The directed Physarum dynamics is known to solve positive linear programs: minimize @math subject to @math and @math for a positive cost vector @math . The directed Physarum dynamics evolves a positive vector @math according to the dynamics @math . Here @math is the solution to @math that minimizes the "energy" @math . In this paper, we study the non-uniform directed dynamics @math , where @math is a positive diagonal matrix. The non-uniform dynamics is more complex than the uniform dynamics (with @math being the identity matrix), as it allows each component of @math to react with different speed to the differences between @math and @math . Our contribution is to show that the non-uniform directed dynamics solves positive linear programs.
Nakagaki et.-al. @cite_3 pointed out that different edge may react with different speed to the differences between flow and capacity. For example, Physarum prefers darkness over bright light and hence edges in a bright environment react differently than edges in darkness. This let to the non-uniform dynamics where @math is an indicator for the reactivity of an edge.
@cite_3
2031275178
Our goal is to turn an intensity image into its perceived luminance without parsing it into depths, surfaces, or scene illuminations. We start with jarring intensity differences at two scales mixed according to edges, identified by a pixel-centric edge detector. We propose angular embedding as a more robust, efficient, and versatile alternative to LS, LLE, and NCUTS for obtaining a global brightness ordering from local differences. Our model explains a variety of brightness illusions with a single algorithm. Brightness of a pixel can be understood locally as its intensity deviating in the gradient direction and globally as finding its rank relative to others, particularly the lightest and darkest ones. This paper proposes an edge-based robust RGB-D visual odometry (VO)using 2-D edge divergence minimization. Our approach focuses on enabling the VO to operate in more general environments subject to low texture and changing brightness, by employing image edge regions and their image gradient vectors within the iterative closest points (ICP)framework. For more robust and stable ICP-based optimization, we propose a robust edge matching criterion with image gradient vectors. In addition, to reduce a bad effect of outlier residuals, we propose an improved edge registration problem of 2-D edge divergence minimization in the manner of an iterative reweight least squares (IRLS)motion estimation. To accelerate the proposed approach, a pixel sub-sampling method is employed. We evaluate estimation performance of our method in changing brightness conditions and low-textured scenes. Our approach shows more robust motion estimation than state-of-the-art methods while maintaining comparable accuracy in challenging image sequences at real-time (25 Hz)operation. Let H d (n,p) signify a random d -uniform hypergraph with n vertices in which each of the @math possible edges is present with probability p = p(n) independently, and let H d (n,m) denote a uniformly distributed d -uniform hypergraph with n vertices and m edges. We derive local limit theorems for the joint distribution of the number of vertices and the number of edges in the largest component of H d (n,p) and H d (n,m) in the regime @math , resp. d(d−1)m n >1+ϵ, where ϵ>0 is arbitrarily small but fixed as n → ∞. The proofs are based on a purely probabilistic approach. Summary Many animals rely on visual motion detection for survival. Motion information is extracted from spatiotemporal intensity patterns on the retina, a paradigmatic neural computation. A phenomenological model, the Hassenstein-Reichardt correlator (HRC), relates visual inputs to neural activity and behavioral responses to motion, but the circuits that implement this computation remain unknown. By using cell-type specific genetic silencing, minimal motion stimuli, and in vivo calcium imaging, we examine two critical HRC inputs. These two pathways respond preferentially to light and dark moving edges. We demonstrate that these pathways perform overlapping but complementary subsets of the computations underlying the HRC. A numerical model implementing differential weighting of these operations displays the observed edge preferences. Intriguingly, these pathways are distinguished by their sensitivities to a stimulus correlation that corresponds to an illusory percept, "reverse phi," that affects many species. Thus, this computational architecture may be widely used to achieve edge selectivity in motion detection. In this paper, we describe an interest point detector using edge foci. Unlike traditional detectors that compute interest points directly from image intensities, we use normalized intensity edges and their orientations. We hypothesize that detectors based on the presence of oriented edges are more robust to non-linear lighting variations and background clutter than intensity based techniques. Specifically, we detect edge foci, which are points in the image that are roughly equidistant from edges with orientations perpendicular to the point. The scale of the interest point is defined by