Shape representation via elementary symmetric polynomials: A complete invariant inspired by the bispectrum

We address the representation of two-dimensional shape in its most general form, i.e., arbitrary sets of points, that may arise in multiple situations, e.g., sparse sets of specific landmarks, or dense sets of image edge points. Our goal are recognition tasks, where the key is balancing two contradicting demands: shapes that differ by rigid transformations or point re-labeling should have the same representation (invariance) but geometrically distinct shapes should have different representations (completeness). In the paper, we introduce a new shape representation that marries properties of the elementary symmetric polynomials and the bispectrum. Like the power spectrum, the bispectrum is insensitive to signal shifts; however, unlike the power spectrum, the bispectrum is complete. The elementary symmetric polynomials are complete and invariant to variable relabeling. We show that the elementary symmetric polynomials of the shape points depend on the shape orientation in a way that enables interpreting them in the frequency domain and building from them a bispectrum. The result is a shape representation that is complete and invariant to rigid transformations and point-relabeling. The paper also reports experiments that illustrate the proved properties.