I'm trying to get the Trajectory from Voronoi Diagram using the library voronoi from Matlab. I'm using this code:
vo = (all the obstacles from a binary picture, plotted in a figure), where:
vo(1,:) : x-axis points
vo(2,:) : y-axis points
Code:
figure; hold on;
plot(vo(1,:),vo(2,:),'sr');
[vx,vy] = voronoi(vo(1,:),vo(2,:));
plot(vx,vy,'-b');
Obtaining:
In other words, how can I separate all the useless lines from the real trajectory?
The distinguishing property of the useless lines in this case is that they contain at least one vertex inside the polygonal obstacles.
There are many ways you might choose to decide if a vertex satisfies this condition but given that the coordinates come from a binary image, one of the most straightforward might be to check if the vertex comes within a pixel's distance of any point in vo:
[~,D] = knnsearch(vo,[vx(:),vy(:)]);
inObstacle = any(reshape(D,size(vx)) < 1);
plot(vx(:,~inObstacle),vy(:,~inObstacle),'-b');
If you cannot rely on the obstacles being filled with pixels (e.g. they may be hollow) then you probably need to determine which pixels belong to the same obstacle (perhaps using kmeans or bwconncomp) and then eliminate Voronoi edges that intrude into each object's convex hull. For a given obstacle made up of the pixels in vo listed by the linear indices idx:
K = convhull(vo(idx,1),vo(idx,2));
inObstacle = inpolygon(vx,vy,vo(idx(K),1),vo(idx(K),2));
Related
I have a surface by the code below and another surface which is created by the exact same code. I want to see the height differences in another figure. How am I able to do that? Already operated with the Minus-operator but this won't work.
Furthermore the matrices have NOT the same size!
Appreciate your help!
x1 = Cx1;
y1 = Cy1;
z1 = Cz1;
tri1 = delaunay(x1,y1);
fig1 = figure%('units','normalized','outerposition',[0 0 1 1]);
trisurf(tri1,x2,y2,z2)
xlabel('x [mm] ','FontSize',30)
ylabel('y [mm] ','FontSize',30)
zlabel('z [mm] ','FontSize',30)
The simplest way to solve this problem is to interpolate from one mesh onto the other one. Such an approach works well when one is more highly resolved than the other, or when you're not as concerned with results at individual nodes, but rather the overall pattern across elements. If that's not the case, then you have a very complicated problem because you need to create a polygonal surface that fully captures all nodes and edges of both triangulations. Consider the following pair of triangular patterns:
A surface that captured all the variations would need to have all the vertices and edges that make up both of them, which is not a purely triangular surface. So, let us instead assume the easier case. To map results from one triangulation to the other, you simply need to formulate functions that define how the values vary along the triangles, which are more broadly called basis functions. It is often assumed that values betweeen the nodes (i.e. vertices) of the triangles vary linearly along the surfaces of the triangles. You can do it differently if you want, it just requires defining new basis functions. If we go for linear functions, then the equations in 2D are pretty simple. Let's say you make an array trimap that has which triangle each of the vertices of the other triangulation is inside of. This can be accomplished using the info here. Then, we set the coordinates of the vertices of the current triangle to (x1,y1), (x2,y2), and (x3,y3), and then do the math:
for cnt1=1,npoints
x1=x(tri1(trimap(cnt1),1));
x2=x(tri1(trimap(cnt1),2));
x3=x(tri1(trimap(cnt1),3));
y1=y(tri1(trimap(cnt1),1));
y2=y(tri1(trimap(cnt1),2));
y3=y(tri1(trimap(cnt1),3));
delta=x2*y3+x1*y2+x3*y1-x2*y1-x1*y3-x3*y2;
delta1=(x2*y3-x3*y2+xstat(cnt1)*(y2-y3)+ystat(cnt1)*(x3-x2));
delta2=(x3*y1-x1*y3+xstat(cnt1)*(y3-y1)+ystat(cnt1)*(x1-x3));
delta3=(x1*y2-x2*y1+xstat(cnt1)*(y1-y2)+ystat(cnt1)*(x2-x1));
weights(cnt1,1)=delta1/delta;
weights(cnt1,2)=delta2/delta;
weights(cnt1,3)=delta3/delta;
z1=z(tri1(trimap(cnt1),1));
z2=z(tri1(trimap(cnt1),2));
z3=z(tri1(trimap(cnt1),3));
valinterp(cnt1)=sum(weights(cnt1,:).*[z1,z2,z3]);
end
valinterp is the interpolated value for each point. Here and here are some nice slides explaining the mathematics behind all this. Note that I've not tested any of this code. Note also that you will need to do something to assign to values outside of the triangulation. Perhaps a null value, or an inverse-distance weighted value.
