Application of Hurwitz-Radon matrices in curve interpolation and almost-smoothing

摘要

Dedicated methods for dealing with curve interpolation and curve smoothing have been developed. One such method, Hurwitz-Radon Matrices (MHR), can be used in reconstruction and interpolation of curves in the plane. The method is based on a family of Hurwitz-Radon (HR) matrices. The matrices are skew-symmetric and possess columns composed of orthogonal vectors. The of Hurwitz-Radon Operator (OHR), built from these matrices, is described. It is shown how to create the orthogonal and discrete OHR and how to use it in a process of curve interpolation and modelling. The method needs suitable choice of nodes, i.e. points of the curve to be reconstructed: nodes should be settled at each local extremum and nodes should be monotonic in one of coordinates (for example equidistance). Application of MHR gives a very good interpolation accuracy in the process of modeling and reconstruction of the curve. Created from the family of N-1 HR matrices and completed with the identity matrix, the system of matrices is orthogonal only for vector spaces of dimensions N = 2, 4 or 8. Orthogonality of columns and rows is very important and significant for stability and high precision of calculations. The MHR method models the curve point by point without using any formula or function. Main features of the MHR method are: the accuracy of curve reconstruction depends on the number of nodes and method of choosing nodes, interpolation of L points of the curve has a computational cost of rank O(L), and the smoothing of the curve depends on the number of OHR operators used to build the average matrix operator. The problem of curve length estimation is also considered. Algorithms and numerical results are presented.