It is shown how some geometrical insights can be used to provide penalties for the various edge configurations in a way that is roughly independent of the pixel discretization. The penalties are consistent over pixels of different sizes, shapes, and orientations, event if these occur in the same pattern. The cases of square, rectangular, hexagonal, and irregular pixels are considered. In an experiment, the penalties perform substantially better than those previously proposed.>

The problem of sequentially determining a mixture proportion is discussed, and the "probabilistic teacher" method, where successive observations are randomly classified according to current posterior probabilities, is investigated. The method is shown to be consistent and expressions for its statistical efficiency are obtained.

A highly efficient algorithm for computing the vertices of the convex hull of a set of points in the plane is described and compared with existing methods. Empirical timings of a FORTRAN implementation of the algorithm show it to be faster than previous methods for most configurations of points. Careful consideration is given throughout to ensure that degenerate configurations, containing collinear or near-collinear points, are handled correctly.