I have some binary images that want to classify them base on shape of them in MATLAB. If they have circular or elliptical shape they belong to class one,if they have elliptical shape with dent in their boundary they belong to class two. I dont know how can I use this feature. Can any body help me with this?
You can use the eccentricity property in regionprops. From MATLAB documentation of eccentricity:
The eccentricity is the ratio of the distance between the foci of the ellipse and its major axis length. The value is between 0 and 1. (0 and 1 are degenerate cases. An ellipse whose eccentricity is 0 is actually a circle, while an ellipse whose eccentricity is 1 is a line segment.)
So as the value of eccentricity increases , the ellipse starts becoming flatter. Hence, at its maximum value = 1, it is a line segment.
To check if there is a dent in the ellipse, you can use check for convexity. Whenever there is a dent in an ellipse, it will be non-convex. In other words, if you try to fit a convex polygon, it won't be able to approximate the shape well enough. You can use convexArea property to check the same. From MATLAB documentation of convexArea:
Returns a p-by-2 matrix that specifies the smallest convex polygon that can contain the region. Each row of the matrix contains the x- and y-coordinates of one vertex of the polygon. Only supported for 2-D label matrices.
So you use bwlabel to create a 2-D label matrix from your binary image and then check the difference between the area of your binary image and the area of the fitted convex polygon. Measuring area could be as simple as counting pixels. You already know that the number pixels of your fitted convex polygon = p. Just take the absolute difference between p and the number of pixels in your original binary image. You should be able to easily set a threshold to classify into one of the two classes.
I think you can write the code for this. Hope this helps.
I have a list of triangles in 3D that form a surface (ie a triangulation). The structure is a deformed triangular lattice. I want to know the change in area of the deformed hexagons of the voronoi tessalation of the lattice with respect to the rest area of the undeformed lattice cells (ie with respect to a regular hexagon). In fact, I really want the sum of the squared change in area of the hexagonal unit cells associated with those triangles.
Background/Math details:
I'm approximating a curved elastic sheet by a triangular lattice. One way to tune the poisson ratio (elastic constant) of the sheet is by adding a 'volumetric' strain energy term to the energy. I'm trying to compute a 'volumetric' strain energy of a deformed, elastic, triangular lattice, defined as: U_volumetric = 1/2 T (e_v)^2, where e_v=deltaV/V is determined by the change in area of a voronoi cell with respect to its reference area, which is a known constant.
Reference: https://www.researchgate.net/publication/265853755_Finite_element_implementation_of_a_non-local_particle_method_for_elasticity_and_fracture_analysis
Want:
Sum[ (DeltaA/ A).^2 ] over all hexagonal cells.
My data is stored in the variables:
xyz = [ x1,y1,z1; x2,y2,z2; etc] %the vertices/particles in 3D
TRI = [ vertex0, vertex1, vertex2; etc] %
where vertex0 is the row of xyz for the particle sitting at vertex 0 of the first triangle.
NeighborList = [ p1n1, p1n2, p1n3, p1n4, p1n5,p1n6 ; p2n1...]
% where p1n1 is particle 1's first nearest neighbor as a row index for xyz. For example, xyz(NL(1,1),:) returns the xyz location of particle 1's first neighbor.
AreaTRI = [ areaTRI1; areaTRI2; etc]
I am writing this in MATLAB.
