Alpha Functions

Piecewise-linear Approximation from Noisy and Hermite Data

We introduce 𝛼-functions, providing piecewise linear approximation to given data as the difference of two convex functions. The parameter 𝛼 controls the shape of a paraboloid that is probing the data and may be used to filter out noise in the data. The use of convex functions enables tools for efficient approximation to the data, adding robustness to outliers, and dealing with gradient information. It also allows using the approach in higher dimension. We show that 𝛼-functions can be efficiently computed and demonstrate their versatility at the example of surface reconstruction from noisy surface samples.