the distance between the edge foci and the edges. We quantify the performance of our detector using the interest point's repeatability, uniformity of spatial distribution, and the uniqueness of the resulting descriptors. Results are found using traditional datasets and new datasets with challenging non-linear lighting variations and occlusions. The LGMD2 belongs to a group of giant movement-detecting neurones which have fan-shaped arbors in the lobula of the locust optic lobe and respond to movements of objects. One of these neurones, the LGMD1, has been shown to respond directionally to movements of objects in depth, generating vigorous, maintained spike discharges during object approach. Here we compare the responses of the LGMD2 neurone with those of the LGMD1 to simulated movements of objects in depth and examine different image cues which could allow the LGMD2 to distinguish approaching from receding objects. In the absence of stimulation, the LGMD2 has a resting discharge of 10–40 spikes s−1 compared with <1 spike s−1 for the LGMD1. The most powerful excitatory stimulus for the LGMD2 is a dark object approaching the eye. Responses to approaching objects are suppressed by wide field movements of the background. Unlike the LGMD1, the LGMD2 is not excited by the approach of light objects; it specifically responds to movement of edges in the light to dark direction. Both neurones rely on the same monocular image cues to distinguish approaching from receding objects: an increase in the velocity with which edges of images travel over the eye; and an increase in the extent of edges in the image during approach. This paper presents a novel image feature representation method, namely color difference histograms (CDH), for image retrieval. This method is entirely different from the existing histograms; most of the existing histogram techniques merely count the number or frequency of pixels. However, the unique characteristic of CDHs is that they count the perceptually uniform color difference between two points under different backgrounds with regard to colors and edge orientations in L*a*b* color space. This method pays more attention to color, edge orientation and perceptually uniform color differences, and encodes color, orientation and perceptually uniform color difference via feature representation in a similar manner to the human visual system. The method can be considered as a novel visual attribute descriptor combining edge orientation, color and perceptually uniform color difference, as well as taking the spatial layout into account without any image segmentation, learning processes or clustering implementation. Experimental results demonstrate that it is much more efficient than the existing image feature descriptors that were originally developed for content-based image retrieval, such as MPEG-7 edge histogram descriptors, color autocorrelograms and multi-texton histograms. It has a strong discriminative power using the color, texture and shape features while accounting for spatial layout. Suppose we are given a bipartite graph with vertex set X, Y, |X| = n , |Y| = N , each point in X (Y) has degree D(d) fixed, respectively, moreover, each pair of points x, x' ∈ X has at most D (log n) 3 (common) neighbours. Let t (X, Y) denote the minimum number of vertices of Y needed to cover all vertices of X. We prove (Theorem 1.1) that t (X, Y) d n tends to 1 as n tends to infinity. This result has many applications: Theorem [5]. Suppose k > r > 1 are fixed, n → ∞. Then there exists a collection of (1 + o (1)) × ( n r ) ( k r ) k -subsets of an n -set so that each r -subset is contained in at least one member of the collection. Analogues and strengthenings of this result are deduced. E.g. for vector spaces, orthogonal or simplectic geometries, random collections of k -sets with constant probabilities, etc. Theorem 3.3. Suppose G is a graph on v vertices and e edges and ℛ is a random graph on n vertices and edge probability e ( v 2 ). Then there exists a collection of (1 + o (1))( n 2 ) ( v 2 ) induced subgraphs of ℛ on v vertices, isomorphic to G and such that each edge (non-edge) of ℛ is covered by an edge (non-edge) of a graph in the collection. We present a novel method for aligning images in an HDR (high-dynamic-range) image stack to produce a new exposure stack where all the images are aligned and appear as if they were taken simultaneously, even in the case of highly dynamic scenes. Our method produces plausible results even where the image used as a reference is either too dark or bright to allow for an accurate registration. Real-time traffic queue detection can directly aid in dynamic traffic light control at road junctions. In this paper, we propose an efficient technique to detect vehicle queue lengths at traffic junctions based on progressive block based image processing. We also propose a two-step approach for vehicle detection that relies on edges and dark features in the image. It is shown that this vehicle detection approach is robust to heavy and light shadows. Further, the threshold adapts itself dynamically to handle varying light conditions. Evaluation of the proposed method using over 45 real video