My issue is this: I have a cloud of 3D points from a CT scan. I know that all these points belong to one incomplete surface so I need to add the rest of the points that fit the surface but to do that I need to know the normal vector so that the points I am adding are in the same direction.
Is it safe to assume that the eigenvector corresponding to the largest eigenvalue of a point is the same as the normal vector?
Without grasping the complete problem from your statement, some RANSAC-related algorithm could perform well, even if there are outliers. RANSAC would give you the most likely plane (e.g. represented as one point on the plane plus the plane normal vector if you formulate the search accordingly), and you are free to assign points from your cloud to the obtained plane in a second step.
The thing is I am not using a covariance matrix but a hessian matrix of the image ( more precisely, it's the convolution of the image with a second derivative gaussian kernel in order to smooth the image and eliminate the noise). So this hessian matrix gives good information about surface's directionality . Does this thing changes the perspective ?
RANSAC assumes that there is a model behind the data, a model whose parameters you have to find. The issue here is that the surface that i'm trying to obtain hasn't a regular shape so I don't know the model. The Least Square is not appropriate for my situation because I have some ugly outliers
For both ransac and least squares fitting, you need a model of the surface. If that is not available, then the number of surfaces that 'fit' your points is HUGE.
I can recommend Powercrust; software written by Nina Amenta:
http://www.cs.ucdavis.edu/~amenta/powercrust.html
The heuristics of Powercrust did a pretty good job triangulating my incomplete dataset of 3D CT bladder-wall points.
If your surface does not follow a planar model, the Hessian approach (with an appropriate scale) may not be bad, actually. We found the direction vectors extracted from Gaussian derivative filters to be quite reliable (in the presence of good contrast, and a local image geometry that matches the implicit model of the filter at the given scale, that is).
The statement "In general, the direction of maximum curvature does not coincide with the surface normal." is correct, of course, but the "curvature" you (Oana) are talking about here is the curvature of the gray values (i.e. of a lifted 3D surface in 4D, if anyone can imagine that), and must not be confused with the curvature of the surface you are trying to detect here.
Hi Hans, I'm glad you join the discussion.
Oana does not have a surface (yet); she has 3D points (probably a lot) and wants to create a surface from them. Of course there is no unique solution.
I faced similar problems when trying to segment the prostate or the bladder in CT-scans. I get patches of those organs right (vertices and triangles), but other parts of the surface are missing. I would like to interpolate the missing parts by some kind of curvature extrapolation.
Does Mevis have a solution, or could you ask Stefan?
I meant the surface in the image, i.e. an organ surface, may not follow the planar model. I just wanted to encourage Oana to use the direction information from the Hessian matrix in that case, which – given that the filter scale / estimation neighborhood is not too small – may indeed give usable, local surface orientation.
MeVisLab may be a nice environment to try something like this out, but I cannot point you to a ready solution, sorry. (And I don't regularly take part in such discussions on RG, just sayin' in case that I disappear again. ;-) )
I like Pablo's answer, since the PCL link contains some background explanations. Since Oana is dealing with CT images, I suggest not to use the "viewpoint", but the local gradient direction for orienting the normals.
Finally, one should mention that a point cloud + normal vectors is a different approach than reconstructing the surface, and then computing the surface normal. The latter approach may be more difficult, but would have other advantages.
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