Multigrid adaptive image processing

We consider a general weighted least squares approximation problem with a membrane spline regularization term. The key parameters in this formulation are the weighting factors which provide the possibility of a spatial adaptation. We prove that the corresponding space-varying variational problem is well posed, and propose a novel multigrid computational solution. This multiresolution relaxation scheme uses three image pyramids (input data, weights, and current solution) and allows for a very efficient computation with an effective O(N) complexity, where N is the number of pixels. This general multigrid solver can be useful for a variety of image processing tasks. In particular, we propose new multigrid solutions for noise reduction in images (adaptive smoothing spline), interpolation/reconstruction of missing image data, and image segmentation using an adaptive extension of the K-means clustering algorithm.