As of now, I am approximating the amount of area attributed to each vertex as 1/3 of the triangle's area, then summing over the 6 nearest neighbor triangles. But a voronoi cell area will NOT be exactly equal to Sum_(i=0,1,...5) 1/3* areaTRI_i, so this is a bad approximation. See the image in the link above, which I think makes this clearer.
You can do this using the DUALMESH-submission on the file exchange:
DUALMESH is a toolbox of mesh processing routines that allow the construction of "dual" meshes based on underlying simplicial triangulations. Support is provided for various planar and surface triangulation types, including non-Delaunay and non-manifold types.
Simply use the following commands to generate a vector areas of all the dual elements' areas. The ordering will correspond to the nodes xyz.
[cp,ce,pv,ev] = makedual2(xyz, TRI);
[~,areas(cp(:,1))] = geomdual2(cp,ce,pv,ev);
You might want to have a look at the boundary areas using:
trisurf(TRI, xyz(:,1), xyz(:,2), areas);
The dual cells of boundary nodes theoretically are unbounded and thus should have infinite area. This submission handles it differently however: Instead of an unbounded cell it will return the intersection of the unbounded cell with the original mesh.
Also mind that your question is not well defined if the mesh you are working with is not planar, as the dual mesh cells will be planar and won't scale the same way as the triangles. So this solution will probably only work correctly if your mesh is really 2D. (From what I can tell, the paper you mention is also only for the 2D-case.)
I have a set of points which I want to propagate on to the edge of shape boundary defined by a binary image. The shape boundary is defined by a 1px wide white edge.
I have the coordinates of these points stored in a 2 row by n column matrix. The shape forms a concave boundary with no holes within itself made of around 2500 points. I have approximately 80 to 150 points that I wish to propagate on the shape boundary.
I want to cast a ray from each point from the set of points in an orthogonal direction and detect at which point it intersects the shape boundary at. The orthogonal direction has already been determined. For the required purposes it is calculated taking the normal of the contour calculated for point, using point-1 and point+1.
What would be the best method to do this?
Are there some sort of ray tracing algorithms that could be used?
Thank you very much in advance for any help!
EDIT: I have tried to make the question much clearer and added a image describing the problem. In the image the grey line represents the shape contour, the red dots the points
I want to propagate and the green line an imaginary orthongally cast ray.
alt text http://img504.imageshack.us/img504/3107/orth.png
ANOTHER EDIT: For clarification I have posted the code used to calculate the normals for each point. Where the xt and yt are vectors storing the coordinates for each point. After calculating the normal value it can be propagated by using the linspace function and the requested length of the orthogonal line.
%#derivaties of contour
dx=[xt(2)-xt(1) (xt(3:end)-xt(1:end-2))/2 xt(end)-xt(end-1)];
dy=[yt(2)-yt(1) (yt(3:end)-yt(1:end-2))/2 yt(end)-yt(end-1)];
%#normals of contourpoints
l=sqrt(dx.^2+dy.^2);
nx = -dy./l;
ny = dx./l;
normals = [nx,ny];
It depends on how many unit vectors you want to test against one shape. If you have one shape and many tests, the easiest thing to do is probably to convert your shape coordinates to polar coordinates which implicitly represent your solution already. This may not be a very effective solution however if you have different shapes and only a few tests for every shape.
Update based on the edited question:
If the rays can start from arbitrary points, not only from the origin, you have to test against all the points. This can be done easily by transforming your shape boundary such that your ray to test starts in the origin in either coordinate direction (positive x in my example code)
% vector of shape boundary points (assumed to be image coordinates, i.e. integers)
shapeBoundary = [xs, ys];
% define the start point and direction you want to test
startPoint = [xsp, ysp];
testVector = unit([xv, yv]);
% now transform the shape boundary
shapeBoundaryTrans(:,1) = shapeBoundary(:,1)-startPoint(1);
shapeBoundaryTrans(:,2) = shapeBoundary(:,2)-startPoint(2);
rotMatrix = [testVector(2), testVector(1); ...
testVector(-1), testVector(2)];
% somewhat strange transformation to keep it vectorized
shapeBoundaryTrans = shapeBoundaryTrans * rotMatrix';
% now the test is easy: find the points close to the positive x-axis
selector = (abs(shapeBoundaryTrans(:,2)) < 0.5) & (shapeBoundaryTrans(:,1) > 0);
shapeBoundaryTrans(:,2) = 1:size(shapeBoundaryTrans, 1)';
shapeBoundaryReduced = shapeBoundaryTrans(selector, :);
if (isempty(shapeBoundaryReduced))
[dummy, idx] = min(shapeBoundaryReduced(:, 1));
collIdx = shapeBoundaryReduced(idx, 2);
% you have a collision with point collIdx of your shapeBoundary
else
% no collision
end
This could be done in a nicer way probably, but you get the idea...
