Interpolation with Splines in Tension: A Green's Function Approach

Interpolation and gridding of data are procedures in the physical sciences and are accomplished typically using an averaging or finite difference scheme on an equidistant grid. Cubic splines are popular because of their smooth appearances: however, these functions can have undesirable oscillations between data points. Adding tension to the spline overcomes this deficiency. Here, we derive a technique for interpolation and gridding in one, two, and three dimensions using Green's functions for splines in tension and examine some of the properties of these functions. For moderate amounts of data, the Green's function technique is superior to conventional finite-difference methods because (1) both data values and directional gradients can be used to constrain the model surface, (2) noise can be suppressed easily by seeking a least-squares fit rather than exact interpolation, and (3) the model can be evaluated at arbitrary locations rather than only on a rectangular grid. We also show that the inclusion of tension greatly improves the stability of the method relative to gridding without tension. Moreover, the one-dimensional situation can be extended easily to handle parametric curve fitting in the plane and in space. Finally, we demonstrate the new method on both synthetic and real data and discuss the merits and drawbacks of the Green's function technique.