We present a general framework that allows image filtering by minimization of a functional using a linear and positive definite partial differential equation (PDE) while also permitting to control the weight of each pixel individually. Linearity and positive definiteness allow to use fast algorithms to calculate the solution. Pixel weighting allows to enforce the preservation of edge information without the need for nonlinear diffusion by making use of information coming from an external source. The proof of existence and uniqueness of the solution is outlined and based on that a numerical scheme for finding the solution is introduced. Using this framework we developed two applications. The first is simple and fast denoising, which incorporates an edge detection algorithm. In this case the functional is designed to enhance the weight of the approximation term over the smoothing term at those places where an edge is detected. The second application is a background suppression algorithm that is robust against noise, shadows thrown by the object, and on the background and varying illumination. The results are qualitatively not quite as good as the ones obtained with nonlinear PDEs, but this disadvantage is compensated by the processing speed, which allows analysis of a 320x240 color frame in about 0.3s on a standard PC. keywords: linear PDE, image smoothing, background subtraction
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