Mathematics

Abstract. In recent years the problem of obtaining a reversible dis-crete surface polyhedrization (DSP) is attracting an increasing interest within the discrete geometry community. In this paper we propose the first algorithm for obtaining a reversible polyhedrization with a guaran- ...

In this paper we define and study digital manifolds of arbitrary dimension, and provide (in particular) a general theoretical basis for curve or surface tracing in picture analysis. The studies involve properties such as one-dimensionality of digital curves and (n−1)-dimensionality of digital hypersurfaces that makes them discrete analogs of corresponding notions in topology. The presented approach is fully based on good pairs of adjacency relations and complements the concept of dimension as common in combinatorial topology. This work appears to be the first one on digital manifolds based on a graph-theoretical definition of dimension. In particular, a digital hypersurface in nD is an (n − 1)-dimensional object, as it is in the case of continuous hypersurfaces. Relying on the obtained properties of digital hypersurfaces, we propose a uniform approach for studying good pairs defined by separations and obtain a classification of good pairs in arbitrary dimension.

In this paper we study discretizations of objects in higher dimensions. We introduce a large class of object discretizations, called kdiscretizations. This class is natural and quite general, including as special cases some known discretizations, like the standard covers and the naive discretizations. Various results are obtained in the proposed general setting.

Given a set S ⊆ R 2 , denote S Z = S ∩ Z 2. We obtain bounds for the number of vertices of the convex hull of S Z , where S ⊆ R 2 is a convex region bounded by two circular arcs. Two of the bounds are tight bounds-in terms of arc length and in terms of the width of the region and the radii of the circles, respectively. Moreover, an upper bound is given in terms of a new notion of "set oblongness." The results complement the well-known O(r 2/3) bound [2] which applies to a disc of radius r.

Digital planarity is defined by digitizing Euclidean planes in the three-dimensional digital space of voxels; voxels are given either in the grid-point or the grid-cube model. The paper summarizes results (also including most of the proofs) about different aspects of digital planarity, such as supporting or separating Euclidean planes, characterizations in arithmetic geometry, periodicity, connectivity, and algorithmic solutions. The paper provides a uniform presentation, which further extends and details a recent book chapter in [R. Klette, A. Rosenfeld, Digital Geometry-Geometric Methods for Digital Picture Analysis,

In this paper we define the notion of gap in an arbitrary digital picture S in a digital space of arbitrary dimension. As a main result, we obtain an explicit formula for the number of gaps in S of maximal dimension. We also derive a combinatorial relation for a digital curve.

In this paper we investigate the advantages of using hexagonal grids in raster and volume graphics. In 2D, we present a hexagonal graphical model based on a hexagonal grid. In 3D, we introduce two honeycomb graphical models in which the voxels are hexagonal prisms, and we show that these are the only possible models under certain reasonable conditions. In the framework of the proposed models, we design two-and three-dimensional analytical honeycomb geometry of linear objects, as well as of circles and spheres. We demonstrate certain advantages of the honeycomb models and address algorithmic and complexity issues.

Studying connectivity of discrete objects is a major issue in discrete geometry and topology. In the present work we deal with connectivity of discrete planes in the framework of Reveillès analytical definition [11]. Accordingly, a discrete plane is a set P (a, b, c, µ, ω) of integer points (x, y, z) satisfying the Diophantine inequalities 0 ≤ ax + by + cz + µ < ω. The parameter µ ∈ Z estimates the plane intercept while ω ∈ N is the plane thickness. Given three integers (plane coefficients) a, b, and c with 0 ≤ a ≤ b ≤ c, one can seek the maximal ω for which the discrete plane P (a, b, c, µ, ω) is disconnected. We call this remarkable topological invariable the connectivity number of P (a, b, c, µ, ω) and denote it Ω(a, b, c). Despite several attempts over the last ten years to determine the connectivity number, this is still an open question. In the present paper we propose a solution to the problem. For this, we first investigate some combinatorial properties of discrete planes. These structural results facilitate the deeper understanding of the discrete plane structure. On this basis, we obtain a series of results, in particular, we provide an explicit solution to the problem under certain conditions. We also obtain exact upper and lower bounds on Ω(a, b, c) and design an O(a log b) algorithm for its computation.

