Least squares contour alignment

The contour alignment problem, considered in this paper, is to compute the minimal distance in a least squares sense, between two explicitly represented contours, specified by corresponding points, after arbitrary rotation, scaling, and translation of one of the contours. This is a constrained nonlinear optimization problem with respect to the translation, rotation and scaling parameters, however, it is transformed into an equivalent linear least squares problem by a nonlinear change of variables. Therefore, a global solution of the contour alignment problem can be computed efficiently. It is shown that a normalization of the cost function minimum value is invariant to ordering and affine transformation of the contours and can be used as a measure for the distance between the contours. A solution is also proposed to the problem of finding a point correspondence between the contours.