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Fréchet Distance
In mathematics, the Fréchet distance is a measure of similarity between curves that takes into account the location and ordering of the points along the curves. It is named after Maurice Fréchet. Intuitive definition Imagine a person traversing a finite curved path while walking their dog on a leash, with the dog traversing a separate finite curved path. Each can vary their speed to keep slack in the leash, but neither can move backwards. The Fréchet distance between the two curves is the length of the shortest leash sufficient for both to traverse their separate paths from start to finish. Note that the definition is symmetric with respect to the two curves—the Fréchet distance would be the same if the dog were walking its owner. Formal definition Let S be a metric space. A curve A in S is a continuous map from the unit interval into S, i.e., A : ,1\rightarrow S. A reparameterization \alpha of ,1/math> is a continuous, non-decreasing, surjection \alpha: ,1\rightarro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Of Similarity
In statistics and related fields, a similarity measure or similarity function or similarity metric is a real-valued function that quantifies the similarity between two objects. Although no single definition of a similarity exists, usually such measures are in some sense the inverse of distance metrics: they take on large values for similar objects and either zero or a negative value for very dissimilar objects. Though, in more broad terms, a similarity function may also satisfy metric axioms. Cosine similarity is a commonly used similarity measure for real-valued vectors, used in (among other fields) information retrieval to score the similarity of documents in the vector space model. In machine learning, common kernel functions such as the RBF kernel can be viewed as similarity functions. Use in clustering In spectral clustering, a similarity, or affinity, measure is used to transform data to overcome difficulties related to lack of convexity in the shape of the data distribut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polygonal Chain
In geometry, a polygonal chain is a connected series of line segments. More formally, a polygonal chain is a curve specified by a sequence of points (A_1, A_2, \dots, A_n) called its vertices. The curve itself consists of the line segments connecting the consecutive vertices. Name A polygonal chain may also be called a polygonal curve, polygonal path, polyline,. piecewise linear curve, broken line or, in geographic information systems, a linestring or linear ring. Variations A simple polygonal chain is one in which only consecutive (or the first and the last) segments intersect and only at their endpoints. A closed polygonal chain is one in which the first vertex coincides with the last one, or, alternatively, the first and the last vertices are also connected by a line segment. A simple closed polygonal chain in the plane is the boundary of a simple polygon. Often the term "polygon" is used in the meaning of "closed polygonal chain", but in some cases it is important to dr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carola Wenk
Carola Wenk (born 1973) is a German-American computer scientist known for her research on algorithms for finding similarities between geometric shapes, such as Map matching, matching vehicle trajectories to road networks, comparing trajectories with each other using Fréchet distance, or testing similarity for Two-dimensional gel electrophoresis, gel electrophoresis data. Her work has also involved biomedical applications of geometric algorithms, including the use of virtual reality to diagnose glaucoma. She is a professor of computer science at Tulane University. Education and career Wenk is originally from Berlin. She earned a diploma in mathematics in 1998 at the Free University of Berlin with the thesis ''Algorithmen für das Crossdating in der Dendrochronologie (Algorithms for Crossdating in Dendrochronology)'' supervised by Helmut Alt. She continued to work with Alt at the Free University of Berlin and in 2002 completed a doctorate in computer science (Dr. rer. nat.) with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line". The noun '' geodesic'' and the adjective ''geodetic'' come from ''geodesy'', the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In a Riemannian manifold or submanifold, geodesics are characterised by the property of having vanishin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shortest Path
In graph theory, the shortest path problem is the problem of finding a path between two vertices (or nodes) in a graph such that the sum of the weights of its constituent edges is minimized. The problem of finding the shortest path between two intersections on a road map may be modeled as a special case of the shortest path problem in graphs, where the vertices correspond to intersections and the edges correspond to road segments, each weighted by the length of the segment. Definition The shortest path problem can be defined for graphs whether undirected, directed, or mixed. It is defined here for undirected graphs; for directed graphs the definition of path requires that consecutive vertices be connected by an appropriate directed edge. Two vertices are adjacent when they are both incident to a common edge. A path in an undirected graph is a sequence of vertices P = ( v_1, v_2, \ldots, v_n ) \in V \times V \times \cdots \times V such that v_i is adjacent to v_ for 1 \leq i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polyhedral Terrain
In computational geometry, a polyhedral terrain in three-dimensional Euclidean space is a polyhedral surface that intersects every line parallel to some particular line in a connected set (i.e., a point or a line segment) or the empty set. Without loss of generality, we may assume that the line in question is the ''z''-axis of the Cartesian coordinate system. Then a polyhedral terrain is the image of a piecewise-linear function in ''x'' and ''y'' variables. The polyhedral terrain is a generalization of the two-dimensional geometric object, the monotone polygonal chain. As the name may suggest, a major application area of polyhedral terrains include geographic information systems to model real-world terrain Terrain or relief (also topographical relief) involves the vertical and horizontal dimensions of land surface. The term bathymetry is used to describe underwater relief, while hypsometry studies terrain relative to sea level. The Latin wo ...s. Representation A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grid Graph