sequences shows nearly 100 accuracy in vehicle detection and queue length estimation. Physarum polycephalum is a unicellular, multinucleate slime mold in the Mycetozoa group that exhibits remarkable capabilities, including the ability to construct efficient networks when foraging food [1]. The Physarum cell grows, as long as nutrition is abundant. When nutrition is limited, Physarum forms a network of interconnected veins; the veins are gel-like tubes in which the cytoplasm flows. It has been experimentally observed that, with two food sources, Physarum's tubular network often retracts to the shortest path between the food sources. Tero, Kobayashi and Nakagaki [2] proposed a model for the dynamics of the Physarum, which in the computer simulations converged to the shortest path on any initial network. We analytically prove that, under this model, the mass of the mold has to eventually converge to the shortest path in the initial network between the two food sources, independently of the structure of the initial network or of the initial mass distribution. This presentation is based on the work in [3,4] and is supported by the Italian Flagship Initiative "InterOmics" (PB.P05). References [1] Toshiyuki Nakagaki, Hiroyasu Yamada, Agota Toth: Maze-solving by an amoeboid organism. Nature 407:470 (2000) [2] Atsushi Tero, Ryo Kobayashi, Toshiyuki Nakagaki: A mathematical model for adaptive transport network in path finding by true slime mold. Journal of Theoretical Biology 244: 553-564 (2007) [3] Vincenzo Bonifaci, Kurt Mehlhorn, Girish Varma: Physarum can compute shortest paths. Journal of Theoretical Biology 309: 121-133 (2012) [4] Vincenzo Bonifaci: Physarum can compute shortest paths: A short proof. Information Processing Letters 113(1-2): 4-7 (2013) Edge and vertex connectivity are fundamental concepts in graph theory. While they have been thoroughly studied in the case of undirected graphs, surprisingly not much has been investigated for directed graphs. In this paper we study @math -edge connectivity problems in directed graphs and, in particular, we consider the computation of the following natural relation: We say that two vertices @math and @math are @math -edge-connected if there are two edge-disjoint paths from @math to @math and two edge-disjoint paths from @math to @math . This relation partitions the vertices into blocks such that all vertices in the same block are @math -edge-connected. Differently from the undirected case, those blocks do not correspond to the @math -edge-connected components of the graph. We show how to compute this relation in linear time so that we can report in constant time if two vertices are @math -edge-connected. We also show how to compute in linear time a sparse certificate for this relation, i.e., a subgraph of the input graph that has @math edges and maintains the same @math -edge-connected blocks as the input graph. Given an undirected graph @math , it is known that its edge-connectivity @math can be computed by solving @math max-flow problems. The best time bounds known for the problem are @math , due to Matula (28th IEEE Symposium on the Foundations of Computer Science, 1987, pp. 249–251) if G is simple, and @math , due to Even and Tarjan (SIAM J. Comput., 4 (1975), pp. 507–518) if G is multiple.An @math time algorithm for computing the edge-connectivity @math of a multigraph @math , where @math is the number of pairs of nodes between which G has an edge, is proposed. This algorithm does not use any max-flow algorithm but consists only of @math times of graph searches and edge contractions. This method is then extended to a capacitated network to compute its minimum cut capacity in @math time. The lack of an accurate representation of the Internet topology at the Autonomous System (AS) level is a limiting factor in the design, simulation, and modeling efforts in inter-domain routing protocols. In this paper, we design and implement a framework for identifying AS links that are missing from the commonly-used Internet topology snapshots. We apply our framework and show that the new links that we find change the current Internet topology model in a non-trivial way. First, in more detail, our framework provides a large-scale comprehensive synthesis of the available sources of information. We cross-validate and compare BGP routing tables, Internet Routing Registries, and traceroute data, while we extract significant newinformation from the less-studied Internet Exchange Points (IXPs). We identify 40 more edges and approximately 300 more peer-to-peer edges compared to commonly used data sets. Second, we identify properties of the new edges and quantify their effects on important topological properties. Given the new peer-to-peer edges, we find that for some ASes more than 50 of their paths stop going through their ISP providers assuming policy-aware routing. A surprising observation is that the degree of a node may be a poor indicator of which ASes it will peer with: the two degrees differ by a factor of four or more in 50 of the peer-to-peer links. Finally, we attempt to estimate the number of edges we may still be missing. Abstract Let S be a set of n ⩾ 4 points in general