An important concept in combinatorial image analysis is that of gap. In this paper we derive a simple formula for the number of gaps in a 2D binary picture. Our approach is based on introducing the notions of free vertex and free edge and studying their properties from point of view of combinatorial topology. The number of gaps characterizes the topological structure of a binary picture and is of potential interest in property-based image analysis.

This special issue is devoted to topics related to mathematics for applications in imaging – a field with increasing importance employed in areas as diverse as medicine, robotics, defense, and security, environmental studies, astronomy, material science, and manufacturing. The papers either contribute to the theoretical foundations of the field or develop mathematical models and methods for solving such problems. As a rule, the processed data are discrete and therefore, in most of the cases a discrete approach is applied, which often features various advantages (in terms of efficiency and accuracy) over the more traditional approaches based on continuous models requiring numeric computation. The works have initially been presented at two related conferences: IWCIA 2015 held in Kolkata, India, November 24–27, 2015 and CompIMAGE 2016 held in Niagara Falls, USA, September 21–23, 2016. Of over 55 works presented at both conferences, after a regular review process, five substantially ext...

Climate variability and human activities interact to increase the abundance of woody plants in arid and semi-arid ecosystems worldwide. How woody plants interact with rainfall to influence patterns of soil moisture through time, at different depths in the soil profile and between neighboring landscape patches is poorly known. In a semi-arid mesquite savanna, we deployed a paired array of sensors beneath a mesquite canopy and in an adjacent open area to measure volumetric soil water content (Â) every 30 min at several depths between 2004 and 2007. In addition, to quantify temporally dynamic variation in soil moisture between the two microsites and across soil depths we analysed  time-series using fast Fourier transforms (FFT). FFT analyses were consistent with the prediction that by reducing evaporative losses through shade and reducing rainfall inputs through canopy interception of small rainfall events, the mesquite canopy was associated with a decline in high-frequency (hour-to-hour and day-to-day) variation in shallow Â. Finally, we found that, in both microsites, high-frequency  variation declined with increasing soil depth as the influence of evaporative losses and inputs associated with smaller rainfall events declined. In this case, we argue that the buffering of shallow soil moisture against high-frequency variations can enhance nutrient cycling and alter the carbon cycle in dryland ecosystems.

It follows from the theory of trace identities developed by Procesi and Razmyslov that the trace cocharacters arising from the trace identities of the algebra M r (F ) of r × r matrices over a field F of characteristic zero are given by TC r,n = λ∈Λ r (n) χ λ ⊗ χ λ where χ λ ⊗ χ λ denotes the Kronecker product of the irreducible characters of the symmetric group associated with the partition λ with itself and Λ r (n) denotes the set of partitions of n with r or fewer parts, i.e. the set of partitions λ = (λ 1 · · · λ k ) with k r. We study the behavior of the sequence of trace cocharacters TC r,n . In particular, we study the behavior of the coefficient of χ (ν,n−m) in TC r,n as a function of n where ν = (ν 1 · · · ν k ) is some fixed partition of m and n − m ν k . Our main result shows that such coefficients always grow as a polynomial in n of degree r − 1.

Classification and extraction of geospatial features from high spatial resolution imageries approved is one of the most significant steps for spatial database acquisition and updating in GIS. However, the conventional method of human interpretation and digitizing cannot fulfill the requirements of large area and fast paced spatial data collections owing to the natures of labor intensive and time consuming. This research is to explore the methodologies of recognizing shape and elevation characteristics of spatial features on the remote sensed images. We focus on the road network classification and extraction among various features on the ground because it carries unique characteristic of elongation. We combined both a pattern recognition model of connected component labeling (CCL) in two dimensional image processing, and a three dimensional DEM elevation filtering model to extract the road and street features from high resolution imageries. Samples of digital aerial photographs in Erie County, New York were used to test the methodology. The results indicated that the correctness of road extraction in rural areas can reach 69.1%; that of completeness is 74.9%; and the overall quality is 73.1%. By contrast, the correctness in urban high-rise region is only 39.5%; that of completeness is 42.8%; and the overall quality is 32.8%.