Detecting Weakly Simple Polygons

A closed curve in the plane is weakly simple if it is the limit (in the Fr\'echet metric) of a sequence of simple closed curves. We describe an algorithm to determine whether a closed walk of length n in a simple plane graph is weakly simple in O(n log n) time, improving an earlier O(n^3)-time algorithm of Cortese et al. [Discrete Math. 2009]. As an immediate corollary, we obtain the first efficient algorithm to determine whether an arbitrary n-vertex polygon is weakly simple; our algorithm runs in O(n^2 log n) time. We also describe algorithms that detect weak simplicity in O(n log n) time for two interesting classes of polygons. Finally, we discuss subtle errors in several previously published definitions of weak simplicity.Comment: 25 pages and 13 figures, submitted to SODA 201

Reconstruction of Weakly Simple Polygons from their Edges

Given n line segments in the plane, do they form the edge set of a weakly simple polygon; that is, can the segment endpoints be perturbed by at most epsilon, for any epsilon > 0, to obtain a simple polygon? While the analogous question for simple polygons can easily be answered in O(n log n) time, we show that it is NP-complete for weakly simple polygons. We give O(n)-time algorithms in two special cases: when all segments are collinear, or the segment endpoints are in general position. These results extend to the variant in which the segments are directed, and the counterclockwise traversal of a polygon should follow the orientation. We study related problems for the case that the union of the n input segments is connected. (i) If each segment can be subdivided into several segments, find the minimum number of subdivision points to form a weakly simple polygon. (ii) If new line segments can be added, find the minimum total length of new segments that creates a weakly simple polygon. We give worst-case upper and lower bounds for both problems

Salient Objects in Clutter: Bringing Salient Object Detection to the Foreground

We provide a comprehensive evaluation of salient object detection (SOD) models. Our analysis identifies a serious design bias of existing SOD datasets which assumes that each image contains at least one clearly outstanding salient object in low clutter. The design bias has led to a saturated high performance for state-of-the-art SOD models when evaluated on existing datasets. The models, however, still perform far from being satisfactory when applied to real-world daily scenes. Based on our analyses, we first identify 7 crucial aspects that a comprehensive and balanced dataset should fulfill. Then, we propose a new high quality dataset and update the previous saliency benchmark. Specifically, our SOC (Salient Objects in Clutter) dataset, includes images with salient and non-salient objects from daily object categories. Beyond object category annotations, each salient image is accompanied by attributes that reflect common challenges in real-world scenes. Finally, we report attribute-based performance assessment on our dataset.Comment: ECCV 201