Completion energies and scale

The detection of smooth curves in images and their completion over gaps are two important problems in perceptual grouping. In this study, we examine the notion of completion energy of curve elements, showing, and exploiting its intrinsic dependence on length and width scales. We introduce a fast method for computing the most likely completion between two elements, by developing novel analytic approximations and a fast numerical procedure for computing the curve of least energy. We then use our newly developed energies to find the most likely completions in images through a generalized summation of induction fields. This is done through multiscale procedures, i.e., separate processing at different scales with some interscale interactions. Such procedures allow the summation of all induction fields to be done in a total of only $O(N \log N)$ operations, where $N$ is the number of pixels in the image. More important, such procedures yield a more realistic dependence of the induction field on the length and width scales: The field of a long element is very different from the sum of the fields of its composing short segments.