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调和矩阵
来自维基百科,自由的百科全书
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在图论中,调和矩阵(harmonic matrix),也称拉普拉斯矩阵或拉氏矩阵(Laplacian matrix)、离散拉普拉斯(discrete Laplacian),是图的矩阵表示。[1]
此條目翻譯品質不佳。 (2020年2月17日) |
定义
若G是简单图,G有n个顶点,A是邻接矩阵,D是度数矩阵,则调和矩阵是[1]
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这跟拉普拉斯算子有什么关系?若f 是加权图G的顶点函数,则[2]
w是边的权重函数。u、v是顶点。f = (f(1), ..., f(n)) 是n维的矢量。上面泛函也称为Dirichlet泛函。[3]
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而且若K是接续矩阵(incidence matrix),则[2]
Kf 是f 的图梯度。另外,特征值满足
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举例
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其他形式
注意[4]
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动力学和微分方程
使用矩阵矢量
解是
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当的时候,
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N = 20;%The number of pixels along a dimension of the image
A = zeros(N, N);%The image
Adj = zeros(N*N, N*N);%The adjacency matrix
%Use 8 neighbors, and fill in the adjacency matrix
dx = [-1, 0, 1, -1, 1, -1, 0, 1];
dy = [-1, -1, -1, 0, 0, 1, 1, 1];
for x = 1:N
for y = 1:N
index = (x-1)*N + y;
for ne = 1:length(dx)
newx = x + dx(ne);
newy = y + dy(ne);
if newx > 0 && newx <= N && newy > 0 && newy <= N
index2 = (newx-1)*N + newy;
Adj(index, index2) = 1;
end
end
end
end
%%%BELOW IS THE KEY CODE THAT COMPUTES THE SOLUTION TO THE DIFFERENTIAL
%%%EQUATION
Deg = diag(sum(Adj, 2));%Compute the degree matrix
L = Deg - Adj;%Compute the laplacian matrix in terms of the degree and adjacency matrices
[V, D] = eig(L);%Compute the eigenvalues/vectors of the laplacian matrix
D = diag(D);
%Initial condition (place a few large positive values around and
%make everything else zero)
C0 = zeros(N, N);
C0(2:5, 2:5) = 5;
C0(10:15, 10:15) = 10;
C0(2:5, 8:13) = 7;
C0 = C0(:);
C0V = V'*C0;%Transform the initial condition into the coordinate system
%of the eigenvectors
for t = 0:0.05:5
%Loop through times and decay each initial component
Phi = C0V.*exp(-D*t);%Exponential decay for each component
Phi = V*Phi;%Transform from eigenvector coordinate system to original coordinate system
Phi = reshape(Phi, N, N);
%Display the results and write to GIF file
imagesc(Phi);
caxis([0, 10]);
title(sprintf('Diffusion t = %3f', t));
frame = getframe(1);
im = frame2im(frame);
[imind, cm] = rgb2ind(im, 256);
if t == 0
imwrite(imind, cm, 'out.gif', 'gif', 'Loopcount', inf, 'DelayTime', 0.1);
else
imwrite(imind, cm, 'out.gif', 'gif', 'WriteMode', 'append', 'DelayTime', 0.1);
end
end

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