同态滤波

公式

$$H(u,v)=({\gamma }_H-{\gamma }_L)[1-e^{-c[D^2(u,v)/D_0^2]}]-{\gamma }_L$$

其中，${\gamma }_H$的值为，${\gamma }_L$的值为
D(u, v)的定义如下：
$$D(u,v)=[(u-\frac{M}{2}{)}^2+(v-\frac{N}{2}{)}^2{]}^{\frac{1}{2}}$$
即在频率域中的点距离中心的距离。(通过对空域做中心变换后，我们在经过快速傅里叶变换之后得到的图像的中心将会是$(\frac{M}{2},\frac{N}{2})$，中心变换操作可以通过和$(-1{)}^{x+y}$做相关得到，可以通过傅里叶变换的平移性得到)

MATLAB 步骤

（1）将图像的灰度值取自然对数，将入射分量和反射分量变成相加的形式，这样就可以利用傅里叶变化的线性特征变换到频域进行图像处理

% Homomophic filtering
[YUV] = imread('office.jpg','jpg');
[height, width, channel] = size(f);
U = YUV(1:height, 1:width, 2);
V = YUV(1:height, 1:width, 3);
f = YUV(1:height, 1:width, 1);
%limit frequency
D0 = 1;
% uint8 to double
f = im2double(f);
% ln
for x = 1:height
    for y = 1:width
        f(x, y) = log(f(x, y) + 1);
    end
end

（2）进行DFT变换

% discrete fourier transfer
F = zeros(height, width);
temp = zeros(height, width);
for y = 1:width
    for u = 1:height
        sum = 0;
        for x = 1:height
            sum = sum + exp(-1j * 2 * pi * u * x / height) * f(x, y);
        end
        temp(u, y) = sum;
    end
end
for u = 1:height
    for v = 1:width
        sum = 0;
        for y = 1:width
            sum = sum + exp(-1j * 2 * pi * v * y / width) * temp(u, y);
        end
        F(u, v) = sum;
    end
end

（3）进行同态滤波

% generate Homomophic filtering kernel
kernel = zeros(height, width);
for u = 1:height
    for v = 1:width
        D = sqrt((u - height / 2)^2 + (v - width / 2)^2);
        % 同态滤波
        kernel(u, v) = 1.75 * (1 - exp(-D^2 / D0^2)) + 0.25;
        % Butterworth 高通滤波
        %kernel(u, v) = 1 / (1 - (D0 / D)^2);
    end
end
imwrite(kernel, 'kernel_.png', 'png');
F = F.*kernel;

（4）进行IDFT变换

% Inverse DFT
% Inverse discrete fourier transfer
temp = zeros(height, width);
for y = 1:width
    for u = 1:height
        sum = 0;
        for x = 1:height
            sum = sum + exp(1j * 2 * pi * u * x / height) * F(x, y);
        end
        temp(u, y) = sum;
    end
end
for u = 1:height
    for v = 1:width
        sum = 0;
        for y = 1:width
            sum = sum + exp(1j * 2 * pi * v * y / width) * temp(u, y);
        end
        cent(u, v) = sum / (width * height);
    end
end

（5）进行幂运算，基数为自然系数

% exp
for u = 1:height
    for v = 1:width
        f(u, v) = exp(f(u, v)) - 1;
    end
end

滤波效果