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Phase Field Models On Graphs
Phase-field models on graphs are a discrete analogue to phase-field models, defined on a graph. They are used in image analysis (for feature identification) and for the segmentation of social networks. Graph Ginzburg–Landau functional For a graph with vertices ''V'' and edge weights \omega_, the graph Ginzburg–Landau functional of a map u:V\to \mathbb is given by :F_\varepsilon(u) = \frac\varepsilon2 \sum_ \omega_ (u_i-u_j)^2 + \frac1\varepsilon \sum_ W(u_i), where ''W'' is a double well potential, for example the quartic potential ''W''(''x'') = ''x''2(1 − ''x''2). The graph Ginzburg–Landau functional was introduced by Bertozzi and Flenner. In analogy to continuum phase-field models, where regions with ''u'' close to 0 or 1 are models for two phases of the material, vertices can be classified into those with ''u''''j'' close to 0 or close to 1, and for small \varepsilon, minimisers of F_\varepsilon will satisfy that ''u''''j'' is close to 0 or 1 for most nodes, splitti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase-field Models
A phase-field model is a mathematical model for solving interfacial problems. It has mainly been applied to solidification dynamics, but it has also been applied to other situations such as viscous fingering, fracture mechanics, hydrogen embrittlement, and vesicle dynamics. The method substitutes boundary conditions at the interface by a partial differential equation for the evolution of an auxiliary field (the phase field) that takes the role of an order parameter. This phase field takes two distinct values (for instance +1 and −1) in each of the phases, with a smooth change between both values in the zone around the interface, which is then diffuse with a finite width. A discrete location of the interface may be defined as the collection of all points where the phase field takes a certain value (e.g., 0). A phase-field model is usually constructed in such a way that in the limit of an infinitesimal interface width (the so-called sharp interface limit) the correct interfacial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph (discrete Mathematics)
In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". The objects correspond to mathematical abstractions called '' vertices'' (also called ''nodes'' or ''points'') and each of the related pairs of vertices is called an ''edge'' (also called ''link'' or ''line''). Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. Graphs are one of the objects of study in discrete mathematics. The edges may be directed or undirected. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person ''A'' can shake hands with a person ''B'' only if ''B'' also shakes hands with ''A''. In contrast, if an edge from a person ''A'' to a person ''B'' means that ''A'' owes money to ''B'', th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Image Analysis
Image analysis or imagery analysis is the extraction of meaningful information from images; mainly from digital images by means of digital image processing techniques. Image analysis tasks can be as simple as reading bar coded tags or as sophisticated as identifying a person from their face. Computers are indispensable for the analysis of large amounts of data, for tasks that require complex computation, or for the extraction of quantitative information. On the other hand, the human visual cortex is an excellent image analysis apparatus, especially for extracting higher-level information, and for many applications — including medicine, security, and remote sensing — human analysts still cannot be replaced by computers. For this reason, many important image analysis tools such as edge detectors and neural networks are inspired by human visual perception models. Digital Digital Image Analysis or Computer Image Analysis is when a computer or electrical device au ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Social Network
A social network is a social structure made up of a set of social actors (such as individuals or organizations), sets of dyadic ties, and other social interactions between actors. The social network perspective provides a set of methods for analyzing the structure of whole social entities as well as a variety of theories explaining the patterns observed in these structures. The study of these structures uses social network analysis to identify local and global patterns, locate influential entities, and examine network dynamics. Social networks and the analysis of them is an inherently interdisciplinary academic field which emerged from social psychology, sociology, statistics, and graph theory. Georg Simmel authored early structural theories in sociology emphasizing the dynamics of triads and "web of group affiliations". Jacob Moreno is credited with developing the first sociograms in the 1930s to study interpersonal relationships. These approaches were mathematically for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steepest Descent
In mathematics, gradient descent (also often called steepest descent) is a first-order iterative optimization algorithm for finding a local minimum of a differentiable function. The idea is to take repeated steps in the opposite direction of the gradient (or approximate gradient) of the function at the current point, because this is the direction of steepest descent. Conversely, stepping in the direction of the gradient will lead to a local maximum of that function; the procedure is then known as gradient ascent. Gradient descent is generally attributed to Augustin-Louis Cauchy, who first suggested it in 1847. Jacques Hadamard independently proposed a similar method in 1907. Its convergence properties for non-linear optimization problems were first studied by Haskell Curry in 1944, with the method becoming increasingly well-studied and used in the following decades. Description Gradient descent is based on the observation that if the multi-variable function F(\mathbf) is d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Allen–Cahn Equation
