Determining the covariance of point clouds in real-time

Question

So basically I have a set of multidimensional data that I need to determine the covariance of between dimensions in real-time. Each point that comes in is a vector. I have gotten the mean and variance per dimension using this answer: Determining the mean and standard deviation in real time. But I also need to find the covariance between dimensions, resulting in a covariance matrix. Each vector is 3 dimensional and is a point in space.

Does anyone know how to do this and does this?

To explain further, I want the covariance matrix so I can then use the eigenvalues and eigenvectors to characterize a point cloud. The point clouds are segments of a scanned environment using the Kinect. I need to make sure the segments are labelled consistently over frames. I will do this by matching the mean and covariance eigenvectors/eigenvalues of segments between frames. I am working with unity.

Edit:

Or what would be equally useful would be how to get the covariance from just the variances and means that I already have.

Answer 1

To make the Online variance algorithm work for vectors, change one line:

M2 += np.outer( delta, delta2 )  # deltas 3-vecs, M2 3 x 3

# outer( x, y ) is pairs x[i] * y[j] --
# [[ x0 y0, x0 y1, x0 y2 ],
#  [ x1 y0, x1 y1, x1 y2 ],
#  [ x2 y0, x2 y1, x2 y2 ]]

A small python class for this is under gist.github.com/denis-bz . It can covariance points with uniform weights (FIR), or up-weight newer data (IIR), or both.

Is there a tutorial on your application with 3d point clouds ?