)=\frac{1}{1+[D_0/D(u,v){]}^{2n}}$$

• Gaussian Filter：

$$H(u,v)=1-exp(\frac{-D^2(u,v)}{2D_0^2})$$

6.1 A Function for Highpass Filtering

根据上述获得的三个高通滤波转化函数，我们可以自定义一个函数实现上述要求，这里不需要太麻烦，可以只可调用function lpfilter，最后结果用1减去即可。

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:
% 
%   Valid value 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.
% 
%   H is of floating point class single. It is returned uncentered
%   for consistency with filtering function dftfilt. To view H as an
%   image or mesh plot, it should be centered using Hc = fftshift (H).
% 
%  The transfer function Hhp of a highpass fílter 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;

Example Highpass filters：

Ideal Filter绘制：

>> subplot(2,3,1);
>> H = fftshift(hpfilter('ideal', 500, 500, 50));
>> mesh(double(H(1:10:500,1:10:500)));
>> axis tight
>> colormap([0 0 0])
>> axis off
>> title('ideal')
>> subplot(2,3,4)
>> imshow(H, []);
>> title('ideal image')

Butterworth Filter绘制：

>> subplot(2,3,2)
>> btwH = fftshift(hpfilter('btw', 500, 500, 50));
>> mesh(double(btwH(1:10:500,1:10:500)));
>> axis tight
>> axis off
>> title('butterworth')
>> subplot(2,3,5)
>> imshow(btwH, []);
>> title('butterworth image')

Gaussian Filter绘制：

>> subplot(2,3,3)
>> gH = fftshift(hpfilter('gaussian', 500, 500, 50));
>> mesh(double(gH(1:10:500,1:10:500)));
>> axis tight
>> axis off
>> title('gaussin')
>> subplot(2,3,6)
>> imshow(gH, []);
>> title('gaussin image')

Example Highpass filtering：利用Gaussian高通滤波函数再频域内处理图片

>> PQ = paddedsize(size(f));
>> D0 = 0.05*PQ(1);
>> H = hpfilter('gaussian', PQ(1), PQ(2), D0);
>> g = dftfilt(f, H);
>> subplot(1,2,1);imshow(f);
>> subplot(1,2,2);imshow(g);

6.2 High-Frequency Emphasis Filtering高频强度滤波

Highpass filters偏离了dc的0项，因此减少了image中平均值为0的值。一种补偿方法是给HPF加上偏移量。当偏移量与滤波器乘以某个大于1的常数结合起来时，可以高亮出图像的高频区域，所以称这种方法为高频强度滤波。

同时乘数的大小也增强了低频的部分，但只要偏移量与乘数相比较小，低频增强的影响就小于高频增强的影响。

传递函数为：
$$H_{HFE}(u,v)=a+b{H}_{HP}(u,v)$$
其中，a是偏移量offset，b是大于1的常数。

Example Combining High-frequency emphasis and histogram equalization：

>> PQ = paddedsize(size(f));
>> D0 = 0.05*PQ(1);
>> HBW = hpfilter('btw', PQ(1), PQ(2), D0, 2);
>> H = 0.5 + 2*HBW;
>> gbw = dftfilt(f, HBW, 'fltpoint');
>> gbw = gscale(gbw);
>> ghf = dftfilt(f, H, 'fltpoint');
>> ghf = gscale(ghf);
>> ghe = histeq(ghf, 256);
>> subplot(2,2,1);imshow(f);
>> subplot(2,2,2);imshow(gbw);
>> subplot(2,2,3);imshow(ghf);
>> subplot(2,2,4);imshow(ghe);

7 Selective Filtering

分为两类：第一类是带阻(bandreject)和带通(bandpass)滤波器，第二类是陷波带阻(notcherject)和陷波带通(notchpass)滤波器。

7.1 Bandreject and Bandpass Filters

这类滤波器可以很容易用lowpass和highpass滤波器构建。表达式可写为
$$H_{BP}(u,v)=1-H_{BR}(u,v)$$
则，对于上述提到的三个Lowpass filter transfer function，我们可以得到带通带阻滤波器：

• Ideal filter：

$$H(u,v)=\{\begin{array}{l}0,D_0-\frac{W}{2}\le D(u,v)\le D_0+\frac{W}{2}\\ 1,\text{otherwise}\end{array}$$

函数实现为

function H = idealReject(D, D0, W)
RI = D <= D0 - (W/2); % Points of region inside the inner 
                      % boundary of the reject band are labeled 1.
                      % All other points are labeled 0.
RO = D >= D0 - (W/2); % Points of region outside the outer
                      % boundary of the reject band are labeled 1.
                      % All other points are labeled 0.
H = tofloat(RO | RI); % Ideal bandreject filter.

