I'd like to build triangulation for larger that 3 dimensions (4-6) cases. I have points representing non-convex surface. for 2D and 3D cases DelaunayTri is a way to go. What about higher dimensions?
(Original problem is to approximate some non-linear hypersurface with linear hyperplanes)
Regards,
Andrey
Use delaunayn(), check out the matlab documentaion: http://www.mathworks.com/help/techdoc/ref/delaunayn.html it explains it pretty well.
Related
TL;DR - How do I only analyse the surface data points for surface estimation?
I have a 3D object, and I would like to estimate the surface shape.
The problem is: MATLAB curve fitting toolbox takes into account all the data points of the object. See the example of a cylinder and it's approximated polynomial. MATLAB is taking into account all the data points for surface estimation, what can I do to over come this?
Assuming that the outer and inner surface have identical shape, you can first do an imfill to make the object solid and then use morphological skeleton, bwmorph(BW,'skel',Inf) to turn it into a single line, which you then can approximate the shape from.
I have a 3D point cloud that I've transformed into a Delaunay triangulation using Matlab's function DelaunayTri. Now I have a test point in 3D and want to compute, in Matlab, the smallest distance between this point and the triangulation.
So far, I've thought of using the nearestNeighbor(...) member function of the DelaunayTri class in Matlab to find the point in the triangulation closest to my test point and then computing the distance between them. That is something but it is not what I really want.
The closest point on the triangulation to my test point is, in general, not a vertex of the triangulation but somewhere on the face of a triangle. How can I find this point?
Thank you!!!
I've written codes for these things, but they are not on the File Exchange. I can be convinced to give them out by direct mail though.
It is relatively easy to find the distance to a convex hull, but not trivial. A delaunay tessellation is bounded by the convex hull anyway. So you can convert that tessellation into the convex hull easily enough, or just use the convex hull. Note that a convex hull is often a terribly poor approximation for many purposes, especially if you are using this for color mapping, which is perhaps the most common thing I see this used for. In that case, an alpha shape is a far better choice. Alpha shapes also will have a triangulated boundary surface, though in general it will not be convex.
So, to find the nearest point on a convex triangulation:
Convert to the convex boundary surface, i.e., the convex hull. This reduces to finding those triangles which are not shared between pairs of tetrahedra. Internal facets will always appear exactly twice in the list of all facets. This trick also works for non-convex tessellations of course, so for alpha shapes.
Compute a bounding circumcircle for each triangular surface facet. This allows you to know when to stop checking facets.
Get the distances to each point on the surface. Sort each facet by the distance to the nearest point in that facet. Start by looking at the nearest facet in that list.
Compute the distance to the apparently nearest facet found in step 3. A simple solution is found by least distance programming (LDP), which can be converted to a constrained linear least squares. Lawson & Hanson have an algorithm for this.
Repeat step 4 until the current best distance found is less than the distance, comparing it to any of the circumcircles from step 2. This loop will be quite short really, at least for a convex hull. For a more general non-convex hull from an alpha shape, it may take more time.
You can also reduce the search space a bit by excluding the facets from your search that point AWAY from the point in question. Use those facet normals for this test.
I wrote the tool point2trimesh for this problem. It's kind of a "brute force" solution which works also for non-convex surfaces.
I want to know how griddata uses linear interpolation.
Does it just consider like four neighbor points of the point where we want to find the value or uses the whole convex hull?
The docs for griddata will guide you in this. Essentially, it does a delaunay triangulation of your point set, then for any new point, determine which simplex the point falls in. Interpolation within a simplex is linear. Thus in 2 dimensions, three points define a triangle, and 3 points determine a locally planar model for z(x,y).
I have this iPhone app that has an array containing around 50 to 100 points. How do I calculate the smoothest curve that will fit the points? It can be bezier, cubic, quadratic, whatever. It just have to look smooth and fit as much as possible all points (obviously, as I did in my drawing, to create a smooth curve, some points have to be created out of the original set... no problem).
See picture:
Maybe you are looking for a Cubic Spline
Cubic Spline
These are the functions with continous second derivative that interpolate your nodes with the smallest curvature so they oscillate less. And there are lots of examples and algorithms to find these.
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