Robert A. Melter

Metric bases in digital geometry

Abstract Let S be a metric space under the distance function d. A metric basis is a subset B ⊆ S such that d(b, x) = d(b, y) for all b ϵ B implies x = y. It is shown that for Euclidean distance, the minimal metric bases for the digital plane are just the sets of three noncollinear points; but for city block or chessboard distance, the digital plane has no finite metric basis. The sizes of minimal metric bases for upright digital rectangles are also derived, and it is shown that there exist rectangles having minimal metric bases of any size ≥ 3.

Bipartite graph matching for points on a line or a circle

Abstract Given two sets of M points on a line or on a circle, a minimal matching between them is found in O(M log M) time. The circular case, where the distance between two points is the length of the shortest arc connecting them, is shown to have the same complexity as the simpler linear case. Finding the shift of one of the sets, linear or circular, that minimizes the cost of matching is also discussed.

A new characterization of digital lines by least square fits

Abstract In this paper we prove that digital line segments and their least square line fits are in one-to-one correspondence and give a new simple representation ( x 1 , n , b 0 , b 1 ) of a digital line segment, where x 1 and n are the x -coordinate of the left endpoint and the number of digital points, respectively, while b 0 and b 1 are the coefficients of the least square line fit Y = b 0 + b 1 X for the given digital line segment. An O( n log n ) time algorithm for obtaining a digital line segment from its least square line fit is described.

New views of linearity and connectedness in digital geometry

Abstract A digital line is classically defined to be the result of subjecting a continuous line to a particular digitization process. This does not take into account noise in the environment. We present a less rigid concept of linearity based on least squares and the correlation coefficient. A new type of connectedness is also discussed; it is intermediate between the usual 4- and 8-connectedness. An appendix contains the continuation of a series of bibliographies on digital metrics.

Some characterizations of city block distance

Abstract It is shown that in the presence of a simple geometrical constraint city block distance between n -dimensional lattice point is characterized by a condition used by Rosenfeld in an investigation of continuous functions on digital images. For the digital plane such a description is obtained in purely metric terms. In any metric space the relation of points being at unit distance is a tolerance, i.e. it is reflexive and symmetric. Necessary and sufficient algebraic conditions are given for a tolerance to be that associated with planar city block distance. An appendix continues the bibliography of papers related to distance in digital geometry begun in a previous report.

Digital metrics: A graph-theoretical approach

Consider the following two graphs M and N, both with vertex set Z x Z, where Z is the set of all integers. In M, two vertices are adjacent when their euclidean distance is 1, while in N, adjacency is obtained when the distance is either 1 or @/2. By definition, H is a metric subgraph of the graph G if the distance between any two points of H is the same as their distance in G. We determine all the metric subgraphs of M and N. The graph-theoretical distances in M and N are equal respectively to the city block and chessboard matrics used in pattern recognition.

Path generated digital metrics

A new class of metrics, called path-generated metrics, is defined for the digital plane. There are essentially five different metrics in the class, including the classical city block and chessboard metrics.

Recognition and characterization of digitized curves

Abstract We consider the graphs of functions representable in the form h ( x ) = Σ j =1 n a j f j ( x ) where the f j constitute a linearly independent set of functions over R . These graphs are digitized by the set of lattice points ( i , ⌊ h ( i )⌋). An algorithm is presented to determine if a given set of lattice points is part of such a digitization. We also study the digitization of polynomials. An important tool used is the set of differences of the y -coordinates (digital derivatives). For example, if h ( x ) is a polynomial of degree n , then its n -th digital derivative is cyclic and its ( n + 1)st digital derivative has a bound which depends only on n .

Tessellation graph characterization using rosettas

The graphs associated with regular and semi-regular plane tessellations are characterized in terms of certain algebraic structures associated with them. Each of the graphs has associated with it a rosetta, a configuration of lattice points and colored edges which is repeated throughout the plane.

Constant time BSR solutions to L 1 metric and digital geometry problems

In this paper we solve several geometric and image problems using the BSR (broadcasting with selective reduction) model of parallel computation. All of the solutions presented are constant time algorithms. The computational geometry problems are based on city block distance metrics: all nearest neighbors and furthest pairs ofm points in a plane are computed on a two criteria BSR withm processors, the all nearest foreign neighbors and the all furthest foreign pairs ofm points in the plane problems are solved on three criteria BSR withm processors while the area and perimeter ofm isooriented rectangles are found on a one criterion BSR withm2 processors. The problems on ann ×n binary image which are solved here all use BSR withn2 processors and include: histogramming (one criterion), distance transform (one criterion), medial axis transform (three criteria) and discrete Voronoi diagram of labeled images (two criteria).