• Butterworth Filter：

$$H(u,v)=\frac{1}{1+[\frac{WD(u,v)}{D^2(u,v)-D_0^2}{]}^{2n}}$$

函数实现为

function H = btwReject(D, D0, W, n)
H = 1./(1 + (((D*W)./(D.^2 - D0.^2)).^2*n));

• Gaussian Filter：

$$H(u,v)=1-exp(-[\frac{D^2(u,v)-D_0^2}{WD(u,v)}{]}^2)$$

函数实现为

function H = gaussReject(D, D0, W)
H = 1 - exp(-((D.^2 - D0.^2)./(D.*W+eps)).^2);

其中，W是对Ideal filter是带宽，对gaussian filter是粗略的截止频率，W和n结合是对butterworth的通带的宽度。

function H = bandfilter(type, band, M, N, D0, W, n)
%BANDFILTER Computes frequency domain band filters.
% 
%  Parameters used in the filter definitions :
%      M: Number of rows in the filter .
%      N: Number of col umns in the filter .
%      DO : Radius of the center of the band.
%      W: "Width" of the band. W is the true width only for
%         ideal filters. For the other two filters this pa r ameter
%         acts more like a smooth cutoff .
%      n: Order of the Butterworth filter if one is specified. W
%         and n interplay to determine the effective broadness of
%         the reject or pass band. Higher values of both these
%         parameters result in broader bands .
%  Valid values of BAND are :
% 
%      'reject'     Bandreject filter .
%      'pass'       Bandpass filter .
% 
%  One of these two values must. be specified for BAND.
% 
%  H = BANDFILTER(' ideal', BAND, M, N, DO, W) computes an M-by-N :
%  ideal bandpass or bandreject filter, depending on the value of
%  BAND.
% 
%  H = BANDFILTER( 'btw', BAND, M, N, D0, W, n) computes an M-by-N
%  Butterworth filter of order n. The filter is either bandpass or
%  bandreject, depending on the value of BAND. The default value of
%  n is 1.
% 
%  H = BANDFILTER( 'gaussian', BAND, M, N, D0, W) computes an M-by-N
%  gaussian filter. The filter is either bandpass or bandreject,
%  depending on BAND.
% 
% H is of floating point class single. It is returned uncentered
% for consistency with filtering function dftfilt. To view H as an
% image or mesh plot, it should be centered using Hc = fftshift (H) .

% 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 = hypot(U, V);
% Determine if need to use default n.
if nargin < 7
    n = 1; % Default BTW filter order
end

% Begin filter computations. All filters are computed as bandreject
% filters. At the end, they are converted to bandpass if so 
% specified. Use lower (type) to protect against the input being
% capitalized.
switch lower(type)
    case 'ideal'
        H = idealReject(D, D0, W);
    case 'btw'
        H = btwReject(D, D0, W, n);
    case 'gaussin'
        H = gaussReject(D, D0, W);
    otherwise
        error('Unknown filter type');
end

% Generate a bandpass filter if one was specified
if strcmp(band, 'pass')
    H = 1 - H;
end

