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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] |
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] |
Discrete And Computational Geometry
'' Discrete & Computational Geometry'' is a peer-reviewed mathematics journal published quarterly by Springer. Founded in 1986 by Jacob E. Goodman and Richard M. Pollack, the journal publishes articles on discrete geometry and computational geometry. Abstracting and indexing The journal is indexed in: * ''Mathematical Reviews'' * ''Zentralblatt MATH'' * ''Science Citation Index'' * ''Current Contents''/Engineering, Computing and Technology Notable articles The articles by Gil Kalai with a proof of a subexponential upper bound on the diameter of a polyhedron and by Samuel Ferguson on the Kepler conjecture, both published in Discrete & Computational geometry, earned their author the Fulkerson Prize The Fulkerson Prize for outstanding papers in the area of discrete mathematics is sponsored jointly by the Mathematical Optimization Society (MOS) and the American Mathematical Society (AMS). Up to three awards of $1,500 each are presented at e .... References External link ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Size Pair
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 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] |
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] |
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] |
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] |
Supremum Norm
In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when the supremum is in fact the maximum, the . The name "uniform norm" derives from the fact that a sequence of functions converges to under the metric derived from the uniform norm if and only if converges to uniformly. If is a continuous function on a closed and bounded interval, or more generally a compact set, then it is bounded and the supremum in the above definition is attained by the Weierstrass extreme value theorem, so we can replace the supremum by the maximum. In this case, the norm is also called the . In particular, if is some vector such that x = \left(x_1, x_2, \ldots, x_n\right) in finite dimensional coordinate space, it takes the form: :\, x\, _\infty := \max \left(\left, x_1\ , \ldots , \left, x_n\\right). Metric and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or ''holes'', of a topological space. To define the ''n''-th homotopy group, the base-point-preserving maps from an ''n''-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the ''n''-th homotopy group, \pi_n(X), of the given space ''X'' with base point. Topological spaces with differing homotopy groups are never equivalent ( homeomorphic), but topological spaces that homeomorphic have the same homotopy groups. The notion of homotopy of paths was introduced by Camille Jordan. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homology Group
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Persistent Homology
:''See homology for an introduction to the notation.'' Persistent homology is a method for computing topological features of a space at different spatial resolutions. More persistent features are detected over a wide range of spatial scales and are deemed more likely to represent true features of the underlying space rather than artifacts of sampling, noise, or particular choice of parameters. To find the persistent homology of a space, the space must first be represented as a simplicial complex. A distance function on the underlying space corresponds to a filtration of the simplicial complex, that is a nested sequence of increasing subsets. Definition Formally, consider a real-valued function on a simplicial complex f:K \rightarrow \mathbb that is non-decreasing on increasing sequences of faces, so f(\sigma) \leq f(\tau) whenever \sigma is a face of \tau in K. Then for every a \in \mathbb the sublevel set K_a=f^((-\infty, a]) is a subcomplex of ''K'', and the ordering of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
