Minimization of Energy Functional with Curve-Represented Edges

Until the mid eighties almost all edge detection methods were based on local operators. Since the edges of interest usually exist at a coarser scale than the inner scale (corresponding to the pixel size,) of the image, these operators always incorporate some averaging or similar mechanism for suppressing fine scale variations. As a consequence the detected edges get blurred and may be displaced. These problems can be circumvented by applying some kind of nonlinear best fit technique. Such local methods exist in abundance. However, they typically take only a small number of possible local edge configurations into consideration. For all other configurations the blurring and displacement problems thus remain. Ideally one would of course like to consider all possible edge configurations. But then the best fit technique must be applied to image regions so large that the edges therein can be assumed to be sparse. Otherwise the desired fine detail suppression will be difficult, or even impossible, to achieve. Conceptually the simplest way to make sure that this is the case, is to use a global best fit technique, which is the essence of global edge detection or global segmentation.