My problem involves matching a set of 2d points to a set of 3d points, with known correspondence between the two. Basically I have points on an image, and I need the optimal translation and rotation to fit the points to a known 3d point cloud. Kabsch algorithm is originally meant for finding the best fit of 3d points to another point cloud, and there are implementations out there for 2d to 2d, but not something I can use. I do know it's possible, but just don't know how to go about it. I searched for code out there and came up empty. I'm programming in matlab at the moment, but any language would do.
Thank you.
Edit: The goal is getting a rotation and translation of the 3d point cloud to best match the 2d points when it is projected onto an image plane.
I should also mention that the 3d to 2d projection is done using a weak perspective.
So basically, you have a "plane" or a "line" of points, like the third dimension was 0. You could threat them like this, and use the tipicall kabsh algorithm of squared distance minimisation, don't you?
EDIT: maybe it's a nonsense, but what about projecting the 3d body to 2d coordinates, and do a 2d comparison? Computationally is expensive, so it includes exploring all the angles of the 3d object + projection, but it's easier losing one dimension by applying a projection, that adding a new dimenssion to a 2d point.
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Is there any simple algorithm like Voronoi diagram to divide any rectangular plane to triangles, eventually, using # of pre-defined points.
To be honest, I have to write a very simple fragment shader like this.
Theoretically, this Voronoii shader could be 'upgraded' by Delaunay triangulation
but wanna find the more elegant solution.
The first thing that comes to my mind is to create n random points (with specific seed) to fill a cylinder volume. The triangle points will be intersection of lines between those points and plane going through the axis of cylinder. The animation would be simply done by rotating the plane ...
I see it something like this:
So the neighboring points should be interconnected with each other. Forming tetrahedrons that fills the volume of the cylinder. So create uniform tetrahedron grid and add random noise to the points position (with specific seed).
This whole task is very similar to rendering cross section of 4D mesh see:
4D rendering techniques
As the 4D simplex is also tetrahedron. The only diference is you are in 3D and cutting by 3D plane.
You can reverse-engineer this example shadertoy.com/view/MdfBzl
like I did. Thanks to mattz.
I would like to find the best fit plane from a list of 3D points. It means the plane has the least square distance from all the points. I read the article
Best fit plane by minimizing orthogonal distances
and
3D Least Squares Plane
I fully understand the solutio but it turns other to be impractical in my situation. I need to read a very very large list of 3d points, direcltly impementation would result in ill posed problem. Even I subtract the data with their average,(refere to the document here-> part3 : http://www.geometrictools.com/Documentation/LeastSquaresFitting.pdf) the number is still very large. So what can I do?
Is there an iterative way to implement it ?
I have changed the way to ask the question, I hope may be there are someone can give me more advices on it ?
Given a list of 3D Points
{(x0,y0,z0),
(x1,y1,z1)...
(xn-1,yn-1,zn-1)}
I would like to construct a plane by fitting all the 3D points. In this sense, I mean to find the plane with format (Ax+By+Cy+D = 0), thus its uses four parameters(A,B,C,D) to characterize a plane. The sum of distance between each point and the plane should be minimium.
I do try the menthod provided in the below link
http://www.geometrictools.com/Documentation/LeastSquaresFitting.pdf
But there are two problems:
-During calculation, the above algorithm needs to do summation of all points value, which lead to overflow problem if my number of points increases
-given newly added points, it has to do all the calculation again, is there a way to use the before calculated plane parameter and the newly given points to somehow fine tune the planes parameters?
PS:I am a bit greedy, if we need to involve all the points, it is possible that the plane finally obtained isn't good enough.I am thinking of using random sample consensus(RANSAC), is it the right direction?
If you are expecting a plane then most of the points are not that useful since even a handful should give you a good approximation of the final solution (module a bit more noise).
So here's the solution. Sample down your data set to something that works and run the smaller set through the fitting algorithm.
If you are not expecting that the points are on a plane then sub-sampling should still work, but you must consider error ranges for any solution (since they will likely be fairly big).
I have surfed the web looking for 3D worlds triangulate code for use with three.js. I know there is the shape object, but it only can use 2D and paths. I'd have outer polygon, points, holes and 'forced' polylines.
How to deal with this and three? Do I have to use another JS framework?
Ideas would be appreciated, thanks.
Are you trying to triangulate a near-planar or planar 3d polygon?
If you have a near-planar or planar 3d-polygon, it can be triangulated by 'projecting' it down to 2d, running the triangulation, then 'de-projecting' back to 3d. Many libraries (such as CGAL, http://www.cgal.org ) do this by choosing one simple 'plane', the xy, yz, or xz plane, projecting down, doing the triangulation, and then 'projecting back up'.
The trick is that if you aren't adding any new points, you don't actually have to 'project back up', you just have to list your old 3d points in a new data structure, indicating which belong to which triangles. And if your code is very clever, you don't even need to 'convert' to 2d, you just need to fool your triangulation algorithm into thinking, for example, that y is x and z is y, (assuming you have projected onto the yz plane).
Thus, if you have a near-planar 3d polygon, and a good 2d-triangulation algorithm (three.js' shape might be sufficient, but if not, maybe some combination of Angus Johnson's Clipper might help, especially if you want to integrate stray points, lines, etc.), you just need to write a bit of 'glue' code to do the projection 3d->2d, and de-projection 2d->3d. Orientation (clockwise vs counterclockwise) of output can be a bit tricky, but it's not the most difficult thing in the world to program.
Good luck.
I have a 3D Cartesian cube. For each point in this cube there is a corresponding density value. When the density changes suddenly it means that there is a cavity. Now to find the cavity I calculate the gradient at each point in the cube. This gives me a point cloud on the surface of the cavity. I would now like to mesh the surface of the cavity given the point cloud.
Unfortunately I don't have any experience with surface reconstruction and was wondering if someone can recommend a suitable algorithm which will produce a closed surface of the cavity?
The cube is quite big so a point cloud of the surface of a cavity can easily be 500.000 points or more. I have read this post: robust algorithm for surface reconstruction from 3D point cloud? which I find useful. However it seems that the problem I am facing is simpler, given that:
The coordinates of the points are always integer
The point distribution is even
The distance from one point to its closest neighbour is either 1, sqrt(2) or sqrt(3)
You probably want the marching cubes algorithm.
The Marching Cubes algorithm will do exactly what you want. For a working implementation (using Three.js for rendering the graphics), check out:
http://stemkoski.github.com/Three.js/Marching-Cubes.html
For more details on the theory, I think the best article is the website:
http://paulbourke.net/geometry/polygonise/
We have a grid with red squares on it. Meaning we have an array of 3 squares (with angles == 90 deg) which as we know have same size, lying on the same plane and with same rotation relative to the plane they are lying on, and are not situated on same line on plane.
We have a projection of the space which contains the plane with squares.
We want to turn our plane projection with squares so that we would see it like it's facing us, in general we need a formula for turning each point of that original plane projection so that it would be facing us like on the image below.
What formulas can be used for solving such problem, how to solve it, has any one faced something like this before?
This is a special case of finding mappings between quadrilaterals that preserve straight lines. These are generally called homographic transforms. Here, one of the quads is a square, so this is a popular special case. You can google these terms ("quad to quad", etc) to find explanations and code, but here are some for you.
Perspective Transform Estimation
a gaming forum discussion
extracting a quadrilateral image to a rectangle
Projective Warping & Mapping
ProjectiveMappings for ImageWarping by Paul Heckbert.
The math isn't particularly pleasant, but it isn't that hard either. You can also find some code from one of the above links.