The world of digital signal processing, in particular computer graphics, vision and image processing, use linear and non-linear, explicit and implicit filtering extensively to analyze, process and synthesize images. Given nowadays high-resolution sensors, these operations are often very time consuming and are limited to devices with high-CPU power.Traditional linear translation-invariant (LTI) transformations, executed using convolution, requires O(N^2) operations. This can be lowered to O(N \log N) via FFT over suitable domains. There are very few sets of filters to which optimal, linear-time, procedures are known. This situation is more complicated in the newly-emerging domain of non-linear spatially-varying filters. Exact application of such filter requires O(N^2) operations and acceleration methods involve higher space dimension introducing severe memory cost and truncation errors.In this research proposal we intend to derive fast, linear-time, procedures for different types of LTI filters by exploiting a deep connection between convolution, spatially-homogeneous elliptic equations and the multigrid method for solving such equations. Based on this circular connection we draw novel prospects for deriving new multiscale filtering procedures.A second part of this research proposal is devoted to deriving efficient explicit and implicit non-linear spatially-varying edge-aware filters. One front consists of the derivation of novel multi-level image decomposition that mimics the action of inhomogeneous diffusion operators. The idea here is, once again, to bridge the gap with numerical analysis and use ideas from multiscale matrix preconditioning for the design of new biorthogonal second-generation wavelets.Moreover, this proposal outlines a new multiscale preconditioning paradigm combining ideas from algebraic multigrid and combinatorial matrix preconditioning. This intermediate approach offers new ways for overcoming fundamental shortcomings in this domain.