If I understand your problem correctly (project each point onto the closest point of the shape boundary), you can
use sub2ind to convert the "2 row by n column matrix" description to a BW image with white pixels, something like
myimage=zeros(imagesize);
myimage(imagesize, x_coords, y_coords) = 1
use imfill to fill the outside of the boundary
run [D,L] = bwdist(BW) on the resulting image, and just read the answers from L.
Should be fairly straightforward.
I have a polyhedron, with a list of vertices (v) and surfaces (s). How do I break this polyhedron into a series of tetrahedra?
I would particularly like to know if there are any built-in MATLAB commands for this.
For the convex case (no dents in the surface which cause surfaces to cover each other) and a triangle mesh, the simple solution is to calculate the center of the polyhedron and then connect the three corners of every face with the new center.
If you don't have a triangle mesh, then you must triangulate, first. Delaunay triangulation might help.
If there are holes or caves, this can be come arbitrarily complex.
I'm not sure the OP wanted a 'mesh' (Steiner points added) or a tetrahedralization (partition into tetrahedra, no Steiner points added). for a convex polyhedron, the addition of Steiner points (e.g. the 'center' point) is not necessary.
Stack overflow will not allow me to comment on gnovice's post (WTF, SO?), but the proof of the statement "the surfaces of a convex polyhedron are constraints in a Delaunay Tesselation" is rather simple: by definition, a simplex or subsimplex is a member in the Delaunay Tesselation if and only if there is a n-sphere circumscribing the simplex that strictly contains no point in the point set. for a surface triangle, construct the smallest circumscribing sphere, and 'puff' it outwards, away from the polyhedron, towards 'infinity'; eventually it will contain no other point. (in fact, the limit of the circumscribing sphere is a half-space; thus the convex hull is always a subset of the Delaunay Tesselation.)
for more on DT, see Okabe, et. al, 'Spatial Tesselations', or any of the papers by Shewchuk
(my thesis was on this stuff, but I remember less of it than I should...)
I would suggest trying the built-in function DELAUNAY3. The example given in the documentation link resembles Aaron's answer in that it uses the vertices plus the center point of the polyhedron to create a 3-D Delaunay tessellation, but shabbychef points out that you can still create a tessellation without including the extra point. You can then use TETRAMESH to visualize the resulting tetrahedral elements.
Your code might look something like this (assuming v is an N-by-3 matrix of vertex coordinate values):
v = [v; mean(v)]; %# Add an additional center point, if desired (this code
%# adds the mean of the vertices)
Tes = delaunay3(v(:,1),v(:,2),v(:,3)); %# Create the triangulation
tetramesh(Tes,v); %# Plot the tetrahedrons
Since you said in a comment that your polyhedron is convex, you shouldn't have to worry about specifying the surfaces as constraints in order to do the triangulation (shabbychef appears to give a more rigorous and terse proof of this than my comments below do).
NOTE: According to the documentation, DELAUNAY3 will be removed in a future release and DelaunayTri will effectively take its place (although currently it appears that defining constrained edges is still limited to only 2-D triangulations). For the sake of completeness, here is how you would use DelaunayTri and visualize the convex hull (i.e. polyhedral surface) as well:
DT = DelaunayTri(v); %# Using the same variable v as above
tetramesh(DT); %# Plot the tetrahedrons
figure; %# Make new figure window
ch = convexHull(DT); %# Get the convex hull
trisurf(ch,v(:,1),v(:,2),v(:,3),'FaceColor','cyan'); %# Plot the convex hull