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Von Neumann Neighborhood
In cellular automata, the von Neumann neighborhood (or 4-neighborhood) is classically defined on a two-dimensional square lattice and is composed of a central cell and its four adjacent cells. The neighborhood is named after John von Neumann, who used it to define the von Neumann cellular automaton and the von Neumann universal constructor within it. It is one of the two most commonly used neighborhood types for two-dimensional cellular automata, the other one being the Moore neighborhood. This neighbourhood can be used to define the notion of 4-connected pixels in computer graphics.. The von Neumann neighbourhood of a cell is the cell itself and the cells at a Manhattan distance of 1. The concept can be extended to higher dimensions, for example forming a 6-cell octahedral neighborhood for a cubic cellular automaton in three dimensions. Von Neumann neighborhood of range ''r'' An extension of the simple von Neumann neighborhood described above is to take the set of p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Graphics
Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. A great deal of specialized hardware and software has been developed, with the displays of most devices being driven by graphics hardware, computer graphics hardware. It is a vast and recently developed area of computer science. The phrase was coined in 1960 by computer graphics researchers Verne Hudson and William Fetter of Boeing. It is often abbreviated as CG, or typically in the context of film as Computer-generated imagery, computer generated imagery (CGI). The non-artistic aspects of computer graphics are the subject of Computer graphics (computer science), computer science research. Some topics in computer graphics include user interface design, Sprite (computer graphics), sprite graphics, raster graphics, Rendering (computer graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pixel Connectivity
In image processing, pixel connectivity is the way in which pixels in 2-dimensional (or hypervoxels in n-dimensional) images relate to their neighbors. Formulation In order to specify a set of connectivities, the dimension and the width of the neighborhood , must be specified. The dimension of a neighborhood is valid for any dimension n\geq1. A common width is 3, which means along each dimension, the central cell will be adjacent to 1 cell on either side for all dimensions. Let M_N^n represent a N-dimensional hypercubic neighborhood with size on each dimension of n=2k+1, k\in\mathbb Let \vec represent a discrete vector in the first orthant from the center structuring element to a point on the boundary of M_N^n. This implies that each element q_i \in \ ,\forall i \in \ and that at least one component q_i = k Let S_N^d represent a N-dimensional hypersphere with radius of d=\left \Vert \vec \right \Vert. Define the amount of elements on the hypersphere S_N^d within the nei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice Graph
In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space , forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense. Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid. Moreover, these terms are also commonly used for a finite section of the infinite graph, as in "an 8 × 8 square grid". The term lattice graph has also been given in the literature to various other kinds of graphs with some regular structure, such as the Cartesian product of a number of complete graphs. Square grid graph A common type of lattice graph (known under different names, such as grid graph or square grid graph) is the graph whose vertices corresp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taxicab Geometry
Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two points is instead defined to be the sum of the absolute differences of their respective Cartesian coordinates, a distance function (or metric) called the ''taxicab distance'', ''Manhattan distance'', or ''city block distance''. The name refers to the island of Manhattan, or generically any planned city with a rectangular grid of streets, in which a taxicab can only travel along grid directions. In taxicab geometry, the distance between any two points equals the length of their shortest grid path. This different definition of distance also leads to a different definition of the length of a curve, for which a line segment between any two points has the same length as a grid path between those points rather than its Euclidean length. The taxicab distance is also sometimes known as ''rectilinear distance'' or distance (see ''Lp'' space). This geometry ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neighbourhood (graph Theory)
In graph theory, an adjacent vertex of a vertex (graph theory), vertex in a Graph (discrete mathematics), graph is a vertex that is connected to by an edge (graph theory), edge. The neighbourhood of a vertex in a graph is the subgraph of induced subgraph, induced by all vertices adjacent to , i.e., the graph composed of the vertices adjacent to and all edges connecting vertices adjacent to . The neighbourhood is often denoted or (when the graph is unambiguous) . The same neighbourhood notation may also be used to refer to sets of adjacent vertices rather than the corresponding induced subgraphs. The neighbourhood described above does not include itself, and is more specifically the open neighbourhood of ; it is also possible to define a neighbourhood in which itself is included, called the closed neighbourhood and denoted by . When stated without any qualification, a neighbourhood is assumed to be open. Neighbourhoods may be used to represent graphs in computer algori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delannoy Number
