Isoperimetric Normalization of Planar Curves

This paper presents an algorithm for transforming closed planar curves into a canonical form, independent of the viewpoint from which the original image of the contour was taken. The transformation that takes the contour to its canonical form is a member of the projective group PGL(2), chosen because PGL(2) contains all possible transformations of a plane curve under central projection onto another plane. The scheme relies on solving computationally an “isoperimetric” problem in which a transformation is sought which maximises the area of a curve given unit perimeter. In the case that the transformation is restricted to the affine subgroup there is a unique extremising transformation for any piecewise smooth closed curve. Uniqueness holds, almost always, even for curves that are not closed. In the full projective case, isoperimetric normalization is well-defined only for closed curves. The question of uniqueness is more complex: we have found computational counterexamples for which there is more than one extrema1 transformation. Numerical algorithms are described and demonstrated both for the affine and the projective cases. Once a canonical curve is obtained, its isoperimetric area can be regarded as an invariant descriptor of shape. Methods for discrimination of nonconvex shapes are already known. Our invariant descriptor is the first example, to our knowledge, of practical discrimination, up to projectivity, of convex, closed curves.