Periodic quasi-orthogonal spline bases and applications to least-squares curve fitting of digital images

In this paper, we present a new covariant basis, dubbed the quasi-orthogonal Q-spline basis, for the space of n-degree periodic uniform splines with k knots, This basis is obtained analogously to the B-spline basis by scaling and periodically translating a single spline function of bounded support, The construction hinges on an important theorem involving the asymptotic behavior (in the dimension) of the inverse of banded Toeplitz matrices, We show that the Gram matrix for this basis is nearly diagonal, hence, the name ''quasi-orthogonal.'' The new basis is applied to the problem of approximating closed digital curves in 2-D images by least-squares fitting, Since the new spline basis is almost orthogonal, the least-squares solution can be approximated by decimating a convolution between a resolution-dependent kernel and the given data, The approximating curve is expressed as a linear combination of the new spline functions and new ''control points.'' Another convolution maps these control points to the classical B-spline control points, A generalization of the result has relevance to the solution of regularized fitting problems.