【图像去噪的扩散滤波】图像线性扩散滤波、边缘增强线性和非线性各向异性滤波（Matlab代码实现）

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📋📋📋本文目录如下：🎁🎁🎁

目录

💥1 概述

📚2 运行结果

2.1 non_aniso

2.2 inhomo_iso

2.3 heat_imp

2.4 heat_explicit

🎉3 参考文献

🌈4 Matlab代码实现

💥1 概述

【图像去噪的扩散滤波】图像线性扩散滤波、边缘增强线性和非线性各向异性滤波

各种基于扩散的图像滤波方法：
1.使用热方程的线性扩散滤波 - 使用隐式和显式欧拉方法求解。
2. 边缘增强线性各向异性扩散滤波。
3. 边缘增强非线性各向异性扩散滤波。

在图像处理领域，去除图像中的噪点和提升细节是非常重要的任务。为此，发展了各种基于扩散的图像滤波方法，旨在改善图像质量和增强图像特征。以下是其中几种常见的滤波方法：

1. 线性扩散滤波：使用热方程模型来进行线性扩散滤波。这种方法可以使用隐式和显式欧拉方法来求解，以改善图像的平滑度和减少噪声。线性扩散滤波能够通过自适应地调整邻域像素的亮度值，实现图像的去噪和平滑。

2. 边缘增强线性各向异性扩散滤波：这种方法结合了线性扩散滤波和各向异性过滤的思想，在保持图像平滑的同时突出了图像中的边缘特征。通过对邻域像素的梯度进行加权处理，边缘增强线性各向异性扩散滤波能够有效地提升图像的边缘细节，使图像更加清晰和锐利。

3. 边缘增强非线性各向异性扩散滤波：与线性滤波相比，非线性滤波方法在处理图像时更加灵活和精确。边缘增强非线性各向异性扩散滤波通过考虑像素之间的非线性关系，使得处理后的图像在细节和边缘保留方面更具优势。这种方法能够在去除噪点的同时保持图像的细节和纹理信息。

这些基于扩散的图像滤波方法为图像去噪和增强提供了强大的工具。根据不同的应用需求和图像特征，选择合适的滤波方法能够显著改善图像质量，并提高图像的可视化效果。无论是在科学研究、医学影像还是图形设计领域，这些滤波方法都发挥着重要作用，为用户带来更好的图像处理体验。

📚2 运行结果

2.1 non_aniso

2.2 inhomo_iso

2.3 heat_imp

2.4 heat_explicit

部分代码：

% set up finite difference parameters
alpha =.5;
k = 1;
h = 1;

lambda = (alpha^2)*(k/(h^2));

[m n] = size(w);
%stick image into a vector
w_vec = reshape(w,n*n,1);
w_old = w;
w_new = w;

%smooth the image
im_smth = filter_function(w,1);
im_smth = im_smth';
% required for g() calculation
[dx_im_smth dy_im_smth] = gradient(im_smth);
gr_im_smth = dx_im_smth.^2 + dy_im_smth.^2;

[mmm nnn] = size(gr_im_smth);
% g() calculation
for i=1:mmm
    for j=1:nnn
        g(i,j) = 1/(1+(gr_im_smth(i,j)/32));
    end
end
%g = g';
jj=1;

[im_sx im_sy]  = gradient(im_smth);
% diffusion tensor D is preconputed here
D = zeros(n*n,2,2);
for i=1:n*n
    row = ceil(i/n);
    col = i - (row-1) * n;
  %  if((col > 1) && (col < n) && (row > 1) && (row < n))
        
        % choose eigenvector parallel and perpendicular to gradient
        eigen_vec = [im_sx(row,col) im_sy(row,col); im_sy(row,col) -im_sx(row,col) ];
        %choose eigenvalues
        eigen_vec(:,1) = eigen_vec(:,1) ./ norm(eigen_vec(:,1));
        eigen_vec(:,2) = eigen_vec(:,2) ./ norm(eigen_vec(:,2));
        eigen_val = [g(row,col) 0;0 1];
        
