I have installed Point Cloud Library(PCL) package for using Iterative closest point (icp) my question is: this package could be used for 2D data or not ?I want to align two TSNE data which are 2D.Link of the icp method
It is not a problem even if it supported only 3D. You can add a z=0 coordinate to your data points and the z component will always be zero in your solution.
PCL does not have a paired data ICP to the best of my knowledge, but it should be pretty trivial to write as the cost function just considers squared distance between each pair.
I would create clouds of the two sets of points, setting z to zero, write your cost function (sum of squared distances of each pair), and then cycle the icp through xy translation steps and z rotation steps using reverse half-ing style icp stepping.
Related
Say I had a point cloud with n number of points in 3d space(relatively densely packed together). What is the most efficient way to create a surface that goes contains every single point in it and lets me calculate values such as the normal and curvature at some point on the surface that was created? I also need to be able to create this surface as fast as possible(a few milliseconds hopefully working with python) and it can be assumed that n < 1000.
There is no "most efficient and effective" way (this is true of any problem in any domain).
In the first place, the surface you have in mind is not mathematically defined uniquely.
A possible approach is by means of the so-called Alpha-shapes, implemented either from a Delaunay tetrahedrization, or by the ball-pivoting method. For other methods, lookup "mesh reconstruction" or "surface reconstruction".
On another hand, normals and curvature can be computed locally, from neighbors configurations, without reconstructing a surface (though there is an ambiguity on the orientation of the normals).
I could suggest Nina Amenta's Power Crust algorithm (link to code), or also meshlab suite, which can compute the curvatures too.
Is there a library in python or c++ that is capable of estimating normals of point clouds in a consistent way?
In a consistent way I mean that the orientation of the normals is globally preserved over the surface.
For example, when I use python open3d package:
downpcd.estimate_normals(search_param=o3d.geometry.KDTreeSearchParamHybrid(
radius=4, max_nn=300))
I get an inconsistent results, where some of the normals point inside while the rest point outside.
many thanks
UPDATE: GOOD NEWS!
The tangent plane algorithm is now implemented in Open3D!
The source code and the documentation.
You can just call pcd.orient_normals_consistent_tangent_plane(k=15).
And k is the knn graph parameter.
Original answer:
Like Mark said, if your point cloud comes from multiple depth images, then you can call open3d.geometry.orient_normals_towards_camera_location(pcd, camera_loc) before concatenating them together (assuming you're using python version of Open3D).
However, if you don't have that information, you can use the tangent plane algorithm:
Build knn-graph for your point cloud.
The graph nodes are the points. Two points are connected if one is the other's k-nearest-neighbor.
Assign weights to the edges in the graph.
The weight associated with edge (i, j) is computed as 1 - |ni ⋅ nj|
Generate the minimal spanning tree of the resulting graph.
Rooting the tree at an initial node,
traverse the tree in depth-first order, assigning each node an
orientation that is consistent with that of its parent.
Actually the above algorithm comes from Section 3.3 of Hoppe's 1992
SIGGRAPH paper Surface Reconstruction from Unorganized Points. The algorithm is also open sourced.
AFAIK the algorithm does not guarantee a perfect orientation, but it should be good enough.
If you know the viewpoint from where each point was captured, it can be used to orient the normals.
I assume that this not the case - so given your situation, which seems rather watertight and uniformly sampled, mesh reconstruction is promising.
PCL library offers many alternatives in the surface module. For the sake of normal estimation, I would start with either:
ConcaveHull
Greedy projection triangulation
Although simple, they should be enough to produce a single coherent mesh.
Once you have a mesh, each triangle defines a normal (the cross product). It is important to note that a mesh isn't just a collection of independent faces. The faces are connected and this connectivity enforces a coherent orientation across the mesh.
pcl::PolygonMesh is an "half edge data structure". This means that every triangle face is defined by an ordered set of vertices, which defines the orientation:
order of vertices => order of cross product => well defined unambiguous normals
You can either use the normals from the mesh (nearest neighbor), or calculate a low resolution mesh and just use it to orient the cloud.
I made an object tracker that calculates the position of an object recorded in a live camera feed using stereoscopic cameras. The math was simple, once you know the camera distance and orientation. However, now I thought it would be nice to allow me to quickly extract all these parameters, so when I change my setup or cameras I will be able to quickly calibrate it again.
To calculate the object position I made some simplifications/assumptions, which made the math easier: the cameras are in the same YZ plane, so there is only a distance in x between them. Their tilt is also just in the XY plane.
To reverse the triangulation I thought a test pattern (square) of 4 points of which I know the distances to each other would suffice. Ideally I would like to get the cameras' positions (distances to test pattern and each other), their rotation in X (and maybe Y and Z if applicable/possible), as well as their view angle (to translate pixel position to real world distances - that should be a camera constant, but in case I change cameras, it is quite a bit to define accurately)
I started with the same trigonometric calculations, but always miss parameters. I am wondering if there is an existing solution or a solid approach. If I need to add parameter (like distances, they are easy enough to measure), it's no problem (my calculations didn't give me any simple equations with that possibility though).