In graph theory, a lattice graph, mesh graph, or grid graph is a graph whose drawing, embedded in some Euclidean space , forms a regular tiling. This implies that the group of bijective transformations that send the graph to itself is a lattice in the group-theoretical sense. Typically, no clear distinction is made between such a graph in the more abstract sense of graph theory, and its drawing in space (often the plane or 3D space). This type of graph may more shortly be called just a lattice, mesh, or grid. Moreover, these terms are also commonly used for a finite section of the infinite graph, as in "an 8 × 8 square grid". The term lattice graph has also been given in the literature to various other kinds of graphs with some regular structure, such as the Cartesian product of a number of complete graphs. Square grid graph A common type of a lattice graph (known under different names, such as square grid graph) is the graph whose vertices correspond to the p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Widest Path Problem
In graph algorithms, the widest path problem is the problem of finding a path between two designated vertices in a weighted graph, maximizing the weight of the minimum-weight edge in the path. The widest path problem is also known as the maximum capacity path problem. It is possible to adapt most shortest path algorithms to compute widest paths, by modifying them to use the bottleneck distance instead of path length. However, in many cases even faster algorithms are possible. For instance, in a graph that represents connections between routers in the Internet, where the weight of an edge represents the bandwidth of a connection between two routers, the widest path problem is the problem of finding an end-to-end path between two Internet nodes that has the maximum possible bandwidth. The smallest edge weight on this path is known as the capacity or bandwidth of the path. As well as its applications in network routing, the widest path problem is also an important component of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generative Adversarial Network
A generative adversarial network (GAN) is a class of machine learning frameworks designed by Ian Goodfellow and his colleagues in June 2014. Two neural networks contest with each other in the form of a zero-sum game, where one agent's gain is another agent's loss. Given a training set, this technique learns to generate new data with the same statistics as the training set. For example, a GAN trained on photographs can generate new photographs that look at least superficially authentic to human observers, having many realistic characteristics. Though originally proposed as a form of generative model for unsupervised learning, GANs have also proved useful for semi-supervised learning, fully supervised learning, and reinforcement learning. The core idea of a GAN is based on the "indirect" training through the discriminator, another neural network that can tell how "realistic" the input seems, which itself is also being updated dynamically. This means that the generator is not trai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fréchet Inception Distance
The Fréchet inception distance (FID) is a metric used to assess the quality of images created by a generative model, like a generative adversarial network (GAN). Unlike the earlier inception score (IS), which evaluates only the distribution of generated images, the FID compares the distribution of generated images with the distribution of a set of real images ("ground truth"). The FID metric was introduced in 2017, and is the current standard metric for assessing the quality of generative models as of 2020. It has been used to measure the quality of many recent models including the high-resolution StyleGAN1 and StyleGAN2 networks. Definition For any two probability distributions \mu, \nu over \R^n having finite mean and variances, their Fréchet distance isd_F (\mu, \nu):=\left( \inf_ \int_ \, x-y\, ^2 \, \mathrm \gamma (x, y) \right)^,where \Gamma(\mu, \nu) is the set of all measures on \R^n \times \R^n with marginals ''\mu'' and ''\nu'' on the first and second factors respe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Distributions
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of would take the value 0.5 (1 in 2 or 1/2) for , and 0.5 for (assuming that the coin is fair). Examples of random phenomena include the weather conditions at some future date, the height of a randomly selected person, the fraction of male students in a school, the results of a survey to be conducted, etc. Introduction A probability distribution is a mathematical description of the probabilities of events, subsets of the sample space. The sample space, often denoted by \Omega, is the set of all possible outcomes of a random phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmut Alt
Helmut Alt (born 1950) is a German computer scientist whose research concerns graph algorithms and computational geometry. He is known for his work on matching geometric shapes, including methods for efficiently computing the Fréchet distance between shapes. He was also the first to use the German phrase "Algorithmische Geometrie" lgorithmic geometryto refer to computational geometry. He is a professor of computer science at the Free University of Berlin. Education and career Alt was born in 1950 in Wolfersweiler, a town in Saarland that later became incorporated into Nohfelden. He became a student of Kurt Mehlhorn at Saarland University, where he completed his Ph.D. in 1976 on algorithms for parsing context-free languages. At the Free University of Berlin, he became the doctoral advisor of many successful students, including Otfried Cheong (1992), Johannes Blömer (1993), Christian Knauer (2002), Carola Wenk (2002), and Maike Buchin (2007). Recognition The Free University o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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