Locally supported bivariate splines in piecewise constant tension

We derive a means to construct a locally supported basis for the truly bivariate cardinal spline functions under the influence of parameter called tension being a different constant in each grid cell. First, the functions are formulated as a solution for a functional minimization problem. Then the associated differential equation is analyzed in the state-space representation to derive formulae for transitions of the state vectors. Finally, the locally supported basis, which is indispensable for practical applications, are constructed by the technique of dead-beat control of the state vectors. Adaptively varying the tension in each grid cell, we can reduce the ringing artifacts when applied to interpolation of image signals.