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8-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 N and the width of the neighborhood n , 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 withi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image Processing
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensional picture, that resembles a subject. In the context of signal processing, an image is a distributed amplitude of color(s). In optics, the term “image” may refer specifically to a 2D image. An image does not have to use the entire visual system to be a visual representation. A popular example of this is of a greyscale image, which uses the visual system's sensitivity to brightness across all wavelengths, without taking into account different colors. A black and white visual representation of something is still an image, even though it does not make full use of the visual system's capabilities. Images are typically still, but in some cases can be moving or animated. Characteristics Images may be two or three-dimensional, such as a pho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moore Neighborhood
In cellular automata, the Moore neighborhood is defined on a two-dimensional square lattice and is composed of a central cell and the eight cells that surround it. Name The neighborhood is named after Edward F. Moore, a pioneer of cellular automata theory. Importance It is one of the two most commonly used neighborhood types, the other one being the von Neumann neighborhood, which excludes the corner cells. The well known Conway's Game of Life, for example, uses the Moore neighborhood. It is similar to the notion of 8-connected pixels in computer graphics. The Moore neighbourhood of a cell is the cell itself and the cells at a Chebyshev distance of 1. The concept can be extended to higher dimensions, for example forming a 26-cell cubic neighborhood for a cellular automaton in three dimensions, as used by 3D Life. In dimension ''d,'' where 0 \le d, d \in \mathbb, the size of the neighborhood is 3''d'' − 1. In two dimensions, the number of cells in an ''ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grid Cell Topology
The grid cell topology is studied in digital topology as part of the theoretical basis for (low-level) algorithms in computer image analysis or computer graphics. The elements of the ''n''-dimensional grid cell topology (''n'' ≥ 1) are all ''n''-dimensional grid cubes and their ''k''-dimensional faces ( for 0 ≤ ''k'' ≤ ''n''−1); between these a partial order ''A'' ≤ ''B'' is defined if ''A'' is a subset of ''B'' (and thus also dim(''A'') ≤ dim(''B'')). The grid cell topology is the Alexandrov topology (open sets are up-sets) with respect to this partial order. (See also poset topology.) Alexandrov and Hopf first introduced the grid cell topology, for the two-dimensional case, within an exercise in their text ''Topologie'' I (1935). A recursive method to obtain ''n''-dimensional grid cells and an intuitive definition for grid cell manifolds can be found in Chen, 2004. It is related to digital manifolds. See also * Pixel connectivity In image processing, p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartesian Coordinate System
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference coordinate line is called a ''coordinate axis'' or just ''axis'' (plural ''axes'') of the system, and the point where they meet is its ''origin'', at ordered pair . The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. One can use the same principle to specify the position of any point in three-dimensional space by three Cartesian coordinates, its signed distances to three mutually perpendicular planes (or, equivalently, by its perpendicular projection onto three mutually perpendicular lines). In general, ''n'' Cartesian coordinates (an element of real ''n''-space) specify the point in an ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stretcher Bond
Brickwork is masonry produced by a bricklayer, using bricks and mortar. Typically, rows of bricks called '' courses'' are laid on top of one another to build up a structure such as a brick wall. Bricks may be differentiated from blocks by size. For example, in the UK a brick is defined as a unit having dimensions less than and a block is defined as a unit having one or more dimensions greater than the largest possible brick. Brick is a popular medium for constructing buildings, and examples of brickwork are found through history as far back as the Bronze Age. The fired-brick faces of the ziggurat of ancient Dur-Kurigalzu in Iraq date from around 1400 BC, and the brick buildings of ancient Mohenjo-daro in Pakistan were built around 2600 BC. Much older examples of brickwork made with dried (but not fired) bricks may be found in such ancient locations as Jericho in Palestine, Çatal Höyük in Anatolia, and Mehrgarh in Pakistan. These structures have survived from the Stone Age ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagonal Tiling
In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of or (as a truncated triangular tiling). English mathematician John Conway called it a hextille. The internal angle of the hexagon is 120 degrees, so three hexagons at a point make a full 360 degrees. It is one of three regular tilings of the plane. The other two are the triangular tiling and the square tiling. Applications The hexagonal tiling is the densest way to arrange circles in two dimensions. The honeycomb conjecture states that the hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making honeycomb (or rather, soap bubbles) was investigated by Lord Kelvin, who believed that the Kelvin structure (or body-centered cubic lattice) is optimal. However, the less regular Weaire–Phel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connectivity (graph Theory)
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. Connected vertices and graphs In an undirected graph , two '' vertices'' and are called connected if contains a path from to . Otherwise, they are called disconnected. If the two vertices are additionally connected by a path of length , i.e. by a single edge, the vertices are called adjacent. A graph is said to be connected if every pair of vertices in the graph is connected. This means that there is a path between every pair of vertices. An undirected graph that is not connected is called disconnected. An undirected graph ''G'' is therefore disconnected if there exist two vertices ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manhattan Distance
A taxicab geometry or a Manhattan geometry is a geometry whose usual distance function or Metric (mathematics), metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates. The taxicab metric is also known as rectilinear distance, ''L''1 distance, ''L''1 distance or \ell_1 norm (see Lp space, ''Lp'' space), Snake (video game), snake distance, city block distance, Manhattan distance or Manhattan length. The latter names refer to the rectilinear street layout on the island of Manhattan, where the shortest path a taxi travels between two points is the sum of the absolute values of distances that it travels on avenues and on streets. The geometry has been used in regression analysis since the 18th century, and is often referred to as Lasso (statistics), LASSO. The geometric interpretation dates to non-Euclidean geometry of the 19th century and is due to Hermann Minkowski. In \mat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pixels
In digital imaging, a pixel (abbreviated px), pel, or picture element is the smallest addressable element in a raster image, or the smallest point in an all points addressable display device. In most digital display devices, pixels are the smallest element that can be manipulated through software. Each pixel is a sample of an original image; more samples typically provide more accurate representations of the original. The intensity of each pixel is variable. In color imaging systems, a color is typically represented by three or four component intensities such as red, green, and blue, or 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 '' sensel'' is sometimes used), while in yet other contexts (like MRI) it may refer to a set of component intensities for a spatial position. Etymology Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multinomial Theorem
In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. Theorem For any positive integer and any non-negative integer , the multinomial formula describes how a sum with terms expands when raised to an arbitrary power : :(x_1 + x_2 + \cdots + x_m)^n = \sum_ \prod_^m x_t^\,, where : = \frac is a multinomial coefficient. The sum is taken over all combinations of nonnegative integer indices through such that the sum of all is . That is, for each term in the expansion, the exponents of the must add up to . Also, as with the binomial theorem, quantities of the form that appear are taken to equal 1 ( even when equals zero). In the case , this statement reduces to that of the binomial theorem. Example The third power of the trinomial is given by :(a+b+c)^3 = a^3 + b^3 + c^3 + 3 a^2 b + 3 a^2 c + 3 b^2 a + 3 b^2 c + ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-sphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, called the ''center''. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit -sphere or simply the -sphere for brevity. In terms of the standard norm, the -sphere is defined as : S^n = \left\ , and an -sphere of radius can be defined as : S^n(r) = \left\ . The dimension of -sphere is , and must not be confused with the dimension of the Euclidean space in which it is naturally embedded. An -sphere is the surface or boundary of an -dimensional ball. In particular: *the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere, *a circle, which i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
