On globally optimal local modeling: From moving least squares to over-parametrization

This paper discusses a variational methodology, which involves locally modeling of data from noisy samples, combined with global model parameter regularization. We show that this methodology encompasses many previously proposed algorithms, from the celebrated moving least squares methods to the globally optimal over-parametrization methods recently published for smoothing and optic flow estimation. However, the unified look at the range of problems and methods previously considered also suggests a wealth of novel global functionals and local modeling possibilities. Specifically, we show that a new non-local variational functional provided by this methodology greatly improves robustness and accuracy in local model recovery compared to previous methods. The proposed methodology may be viewed as a basis for a general framework for addressing a variety of common problem domains in signal and image processing and analysis, such as denoising, adaptive smoothing, reconstruction and segmentation.