7.2 Notchreject and Notchpass Filters

陷波滤波器是更有用的选择滤波器。NF事先定义关于频率矩形的中心的邻域内的频率。零相移滤波器(Zero-phase-shift)必须关于远点对称。

以频率(u0,v0)为中心开的notch必须有相应的位于(-u0,-v0)开的notch。

Notchreject Filter可以用中心被平移到陷波滤波器中心的HP的乘积来构造。包括一对开槽Q的general form为
$$H_{NR}(u,v)=\prod _{k=1}^Q{H}_k(u,v)H_{-k}(u,v)$$
其中，$H_k(u,v)$和$H_{-k}(u,v)$是分别以$(u_k,v_k)$和$(-u_k,-v_k)$为中心的HPF，这些中心是以频率矩形的中心(M/2, N/2)来确定的，所以，每个Filter的distance function有：
$$\begin{gathered} D_k(u,v)=[(u-M/2-u_k{)}^2+(v-N/2-v_k{)}^2{]}^{\frac{1}{2}} \\ {D}_{-k}(u,v)=[(u-M/2+u_k{)}^2+(v-N/2+v_k{)}^2{]}^{\frac{1}{2}} \end{gathered}$$
类似于带阻和带通滤波器，我们可以将陷波带阻和带通滤波器的表达式写为
$$H_{NP}(u,v)=1-H_{NR}(u,v)$$
Example 用陷波滤波器减少波纹模式：

>> f = imread('Fig0321(a).tif');
>> subplot(2,3,1);imshow(f);
>> [M N] = size(f);
>> [f, revertClass] = tofloat(f);
>> F = fft2(f);
>> S = gscale(log(1 + abs(fftshift(F)))); % Spectrum
>> title('Image with moiré patterns');
>> subplot(2,3,2);imshow(S);title('Spectrum');
>> % Use function imtool to obtain the coordinates of the
>> % spikes interactively.
>> C1 = [99 154; 128 163];
>> % Notch filter:
>> H1 = cnotch('gaussian', 'reject', M, N, C1, 5);
>> H1 = cnotch('gaussian', 'reject', M, N, C1, 5);
>> % Compute spectrum of the filtered transform and shot it as
>> % an image.
>> P1 = gscale(fftshift(H1).*(tofloat(S)));
>> subplot(2,3,3);imshow(P1);title('Spectrum low-frequency bursts with moiré patterns');
>> % Filter Image
>> g1 = dftfilt(f, H1);
>> g1 = revertClass(g1);
>> subplot(2,3,4);imshow(g1);title('Notch Filter Result');
>> % Repeat with the following C2 to reduce the higher
>> % frequency interference components.
>> C2 = [99 154; 128 163; 49 160; 133 233; 55 132; 108 225; 112 74];
>> H2 = cnotch('gaussian', 'reject', M, N, C2, 5);
>> % Compute the spectrum of the filtered transform and show
>> % it as an image.
>> P2 = gscale(fftshift(H2).*(tofloat(S)));
>> subplot(2,3,5);imshow(P2);title('Spectrum eliminae higher frequency noise with more filters');
>> % Filter image
>> g2 = dftfilt(f, H2);
>> g2 = revertClass(g2);
>> subplot(2,3,6);imshow(g2);title('Filtered result with less noise');

陷波滤波的特殊情况包括沿着DFT的轴的取值范围进行滤波，我们可以设计一个recnotch函数使用放置在轴上的矩形完成计算。

Example Using Notch filtering to reduce periodic interference caused by malfunctioning imaging equipment：

本例中的original image附加了水平方向上的干扰信号。

>> f = imread('Fig0322(a).tif');
>> subplot(2,2,1);imshow(f);title('image');
>> [M N] = size(f);
>> [f, revertClass] = tofloat(f);
>> F = fft2(f);
>> S = gscale(log(1 + abs(fftshift(F))));
>> subplot(2,2,2);imshow(S);title('Spectrum');
>> H = recnotch('reject', 'vertical', M, N, 3, 15, 15);
>> subplot(2,2,3);imshow(fftshift(H));title('Result: NRF multiples DFT');
>> g = dftfilt(f, H);
>> g = revertClass(g);
>> subplot(2,2,4);imshow(g);title('Result: Computing IDFT');

从实验中我们可以看出，干扰是一个近似关于垂直方向周期的，所以我们期望在谱的垂直轴上找到存在的能量脉冲。我们采用窄的、矩形的陷波滤波器进行处理，结果如图三所示。最后，我们通过IDFT的计算获得改进后的结果。

为了得到空间干扰自身，我们可以使用NPF代替NRF，在vertical上隔离出干扰的频率。然后，利用IDFT得到空间干扰模式本身：

>> Hrecpass = recnotch('pass', 'vertical', M, N, 3, 15, 15);
>> interference = dftfilt(f, Hrecpass);
>> subplot(1,2,1);imshow(fftshift(Hrecpass));
>> interference = gscale(interference);
>> subplot(1,2,2);imshow(interference);