I have a set of 2D points of a known density I want to mesh by taking the holes in account. Basically, given the following input:
I want something link this:
I tried PCL ConcaveHull, but it doens't handle the holes and splitted mesh very well.
I looked at CGAL Alpha shapes, which seems to go in the right direction (creating a polygon from a point cloud), but I don't know how to get triangles after that.
I though of passing the resulting polygons to a constrained triangulation algorithm and mark domains, but I didn't find how to get a list of polygons.
The resulting triangulated polygon is about a two step process at the least. First you need to triangulate your 2D points (using something like a Delaunay2D algorithm). There you can set the maximum length for the triangles and get the the desired shape. Then you can decimate the point cloud and re-triangulate. Another option is to use the convex hull to get the outside polygon, then extract the inside polygon through a TriangulationCDT algorithm, the apply some PolygonBooleanOperations, obtain the desired polygon, and finaly re-triangulate.
I suggest you look into the Geometric Tools library and specifically the Geometric Samples. I think everything you need is in there, and is much less library and path heavy than CGAL (the algorithms are not free for this type of work unless is a school project) or the PCL (I really like the library for segmentation, but their triangulation breaks often and is slow).
If this solves your problem, please mark it as your answer. Thank you!
Related
I'm still trying to density control (grade) meshes in CGAL. Specifically tet-meshing a polygon surface (or multiple surface manifolds) that I simply load as OFF files. I can also load lists of selected faces or face nodes too.
But I can't seem to get to first base on this with the polygon tet-mesher. All I want to do is assign and enforce a mesh density/size at selected faces in the OFF file.
I CAN get some kinds of mesh density working by inserting 1-D features with volumetric data meshing, but for CAD and 3D printing purposes it has to be computed from an STL-like triangular surface manifold, so volume-based meshing is not do-able.
Is what I'm trying to do even possible in CGAL? It feels to me like it must be, and I'm just missing something obvious.
I really hope someone can help here. FYI i'm mostly working with the Mesh3 example using v4.14.
Thanks very much.
Look at the Mesh_facet_criteria and in particular this constructor where SizingField is where you can control the size. For locating the point wrt a face, you can use the AABB-tree function closest_point_and_primitive().
I have various point clouds defining RT-STRUCTs called ROI from DICOM files. DICOM files are formed by tomographic scanners. Each ROI is formed by point cloud and it represents some 3D object.
The goal is to get 2D curve which is formed by plane, cutting ROI's cloud point. The problem is that I can't just use points which were intersected by plane. What I probably need is to intersect 3D concave hull with some plane and get resulting intersection contour.
Is there any libraries which have already implemented these operations? I've found PCL library and probably it should be able to solve my problem, but I can't figure out how to achieve it with PCL. In addition I can use Matlab as well - we use it through its runtime from C++.
Has anyone stumbled with this problem already?
P.S. As I've mentioned above, I need to use a solution from my C++ code - so it should be some library or matlab solution which I'll use through Matlab Runtime.
P.P.S. Accuracy in such kind of calculations is really important - it will be used in a medical software intended for work with brain tumors, so you can imagine consequences of an error (:
You first need to form a surface from the point set.
If it's possible to pick a 2d direction for the points (ie they form a convexhull in one view) you can use a simple 2D Delaunay triangluation in those 2 coordinates.
otherwise you need a full 3D surfacing function (marching cubes or Poisson)
Then once you have the triangles it's simple to calculate the contour line that a plane cuts them.
See links in Mesh generation from points with x, y and z coordinates
Perhaps you could just discard the points that are far from the plane and project the remaining ones onto the plane. You'll still need to reconstruct the curve in the plane but there are several good methods for that. See for instance http://www.cse.ohio-state.edu/~tamaldey/curverecon.htm and http://valis.cs.uiuc.edu/~sariel/research/CG/applets/Crust/Crust.html.
Hey so i was told in a previous answer that to make concave shapes out of multiple convex ones i do the following:
If you don't have a convex hull, perform a package wrapping algorithm
to get a convex border that encompasses all your points (again quite
fast). en.wikipedia.org/wiki/Gift_wrapping_algorithm
Choose a point that is on the boarder as a starter point for the algorithm.
Now, itterate through the following points that are on your shape,
but aren't on the convex border.
When one is found, create a new shape with the vertices from
the starter point to the found non-border point.
Finally set the starter point to be the the found off-border point
Recursion is now your friend: do the exact same process on each new
sub-shape you make.
I'm confused on one thing though. What do you do when two vertices in a row are off-border? After reaching the first one you connect the starter point to it, but then you immediatly run into another off-border point after you start itterating again, leaving you with only 2 vertices to work with: the starter point and new off-border point. What am i missing?
