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Matching Distance
In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ..., the matching distanceMichele d'Amico, Patrizio Frosini, Claudia Landi, ''Using matching distance in Size Theory: a survey'', International Journal of Imaging Systems and Technology, 16(5):154–161, 2006.Michele d'Amico, Patrizio Frosini, Claudia Landi, ''Natural pseudo-distance and optimal matching between reduced size functions'', Acta Applicandae Mathematicae, 109(2):527-554, 2010. is a metric (mathematics), metric on the space of size functions. The core of the definition of matching distance is the observation that the information contained in a size function can be combinatorially stored in a formal series of lines and points of the plane, called respectively ''size function, cornerlines'' and ''size function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric (mathematics)
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane. A metric may correspond to a metaphorical, rather than physical, notion of distance: for example, the set of 100-character Unicode strings can be equipped with the Hamming distance, which measures the number of characters that need to be changed to get from one string to another. Since they are very general, metric spaces are a tool used in many different branches of mathematics. Many types of mathematical objects have a natural notion of distance and t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Size Function
Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane x Formal definition In , the size function associated with the[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Size Theory
In mathematics, size theory studies the properties of topological spaces endowed with \mathbb^k-valued functions, with respect to the change of these functions. More formally, the subject of size theory is the study of the natural pseudodistance between size pairs. A survey of size theory can be found in .Silvia Biasotti, Leila De Floriani, Bianca Falcidieno, Patrizio Frosini, Daniela Giorgi, Claudia Landi, Laura Papaleo, Michela Spagnuolo, Describing shapes by geometrical-topological properties of real functions, ACM Computing Surveys, vol. 40 (2008), n. 4, 12:1–12:87. History and applications The beginning of size theory is rooted in the concept of size function, introduced by Frosini.Patrizio Frosini, ''A distance for similarity classes of submanifolds of a Euclidean space'', Bulletin of the Australian Mathematical Society, 42(3):407–416, 1990. Size functions have been initially used as a mathematical tool for shape comparison in computer vision and pattern reco ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Size Function
Size functions are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane x Formal definition In , the size function associated with the[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Size Functor
Given a size pair (M,f)\ where M\ is a manifold of dimension n\ and f\ is an arbitrary real continuous function defined on it, the i-th size functor, with i=0,\ldots,n\ , denoted by F_i\ , is the functor in Fun(\mathrm,\mathrm)\ , where \mathrm\ is the category of ordered real numbers, and \mathrm\ is the category of Abelian groups, defined in the following way. For x\le y\ , setting M_x=\\ , M_y=\\ , j_\ equal to the inclusion from M_x\ into M_y\ , and k_\ equal to the morphism in \mathrm\ from x\ to y\ , * for each x\in\R\ , F_i(x)=H_i(M_x);\ * F_i(k_)=H_i(j_).\ In other words, the size functor studies the process of the birth and death of homology classes as the lower level set changes. When M\ is smooth and compact and f\ is a Morse function, the functor F_0\ can be described by oriented trees, called H_0\ − trees. The concept of size functor was introduced as an extension to homology theory and category theory of the idea of size function. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Size Homotopy Group
The concept of size homotopy group is analogous in size theory of the classical concept of homotopy group. In order to give its definition, let us assume that a size pair (M,\varphi) is given, where M is a closed manifold of class C^0\ and \varphi:M\to \mathbb^k is a continuous function. Consider the lexicographical order \preceq on \mathbb^k defined by setting (x_1,\ldots,x_k)\preceq(y_1,\ldots,y_k)\ if and only if x_1 \le y_1,\ldots, x_k \le y_k. For every Y\in\mathbb^k set M_=\. Assume that P\in M_X\ and X\preceq Y\ . If \alpha\ , \beta\ are two paths from P\ to P\ and a homotopy from \alpha\ to \beta\ , based at P\ , exists in the topological space M_\ , then we write \alpha \approx_\beta\ . The first size homotopy group of the size pair (M,\varphi)\ computed at (X,Y)\ is defined to be the quotient set of the set of all paths from P\ to P\ in M_X\ with respect to the equivalence relation \approx_\ , endowed with the operation induced by the usual composition o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Pseudodistance
In size theory, the natural pseudodistance between two size pairs (M,\varphi:M\to \mathbb)\ , (N,\psi:N\to \mathbb)\ is the value \inf_h \, \varphi-\psi\circ h\, _\infty\ , where h\ varies in the set of all homeomorphisms from the manifold M\ to the manifold N\ and \, \cdot\, _\infty\ is the supremum norm. If M\ and N\ are not homeomorphic, then the natural pseudodistance is defined to be \infty\ . It is usually assumed that M\ , N\ are C^1\ closed manifolds and the measuring functions \varphi,\psi\ are C^1\ . Put another way, the natural pseudodistance measures the infimum of the change of the measuring function induced by the homeomorphisms from M\ to N\ . The concept of natural pseudodistance can be easily extended to size pairs where the measuring function \varphi\ takes values in \mathbb^m\ .Patrizio Frosini, Michele Mulazzani, ''Size homotopy groups for computation of natural size distances'', Bulletin of the Belgian Mathematical Society, 6:455-464, 1999. Whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
