A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open segment between them does not contain any other point of the set. We study the recognition problem for point visibility graphs: given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals. Hence the problem is as hard as deciding the existence of a real solution to a system of polynomial inequalities. The proof involves simple substructures forcing collinearities in all realizations of some visibility graphs, which are applied to the algebraic universality constructions of Mn "ev and Richter-Gebert. This solves a longstanding open question and paves the way for the analysis of other classes of visibility graphs. Furthermore, as a corollary of one of our construction, we show that there exist point visibility graphs that do not admit any geometric realization with points having integer coordinates.
A linear-time recognition algorithm has been proposed by Ghosh and Roy for planar point visibility graphs @cite_16 . For general point visibility graphs they showed that recognition problem lies in @math . More recently, Roy @cite_9 published an ingenious and rather involved NP-hardness proof for recognition of arbitrary point visibility graphs. Our result clearly implies NP-hardness as well, and, in our opinion, has a more concise proof.
@cite_9 @cite_16
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In this paper, we present two necessary conditions for recognizing point visibility graphs. We show that this recognition problem lies in PSPACE. We state new properties of point visibility graphs along with some known properties that are important in understanding point visibility graphs. For planar point visibility graphs, we present a complete characterization which leads to a linear time recognition and reconstruction algorithm. A graph-based method for 3D view-invariant human action recognition is proposed.An action is represented as a sequence of graphs.The vertices can either be tracked skeleton joints or spatio-temporal keypoints.A spectral graph wavelet transform is leveraged to extract features from the graphs.The method is useful for both single- and multi-view action recognition tasks. Display Omitted We present a method for view-invariant action recognition from depth cameras based on graph signal processing techniques. Our framework leverages a novel graph representation of an action as a temporal sequence of graphs, onto which we apply a spectral graph wavelet transform for creating our feature descriptor. We evaluate two view-invariant graph types: skeleton-based and keypoint-based. The skeleton-based descriptor captures the spatial pose of the subject, whereas the keypoint-based is able to capture complementary information about human-object interaction and the shape of the point cloud. We investigate the effectiveness of our method by experiments on five publicly available datasets. By the graph structure, our method captures the temporal interaction between depth map interest points and achieves a 19.8 increase in performance compared to state-of-the-art results for cross-view action recognition, and competing results for frontal-view action recognition and human-object interaction. Namely, our method results in 90.8 accuracy on the cross-view N-UCLA Multiview Action3D dataset and 91.4 accuracy on the challenging MSRAction3D dataset in the cross-subject setting. For human-object interaction, our method achieves 72.3 accuracy on the Online RGBD Action dataset. We also achieve 96.0 and 98.8 accuracy on the MSRActionPairs3D and UCF-Kinect datasets, respectively. In this paper we address the problem of illumination invariant face recognition. Using a fundamental concept that in general, patterns from a single object class lie on a linear subspace [2], we develop a linear model representing a probe image as a linear combination of class-specific galleries. In the presence of noise, the well-conditioned inverse problem is solved using the robust Huber estimation and the decision is ruled in favor of the class with the minimum reconstruction error. The proposed Robust Linear Regression Classification (RLRC) algorithm is extensively evaluated for two standard databases and has shown good performance index compared to the state-of-art robust approaches. We show that for every fixed k, there is a linear time algorithm that decides whether or not a given graph has crossing number at most k, and if this is the case, computes a drawing of the graph in the plane with at most k crossings. This answers the question posed by Grohe (STOC'01 and JCSS 2004). Our algorithm can be viewed as a generalization of the seminal result by Hopcroft and Tarjan lin1, which determines if a given graph is planar in linear time. Our algorithm can also be compared to the algorithms by Mohar (STOC'96 and Siam J. Discrete Math 2001), for testing the embeddability of an input graph in a fixed surface. For each surface s, Mohar describes an algorithm which yields either an embedding of G in s or a minor of G which is not embeddable in s and is minimal with this property. The same approach allows us to obtain linear time algorithms for the same question for a variety of other crossing numbers. We can also determine in linear time if an input graph can be made planar by the deletion of k edges (for fixed k).