Technical Report 2021-002

A omega-Circulant Regularization for Linear Systems Arising in Interpolation with Subdivision Schemes

Rafael Diaz Fuentes, Mariarosa Mazza, and Stefano Serra-Capizzano

March 2021

In the curve interpolation with primal and dual form of stationary subdivision schemes, the computation of the relevant parameters amounts in solving special banded circulant linear systems, whose coefficients are related to quantities arising from the used stationary subdivision schemes. In some important cases it happens that the associated generating function, which is a special Laurent polynomial called symbol, has zeros on the unit complex circle of the form exp(2pi i j/n), where n is the size of the matrix, i2=-1, and j is a non-negative integer bounded by n-1. When this situation occurs the discrete problem is ill-posed simply because the circulant coefficient matrix is singular and the problem has no solution (or infinitely many). Standard and nonstandard regularization techniques such as least square solutions or Tikhonov regularization have been tried, but the quality of the solution is not good enough. In this work we propose a structure preserving regularization in which the circulant matrix is replaced by the omega-circulant counterpart, with omega being a complex parameter. A careful choice of omega close to 1 (recall that the set of 1-circulants coincides with standard circulant matrices) allows to solve satisfactorily the problem of the ill-posedness, even if the quality of the reconstruction is reasonable only in a restricted number of cases. Numerical experiments and further algorithmic proposals are presented and critically discussed.

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