How to convert 2D triangulation to binary image?

Does anyone know of an easy way to convert a 2D triangulation to a binary image, which represents the inside?

Let my clarify my question with a picture: http://i41.tinypic.com/2gvvvnl.jpg The goal is to go from left to right...

My triangulation is given its Points and the ConnectivityList (i.e., it was created with the matlab function "triangulation").

It is important that no holes get filled and all features are preserved.

BEST ANSWER SO FAR:

% Make some data
pts = rand(50,2);
DT = delaunayTriangulation(pts);
% Find a central triangle to remove
[~,holeTrias] = min(sum(abs(DT.incenter - 0.5),2));
holeTrias = [holeTrias DT.neighbors(holeTrias)];
triasMask = ~ismember(1:DT.size(1), holeTrias);
T = triangulation(DT.ConnectivityList(triasMask,:), DT.Points);

% Define query points
X = 0:0.05:1;
Y = 0:0.05:1;
[xMat,yMat] = meshgrid(X,Y);

IN = false(size(xMat));
for t = 1:size(T,1)
    vertsXY = T.Points(T.ConnectivityList(t,:),:);
    IN = IN | inpolygon(xMat,yMat, vertsXY(:,1), vertsXY(:,2));
end

% Show the result
figure
patch('faces',T.ConnectivityList,'vertices',T.Points,'FaceColor','g')
hold on
plot(xMat(IN),yMat(IN),'b.',xMat(~IN),yMat(~IN),'r.')

Answer 1

[Sven]

Hi Geert,

Here's something that does what you want. It's not optimised for speed as it simply iterates naively through triangles, but it's at least correct.

% Make some data
pts = rand(50,2);
DT = delaunayTriangulation(pts);
% Find a central triangle to remove
[~,holeTrias] = min(sum(abs(DT.incenter - 0.5),2));
holeTrias = [holeTrias DT.neighbors(holeTrias)];
triasMask = ~ismember(1:DT.size(1), holeTrias);
T = triangulation(DT.ConnectivityList(triasMask,:), DT.Points);

% Define query points
X = 0:0.05:1;
Y = 0:0.05:1;
[xMat,yMat] = meshgrid(X,Y);

% Query each point
IN = false(size(xMat));
for i = 1:numel(IN)
    for t = 1:size(T,1)
        vertsXY = T.Points(T.ConnectivityList(t,:),:);
        if inpolygon(xMat(i),yMat(i), vertsXY(:,1), vertsXY(:,2))
            IN(i) = true;
            break;
        end
    end
end

% Show the result
figure
patch('faces',T.ConnectivityList,'vertices',T.Points,'FaceColor','g')
hold on
plot(xMat(IN),yMat(IN),'b.',xMat(~IN),yMat(~IN),'r.')

How does that work for you?

[Sven]

One little speedup can be gained by not considering points outside the convex hull:

% Query each point
IN = false(size(xMat));
k = convhull(T.Points);
for i = find(inpolygon(xMat,yMat, T.Points(k,1), T.Points(k,2)))'
    for t = 1:size(T,1)
        vertsXY = T.Points(T.ConnectivityList(t,:),:);
        if inpolygon(xMat(i),yMat(i), vertsXY(:,1), vertsXY(:,2))
            IN(i) = true;
            break;
        end
    end
end

[Matt Kindig]

Another speedup can probably be realized by using the FEX contribution inpoly ( http://www.mathworks.com/matlabcentral/fileexchange/10391-fast-points-in-polygon-test). In my experience, it is MUCH faster than the built-in inpolygon(), in addition to being more versatile.

[Sven]

Oh, yep, that's true Matt. That submission is indeed faster than the built-in inpolygon (which has a few extra things inside such as allowing for NaN coordinates to define breaks in polygons... I think Mathworks should probably mex-ify it anyway).

[Geert]

Thanks you guys, nice work!

I was able to improve the speed even more (by a factor 10), by applying some vectorization:

% Query each point
IN = false(size(xMat));
for t = 1:size(T,1)
    vertsXY = T.Points(T.ConnectivityList(t,:),:);
    IN = IN | inpolygon(xMat,yMat, vertsXY(:,1), vertsXY(:,2));
end

Speeds can be compared with the following code:

% Make some data
pts = rand(50,2);
DT = delaunayTriangulation(pts);
% Find a central triangle to remove
[~,holeTrias] = min(sum(abs(DT.incenter - 0.5),2));
holeTrias = [holeTrias DT.neighbors(holeTrias)];
triasMask = ~ismember(1:DT.size(1), holeTrias);
T = triangulation(DT.ConnectivityList(triasMask,:), DT.Points);

% Define query points
X = 0:0.05:1;
Y = 0:0.05:1;
[xMat,yMat] = meshgrid(X,Y);

tic
% Query each point
IN1 = false(size(xMat));
k = convhull(T.Points);
for i = find(inpolygon(xMat,yMat, T.Points(k,1), T.Points(k,2)))'
    for t = 1:size(T,1)
        vertsXY = T.Points(T.ConnectivityList(t,:),:);
        if inpolygon(xMat(i),yMat(i), vertsXY(:,1), vertsXY(:,2))
            IN1(i) = true;
            break;
        end
    end
end
disp(['First method elapsed time  = ', num2str(toc)])

tic
IN2 = false(size(xMat));
for t = 1:size(T,1)
    vertsXY = T.Points(T.ConnectivityList(t,:),:);
    IN2 = IN2 | inpolygon(xMat,yMat, vertsXY(:,1), vertsXY(:,2));
end
disp(['Second method elapsed time = ', num2str(toc)])

% Show the result
figure
subplot(1,2,1)
patch('faces',T.ConnectivityList,'vertices',T.Points,'FaceColor','g')
hold on
plot(xMat(IN1),yMat(IN1),'b.',xMat(~IN1),yMat(~IN1),'r.')
title('first result: IN1')
subplot(1,2,2)
patch('faces',T.ConnectivityList,'vertices',T.Points,'FaceColor','g')
hold on
plot(xMat(IN2),yMat(IN2),'b.',xMat(~IN2),yMat(~IN2),'r.')
title('second result: IN2')

Once again: many thanks!

[T. Dunker]

with R2018b and later this simplifies to

%% Make some data
pts = rand(50,2);
DT = delaunayTriangulation(pts);
% Find a central triangle to remove
[~,holeTrias] = min(sum(abs(DT.incenter - 0.5),2));
holeTrias = [holeTrias DT.neighbors(holeTrias)];
triasMask = ~ismember(1:DT.size(1), holeTrias);
T = triangulation(DT.ConnectivityList(triasMask,:), DT.Points);
% Define query points
X = 0:0.02:1;
Y = 0:0.02:1;

%% use polyshape
tic
[xMat,yMat] = meshgrid(X,Y);
bp = boundaryshape(T);
IN = reshape(isinterior(bp, xMat(:), yMat(:)), size(xMat));
toc

Elapsed time is 0.178962 seconds.

%%
colormap(gray(2))
imagesc(X, Y, IN)
hold on;
patch('faces', T.ConnectivityList, 'vertices', T.Points, ...
    'FaceColor', 'r', 'FaceAlpha', .2)
hold off;
axis equal;
axis tight;