Valentin E. Brimkov

Digital planarity-A review

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, Morgan Kaufmann, San Francisco, 2004].

Object discretizations in higher dimensions

Abstract In this paper we study discretizations of objects in higher dimensions. We introduce a large class of object discretizations, called k-discretizations. 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.

Analytical Honeycomb Geometry for Raster and Volume Graphics

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 the 2D and 3D 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.

Editorial: Computational modeling of objects represented in images

This special issue of Graphical Models contains six articles, which are substantial extensions of papers presented at the International Symposium ‘‘Computational Modeling of Objects Represented in Images. Fundamentals, Methods and Applications’’ (CompIMAGE 2010) held in Buffalo, NY, May 5–7, 2010. The purpose of CompIMAGE 2010 was to provide a common forum for researchers, scientists, engineers, and practitioners around the world to present their latest research findings, ideas, developments, and applications in the area. CompIMAGE 2010 received 77 submissions. After a rigorous review process involving three to four independent double-blind reviews, 28 papers were accepted for presentation at the symposium and included in the symposium proceedings published in the Springer’s LNCS series. After the event, the authors of 10 papers which received highest scores by reviewers and whose topics were relevant to Graphical Models were invited to submit extensions of their works to a special issue of the journal. The extension was required to be very essential— to the extent of a new paper. After several reviewing rounds, typically by at least four referees, six papers were accepted for publication in the present special GMOD issue. We believe that as a result of the long and rigorous selection process it contains only papers of very high quality. In the first paper ‘‘Surface-based analysis methods for high-resolution functional magnetic resonance imaging’’ by Rez Khan, Qin Zhang, Shayan Darayan, Sankari Dhandapani, Sucharit Katyal, Clint Greene, Chandra Bajaj, and David Ress, the authors present a set of surface-based methods to exploit the use of highresolution fMRI for depth analysis of the brain tissue. The proposed methods provide averaging schemes that can increase contrast-to-noise ratio and permit the direct analysis of depth profiles of functional activity in the human brain. In the next paper ‘‘Connected distance-based rasterization of objects in arbitrary dimension,’’ the authors Valentin E. Brimkov, Reneta P. Barneva, and Boris Brimkov investigate an approach of constructing a digital curve by taking the integer points within an offset of a certain

Plane digitization and related combinatorial problems

In this paper we propose a simple scheme for obtaining plane digitizations. We study digital plane periodicity and consider various issues related to two-dimensional (2D) Sturmian words. Concepts and results, already known for one-dimensional words, are extended to 2D words. In particular, we address a conjecture by Maurice Nivat for the case of digital 2D rays. Our approach is based in part on extending periodicity studies in theory of words to 2D words based on (Proceedings of the Third ACM-SIAM Symposium on Discrete Algorithms, 1992, pp. 440-452; Proceedings of the 33rd IEEE Symposium on Foundations in Computer Science, 1992, pp. 247-250).

Curves, hypersurfaces, and good pairs of adjacency relations

In this paper we propose several equivalent definitions of digital curves and hypersurfaces in arbitrary dimension. The definitions involve properties such as one-dimensionality of curves and (n – 1)-dimensionality of hypersurfaces that make them discrete analogs of corresponding notions in topology. Thus this work appears to be the first one on digital manifolds where the definitions involve the notion 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 clssification of good pairs in arbitrary dimension.

Border and SurfaceTracing - Theoretical Foundations

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 the one-dimensionality of digital curves and (n - 1)-dimensionality of digital hypersurfaces that makes them discrete analogs of corresponding notions in continuous topology. The presented approach is fully based on the concept of adjacency relation 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, in the n-dimensional digital space, a digital curve is a one-dimensional object and a digital hypersurface is an (n - 1)-dimensional object, as it is in the case of curves and hypersurfaces in the Euclidean space. 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. We also discuss possible applications of the presented definitions and results.

Guarding a set of line segments in the plane

We consider the following problem: Given a finite set of straight line segments in the plane, find a set of points of minimum size, so that every segment contains at least one point in the set. This problem can be interpreted as looking for a minimum number of locations of policemen, guards, cameras or other sensors, that can observe a network of streets, corridors, tunnels, tubes, etc. We show that the problem is strongly NP-complete even for a set of segments with a cubic graph structure, but in P for tree structures.

Computational aspects of digital plane and hyperplane recognition

In these note we review some basic approaches and algorithms for discrete plane/hyperplane recognition. We present, analyze, and compare related theoretical and experimental results and discuss on the possibilities for creating algorithms with higher efficiency.

The number of gaps in binary pictures

This paper identifies the total number of gaps of object pixels in a binary picture, which solves an open problem in 2D digital geometry (or combinatorial topology of binary pictures). We obtain a formula for the total number of gaps as a function of the number of object pixels (grid squares), vertices (corners of grid squares), holes, connected components, and 2 × 2 squares of pixels. It can be used to test a binary picture (or just one region: e.g., a digital curve) for gap-freeness.