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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.
