On multiscale quasi-interpolation of scattered scalar- and manifold-valued functions

We address the problem of approximating an unknown function from its discrete samples given at arbitrarily scattered sites. This problem is essential in numerical sciences, where modern applications also highlight the need for a solution to the case of functions with manifold values. In this paper, we introduce and analyze a combination of kernel-based quasi-interpolation and multiscale approximations for both scalar- and manifold-valued functions. While quasi-interpolation provides a powerful tool for approximation problems if the data is defined on infinite grids, the situation is more complicated when it comes to scattered data. Here, higher-order quasi-interpolation schemes either require derivative information or become numerically unstable. Hence, this paper principally studies the improvement achieved by combining quasi-interpolation with a multiscale technique. The main contributions of this paper are as follows. First, we introduce the multiscale quasi-interpolation technique for scalar-valued functions. Second, we show how this technique can be carried over using moving least-squares operators to the manifold-valued setting. Third, we give a mathematical proof that converging quasi-interpolation will also lead to converging multiscale quasi-interpolation. Fourth, we provide ample numerical evidence that multiscale quasi-interpolation has superior convergence to quasi-interpolation. In addition, we will provide examples showing that the multiscale quasi-interpolation approach offers a powerful tool for many data analysis tasks, such as denoising and anomaly detection. It is especially attractive for cases of massive data points and high dimensionality.