如何从多边形的离散细胞表示中生成多边形顶点的有序序列

Question

这是多边形的细胞表示：-

这是提取的多边形。

我们的输入是一个矩阵polM，它为粉色单元格存储 1，为黄色单元格存储 2。我们要从黄色单元格中创建一个多边形。这也可以使用从here 派生的以下代码来完成。

clear;
close all;
A=ones(25,25);
indices=[64,65,88,89,90,91,111,112,113,114,115,116,117,135,136,137,138,139,140,141,142,143,158,159,160,161,162,163,164,165,166,167,168,169,182,183,184,185,186,187,188,189,190,191,192,193,194,195,207,208,209,210,211,212,213,214,215,216,217,218,219,220,221,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,282,283,284,285,286,287,288,289,290,291,292,293,294,295,307,308,309,310,311,314,315,316,317,318,319,320,341,342,343,344,345,367,368,369,393,394];
A(indices)=2;

figure
hold on
axis equal
axis ij
xx=10:20:500;
yy=10:20:500;
imagesc(xx,yy,A);
colormap([1.0000         0    0.7931;
    1.0000    0.8276         0]);
rt=[0,500];
gcs=0:20:500;
gds=ones(size(rt,2),size(gcs,2));
gds = bsxfun(@times,gds,gcs);
plot(rt,gds,'Color','k','LineStyle','-');
plot(gds,rt,'Color','k','LineStyle','-');
axis([0 350 100 450])

A=A-1;
% we identify patterns with edges over 2x2 patches, describing with
% the first 4 binary values what pixels are set, and with the next 2
% the edge with 2 indices over the 2x2 patch
patterns = [
    0,1,0,1,  3,4 % vertical edge at rhe right
    1,0,1,0,  1,2 % vertical edge at the left
    0,0,1,1,  2,4 % horizontal edge at the bottom
    1,1,0,0,  1,3 % horizontal edge at the top
    1,0,0,1,  1,4 % diagonal edge
    0,1,1,0,  2,3 % diagonal edge
    1,0,1,1,  1,4 % diagonal edge, extra pixel set
    1,1,0,1,  1,4 % diagonal edge, extra pixel set
    1,1,1,0,  2,3 % diagonal edge, extra pixel set
    0,1,1,1,  2,3 % diagonal edge, extra pixel set
    ];

% 2x2 patches (matrix form)
P00 = A(1:end-1,1:end-1);
P10 = A(2:end,1:end-1);
P01 = A(1:end-1,2:end);
P11 = A(2:end,2:end);

% edge unique identifier using powers of 2
id = @(p00,p01,p10,p11) 1*p00 + 2*p10 + 4*p01 + 8*p11;
P = id(P00,P01,P10,P11); % vectorized pattern identification

% edges
e0 = []; % from (i,j)
e1 = []; % to (i,j)
for i = 1:size(patterns, 1) % small loop over the 10 patterns
    p = patterns(i, :);
    E = (P == id(p(1),p(2),p(3),p(4))); % pattern search, vectorized
    [c,r] = ind2sub(size(E), find(E));
    [c0,r0] = ind2sub([2,2], p(5));
    [c1,r1] = ind2sub([2,2], p(6));
    e0 = [e0; c+c0, r+r0];
    e1 = [e1; c+c1, r+r1];
end

X = [e0(:,2) e1(:,2)]';
Y = size(A,1) - [e0(:,1) e1(:,1)]';

figure
hold on
axis equal
axis ij
X=X*20;
Y=Y*20; Y=500-Y;
plot(X, Y, '.-')
axis([0 350 100 450])

但是代表多边形的这段代码的输出由数组X 和Y 给出。我想以 nx2 矩阵形式表示输出，如果最终多边形和顶点从第一个顶点开始按顺时针或逆时针顺序排列，则所有行都表示一个顶点。有一些方法here 可以将混乱的顶点序列后处理为有序序列，但我想要一种算法来确定来自输入矩阵polM 的多边形顶点仅在有序序列中。 SO 输入是polM，输出是一个 nx2 多边形顶点矩阵，只有 cw 或 ccw 顺序。

有人可以建议对算法进行有效的更改吗？

编辑：

如果你添加这段代码假设已经得到了有序的结果，你会得到错误：-

newp=[X(1,:)',Y(1,:)'];
figure
hold on
axis([0 350 100 450])
axis equal
axis ij
line(newp(:,1), newp(:,2));

基本上我想解决这个错误。

Answer 1

X 和Y 都是2xn 矩阵，指定n 无序顶点，即应该在[X(1, i) Y(1, i)] 和[X(2, i) Y(2, i)] 之间为每个i=1:n 画一条线。因为所有顶点都是连接的，所以每个点都应该在X 和Y 矩阵中出现两次。因此，可以从多边形上的任意点开始并迭代搜索下一个点。这可以按如下方式完成：

% XY_from: a `nx2` matrix containing all starting points 
% with a third column indicating if this vertex is already used
XY_from = [X(1, :)' Y(1, :)' zeros(size(X, 2), 1)]; 
XY_to = [X(2, :)' Y(2, :)' zeros(size(X, 2), 1)]; % similar matrix for destination points
new_poly = zeros(size(X, 2)+1, 2); % allocate output space
new_poly(1, :) = XY_from(1, 1:2); % select an arbitrary starting point
for i=2:size(new_poly, 1)
  index = find(all([new_poly(i-1, :) 0] == XY_from, 2), 1); % search the previous (unused) point in the starting points
  if isempty(index) 
    index = find(all([new_poly(i-1, :) 0] == XY_to, 2), 1); % if no point was found, search it in the destination points
    new_poly(i, :) = XY_from(index, 1:2); % add the new point (corresponding with the found destination point)
  else
    new_poly(i, :) = XY_to(index, 1:2); % add the new point (corresponding with the found start point)
  end
  XY_from(index, 3) = 1; % indicate that this start point is used
  XY_to(index, 3) = 1; % indicate that this destination point is used
end

hold on
plot(new_poly(:, 1), new_poly(:, 2), '--k', 'LineWidth', 2); % plot on top of the polygon