In mathematics, a Delannoy number D counts the paths from the southwest corner (0, 0) of a rectangular grid to the northeast corner (''m'', ''n''), using only single steps north, northeast, or east. The Delannoy numbers are named after French army officer and amateur mathematician Henri Delannoy. The Delannoy number D(m,n) also counts the global alignments of two sequences of lengths m and n, the points in an ''m''-dimensional integer lattice or cross polytope which are at most ''n'' steps from the origin, and, in cellular automata, the cells in an ''m''-dimensional von Neumann neighborhood of radius ''n''. Example The Delannoy number ''D''(3, 3) equals 63. The following figure illustrates the 63 Delannoy paths from (0, 0) to (3, 3): The subset of paths that do not rise above the SW–NE diagonal are counted by a related family of numbers, the Schröder numbers. Delannoy array The Delannoy array is an infinite matrix of the Delannoy numbers: : In this array, the numbe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedron
In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of irregular octahedra also exist, including both convex set, convex and non-convex shapes. Combinatorially equivalent to the regular octahedron The following polyhedra are combinatorially equivalent to the regular octahedron. They all have six vertices, eight triangular faces, and twelve edges that correspond one-for-one with the features of it: * Triangular antiprisms: Two faces are equilateral, lie on parallel planes, and have a common axis of symmetry. The other six triangles are isosceles. The regular octahedron is a special case in which the six lateral triangles are also equilateral. * Tetragonal bipyramids, in which at least one of the equatorial quadrilaterals lies on a plane. The regular octahedron is a special case in which all thr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manhattan Distance
Taxicab geometry or Manhattan geometry is geometry where the familiar Euclidean distance is ignored, and the distance between two point (geometry), points is instead defined to be the sum of the absolute differences of their respective Cartesian coordinates, a distance function (or Metric (mathematics), metric) called the ''taxicab distance'', ''Manhattan distance'', or ''city block distance''. The name refers to the island of Manhattan, or generically any planned city with a rectangular grid of streets, in which a taxicab can only travel along grid directions. In taxicab geometry, the distance between any two points equals the length of their shortest grid path. This different definition of distance also leads to a different definition of the length of a curve, for which a line segment between any two points has the same length as a grid path between those points rather than its Euclidean length. The taxicab distance is also sometimes known as ''rectilinear distance'' or distanc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pixel
In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a Raster graphics, raster image, or the smallest addressable element in a dot matrix display device. In most digital display devices, pixels are the smallest element that can be manipulated through software. Each pixel is a Sampling (signal processing), sample of an original image; more samples typically provide more accurate representations of the original. The Intensity (physics), intensity of each pixel is variable. In color imaging systems, a color is typically represented by three or four component intensities such as RGB color model, red, green, and blue, or CMYK color model, cyan, magenta, yellow, and black. In some contexts (such as descriptions of camera sensors), ''pixel'' refers to a single scalar element of a multi-component representation (called a ''photosite'' in the camera sensor context, although ''wikt:sensel, sensel'' is sometimes used), while in yet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manhattan Distance R=2
Manhattan ( ) is the most densely populated and geographically smallest of the five boroughs of New York City. Coextensive with New York County, Manhattan is the smallest county by area in the U.S. state of New York. Located almost entirely on Manhattan Island near the southern tip of the state, Manhattan constitutes the center of the Northeast megalopolis and the urban core of the New York metropolitan area. Manhattan serves as New York City's economic and administrative center and has been described as the cultural, financial, media, and entertainment capital of the world. Present-day Manhattan was originally part of Lenape territory. European settlement began with the establishment of a trading post by Dutch colonists in 1624 on Manhattan Island; the post was named New Amsterdam in 1626. The territory came under English control in 1664 and was renamed New York after King Charles II of England granted the lands to his brother, the Duke of York. New York, based in present- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4-connected Neighborhood
In image processing, pixel connectivity is the way in which pixels in 2-dimensional (or hypervoxels in n-dimensional) images relate to their neighbors. Formulation In order to specify a set of connectivities, the dimension and the width of the neighborhood , must be specified. The dimension of a neighborhood is valid for any dimension n\geq1. A common width is 3, which means along each dimension, the central cell will be adjacent to 1 cell on either side for all dimensions. Let M_N^n represent a N-dimensional hypercubic neighborhood with size on each dimension of n=2k+1, k\in\mathbb Let \vec represent a discrete vector in the first orthant from the center structuring element to a point on the boundary of M_N^n. This implies that each element q_i \in \ ,\forall i \in \ and that at least one component q_i = k Let S_N^d represent a N-dimensional hypersphere with radius of d=\left \Vert \vec \right \Vert. Define the amount of elements on the hypersphere S_N^d within the nei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
