Bivariate Splines for Scattered Data Fitting

Give a set of data values over scattered points over a 2D polygonal domain, one would like to find a smooth surface to fit the given data values over the scattered locations. For example, we have a set of data {(x_i,y_i,z_i), i=1, ..., 893}, black dots as shown in the following graph. We used a C^1 (continuously differentiable) quintic bivariate splines to interpolate the given data. See the smooth surface below.

For another example, we use C^1 cubic splines to interpolate the temperature values over various cities over the U.S. continent based on a triangulated quadrangulation.

For another example, in paper by Ming-Jun Lai and Schumaker, we used C^1 quintic splines to find a smooth interpolatory spline surface to interpolate a set of terrain data (15585 locations and heights) based on a domain decomposition technique. This is a powerful method enable us to interpolate and/fit ten of thousand of data values over scattered locations.

There are many bivariate spline data fitting methods:

Interpolatory Methods:

• Minimal Energy Interpolatory Method;
• L1 Spline Interpolatory Method;
• Minimal Triharmonic Interpolatory Method;

Data Fitting Methods:

• Discrete Least Squares Method;
• Penalized Least Squares Method;
• L1 Smoothing Spline Method;
• Minmial Absolute Deviation Method;

Or one simply mix interpolation and fitting requirements together to achieve a good fit surface. Some computational algorithms are described in paper by Awanou, Lai, and Wenston'06 and paper by Lai and Wenston'04. In paper by Ming-Jun Lai, some of these methods and their approximation properties are summarized.