I also read about Homography in opencv, but it seems it applies to 2D space only, or not?
Any help is appreciated!
I start this thread asking for your help in Excel.
The main goal is to determine the coordinates of the intersection point P=(x,y) between two curves (curve A, curve B) modeled by points.
The curves are non-linear and each defining point is determined using complex equations (equations are dependent by a lot of parameters chosen by user, as well as user will choose the number of points which will define the accuracy of the curves). That is to say that each curve (curve A and curve B) is always changing in the plane XY (Z coordinate is always zero, we are working on the XY plane) according to the input parameters and the number of the defining points is also depending by the user choice.
My first attempt was to determine the intersection point through the trend equations of each curve (I used the LINEST function to determine the coefficients of the polynomial equation) and by solving the solution putting them into a system. The problem is that Excel is not interpolating very well the curves because they are too wide, then the intersection point (the solution of the system) is very far from the real solution.
Then, what I want to do is to shorten the ranges of points to be able to find two defining trend equations for the curves, cutting away the portion of curves where cannot exist the intersection.
Today, in order to find the solution, I plot the curves on Siemens NX cad using multi-segment splines with order 3 and then I can easily find the coordinates of the intersection point. Please notice that I am using the multi-segment splines to be more precise with the approximation of the functions curve A and curve B.
Since I want to avoid the CAD tool and stay always on Excel, is there a way to select a shorter range of the defining points close to the intersection point in order to better approximate curve A and curve B with trend equations (Linest function with 4 points and 3rd order spline) and then find the solution?
I attach a picture to give you an example of Curve A and Curve B on the plane:
https://postimg.cc/MfnKYqtk
At the following link you can find the Excel file with the coordinate points and the curve plot:
https://www.mediafire.com/file/jqph8jrnin0i7g1/intersection.xlsx/file
I hope to solve this problem with your help, thank you in advance!
kalo86
Your question gave me some days of thinking and research.
With the help of https://pomax.github.io/bezierinfo/
§ 27 - Intersections (Line-line intersections)
and
§ 28 - Curve/curve intersection
your problem can be solved in Excel.
About the mystery of Excel smoothed lines you find details here:
https://blog.splitwise.com/2012/01/31/mystery-solved-the-secret-of-excel-curved-line-interpolation/
The author of this fit is Dr. Brian T. Murphy, PhD, PE from www.xlrotor.com. You find details here:
https://www.xlrotor.com/index.php/our-company/about-dr-murphy
https://www.xlrotor.com/index.php/knowledge-center/files
=>see Smooth_curve_bezier_example_file.xls
https://www.xlrotor.com/smooth_curve_bezier_example_file.zip
These knitted together you get the following results for the intersection of your given curves:
for the straight line intersection:
(x = -1,02914127711195 / y = 23,2340949174492)
for the smooth line intersection:
(x = -1,02947493047196 / y = 23,2370611219553)
For a full automation of your task you would need to add more details regarding the needed accuracy and what details you need for further processing (and this is actually not the scope of this website ;-).
Intersection of the straight lines:
Intersection of the smoothed lines:
comparison charts:
solution,
Thank you very much for the anwer, you perfectly centered my goal.
Your solution (for the smoothed lines) is very very close to what I determine in Siemens NX.
I'm going to read the documentation at the provided link https://pomax.github.io/bezierinfo/ in order to better understand the math behind this argument.
Then, to resume my request, you have been able to find the coordinates (x,y) of the intersection point between two curves without passing through an advanced CAD system with a very good precision.
I am starting to study now, best regards!
kalo86
I am calculating an dynamic resistance of a diode and I have a lot of measurements and I've created a graph from them. And the question is, how do I find from this graph an exact value of arguments, for example: I want to obtain f(x) value for x=5 where i have measurement for exact value fe. x=10 -> y=213, x=1 y->110, and got a graph curve, but how to find f(5) = ?
This is not trivial: it will depend on your interpolation scheme and Excel does not expose the scheme it uses when drawing a graph.
Unless you tell it otherwise, Excel (I think) uses a Bezier Curve with 2 control points to perform its graphing.
This interpolation scheme transforms, via some linear algebra, to a cubic spline interpolation.
But to use cubic spline interpolation, you need more than two data points.
Since you've only given us two points, the best thing you can do is to interpolate linearly but that will not be what Excel does.
An answer more detailed than this if anything will epitomise the rather broad nature of your question. Do Google any terms that I've used: armed with a bit of time and a good internet connection, you ought to be able to solve this problem adequately.
See https://en.wikipedia.org/wiki/Spline_interpolation, https://en.wikipedia.org/wiki/B%C3%A9zier_curve
I think that you can use a preinstalled add-on named Solver. You have to activate it as shown here.
Then you have to follow one of the tutorial you can find over the Internet (like this one) without finding min o max but finding the exact value you want.