Data Handling With Two Independent Variables and the Bezier Filter

Bezier functions of two independent variables are excellent vehicles for three dimensional data reduction that require preservation of derivative content. They can be applied to both smooth and noisy data. They represent the original data with good fidelity. The approximation is smooth and continuous to a high degree. The coefficients of the approximating Bezier function are obtained through a non iterative algebraic relation. These coefficients can replace the original data leading to significant data reduction. The Bezier function also allows the data over the entire domain to be represented by simple bivariate polynomials in two parameters. This continuous representation allows easy extraction of gradient and higher derivative information about the data. The first two examples demonstrate the technique using smooth data. Using random perturbation on these examples, two additional examples are generated to represent noisy data. It is shown that the underlying original smooth data can be recognized using the same algorithm. This process is recognized as the Bezier filter in this paper. These examples illustrate that the Bezier functions are excellent for data approximation, reduction, transformation, and smoothing. The Bezier function is global over all of the data so that any interpolated data have the same properties as the rest of the data. The examples establish two important properties in the use of Bezier functions for data analysis. The mean of the original data and the approximate data using Bezier function are the same. Large orders of polynomials can be used without significant distortion in the approximation.