Variational Method Integrating Edge Detection andSmoothing in Digital Images

There is a consensus in computer vision about the importance of the scale concept for edge extraction and for image smoothing or representation. In this paper we explore a variational approach that allows to put together edge detection and image smoothing in a unified linear scheme. Basically, the functional proposed by Mumford and Shah is re-written as an energy defined with two arguments: the first one representing smooth versions of the original image and the second one encompassing its edge set. We follow known results in the variational analysis to obtain a numerical scheme to minimize the energy. We apply Fourier analysis to verify that the iterative scheme converges to a low-pass representation of the original image in the first argument and a high-pass signal in the other one. In the experimental results we show that the obtained scheme encourages intraregion image smoothing in preference to interregion blurring with edge localization at a desired scale.