Topological information are the most important kind of qualitative spatial information. Current formalisms for the topological aspect of space focus on relations between regions or properties of regions. This work provides a qualitative model for representing the topological internal structure of complex regions, which could be of multiple pieces and or have holes and islands to any finite level. We propose a layered graph model for representing the internal structure of complex plane regions, where each node represents the closure of a connected component of the interior or the exterior of a complex region. The model provides a complete representation in the sense that the (global) nine-intersections between the interiors, the boundaries, and the exteriors of two complex regions can be determined by the (local) RCC8 topological relations between the associated simple regions. Moreover, this graph model has an inherent hierarchy which could be exploited for map generalization.
@cite_15 defined a complex region as a bounded region @math which has finite components (called there) and required that the intersection of any two components is a finite set. They considered a complex region as a collection of faces. For example, the bounded region shown in Figure is interpreted as having seven faces: one has a hole, the other six are simple regions. How these faces are composed is not mentioned explicitly in their definition. As another example, the two complex regions in Figure have the same set of faces, but different link graphs and internal structures. When restricted to semi-algebraic sets, all bounded regions consider in this paper are complex regions in the sense of @cite_15 .
@cite_15
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The 4-intersection, a model for the representation of topological relations between 2-dimensional objects with connected boundaries and connected interiors, is extended to cover topological relations between 2-dimensional objects with arbitrary holes, called regions with holes. Each region with holes is represented by its generalized region—the union of the object and its holes — and the closure of each hole. The topological relation between two regions with holes, A and B, is described by the set of all individual topological relations between (1) A ’s generalized region and B’s generalized region, (2) A ’s generalized region and each of B’s holes, (3) B’s generalized region with each of A ’s holes, and (4) each of A ’s holes with each of B’s holes. As a side product, the same formalism applies to the description of topological relations between 1-spheres. An algorithm is developed that minimizes the number of individual topological relations necessary to describe a configuration completely. This model of representing complex topological relations is suitable for a multi-level treatment of topological relations, at the least detailed level of which the relation between the generalized regions prevails. It is shown how this model applies to the assessment of consistency in multiple representations when, at a coarser level of less detail, regions are generalized by dropping holes. We consider the non-crossing connectors problem, which is stated as follows: Given n simply connected regions R_1,...,R_n in the plane and finite point sets P_i subset of R_i for i=1,...,n, are there non-crossing connectors y_i for (R_i,P_i), i.e., arc-connected sets y_i with P_i subset of y_i subset of R_i for every i=1,...,n, such that y_i and y_j are disjoint for all i different from j? We prove that non-crossing connectors do always exist if the regions form a collection of pseudo-disks, i.e., the boundaries of every pair of regions intersect at most twice. We provide a simple polynomial-time algorithm if the regions are axis-aligned rectangles. Finally we prove that the general problem is NP-complete, even if the regions are convex, the boundaries of every pair of regions intersect at most four times and P_i consists of only two points on the boundary of R_i for i=1,...,n. We investigate a principle way to progressively mine discriminative object regions using classification networks to address the weakly-supervised semantic segmentation problems. Classification networks are only responsive to small and sparse discriminative regions from the object of interest, which deviates from the requirement of the segmentation task that needs to localize dense, interior and integral regions for pixel-wise inference. To mitigate this gap, we propose a new adversarial erasing approach for localizing and expanding object regions progressively. Starting with a single small object region, our proposed approach drives the classification network to sequentially discover new and complement object regions by erasing the current mined regions in an adversarial manner. These localized regions eventually constitute a dense and complete object region for learning semantic segmentation. To further enhance the quality of the discovered regions by adversarial erasing, an online prohibitive segmentation learning approach is developed to collaborate with adversarial erasing by providing auxiliary segmentation supervision modulated by the more reliable classification scores. Despite its apparent simplicity, the proposed approach achieves 55.0 and 55.7 mean Intersection-over-Union (mIoU) scores on PASCAL VOC 2012 val and test sets, which are the new state-of-the-arts. Let @math be a geometrically bounded symplectic manifold, @math be a closed, regular coisotropic submanifold, and @math be a Hamiltonian diffeomorphism. The main result of this article is that the number of leafwise fixed points of @math is bounded below by the sum of the Betti numbers of @math , provided that the Hofer distance between @math and the identity is small enough and the pair @math is non-degenerate. As an application, I prove a presymplectic non-embedding result. A version of the Arnold-Givental conjecture for coisotropic submanifolds is also discussed.