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Hausdorff Distance
In mathematics, the Hausdorff distance, or Hausdorff metric, also called Pompeiu–Hausdorff distance, measures how far two subsets of a metric space are from each other. It turns the set of non-empty set, non-empty compact space, compact subsets of a metric space into a metric space in its own right. It is named after Felix Hausdorff and Dimitrie Pompeiu. Informally, two sets are close in the Hausdorff distance if every point of either set is close to some point of the other set. The Hausdorff distance is the longest distance someone can be forced to travel by an adversary who chooses a point in one of the two sets, from where they then must travel to the other set. In other words, it is the greatest of all the distances from a point in one set to the closest point in the other set. This distance was first introduced by Hausdorff in his book ''Grundzüge der Mengenlehre'', first published in 1914, although a very close relative appeared in the doctoral thesis of Maurice René Fré ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complete Metric Space
In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. \sqrt is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below). It is always possible to "fill all the holes", leading to the ''completion'' of a given space, as explained below. Definition Cauchy sequence A sequence x_1, x_2, x_3, \ldots of elements from X of a metric space (X, d) is called Cauchy if for every positive real number r > 0 there is a positive integer N such that for all positive integers m, n > N, d(x_m, x_n) < r. Complete space A metric space is complete if any of the following equivalent conditions are satisfied: #Every Cauchy seq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov–Hausdorff Convergence
In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff distance. Gromov–Hausdorff distance The Gromov–Hausdorff distance was introduced by David Edwards in 1975, and it was later rediscovered and generalized by Mikhail Gromov in 1981. This distance measures how far two compact metric spaces are from being isometric. If ''X'' and ''Y'' are two compact metric spaces, then ''dGH'' (''X'', ''Y'') is defined to be the infimum of all numbers ''d''''H''(''f''(''X''), ''g''(''Y'')) for all (compact) metric spaces ''M'' and all isometric embeddings ''f'' : ''X'' → ''M'' and ''g'' : ''Y'' → ''M''. Here ''d''''H'' denotes Hausdorff distance between subsets in ''M'' and the ''isometric embedding'' is understood in the global sense, i.e. it must preserve all distances, not only infinitesimally small ones; for example no c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' meaning "equal", and μέτρον ''metron'' meaning "measure". If the transformation is from a metric space to itself, it is a kind of geometric transformation known as a motion. Introduction Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; the isometry that relates them is either a rigid motion (translation or rotati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Point Nemo
In geography, a pole of inaccessibility is the farthest (or most difficult to reach) location in a given landmass, sea, or other topographical feature, starting from a given boundary, relative to a given criterion. A geographical criterion of inaccessibility marks a location that is the most challenging to reach according to that criterion. Often it refers to the most distant point from the coastline, implying the farthest point into a landmass from the shore, or the farthest point into a body of water from the shore. In these cases, a pole of inaccessibility is the center of a maximally large circle that can be drawn within an area of interest only touching but not crossing a coastline. Where a coast is imprecisely defined, the pole will be similarly imprecise. Northern pole of inaccessibility The northern pole of inaccessibility, sometimes known as the Arctic pole, is located on the Arctic Ocean pack ice at a distance farthest from any landmass. The original position was wro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oceanic Pole Of Inaccessibility
Oceanic may refer to: *Of or relating to the ocean *Of or relating to Oceania **Oceanic climate **Oceanic languages **Oceanic person or people, also called "Pacific Islander(s)" Places * Oceanic, British Columbia, a settlement on Smith Island, British Columbia, Canada *Oceanic, New Jersey, an unincorporated community within Rumson Borough, Monmouth County, New Jersey, United States Ships named Oceanic * , the White Star Line's first ocean liner ** Oceanic-class ocean liner, class of liners based on SS ''Oceanic'' (1870) * , a transatlantic ocean liner built for the White Star Line * , a project of the 1930s * , built as SS ''Independence'' in 1950 * , also named ''Big Red Boat I'' by Premier Cruises Art, entertainment, and media Fictional entities * Oceanic Airlines or Oceanic Airways, often used in disaster movies * Oceanic Flight 815, a flight in the television series ''Lost'' Literature * "Oceanic" (novella), a 1998 sci-fi novella by Greg Egan Music ;Artists * Oceanic ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Level Of Detail (computer Graphics)
In computer graphics, level of detail (LOD) refers to the complexity of a 3D model representation. LOD can be decreased as the model moves away from the viewer or according to other metrics such as object importance, viewpoint-relative speed or position. LOD techniques increase the efficiency of rendering (computer graphics), rendering by decreasing the workload on graphics pipeline stages, usually vertex transformations. The reduced visual quality of the model is often unnoticed because of the small effect on object appearance when distant or moving fast. Although most of the time LOD is applied to Geometry (computer graphics), geometry detail only, the basic concept can be generalized. Recently, LOD techniques also included shader management to keep control of pixel complexity. A form of level of detail management has been applied to texture maps for years, under the name of mipmapping, also providing higher rendering quality. It is commonplace to say that "an object has been '' ... [...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] |
Binary Image
A binary image is a digital image that consists of pixels that can have one of exactly two colors, usually black and white. Each pixel is stored as a single bit — i.e. either a 0 or 1. A binary image can be stored in memory as a bitmap: a packed array of bits. A binary image of 640×480 pixels has a file size of only 37.5 Kibibyte, KiB, and most also compress well with simple Run-length encoding, run-length compression. A binary image format is often used in contexts where it is important to have a small file size for transmission or storage, or due to color limitations on displays or printers. It also has technical and artistic applications, for example in digital image processing and pixel art. Binary images can be interpreted as subsets of the square lattice, two-dimensional integer lattice ''Z''2; the field of Mathematical morphology, morphological image processing was largely inspired by this view. Terminology Binary images are also called ''bi-level'' or ''two-level ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edge Detection
Edge or EDGE may refer to: Technology Computing * Edge computing, a network load-balancing system * Edge device, an entry point to a computer network * Adobe Edge, a graphical development application * Microsoft Edge, a web browser developed by Microsoft * Microsoft Edge Legacy, a discontinued web browser developed by Microsoft * EdgeHTML, the layout engine used in Microsoft Edge Legacy * ThinkPad Edge, a Lenovo laptop computer series marketed from 2010 * Silhouette edge, in computer graphics, a feature of a 3D body projected onto a 2D plane * Explicit data graph execution, a computer instruction set architecture Telecommunication(s) * EDGE (telecommunication), a 2G digital cellular communications technology * Edge Wireless, an American mobile phone provider * Motorola Edge series, a series of smartphones made by Motorola * Samsung Galaxy Note Edge, a phablet made by Samsung * Samsung Galaxy S7 Edge or Samsung Galaxy S6 Edge, smartphones made by Samsung * Ubuntu Edg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computer Vision
Computer vision tasks include methods for image sensor, acquiring, Image processing, processing, Image analysis, analyzing, and understanding digital images, and extraction of high-dimensional data from the real world in order to produce numerical or symbolic information, e.g. in the form of decisions. "Understanding" in this context signifies the transformation of visual images (the input to the retina) into descriptions of the world that make sense to thought processes and can elicit appropriate action. This image understanding can be seen as the disentangling of symbolic information from image data using models constructed with the aid of geometry, physics, statistics, and learning theory. The scientific discipline of computer vision is concerned with the theory behind artificial systems that extract information from images. Image data can take many forms, such as video sequences, views from multiple cameras, multi-dimensional data from a 3D scanning, 3D scanner, 3D point clouds ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
