Beyond connecting the dots: A polynomial-time algorithm for segmentation and boundary estimation with imprecise user input

We propose a polynomial-time algorithm for segmentation and (open) boundary estimation which takes into account a series of user-specified attraction points. In contrast to existing algorithms which impose that the segmenting boundary passes through these points, our algorithm allows an imprecision in the user input. An energy minimization approach imposes that the segmenting boundary optimally passes along high-contrast edges in such a way that at least one point along the computed boundary is as close as possible to any given attraction point. In this sense, the user input can be seen as a soft constraint. We prove that the resulting optimization problem is NP-hard. We prove that in the case that the user attraction points are ordered, then optimal solutions can be computed in polynomial time using a shortest path formulation in an appropriately constructed four-dimensional graph spanned by the image pixels, a set of tangent angles and the user attraction points. Experimental results on a variety of images demonstrate that good quality segmentations can be obtained with a few imprecise user clicks.