The Allen–Cahn equation (after John W. Cahn and Sam Allen) is a reaction–diffusion equation of mathematical physics which describes the process of phase separation in multi-component alloy systems, including order-disorder transitions. The equation describes the time evolution of a scalar-valued state variable \eta on a domain \Omega during a time interval \mathcal, and is given by: :=M_\eta operatorname(\varepsilon^_\nabla\,\eta)-f'(\eta)quad \text \Omega\times\mathcal, \quad \eta=\bar\eta\quad\text\partial_\eta\Omega\times\mathcal, :\quad -(\varepsilon^2_\eta\nabla\,\eta)\cdot m = q\quad\text \partial_q \Omega \times \mathcal, \quad \eta=\eta_o \quad\text \Omega\times\, where M_ is the mobility, f is a double-well potential, \bar\eta is the control on the state variable at the portion of the boundary \partial_\eta\Omega, q is the source control at \partial_q\Omega, \eta_o is the initial condition, and m is the outward normal to \partial\Omega. It is the L2 gradient fl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplacian Matrix
In the mathematical field of graph theory, the Laplacian matrix, also called the graph Laplacian, admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Named after Pierre-Simon Laplace, the graph Laplacian matrix can be viewed as a matrix form of the negative discrete Laplace operator on a graph approximating the negative continuous Laplacian obtained by the finite difference method. The Laplacian matrix relates to many useful properties of a graph. Together with Kirchhoff's theorem, it can be used to calculate the number of spanning trees for a given graph. The sparsest cut of a graph can be approximated through the Fiedler vector — the eigenvector corresponding to the second smallest eigenvalue of the graph Laplacian — as established by Cheeger's inequality. The spectral decomposition of the Laplacian matrix allows constructing low dimensional embeddings that appear in many machine learning applications and determines a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus On Finite Weighted Graphs
In mathematics, calculus on finite weighted graphs is a discrete calculus for functions whose domain is the vertex set of a graph with a finite number of vertices and weights associated to the edges. This involves formulating discrete operators on graphs which are analogous to differential operators in calculus, such as graph Laplacians (or discrete Laplace operators) as discrete versions of the Laplacian, and using these operators to formulate differential equations, difference equations, or variational models on graphs which can be interpreted as discrete versions of partial differential equations or continuum variational models. Such equations and models are important tools to mathematically model, analyze, and process discrete information in many different research fields, e.g., image processing, machine learning, and network analysis. In applications, finite weighted graphs represent a finite number of entities by the graph's vertices, any pairwise relationships between the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Curvature Flow
In the field of differential geometry in mathematics, mean curvature flow is an example of a geometric flow of hypersurfaces in a Riemannian manifold (for example, smooth surfaces in 3-dimensional Euclidean space). Intuitively, a family of surfaces evolves under mean curvature flow if the normal component of the velocity of which a point on the surface moves is given by the mean curvature of the surface. For example, a round sphere evolves under mean curvature flow by shrinking inward uniformly (since the mean curvature vector of a sphere points inward). Except in special cases, the mean curvature flow develops singularities. Under the constraint that volume enclosed is constant, this is called surface tension flow. It is a parabolic partial differential equation, and can be interpreted as "smoothing". Existence and uniqueness The following was shown by Michael Gage and Richard S. Hamilton as an application of Hamilton's general existence theorem for parabolic geometric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thresholding (image Processing)
In digital image processing, thresholding is the simplest method of segmenting images. From a grayscale image, thresholding can be used to create binary images. Definition The simplest thresholding methods replace each pixel in an image with a black pixel if the image intensity I_ is less than a fixed value called the threshold T, or a white pixel if the pixel intensity is greater than that threshold. In the example image on the right, this results in the dark tree becoming completely black, and the bright snow becoming completely white. Automatic thresholding While in some cases, the threshold T can be selected manually by the user, there are many cases where the user wants the threshold to be automatically set by an algorithm. In those cases, the threshold should be the "best" threshold in the sense that the partition of the pixels above and below the threshold should match as closely as possible the actual partition between the two classes of objects represented by those ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Cuts In Computer Vision
As applied in the field of computer vision, graph cut optimization can be employed to efficiently solve a wide variety of low-level computer vision problems (''early vision''), such as image smoothing, the stereo correspondence problem, image segmentation, object co-segmentation, and many other computer vision problems that can be formulated in terms of energy minimization. Many of these energy minimization problems can be approximated by solving a maximum flow problem in a graph (and thus, by the max-flow min-cut theorem, define a minimal cut of the graph). Under most formulations of such problems in computer vision, the minimum energy solution corresponds to the maximum a posteriori estimate of a solution. Although many computer vision algorithms involve cutting a graph (e.g., normalized cuts), the term "graph cuts" is applied specifically to those models which employ a max-flow/min-cut optimization (other graph cutting algorithms may be considered as graph partitioning alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
