Benedek Nagy

Characterization of digital circles in triangular grid

In this paper we present some former results about properties of digital circles defined by neighbourhood sequences in the triangular grid. Das and Chatterji [Inform. Sci. 50 (1990) 123] analyzed the geometric behaviour of two-dimensional periodic neighbourhood sequences. We use a more general definition of neighbourhood sequences, which does not require periodicity [Publ. Math. Debrecen 60 (2002) 405]. We study the development of wave-fronts and grow digital circles from a triangle with general neighbourhood sequences in triangular grid. We present the possible types of polygons, and characterize them by the initial part of the neighbourhood sequences. The symmetry and the convexity analysis of the digital circles is also presented.Besides those who are interested in the underlying theory there may be readers from the pattern recognition or the image processing communities or even the geometric modelling field who could find some of the consequences of the paper of interest.

Digital distance functions on three-dimensional grids

In this paper, we examine five different three-dimensional grids suited for image processing. Digital distance functions are defined on the cubic, face-centered cubic, body-centered cubic, honeycomb, and diamond grids. We give the parameters that minimize an error function that favors distance functions with low rotational dependency. We also give an algorithm for computing the distance transform-the tool by which these distance functions can be applied in image processing applications.

Distances with neighbourhood sequences in cubic and triangular grids

In this paper we compute distances with neighbourhood sequences in the cubic and in the triangular grids. First we give a formula which computes the distance with arbitrary neighbourhood sequence in the three-dimensional digital space. After this, using the injection of the triangular grid to the cubic grid, we modify the formula for Z^3 to the triangular plane. The distances in the triangular grid have some properties which are not present on the square and cubic grids. It may be non-symmetric, and it is possible that the distance depends on the ordering of elements of the initial part of the neighbourhood sequence. The distance depends on the ordering of the initial part (up to the kth element) of the neighbourhood sequence if and only if there is a permutation of these elements such that the distance (up to value k) is non-symmetric using the elements in this new order. This dependence means somehow more flexibility of the distances based on neighbourhood sequences on the triangular grid than in Z^n.

Distances based on neighbourhood sequences in non-standard three-dimensional grids

Properties for distances based on neighbourhood sequences on the face-centred cubic (fcc) and the body-centred cubic (bcc) grids are presented. Formulas to both compute the distances and assure that the distances satisfy the conditions for being metrics are presented and proved to be correct. The formulas are used to calculate the neighbourhood sequences that generates distances with lowest deviation from the Euclidean distance.

NON-TRADITIONAL GRIDS EMBEDDED IN ℤn

The two-dimensional hexagonal grid and the three-dimensional face-centered cubic grid can be described by intersecting ℤ3 and ℤ4 with (hyper)planes. Corresponding grids in higher dimensions are examined. Also, we explain the connection between a number of well-known three-dimensional grids by using this construction. The union of four hyperplanes (in a circular way) gives the bcc grid. Based on these connections, several types of neighborhood structures are introduced on these grids. These structures span from the most natural ones (crystal bonds, Voronoi neighbors) to infinite families. In this paper, we define path-based distance functions on the high-dimensional generalizations of the hexagonal grid.

Distance with generalized neighbourhood sequences in nD and ∞D

In this paper we generalize some former results of Das et al. about distances with n-dimensional periodic neighbourhood sequences. We use a more general definition of neighbourhood sequences, which does not require periodicity. As an extension of the earlier results, we give a formula to calculate the distance between two arbitrary points with general neighbourhood sequences in an arbitrary finite dimension. Moreover we extend the result to the infinite-dimensional digital space.

A family of triangular grids in digital geometry

In this paper we show a new geometric interpretation of the hexagonal and triangular grids. They can be considered as the sets of points of one (see (I. Her, 1995)), respectively two plane(s) in Z/sup 3/. By this approach we can build up a whole family of triangular grids (the so called n-planes triangular grids). The hexagonal and triangular grids are the first two members of this family, moreover, they are duals of each other. We investigate the three-planes grid, the third member of the family, and its dual in detail. We show that for n /spl ges/ 4 on, the n-planes triangular grids are non-planar.

A Connection between Z n and Generalized Triangular Grids

In this paper we show how non-standard three-dimensional grids,such as the face-centered cubic (fcc), the body-centered cubic(bcc), and the diamond grids can be embedded in Z4. Thefcc grid is a hyperplane in Z4, the diamond grid is theunion of two parallel hyperplanes. The union of four hyperplanes(in a circular way) gives the bcc grid. Based on these connections,several types of neighborhood structures are introduced on thesegrids. These structures span from the most natural ones (crystalbonds, Voronoi neighbors) to infinite families.

Approximating euclidean distance using distances based on neighbourhood sequences in non-standard three-dimensional grids

In image processing, it is often of great importance to have small rotational dependency for distance functions. We present an optimization for distances based on neighbourhood sequences for the face-centered cubic (fcc) and body-centered cubic (bcc) grids. In the optimization, several error functions are used measuring different geometrical properties of the balls obtained when using these distances.

Cellular topology and topological coordinate systems on the hexagonal and on the triangular grids

In this paper we use symmetric coordinate systems for the hexagonal and the triangular grids (that are dual of each other). We present new coordinate systems by extending the symmetric coordinate systems that are appropriate to address elements (cells) of cell complexes. Coordinate triplets are used to address the hexagon/triangle pixels, their sides (the edges between the border of neighbour pixels) and the points at the corners of the hexagon/triangle pixels. Properties of the coordinate systems are detailed, lines (zig-zag lines) and lanes (hexagonal stepping lanes) are defined on the triangular (resp. hexagonal) grid by fixing a coordinate value. The bounding relation of the cells can easily be captured by the coordinate values. To illustrate the utility of these coordinate systems some topological algorithms, namely collapses and cuts are presented.