信息处理 - 使用 MATLAB / Octave 对具有高斯噪声的图像进行去噪 - 吾爱随笔录

当观察到的图像是

f = u + v

$f=u+v$ 在哪里

u

$u$ 是恢复的图像（是我想要恢复的图像）和

v

$v$ 是高斯噪声。

恢复 $u$ ，我解决了以下最小化问题：

min_{u \in H^{1} (Ω)} \int_{Ω} γ | \nabla u (x) |^{2} d x + \int_{Ω} (f (x) - u (x))^{2} d x,

$\min_{u \in H^1(\Omega)} \int_\Omega \gamma|\nabla u(x)|^2dx+ \int_\Omega (f(x)-u(x))^2dx,$ 其中是正则化系数，和 size( )。

γ

$\gamma$

Ω = [0, n] \times [0, m]

$\Omega=[0,n]\times[0,m]$

[n, m] =

$[n,m]=$

u

$u$

我想使用 MATLAB 求解 PDE（欧拉-拉格朗日）：

\begin{array}{rcl} d i v (γ \nabla u) + u = f i n Ω \\ \frac{\partial u}{\partial n} = 0 i n \partial Ω \end{array}

$\begin{eqnarray} div(\gamma \nabla u) + u = f \;in \;\Omega \\ \frac{\partial u}{\partial n}=0 \; in \; \partial \Omega \end{eqnarray}$

谁能帮我解决这个问题？谢谢！

我尝试了以下代码：

clear all, close all,clc

uor=imread('gourd.bmp'); % the original image 
u0 = imnoise(uor,'gaussian',0,0.01);
u0=double(u0);
[m n]=size(u0);
uor=double(uor);
u=u0;
c=0.028;
h=1;     
for Iter=1:50, 
    for i=2:m-1,
      for j=2:n-1,
          Lap=0.003*(u(i+1,j)+u(i-1,j)-4*u(i,j)+u(i,j+1)+u(i,j-1));
          u(i,j)=(u0(i,j)+(1/(2*c*h*h))*Lap);
     end
    end
    for i=2:m-1,
          u(i,1)=u(i,2);
          u(i,n)=u(i,n-1);
        end

    for j=2:n-1,
          u(1,j)=u(2,j);
          u(m,j)=u(m-1,j);
        end

        u(1,1)=u(2,2);
        u(1,n)=u(2,n-1); 
        u(m,1)=u(m-1,2);
        u(m,n)=u(m-1,n-1);

en=0.0;  
    for i=2:m-1,
      for j=2:n-1,
      ux=(u(i+1,j)-u(i,j))/h;
      uy=(u(i,j+1)-u(i,j))/h;
      fidelity=(u0(i,j)-u(i,j))*(u0(i,j)-u(i,j));

      en=en+c*fidelity;
      end
    end


Energy(Iter)=en; 

%  Error between uor and u0
 ur=reshape(u,m*n,1);
 uori=reshape(uor,m*n,1);
 residu=norm(ur-uori)/norm(uori);

 [peaksnr, snr] = psnr(uor, u);

disp(['    iter ' num2str(Iter), ' :     Error = ' num2str(residu), ...
    ' ,    Peak-snr ' num2str(-peaksnr), ' ,    SNR ' num2str(snr)]);


  end 

% show the structural similarity index for measuring image quality 
[ssimval, mapssim] = ssim(u,uor);
disp([' the structural similarity index is ' num2str(ssimval)]);
figure,imshow(mapssim,[]); axis square; 

figure,imagesc(u); axis image; axis off; colormap(gray);

原图在这里： https ://www.dropbox.com/s/4bccby1f4lxp4j9/gourd.rar?dl=0

最好的祝福