To illustrate my problem, here's a shape pertaining to this issue: It would be great if someone could draw all over it and walk through the steps of the algorithm using this. And using point 1 as the starting point.
Thanks!
Assuming you really want to take a convex polygon (as you've illustrated) and decompose it into convex parts without introducing new vertices, the usual approach is called "ear clipping" and is described in this Wikipedia article, Polygon triangulation. In this approach the convex pieces are triangles, which are necessarily convex.
This problem has been discussed in connection with the CGAL computational geometry software here in Stackoverflow, C++ 2D tessellation library.
I've got some convex polygons stored as an STL vector of points (more or less). I want to tessellate them really quickly, preferably into fairly evenly sized pieces, and with no "slivers".
I'm going to use it to explode some objects into little pieces. Does anyone know of a nice library to tessellate polygons (partition them into a mesh of smaller convex polygons or triangles)?
I've looked at a few I've found online already, but I can't even get them to compile. These academic type don't give much regard for ease of use.
CGAL has packages to solve this problem. The best would be probably to use the 2D Polygon Partitioning package. For example you could generate y-monotone partition of a polygon (works for non-convex polygons, as well) and you would get something like this:
The runnning time is O(n log n).
In terms of ease of use this is a small example code generating a random polygon and partitioning it (based on this manual example):
typedef CGAL::Exact_predicates_inexact_constructions_kernel K;
typedef CGAL::Partition_traits_2<K> Traits;
typedef Traits::Point_2 Point_2;
typedef Traits::Polygon_2 Polygon_2;
typedef std::list<Polygon_2> Polygon_list;
typedef CGAL::Creator_uniform_2<int, Point_2> Creator;
typedef CGAL::Random_points_in_square_2<Point_2, Creator> Point_generator;
int main( )
{
Polygon_2 polygon;
Polygon_list partition_polys;
CGAL::random_polygon_2(50, std::back_inserter(polygon),
Point_generator(100));
CGAL::y_monotone_partition_2(polygon.vertices_begin(),
polygon.vertices_end(),
std::back_inserter(partition_polys));
// at this point partition_polys contains the partition of the input polygons
return 0;
}
To install cgal, if you are on windows you can use the installer to get the precompiled library, and there are installations guides for every platform on this page. It might not be the simplest to install but you get the most used and robust computational geometry library there is out there, and the cgal mailing list is very helpful to answer questions...
poly2tri looks like a really nice lightweight C++ library for 2D Delaunay triangulation.
As balint.miklos mentioned in a comment above, the Shewchuk's triangle package is quite good. I have used it myself many times; it integrates nicely into projects and there is the triangle++ C++ interface. If you want to avoid slivers, then allow triangle to add (interior) Steiner points, so that you generate a quality mesh (usually a constrained conforming delaunay triangulation).
If you don't want to build the whole of GCAL into your app - this is probably simpler to implement.
http://www.flipcode.com/archives/Efficient_Polygon_Triangulation.shtml
I've just begun looking into this same problem and I'm considering voronoi tessellation. The original polygon will get a scattering of semi random points that will be the centers of the voronoi cells, the more evenly distributed they are the more regularly sized the cells will be, but they shouldn't be in a perfect grid otherwise the interior polygons will all look the same. So the first thing is to be able to generate those cell center points- generating them over the bounding box of the source polygon and a interior/exterior test shouldn't be too hard.
The voronoi edges are the dotted lines in this picture, and are sort of the complement of the delaunay triangulation. All the sharp triangle points become blunted:
Boost has some voronoi functionality:
http://www.boost.org/doc/libs/1_55_0/libs/polygon/doc/voronoi_basic_tutorial.htm
The next step is creating the voronoi polygons. Voro++ http://math.lbl.gov/voro++/ is 3D oriented but it is suggested elsewhere that approximately 2d structure will work, but be much slower than software oriented towards 2D voronoi. The other package that looks to be a lot better than a random academic homepage orphan project is https://github.com/aewallin/openvoronoi.
It looks like OpenCV used to support do something along these lines, but it has been deprecated (but the c-api still works?). cv::distTransform is still maintained but operates on pixels and generates pixel output, not vertices and edge polygon data structures, but may be sufficient for my needs if not yours.
I'll update this once I've learned more.
A bit more detail on your desired input and output might be helpful.
For example, if you're just trying to get the polygons into triangles, a triangle fan would probably work. If you're trying to cut a polygon into little pieces, you could implement some kind of marching squares.
Okay, I made a bad assumption - I assumed that marching squares would be more similar to marching cubes. Turns out it's quite different, and not what I meant at all.. :|
In any case, to directly answer your question, I don't know of any simple library that does what you're looking for. I agree about the usability of CGAL.