        % form diffusion tensor
        D(i,:,:) = eigen_vec * eigen_val * (eigen_vec');        
        % end
end

figure;
for k=1:400 % for each iteration
    for i=1:n*n % solve using Jacobi iterations scheme
        row = ceil(i/n); %compute what row this pixel belongs to in original image
        col = i - (row-1) * n; % compute cols similarly

        %different if conditions handles pixels at different location in
        %the image as depending on their location they may or may not have
        %all their neighbor pixels which will be required for finite
        %differences
        
        if((col > 1) && (col < n) && (row > 1) && (row < n))
            
            s = -lambda * ((D(i,1,1) * w_old(i-1)) + ((D(i,1,1) - D(i,1,2) - D(i,2,1)) * (w_old(i+1))) +  ...
                (D(i,2,2) * w_old(i-n)) + ((D(i,2,2) - D(i,1,2) - D(i,2,1)) * (w_old(i+n))) + ((D(i,1,2) + D(i,2,1)) * w_old(i+n+1)));
            w_new(i) = (-s + w_old(i))/(1 + lambda * (2 * D(i,1,1) + 2 * D(i,2,2) - D(i,1,2) - D(i,2,1)));
            
        elseif((row == 1) && (col > 1) && (col < n))
            s = -lambda * ((D(i,1,1) * w_old(i-1)) + ((D(i,1,1) - D(i,1,2) - D(i,2,1)) * (w_old(i+1))) +  ...
                ((D(i,2,2) - D(i,1,2) - D(i,2,1)) * (w_old(i+n))) + ((D(i,1,2) + D(i,2,1)) * w_old(i+n+1)));
            
            %s = -(lambda) * (w_old(i+1) + w_old(i-1) + w_old(i+n));
            w_new(i) = (-s + w_old(i))/(1+4*lambda);
        elseif((row == n) && (col > 1) && (col < n))
            s = -lambda * ((D(i,1,1) * w_old(i-1)) + ((D(i,1,1) - D(i,1,2) - D(i,2,1)) * (w_old(i+1))) +  ...
                (D(i,2,2) * w_old(i-n)));
            
            %s = -(lambda) * (w_old(i+1) + w_old(i-1) + w_old(i-n));
            w_new(i) = (-s + w_old(i))/(1+4*lambda);
        elseif((col == 1) && (row > 1) && (row < n))
            s = -lambda * (((D(i,1,1) - D(i,1,2) - D(i,2,1)) * (w_old(i+1))) +  ...
                (D(i,2,2) * w_old(i-n)) + ((D(i,2,2) - D(i,1,2) - D(i,2,1)) * (w_old(i+n))) + ((D(i,1,2) + D(i,2,1)) * w_old(i+n+1)));
            
            %s = -(lambda) * (w_old(i+1) + w_old(i+n) + w_old(i-n));
            w_new(i) = (-s + w_old(i))/(1+4*lambda);
        elseif((col == n) && (row > 1) && (row < n))
            s = -lambda * ((D(i,1,1) * w_old(i-1)) +  ...
                (D(i,2,2) * w_old(i-n)) + ((D(i,2,2) - D(i,1,2) - D(i,2,1)) * (w_old(i+n))));            
            %s = -(lambda) * (w_old(i-1) + w_old(i+n) + w_old(i-n));
            w_new(i) = (-s + w_old(i))/(1+4*lambda);
        elseif((col==1) && (row==1))

% set up finite difference parameters
alpha =.5;
k = 1;
h = 1;

lambda = (alpha^2)*(k/(h^2));

[m n] = size(w);
%stick image into a vector
w_vec = reshape(w,n*n,1);
w_old = w;
w_new = w;

%smooth the image
im_smth = filter_function(w,1);
im_smth = im_smth';
% required for g() calculation
[dx_im_smth dy_im_smth] = gradient(im_smth);
gr_im_smth = dx_im_smth.^2 + dy_im_smth.^2;

[mmm nnn] = size(gr_im_smth);
% g() calculation
for i=1:mmm
for j=1:nnn
g(i,j) = 1/(1+(gr_im_smth(i,j)/32));
end
end
%g = g';
jj=1;

