3D Point Cloud - Gaussian Curvature

Hello All.

I have a 3D point cloud data (X [m]; Y[m]; Z[m]). And I want to calculate the value of some curvature (e.g. gaussian) in every point of the point cloud. Based on these curvature information I would select the points, that may belong to some geometric shape in the point cloud (e. g. to some planes, spheres etc.).

Is there a built in function for it in Matlab, like normals() for normal vector estimation? Or does anyone know a way how to do this? It should be fast and dont need to be really precise, because it is only a pre-processing step.

Thanks in Advance,

Richard

Answer 1

[darova]

one way

clc,clear

    % generate some data
r = 3;
t = linspace(0,10)';
x = r*cos(t);
y = r*sin(t);
z = sin(1*t);
t1 = t(2:end) - diff(t)/2;

dv = diff([x y z])./diff([t t t]);      % first derivative at midpoints
d2v= diff(dv)./diff([t1 t1 t1]);        % second derivative at points [2:end-1]
dv = 1/2*( dv(1:end-1,:) + dv(2:end,:) );% first derivative at points [2:end-1]

[dx,dy,dz] = deal( dv(:,1),dv(:,2),dv(:,3) );
[d2x,d2y,d2z] = deal( d2v(:,1),d2v(:,2),d2v(:,3) );

    % curvature
kk = (d2z.*dy - d2y.*dz).^2 + (d2x.*dz - d2z.*dx).^2 + (d2y.*dx - d2x.*dy).^2;
kk = sqrt(kk) ./ (dx.^2+dy.^2+dz.^2).^(3/2);
    % radius
RR1 = 1./kk;

plot(t(2:end-1),RR1)

LINK

[RiHo]

Thank you for your answer darova. It is almost the thing, what I looking for. :)

But the problem with this is that, I dont know, how to determine the t. Because I have a 3D scanned point cloud (e.g. a point cloud from a building scan or something), where you can have lot of shapes, so its not only one curve.

For example:

With a CloudCompare software its possible to calculate the Gaussian curvature in every point of the point cloud, and then you get this:

You can see, that the points belonging to these small spheres are in green color.

I want to get something similar. So some curvature information in every point.

Thanks,

Richard

[darova]

Do you have neighbouring points? Or you have only points (cloud)?

Surface curvate depends on plane

[RiHo]

I have only points, but I can select e.g. 5 neighbouring points. I´m calculating also the normal vectors at each point using a local small plane from 5 closest points.

But I have problem with the curvature.

[darova]

Two different curvatures, depends on defying a plane

[RiHo]

The plane is defined as a contact surface (best fit plane) in the choosen points, and the parameters are calculated using orthogonal regression.

Thats the way how I can calculate the normals also.

[darova]

Can you show plane at which you want to compute curvature?

[RiHo]

Fig. - Points lying on the surface of the sphere in 2D with 2 planes fitted in 2 picked points.

Legend:

Black - points of the cloud, lying on the surface of a sphere;

Blue, Yellow - 2 picked points, with its plane fitted in 5 nearest neighbour;

Red - 5 neighbours for the Blue piced point;

Green - 5 neighbours for the Yellow piced point.

So I want to calculate the curvature in each point of the point cloud. For example using 5 neighboring points for calculation. Or maybe otherwise, idk.

But like I said this point cloud is a complex pointcloud containing lot of shapes, etc. (see the picture from the previous comments).

I hope it's understandable.

[darova]

Can you create a surface from the point cloud? Can you use surfnorm?

[RiHo]

I dont think so, because scanned point clouds have really rugged surface, like when you imagine point cloud of complex objects like buildings and so on. There are lot of complicated surfaces.

That's why I'm fitting a local small plane in every point for point normal vector computation, like in pcnormals.

[darova]

If you can get normals for neighbouring points:

alpha = acos(dot(n1,n2));