Optimal Orie~tation Detection of Linear Symmetry

The problem of optimal detection of orientation in arbitrary neighborhoods is solved in the least squares sense. It is shown that this corresponds to fitting an axis in the Fourier domain of the n-dimensional neighborhood, the solution of which is a well known solution of a matrix eigenvalue problem. The eigenvalues are the variance or inertia with respect to the axes given by their respective eigen vectors. The orientation is taken as the axis given by the least eigenvaJ~e~ Moreover it is shown that the necessary computations can be pursued in the spatial domain without doing a Fourier transformation. An implementation for 2-D is presented. Two certainty measures are given corresponding to the orientation estimate. These are the relative or the absolute distances between the two eigenvalues, revealing whether the fitted axis is much better than an axis orthogonal to it. The result of the implementation is verified by experiments which confirm an accurate orientation estimation and reliable certainty measure in the presence of additive noise at high level as well as low levels.