I want to detect the best rototraslation matrix between two set of points.
The second set of points is the same of the first, but rotated, traslated and affecteb by noise.
I tried to use least squared method by obviously the solution is usually similar to a rotation matrix, but with incompatible structure (for example, where i should get a value that represents the cosine of an angle i could get a value >1).
I've searched for the Constrained Least Squared method but it seems to me that the constrains of a rototraslation matrix cannot be expressed in this form.
In this PDF i've stated the problem more formally:
http://dl.dropbox.com/u/3185608/minquad_en.pdf
Thank you for the help.
The short answer: What you will need here is "Principal Component Analysis".
Apply this to both sets of points centered at their respective centers of mass. The PCA will effectively give you a rotation matrix for each aligned to the data set principal components. Multiplying the inverse matrix of the original set by the new rotation will give you a matrix that takes the old (centered) set to the new. Inverse translations and translations can similarly be applied to the rotation to create a homogeneous matrix that maps the one set to the other.
The book PRINCE, Simon JD. Computer vision: models, learning, and inference. Cambridge University Press, 2012.
gives, in Appendix "B.4 Reparameterization", some info about how to constrain a matrix to be a rotation matrix.
It seems to me that your problem has also a solution based on SVD: see the Kabsch algorithm also described by Olga Sorkine-Hornung and Michael Rabinovich in
Least-Squares Rigid Motion Using SVD and, more practically, by Nghia Kien Ho in FINDING OPTIMAL ROTATION AND TRANSLATION BETWEEN CORRESPONDING 3D POINTS.
Related
I use contract_until_convergence function from CGAL Mean_curvature_flow_skeletonization to produce skeleton from input polygon.
https://doc.cgal.org/latest/Surface_mesh_skeletonization/classCGAL_1_1Mean__curvature__flow__skeletonization.html
In some cases the skeleton creates branches (see top of the image above, skeleton in red color) that does not exist in input polygon. Is there some parameters to set to prevent this ?
using Skeletonization = CGAL::Mean_curvature_flow_skeletonization<Polyhedron>;
Skeletonization mean_curve_skeletonizer(polyhedron);
mean_curve_skeletonizer.contract_until_convergence();
There are two parameters controlling the quality of the skeleton:
quality_speed_tradeoff()
medially_centered_speed_tradeoff()
Also one thing that affect the skeleton is the sampling of the input surface that is used to compute Voronoi poles. In the original papers, it is said: Given a sufficiently good sampling, the Voronoi poles [ACK00] form a provably convergent sampling of the medial axis.
[ACK00]AMENTAN., CHOIS., KOLLURIR. K.: The powercrust, unions of balls, and the medial axis transform. Computational Geometry: Theory and Applications 19(2000), 127–153.3
You can use the function isotropic_remeshing with a sufficiently small target edge length to improve the Voronoi pole computation.
I am trying to extract camera matrix from essential matrix. I found some answers about this.
determine camera rotation and translation matrix from essential matrix
Rotation and Translation from Essential Matrix incorrect
In these answers, they suggest me to use newE where [U,S,V] = svd(E) and newE = U*diag(1,1,0)*Vt. I don't understand why I need to use newE. As I know, singular values are unique. So changing singular values to diag(1,1,0) seems to make E to completely different values.
I read 'Multiple View Geometry in Computer Vision' also, but it just refers to the ideal case, i.e., singular values are (1,1,0). I didn't find the reason of using newE.
Please can anyone explain me why people use newE?
If I understand your question correctly, then since you source data (and thus E) is usually noisy real world data, then by using diag(1,1,0) you are constraining the matrix to be of the correct scale and rank and algebraically enforcing the geometric constraints.
Wikipedia also a has a nice section explaining this.
I am currently creating a feature and patterning it across a flat plane to get the maximum number of features to fit on the plane. I do this frequently enough to warrant building some sort of marcro for this if possible. The issue that I run into is I still have to manually set the spacing between the parts. I want to be able to create a feature and have it determine "best" fit spacing given an area while avoiding overlaps. I have had very little luck finding any resources describing this. Any information or links to potentially helpful resources on this would be much appreciated!
Thank you.
Before, you start the linear pattern bit:
Select the face2 of that feature2, get the outer most loop2 of edges. You can test for that using loop2.IsOuter.
Now:
if the loop has one edge: that means it's a circle and the spacing must superior to the circle's radius
if the loop has more that one edge, that you need to calculate all the distances between the vertices and assume that the largest distance is the safest spacing.
NOTA: If one of the edges is a spline, then you need a different strategy:
You would need to convert the face into a sketch and finds the coordinates of that spline to calculate the highest distances.
