# 2 1 Metrics for finding correspondence between two point cloudsA

2.1. Metrics for finding correspondence between two point cloudsAlthough the simplest method of estimating the surface normal vector is the first order three-dimensional plane fitting , the covariance matrix will be utilised in this paper since the first order plane fitting is equivalent to the eigenvalue problem of the covariance matrix. In addition, the covariance analysis provides additional geometric information such as curvature and its higher order derivatives. Let pi be the coordinates of ith point in a point cloud and note that a bold letter represents a matrix or a vector. The covariance of a point and its k neighbour points is expressed as:COV(pi)=1k��m=1krmrmT=��l=02��lelelT(1)where rm = pi ? pcentorid, pcentroid, pcentroid is the centroid of the k neighbourhood and el is the eigenvector of the (l+1)th smallest eigenvalue.

Since COV(pi) is a real, positive and semi-definite matrix, its eigenvalue are always greater than or equal to zero . The eigenvector of the minimum eigenvalue is the estimated normal vector of the surface formed by pi and its neighbourhood. The other eigenvectors are the tangential vectors of the surface and if the minimum eigenvalues are close to zero, and then the surface consisting of a point and its neighbourhood is geometrically flat. If all eigenvalues are similar, then the surface is a round-shape and locally well distributed. One can find details of other methods based on the covariance analysis for 3D point clouds in .There are many ways to define geometric curvature, e.g.

through Gaussian and mean curvatures or using the eigenvalues of the covariance matrix . It is preferable to estimate curvature directly by using points without any pre-process such as triangulation and surface fitting since it is faster to use the neighbourhood of a point than to utilise the connectivity information provided by triangulation. Hoppe et al.  proposed a covariance analysis method for the estimation of the normal vector with consistent orientation. The covariance analysis method has been also utilised for the estimation of local curvature estimation using the ratio between the minimum eigenvalue and the sum of the eigenvalues. Definition of local curvature proposed by Hoppe et al.  is used in this paper and this method estimates the first order differential of local surface rather than local curvature itself.Each eigenvalue of the covariance matrix represents the spatial variation along the direction of the corresponding eigenvector. The curvature approximation quantifies AV-951 the percentage of variance attributed by surface deviation from the tangential plane formed by e1 and e2.