The algorithm I was thinking of was basically splitting polygons with lines, where the lines are a grid, so you mostly get quads. If you had a polygon-line intersection, the implementation would be simple. Another way to pose this problem is treating the 2d polygon like a function, and overlaying a grid of points. Then you just do something similar to marching cubes.. if all 4 points are in the polygon, make a quad, if 3 are in make a triangle, 2 are in make a rectangle, etc. Probably overkill. If you wanted slightly irregular-looking polygons you could randomize the locations of the grid points.
On the other hand, you could do a catmull-clark style subdivision, but omit the smoothing. The algorithm is basically you add a point at the centroid and at the midpoint of each edge. Then for each corner of the original polygon you make a new smaller polygon that connects the edge midpoint previous to the corner, the corner, the next edge midpoint, and the centroid. This tiles the space, and will have angles similar to your input polygon.
So, lots of options, and I like brainstorming solutions, but I still have no idea what you're planning on using this for. Is this to create destructible meshes? Are you doing some kind of mesh processing that requires smaller elements? Trying to avoid Gouraud shading artifacts? Is this something that runs as a pre-process or realtime? How important is exactness? More information would result in better suggestions.
If you have convex polygons, and you're not too hung up on quality, then this is really simple - just do ear clipping. Don't worry, it's not O(n^2) for convex polygons. If you do this naively (i.e., you clip the ears as you find them), then you'll get a triangle fan, which is a bit of a drag if you're trying to avoid slivers. Two trivial heuristics that can improve the triangulation are to
Sort the ears, or if that's too slow
Choose an ear at random.
If you want a more robust triangulator based on ear clipping, check out FIST.
From My last question: Marching Cube Question
However, i am still unclear as in:
how to create imaginary cube/voxel to check if a vertex is below the isosurface?
how do i know which vertex is below the isosurface?
how does each cube/voxel determines which cubeindex/surface to use?
how draw surface using the data in triTable?
Let's say i have a point cloud data of an apple.
how do i proceed?
can anybody that are familiar with Marching Cube help me?
i only know C++ and opengl.(c is a little bit out of my hand)
First of all, the isosurface can be represented in two ways. One way is to have the isovalue and per-point scalars as a dataset from an external source. That's how MRI scans work. The second approach is to make an implicit function F() which takes a point/vertex as its parameter and returns a new scalar. Consider this function:
float computeScalar(const Vector3<float>& v)
{
return std::sqrt(v.x*v.x + v.y*v.y + v.z*v.z);
}
Which would compute the distance from the point and to the origin for every point in your scalar field. If the isovalue is the radius, you just figured a way to represent a sphere.
This is because |v| <= R is true for all points inside a sphere, or which lives on its interior. Just figure out which vertices are inside the sphere and which ones are on the outside. You want to use the less or greater-than operators because a volume divides the space in two. When you know which points in your cube are classified as inside and outside, you also know which edges the isosurface intersects. You can end up with everything from no triangles to five triangles. The position of the mesh vertices can be computed by interpolating across the intersected edges to find the actual intersection point.
If you want to represent say an apple with scalar fields, you would either need to get the source data set to plug in to your application, or use a pretty complex implicit function. I recommend getting simple geometric primitives like spheres and tori to work first, and then expand from there.
1) It depends on yoru implementation. You'll need to have a data structure where you can lookup the values at each corner (vertex) of the voxel or cube. This can be a 3d image (ie: an 3D texture in OpenGL), or it can be a customized array data structure, or any other format you wish.
2) You need to check the vertices of the cube. There are different optimizations on this, but in general, start with the first corner, and just check the values of all 8 corners of the cube.
3) Most (fast) algorithms create a bitmask to use as a lookup table into a static array of options. There are only so many possible options for this.
4) Once you've made the triangles from the triTable, you can use OpenGL to render them.
Let's say i have a point cloud data of an apple. how do i proceed?
This isn't going to work with marching cubes. Marching cubes requires voxel data, so you'd need to use some algorithm to put the point cloud of data into a cubic volume. Gaussian Splatting is an option here.
Normally, if you are working from a point cloud, and want to see the surface, you should look at surface reconstruction algorithms instead of marching cubes.
If you want to learn more, I'd highly recommend reading some books on visualization techniques. A good one is from the Kitware folks - The Visualization Toolkit.
You might want to take a look at VTK. It has a C++ implementation of Marching Cubes, and is fully open sourced.
As requested, here is some sample code implementing the Marching Cubes algorithm (using JavaScript/Three.js for the graphics):
http://stemkoski.github.com/Three.js/Marching-Cubes.html
For more details on the theory, you should check out the article at
http://paulbourke.net/geometry/polygonise/