Example: The distance between the edges is lower than the distance between summit of the splines. If the linear pattern has the a vertical direction, then spacing has to be superior to the distance between the summit.
When I say distance, I mean the distance projected on the linear pattern direction.
Basically, I have a set of up to 100 co-ordinates, along with the desired tangents to the curve at the first and last point.
I have looked into various methods of curve-fitting, by which I mean an algorithm with takes the inputted data points and tangents, and outputs the equation of the cure, such as the gaussian method and interpolation, but I really struggled understanding them.
I am not asking for code (If you choose to give it, thats acceptable though :) ), I am simply looking for help into this algorithm. It will eventually be converted to Objective-C for an iPhone app, if that changes anything..
EDIT:
I know the order of all of the points. They are not too close together, so passing through all points is necessary - aka interpolation (unless anyone can suggest something else). And as far as I know, an algebraic curve is what I'm looking for. This is all being done on a 2D plane by the way
I'd recommend to consider cubic splines. There is some explanation and code to calculate them in plain C in Numerical Recipes book (chapter 3.3)
Most interpolation methods originally work with functions: given a set of x and y values, they compute a function which computes a y value for every x value, meeting the specified constraints. As a function can only ever compute a single y value for every x value, such an curve cannot loop back on itself.
To turn this into a real 2D setup, you want two functions which compute x resp. y values based on some parameter that is conventionally called t. So the first step is computing t values for your input data. You can usually get a good approximation by summing over euclidean distances: think about a polyline connecting all your points with straight segments. Then the parameter would be the distance along this line for every input pair.
So now you have two interpolation problem: one to compute x from t and the other y from t. You can formulate this as a spline interpolation, e.g. using cubic splines. That gives you a large system of linear equations which you can solve iteratively up to the desired precision.
The result of a spline interpolation will be a piecewise description of a suitable curve. If you wanted a single equation, then a lagrange interpolation would fit that bill, but the result might have odd twists and turns for many sets of input data.
I'm trying to implement a geometry templating engine. One of the parts is taking a prototypical polygonal mesh and aligning an instantiation with some points in the larger object.
So, the problem is this: given 3d point positions for some (perhaps all) of the verts in a polygonal mesh, find a scaled rotation that minimizes the difference between the transformed verts and the given point positions. I also have a centerpoint that can remain fixed, if that helps. The correspondence between the verts and the 3d locations is fixed.
I'm thinking this could be done by solving for the coefficients of a transformation matrix, but I'm a little unsure how to build the system to solve.
An example of this is a cube. The prototype would be the unit cube, centered at the origin, with vert indices:
4----5
|\ \
| 6----7
| | |
0 | 1 |
\| |
2----3
An example of the vert locations to fit:
v0: 1.243,2.163,-3.426
v1: 4.190,-0.408,-0.485
v2: -1.974,-1.525,-3.426
v3: 0.974,-4.096,-0.485
v5: 1.974,1.525,3.426
v7: -1.243,-2.163,3.426
So, given that prototype and those points, how do I find the single scale factor, and the rotation about x, y, and z that will minimize the distance between the verts and those positions? It would be best for the method to be generalizable to an arbitrary mesh, not just a cube.
Assuming you have all points and their correspondences, you can fine-tune your match by solving the least squares problem:
minimize Norm(T*V-M)
where T is the transformation matrix you are looking for, V are the vertices to fit, and M are the vertices of the prototype. Norm refers to the Frobenius norm. M and V are 3xN matrices where each column is a 3-vector of a vertex of the prototype and corresponding vertex in the fitting vertex set. T is a 3x3 transformation matrix. Then the transformation matrix that minimizes the mean squared error is inverse(V*transpose(V))*V*transpose(M). The resulting matrix will in general not be orthogonal (you wanted one which has no shear), so you can solve a matrix Procrustes problem to find the nearest orthogonal matrix with the SVD.
Now, if you don't know which given points will correspond to which prototype points, the problem you want to solve is called surface registration. This is an active field of research. See for example this paper, which also covers rigid registration, which is what you're after.
If you want to create a mesh on an arbitrary 3D geometry, this is not the way it's typically done.
You should look at octree mesh generation techniques. You'll have better success if you work with a true 3D primitive, which means tetrahedra instead of cubes.
If your geometry is a 3D body, all you'll have is a surface description to start with. Determining "optimal" interior points isn't meaningful, because you don't have any. You'll want them to be arranged in such a way that the tetrahedra inside aren't too distorted, but that's the best you'll be able to do.