position in the plane, and let h n be the number of extreme points of S . We show how to construct a 3-connected plane graph with vertex set S , having max ⌈ 3 n 2 ⌉ , n + h − 1 edges, and we prove that there is no 3-connected plane graph on top of S with a smaller number of edges. In particular, this implies that S admits a 3-connected cubic plane graph if and only if n ⩾ 4 is even and h ⩽ n 2 + 1 . The same bounds also hold when 3-edge-connectivity is considered. We also give a partial characterization of the point sets in the plane that can be the vertex set of a cubic plane graph. Network edge data attributes are usually encoded using color, opacity, stroke thickness and stroke pattern, or some combination thereof. In addition to these static variables, it is also possible to animate dynamic particles flowing along the edges. This opens a larger design space of animated edge textures, featuring additional visual encodings that have potential not only in terms of visual mapping capacity but also playfulness and aesthetics. Such animated edge textures have been used in several commercial and design-oriented visualizations, but to our knowledge almost always in a relatively ad hoc manner. We introduce a design space and Web-based framework for generating animated edge textures, and report on an initial evaluation of particle properties - particle speed, pattern and frequency - in terms of visual perception. If G is an arbitrary non-complete graph, let cZ1(G) denote the least number k such that by deleting k appropriately chosen vertices from G (i. e. deleting the k points in question and all edges starting from these points) the resulting graph is not connected. If G is a complete graph of order n, we put c,(G) = n-l. Let cY(G) denote the least number 1 such that by deleting 1 appropriately chosen edges from G the resulting graph is not connected. We may measure the strength of connectedness of G by any of the numbers c],(G), c$,(G) and in a certain sense if G is known to be connected) also by c(G). Evidently one has A k-uniform hypergraph H = (V, E) is called l-orientable, if there is an assignment of each edge e e E to one of its vertices v e e such that no vertex is assigned more than l edges. Let Hn,m,k be a hypergraph, drawn uniformly at random from the set of all k-uniform hypergraphs with n vertices and m edges. In this paper we establish the threshold for the l-orientability of Hn,m,k for all k ≥ 3 and l ≥ 1, i.e., we determine a critical quantity c*k,l such that with probability 1 − o(1) the graph Hn,cn,k has an l-orientation if c c*k,l. Our result has various applications including sharp load thresholds for cuckoo hashing, load balancing with guaranteed maximum load, and massive parallel access to hard disk arrays. We present an algorithm to estimate the rotational motion of an event camera. In contrast to traditional cameras, which produce images at a fixed rate, event cameras have independent pixels that respond asynchronously to brightness changes, with microsecond resolution. Our method leverages the type of information conveyed by these novel sensors (i.e., edges) to directly estimate the angular velocity of the camera, without requiring optical flow or image intensity estimation. The core of the method is a contrast maximization design. The method performs favorably against ground truth data and gyroscopic measurements from an Inertial Measurement Unit, even in the presence of very high-speed motions (close to 1000 deg s). We give an approximation algorithm for non-uniform sparsest cut with the following guarantee: For any @math , given cost and demand graphs with edge weights @math respectively, we can find a set @math with @math at most @math times the optimal non-uniform sparsest cut value, in time @math provided @math . Here @math is the @math 'th smallest generalized eigenvalue of the Laplacian matrices of cost and demand graphs; @math (resp. @math ) is the weight of edges crossing the @math cut in cost (resp. demand) graph and @math is the sparsity of the optimal cut. In words, we show that the non-uniform sparsest cut problem is easy when the generalized spectrum grows moderately fast. To the best of our knowledge, there were no results based on higher order spectra for non-uniform sparsest cut prior to this work. Even for uniform sparsest cut, the quantitative aspects of our result are somewhat stronger than previous methods. Similar results hold for other expansion measures like edge expansion, normalized cut, and conductance, with the @math 'th smallest eigenvalue of the normalized Laplacian playing the role of @math in the latter two cases. Our proof is based on an l1-embedding of vectors from a semi-definite program from the Lasserre hierarchy. The embedded vectors are then rounded to a cut using standard threshold rounding. We hope that the ideas connecting @math -embeddings to Lasserre SDPs will find other applications. Another aspect of the analysis is the adaptation of the column selection paradigm from our earlier work on rounding Lasserre SDPs [GS11] to pick a set of edges rather than vertices. This feature is important in order to extend the algorithms to non-uniform sparsest cut.