General plane curve matching under affine transformations

It is common to use an affine transformation to approximate in dealing with the matching of plane curves under a projective transformation. The plane curve itself can be used as an identity to solve the parameters of an affine transformation. The objective of this paper is to obtain a closed form solution of the parameters using low order derivatives of the plane curve. A unique solution to the parameters of an affine transformation with up to second order derivatives is presented using differential invariants as well as the available global information. The computational time on verification has been significantly reduced. In computer vision, derivatives are obtained by numerical means. Achieving accurate numerical derivatives is an important application issue. Smoothing with a Gaussian filter modified by a linear combination of Hermite polynomials, can preserve the accuracy of continuous polynomials with powers up to the same order as the Hermite polynomials. In the discrete space however, the introduction of Hermite polynomials leads to a choice of a large smoothing scale σ in order to reduce computational errors at the expense of a reduction of local controllability and over-smoothing of the curve. It is shown that using a σ proportional to the order of the derivatives is more reliable in applications