Projective curvature and integral invariants
It is known, that the projective group and its three subgroups namely the Euclidean group \(E(2)\), special affine group \(SA(2)\) and the full affine group \(A(2)\) play an important role in computer vision. The fundamental invariant of the projective group is the projective curvature which characterises curves under a projective transformation. It depends on seventh order derivatives and hence is very sensitive to noise and of little practical use in planar object recognition. To get over the above problem, the authors propose an extension of all Lie group actions on \(\mathbb{R}^2\) to coordinates defined by potentials so that the resulting invariants will depend on integrals rather than derivatives and so will not be sensitive to noise. In this process the authors derive explicit formulas for the extended group action in the case of the full affine group and fail to determine it in the case of the projective group. The determination of the projective integral invariants that are not dependent on derivatives and the incorporation of this in finding an efficient planar object recognition algorithm are problems yet to be resolved.
scientific article; zbMATH DE number 1879601