Nonlinear isotropic diffusion filtering

∂ t u = div ⁡ ( D ( ∇ u σ ) ∇ u ) \partial_{t} u=\operatorname{div}\left(D\left(\nabla u_{\sigma}\right) \nabla u\right) ∂tu=div(D(∇uσ)∇u) D ( ∇ u σ ) = g ( ∣ ∇ u σ ∣ ) I D\left(\nabla u_{\sigma}\right)=g\left(\left|\nabla u_{\sigma}\right|\right) I D(∇uσ)=g(∣∇uσ∣)I g ( s ) = e x p ( − s 5 5 λ 5 ) g(s) = exp(\frac{-s^5}{5\lambda^5}) g(s)=exp(5λ5−s5)

% convolute.m
function rtn = convolute(f, M, N, sigma)rtn = zeros(M,N);if sigma == 0rtn = f;return;elseif sigma == 1K = [0.01, 0.01, 0.01, 0.01, 0.01; ...0.01, 0.07, 0.1, 0.07, 0.01;...0.01, 0.1, 0.16, 0.1, 0.01; ...0.01, 0.07, 0.1, 0.07, 0.01; ...0.01, 0.01, 0.01, 0.01, 0.01];% sum(sum(K))for x = 1:Mfor y = 1:Nfor i = 1:5for j = 1:5tmp1 = mod(x+3-i,M);tmp2 = mod(y+3-j,N);if tmp1 ==0tmp1 = M;endif tmp2 ==0tmp2 = N;endrtn(x,y) = rtn(x,y) + ...f(tmp1,tmp2) * K(i,j);endendendendreturn;elseif sigma == 2K = [0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01; ...0.01, 0.015, 0.02, 0.03, 0.02, 0.015, 0.01; ...0.01, 0.02, 0.04, 0.035, 0.04, 0.02, 0.01;...0.01, 0.03, 0.045, 0.09, 0.045, 0.03, 0.01; ...0.01, 0.02, 0.04, 0.045, 0.04, 0.02, 0.01; ...0.01, 0.015, 0.02, 0.03, 0.02, 0.015, 0.01; ...0.01, 0.01, 0.01, 0.01, 0.01, 0.01, 0.01];% sum(sum(K))for x = 1:Mfor y = 1:Nfor i = 1:7for j = 1:7tmp1 = mod(x+4-i,M);tmp2 = mod(y+4-j,N);if tmp1 ==0tmp1 = M;endif tmp2 ==0tmp2 = N;endrtn(x,y) = rtn(x,y) + ...f(tmp1,tmp2) * K(i,j);endendendendreturn;end
end% % calculating normal distribution using R
% sigma = 2
% M = matrix(0, nrow = 2*sigma+1, ncol = 2*sigma+1)
% for(i in (-sigma):(sigma)){%   for(j in (-sigma):(sigma)){%       M[i+sigma+1,j+sigma+1] =
%       pnorm((i+0.5)/sigma)*pnorm((j+0.5)/sigma)-pnorm(((i+0.5)-1)/sigma)*pnorm((j+0.5)/sigma)-pnorm((i+0.5)/sigma)*pnorm(((j+0.5)-1)/sigma)+pnorm(((i+0.5)-1)/sigma)*pnorm(((j+0.5)-1)/sigma)
%
%   }
% }
% M

%main.m
pic = imread('pic0.jpg');
u = imresize(pic,0.4);
u = rgb2gray(u);
imwrite(u,'pic.jpg');pic = imread('pic.jpg');
u = double(pic);
s = size(u); M = s(1); N = s(2);
n_loop = 10;
sigma = 1; % 卷积的程度，1和2差不多
lambda = 3; % lambda越大，处理得到的图片越光滑% circular boundary value
for x = 2:M-1u(x,1) = u(x,N-1);u(x,N) = u(x,1+1);
end
for y = 2:N-1u(1,y) = u(M-1,y);u(M,y) = u(1+1,y);
enddiv_x = zeros(M,N); div_y = zeros(M,N);
D = zeros(M,N);
for i = 1:n_loopu_sigma = convolute(u,M,N,sigma);for x = 2:M-1for y = 2:N-1s = ((u_sigma(x,y)-u_sigma(x-1,y))^2+(u_sigma(x,y)-u_sigma(x,y-1))^2)^0.5;D(x,y) = exp(-s^5/(5*lambda^5));endendfor x = 2:M-1for y = 2:N-1div_x(x,y) = (u(x,y)-u(x-1,y))*D(x,y);div_y(x,y) = (u(x,y)-u(x,y-1))*D(x,y);endendfor x = 3:M-1for y = 3:N-1u(x,y) = u(x,y)+div_x(x,y)-div_x(x-1,y)+div_y(x,y)-div_y(x,y-1);endenddisp(i);if mod(i,1) == 0string = ['u_',num2str(i),'.jpg'];imwrite(uint8(u),string);end
end

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