Bivariate splines in piecewise constant tension

Abstract

An extension of the bivariate cubic spline on the uniform grid is derived in this paper to have different tensions in different square cells of the grid. The resulting function can be interpreted also as a bivariate extension of the univariate spline in piecewise constant tension which was applied to adaptive interpolation of digital images for their magnification and rotation. The bivariate function will hopefully make it possible to magnify and rotate images better and even to deform images into any shapes. A locally supported basis, which is crucial for the practical use of the bivariate functions, has not been constructed at the moment and its construction is left for the next step of study.