Extension of One-Dimensional Proximity Regions to Higher Dimensions

Proximity maps and regions are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in various fields. The simplest of such maps is the spherical proximity map which maps a point from the class of interest to a disk centered at the same point with radius being the distance to the closest point from the other class in the region. The spherical proximity map gave rise to class cover catch digraph (CCCD) which was applied to pattern classification. Furthermore for uniform data on the real line, the exact and asymptotic distribution of the domination number of CCCDs were analytically available. In this article, we determine some appealing properties of the spherical proximity map in compact intervals on the real line and use these properties as a guideline for defining new proximity maps in higher dimensions. Delaunay triangulation is used to partition the region of interest in higher dimensions. Furthermore, we introduce the auxiliary tools used for the construction of the new proximity maps, as well as some related concepts that will be used in the investigation and comparison of them and the resulting graphs. We characterize the geometry invariance of PCDs for uniform data. We also provide some newly defined proximity maps in higher dimensions as illustrative examples

Space-time percolation and detection by mobile nodes

Consider the model where nodes are initially distributed as a Poisson point process with intensity $\lambda$ over $\mathbb{R}^d$ and are moving in continuous time according to independent Brownian motions. We assume that nodes are capable of detecting all points within distance $r$ of their location and study the problem of determining the first time at which a target particle, which is initially placed at the origin of $\mathbb{R}^d$, is detected by at least one node. We consider the case where the target particle can move according to any continuous function and can adapt its motion based on the location of the nodes. We show that there exists a sufficiently large value of $\lambda$ so that the target will eventually be detected almost surely. This means that the target cannot evade detection even if it has full information about the past, present and future locations of the nodes. Also, this establishes a phase transition for $\lambda$ since, for small enough $\lambda$, with positive probability the target can avoid detection forever. A key ingredient of our proof is to use fractal percolation and multi-scale analysis to show that cells with a small density of nodes do not percolate in space and time.Comment: Published at http://dx.doi.org/10.1214/14-AAP1052 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Surjective cellular automata far from the Garden of Eden

Automata, Logic and SemanticsInternational audienceOne of the first and most famous results of cellular automata theory, Moore's Garden-of-Eden theorem has been proven to hold if and only if the underlying group possesses the measure-theoretic properties suggested by von Neumann to be the obstacle to the Banach-Tarski paradox. We show that several other results from the literature, already known to characterize surjective cellular automata in dimension d, hold precisely when the Garden-of-Eden theorem does. We focus in particular on the balancedness theorem, which has been proven by Bartholdi to fail on amenable groups, and we measure the amount of such failure

Proceedings of JAC 2010. Journées Automates Cellulaires

The second Symposium on Cellular Automata “Journ´ees Automates Cellulaires” (JAC 2010) took place in Turku, Finland, on December 15-17, 2010. The first two conference days were held in the Educarium building of the University of Turku, while the talks of the third day were given onboard passenger ferry boats in the beautiful Turku archipelago, along the route Turku–Mariehamn–Turku. The conference was organized by FUNDIM, the Fundamentals of Computing and Discrete Mathematics research center at the mathematics department of the University of Turku. The program of the conference included 17 submitted papers that were selected by the international program committee, based on three peer reviews of each paper. These papers form the core of these proceedings. I want to thank the members of the program committee and the external referees for the excellent work that have done in choosing the papers to be presented in the conference. In addition to the submitted papers, the program of JAC 2010 included four distinguished invited speakers: Michel Coornaert (Universit´e de Strasbourg, France), Bruno Durand (Universit´e de Provence, Marseille, France), Dora Giammarresi (Universit` a di Roma Tor Vergata, Italy) and Martin Kutrib (Universit¨at Gie_en, Germany). I sincerely thank the invited speakers for accepting our invitation to come and give a plenary talk in the conference. The invited talk by Bruno Durand was eventually given by his co-author Alexander Shen, and I thank him for accepting to make the presentation with a short notice. Abstracts or extended abstracts of the invited presentations appear in the first part of this volume. The program also included several informal presentations describing very recent developments and ongoing research projects. I wish to thank all the speakers for their contribution to the success of the symposium. I also would like to thank the sponsors and our collaborators: the Finnish Academy of Science and Letters, the French National Research Agency project EMC (ANR-09-BLAN-0164), Turku Centre for Computer Science, the University of Turku, and Centro Hotel. Finally, I sincerely thank the members of the local organizing committee for making the conference possible. These proceedings are published both in an electronic format and in print. The electronic proceedings are available on the electronic repository HAL, managed by several French research agencies. The printed version is published in the general publications series of TUCS, Turku Centre for Computer Science. We thank both HAL and TUCS for accepting to publish the proceedings.Siirretty Doriast

Perceptual Image Similarity Metrics and Applications.

This dissertation presents research in perceptual image similarity metrics and applications, e.g., content-based image retrieval, perceptual image compression, image similarity assessment and texture analysis. The first part aims to design texture similarity metrics consistent with human perception. A new family of statistical texture similarity features, called Local Radius Index (LRI), and corresponding similarity metrics are proposed. Compared to state-of-the-art metrics in the STSIM family, LRI-based metrics achieve better texture retrieval performance with much less computation. When applied to the recently developed perceptual image coder, Matched Texture Coding (MTC), they enable similar performance while significantly accelerating encoding. Additionally, in photographic paper classification, LRI-based metrics also outperform pre-existing metrics. To fulfill the needs of texture classification and other applications, a rotation-invariant version of LRI, called Rotation-Invariant Local Radius Index (RI-LRI), is proposed. RI-LRI is also grayscale and illuminance insensitive. The corresponding similarity metric achieves texture classification accuracy comparable to state-of-the-art metrics. Moreover, its much lower dimensional feature vector requires substantially less computation and storage than other state-of-the-art texture features. The second part of the dissertation focuses on bilevel images, which are images whose pixels are either black or white. The contributions include new objective similarity metrics intended to quantify similarity consistent with human perception, and a subjective experiment to obtain ground truth for judging the performance of objective metrics. Several similarity metrics are proposed that outperform existing ones in the sense of attaining significantly higher Pearson and Spearman-rank correlations with the ground truth. The new metrics include Adjusted Percentage Error, Bilevel Gradient Histogram, Connected Components Comparison and combinations of such. Another portion of the dissertation focuses on the aforementioned MTC, which is a block-based image coder that uses texture similarity metrics to decide if blocks of the image can be encoded by pointing to perceptually similar ones in the already coded region. The key to its success is an effective texture similarity metric, such as an LRI-based metric, and an effective search strategy. Compared to traditional image compression algorithms, e.g., JPEG, MTC achieves similar coding rate with higher reconstruction quality. And the advantage of MTC becomes larger as coding rate decreases.PhDElectrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/113586/1/yhzhai_1.pd