[im_sx im_sy] = gradient(im_smth);
% diffusion tensor D is preconputed here
D = zeros(n*n,2,2);
for i=1:n*n
row = ceil(i/n);
col = i - (row-1) * n;
% if((col > 1) && (col < n) && (row > 1) && (row < n))

% choose eigenvector parallel and perpendicular to gradient
eigen_vec = [im_sx(row,col) im_sy(row,col); im_sy(row,col) -im_sx(row,col) ];
%choose eigenvalues
eigen_vec(:,1) = eigen_vec(:,1) ./ norm(eigen_vec(:,1));
eigen_vec(:,2) = eigen_vec(:,2) ./ norm(eigen_vec(:,2));
eigen_val = [g(row,col) 0;0 1];

% form diffusion tensor
D(i,:,:) = eigen_vec * eigen_val * (eigen_vec');
% end
end

figure;
for k=1:400 % for each iteration
for i=1:n*n % solve using Jacobi iterations scheme
row = ceil(i/n); %compute what row this pixel belongs to in original image
col = i - (row-1) * n; % compute cols similarly

%different if conditions handles pixels at different location in
%the image as depending on their location they may or may not have
%all their neighbor pixels which will be required for finite
%differences

if((col > 1) && (col < n) && (row > 1) && (row < n))

s = -lambda * ((D(i,1,1) * w_old(i-1)) + ((D(i,1,1) - D(i,1,2) - D(i,2,1)) * (w_old(i+1))) + ...
(D(i,2,2) * w_old(i-n)) + ((D(i,2,2) - D(i,1,2) - D(i,2,1)) * (w_old(i+n))) + ((D(i,1,2) + D(i,2,1)) * w_old(i+n+1)));
w_new(i) = (-s + w_old(i))/(1 + lambda * (2 * D(i,1,1) + 2 * D(i,2,2) - D(i,1,2) - D(i,2,1)));

elseif((row == 1) && (col > 1) && (col < n))
s = -lambda * ((D(i,1,1) * w_old(i-1)) + ((D(i,1,1) - D(i,1,2) - D(i,2,1)) * (w_old(i+1))) + ...
((D(i,2,2) - D(i,1,2) - D(i,2,1)) * (w_old(i+n))) + ((D(i,1,2) + D(i,2,1)) * w_old(i+n+1)));

%s = -(lambda) * (w_old(i+1) + w_old(i-1) + w_old(i+n));
w_new(i) = (-s + w_old(i))/(1+4*lambda);
elseif((row == n) && (col > 1) && (col < n))
s = -lambda * ((D(i,1,1) * w_old(i-1)) + ((D(i,1,1) - D(i,1,2) - D(i,2,1)) * (w_old(i+1))) + ...
(D(i,2,2) * w_old(i-n)));

%s = -(lambda) * (w_old(i+1) + w_old(i-1) + w_old(i-n));
w_new(i) = (-s + w_old(i))/(1+4*lambda);
elseif((col == 1) && (row > 1) && (row < n))
s = -lambda * (((D(i,1,1) - D(i,1,2) - D(i,2,1)) * (w_old(i+1))) + ...
(D(i,2,2) * w_old(i-n)) + ((D(i,2,2) - D(i,1,2) - D(i,2,1)) * (w_old(i+n))) + ((D(i,1,2) + D(i,2,1)) * w_old(i+n+1)));

%s = -(lambda) * (w_old(i+1) + w_old(i+n) + w_old(i-n));
w_new(i) = (-s + w_old(i))/(1+4*lambda);
elseif((col == n) && (row > 1) && (row < n))
s = -lambda * ((D(i,1,1) * w_old(i-1)) + ...
(D(i,2,2) * w_old(i-n)) + ((D(i,2,2) - D(i,1,2) - D(i,2,1)) * (w_old(i+n))));
%s = -(lambda) * (w_old(i-1) + w_old(i+n) + w_old(i-n));
w_new(i) = (-s + w_old(i))/(1+4*lambda);
elseif((col==1) && (row==1))