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For all kinds of stationary plane grids only the 6-neighborhood is consistent. It admits the definition of a digital curve, in particular a contour, as a set of px pairs. Thus curves become objects of zero thickness and may be coded with one bit per grid step.
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April 2000 · Ergodic Theory and Dynamical Systems
A partially ordered set that is invariantly associated to a subshift is constructed. A property of subshifts, also an invariant of topological conjugacy, is described. If this property is present in a subshift then the constructed partially ordered set is a partially ordered semigroup (with zero). In the description of these invariants the notion of context is instrumental.
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June 2009 · Doklady Mathematics
The marked point method (MPM) for optimal linearly coding multivariate continuous functions is suggested. For the point set of the infinite compact space and a linear continuous mapping, there exists a continuous function such that the linear continuous mapping is equal to zero. The subset of functions satisfying the Lipschitz conditions with a given Lipschitz constant consists of all continuous … [Show full abstract] real-valued functions satisfying the Lipschitz condition. The MPM coding is found to be any mapping taking each continuous function to the vector of its values at the points of the grid. MPM optimization includes estimating the error of reconstruction of a function in the given subset from its values.
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March 1999 · Lecture Notes in Computer Science
A new method of representing a surface in the 3D space as a single digitally continuous sequence of faces is described. The method is based on topological properties of quasi-manifolds. It is realized as tracing the boundary of a growing set of labeled faces. As the result the surface is encoded as a single sequence of mutually adjacent faces. Each face is encoded by one byte. The code of the … [Show full abstract] surface of a three-dimensional object takes much less memory space then the raster representation of the object. The object may be exactly reconstructed from the code. Surfaces of a genus greater that zero (e.g. that of a torus) may also be encoded by a single continuous sequence. The traversal algorithm recognizes the genus of the surface.
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October 2010 · Journal of Mathematical Imaging and Vision
The paper presents a new set of axioms of digital topology, which are easily understandable for application developers. They define a class of locally finite (LF) topological spaces. An important property of LF spaces satisfying the axioms is that the neighborhood relation is antisymmetric and transitive. Therefore any connected and non-trivial LF space is isomorphic to an abstract cell complex. … [Show full abstract] The paper demonstrates that in an n-dimensional digital space only those of the (a, b)-adjacencies commonly used in computer imagery have analogs among the LF spaces, in which a and b are different and one of the adjacencies is the “maximal” one, corresponding to 3n
− 1 neighbors. Even these (a, b)-adjacencies have important limitations and drawbacks. The most important one is that they are applicable only to binary images. The way of easily using LF spaces in computer imagery on standard orthogonal grids containing only pixels or voxels and no cells of lower dimensions is suggested.
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December 1997
A new classification of digital curves into boundary curves and visual curves of different thickness is suggested. A fast algorithm for recognizing digital straight line segments in boundary curves is presented. The algorithm is applied to encode the boundaries of homogeneous regions in digitized images. The code is economical and enables an exact reconstruction of the original image.
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December 2004 · Lecture Notes in Computer Science
The paper presents some algorithms in digital geometry based on the topology of cell complexes. The paper contains an axiomatic justification of the necessity of using cell complexes in digital geometry. Algorithms for solving the following problems are presented: tracing of curves and surfaces, recognition of digital straight line segments (DSS), segmentation of digital curves into longest DSS, … [Show full abstract] recognition of digital plane segments, computing the curvature of digital curves, filling of interiors of n-dimensional regions (n=2,3,4), labeling of components (n=2,3), computing of skeletons (n=2, 3).
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August năm trước · Discrete Mathematics
A set of vertices of a graph is paired-dominating if induces a matching in and dominates all vertices of . A set is locating if for any two distinct vertices , , where and are open neighborhoods of vertices and . We give a complete characterization of locating–paired-dominating sets with minimal density in the infinite square grid .
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