Shape Interpretation of Second-Order Moment Invariants

This paper deals with the following problem: What can be said about the shape of an object if a certain invariant of it is known? Such a, herein called, shape invariant interpretation problem has not been studied/solved for most invariants, and it is also not known to which extent the shape interpretation of certain invariants exists. In this paper, we consider a well-known second-order affine moment invariant. This invariant has been expressed recently by Xu and Li (2008) as the average square area of triangles whose one vertex is the shape centroid while the remaining two vertices vary through the shape considered. The main results of the paper are: (i) ellipses are the shapes that minimize such an average square triangle area, i.e., that minimize the affine invariant considered; (ii) this minimum is 1/(16π²) and is reached only by ellipses. As by-products, we obtain several results including the expression of the second Hu moment invariant in terms of one shape compactness measure and one shape ellipticity measure. This expression further leads to the shape interpretation of the second Hu moment invariant, which is also given in the paper.