Reneta P. Barneva

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.

Thin discrete triangular meshes

Abstract In this paper we present an approach to describe polyhedra by meshes of discrete triangles. The study is based on the theory of arithmetic discrete geometry, (J.-P. Reveilles, Geometrie discrete, calcul en nombres entiers et algorithmique, These d’etat, Universite Louis Pasteur, Strasbourg, December 1991). As distinct from the previous investigations on this topic, the triangles we introduce are parts of the thinnest possible discrete 6-tunnel-free planes, i.e., those that are usually used in practice. Given a plane P in the space, we define a 6-tunnel-free discrete plane, called a regular plane, which appears to be the best approximation to P . Given a mesh of triangles, we propose a method to approximate any triangle by a discrete triangular patch – a portion of a regular plane, and we prove that the resulting triangular mesh is 6-tunnel-free. The properties of the approximation obtained make the suggested approach convenient for practical applications.

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).

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.

“Honeycomb” vs Square and Cubic Models

Abstract In this paper we summarize some observations about the advantages of using hexagonal grids in raster graphics. We initiate a study of honeycomb graphics , whose 2D version is based on a hexagonal grid, while in its 3D counterpart the voxels are hexagonal prisms. We design an analytical honeycomb geometry of linear objects, which parallels similar developments already known in classical raster graphics [6]. We also demonstrate certain advantages of honeycomb graphics, in particular that it provides a better tunnel-free approximation to continuous objects.

Some theoretical challenges in digital geometry: A perspective

In recent years image analysis has become a research field of exceptional significance, due to its relevance to real life problems in important societal and governmental sectors, such as medicine, defense, and security. The explicit purpose of the present Perspective is to suggest a number of strategic objectives for theoretical research, with an emphasis on the combinatorial approach in image analysis. Most of the proposed objectives relate to the need to make the theoretical foundations of combinatorial image analysis better integrated within a number of well-established subjects of theoretical computer science and discrete applied mathematics, such as the theory of algorithms and problem complexity, combinatorial optimization and polyhedral combinatorics, integer and linear programming, and computational geometry.

Graceful Planes and Thin Tunnel-Free Meshes

In this paper we present an approach to describe polyhedra by meshes of discrete triangles. The study is based on the theory of arithmetic geometry [10]. We introduce a new class of discrete planes (respectively lines) which we call graceful planes (respectively graceful lines).We use naive planes and graceful lines to obtain as thin as possible triangular mesh discretization admitting an analytical description. The interiors of the triangles are portions of naive planes, while the sides are graceful lines.

Minimal offsets that guarantee maximal or minimal connectivity of digital curves in nD

In this paper we investigate an approach of constructing a digital curve by taking the integer points within an offset of a certain radius of a continuous curve. Our considerations apply to digitizations of arbitrary curves in arbitrary dimension n. As main theoretical results, we first show that if the offset radius is greater than or equal to √n/2,then the obtained digital curve features maximal connectivity. We also demonstrate that the radius value √n/2 is the minimal possible that always guarantees such a connectivity. Moreover, we prove that a radius length greater than or equal to √n - 1/2 guarantees 0-connectivity, and that this is the minimal possible value with this property. Thus, we answer the question about the minimal offset size that guarantees maximal or minimal connectivity of an offset digital curve.

Object Discretization in Higher Dimensions

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.