Piecewise-linear surface approximation from noisy scattered samples

We consider the problem of approximating a smooth surface f(x,y), based on n scattered samples {(x_i, y_i, z_i)_iⁿ = 1} where the sample values {z_i} are contaminated with noise: z_i = f(x_i, y_i) + ε_i. We present an algorithm that generates a PLS (Piecewise Linear Surface) f′, defined on a triangulation of the sample locations V = {(x_i, y_i)_iⁿ=1}, approximating f well. Constructing the PLS involves specifying both the triangulation of V and the values of f′ at the points of V. We demonstrate that even when the sampling process is not noisy, a better approximation for f is obtained using our algorithm, compared to existing methods. This algorithm is useful for DTM (Digital Terrain Map) manipulation by polygon-based graphics engines for visualization applications.