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Metric (mathematics)

About: Metric (mathematics) is a research topic. Over the lifetime, 42617 publications have been published within this topic receiving 836571 citations. The topic is also known as: distance function & metric.


Papers
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Book ChapterDOI
01 Nov 2014
TL;DR: A new large margin multi-metric learning (LM\(^3\)L) method for face and kinship verification in the wild that jointly learns multiple distance metrics under which the correlations of different feature representations of each sample are maximized.
Abstract: Metric learning has been widely used in face and kinship verification and a number of such algorithms have been proposed over the past decade. However, most existing metric learning methods only learn one Mahalanobis distance metric from a single feature representation for each face image and cannot deal with multiple feature representations directly. In many face verification applications, we have access to extract multiple features for each face image to extract more complementary information, and it is desirable to learn distance metrics from these multiple features so that more discriminative information can be exploited than those learned from individual features. To achieve this, we propose a new large margin multi-metric learning (LM\(^3\)L) method for face and kinship verification in the wild. Our method jointly learns multiple distance metrics under which the correlations of different feature representations of each sample are maximized, and the distance of each positive is less than a low threshold and that of each negative pair is greater than a high threshold, simultaneously. Experimental results show that our method can achieve competitive results compared with the state-of-the-art methods.

168 citations

Patent
23 Jun 2010
TL;DR: In this paper, a colorfulness metric (e.g., saturation) or a contrast metric is used to determine which of, or in which order, different image recognition processes should be invoked in order to present responsive information to a user.
Abstract: Image data, such as from a mobile phone camera, is analyzed to determine a colorfulness metric (e.g., saturation) or a contrast metric (e.g., Weber contrast). This metric is then used in deciding which of, or in which order, plural different image recognition processes should be invoked in order to present responsive information to a user. A great number of other features and arrangements are also detailed.

167 citations

Journal ArticleDOI
TL;DR: A set of guiding principles for the design of metrics to evaluate the confidence afforded by a set of paths and a new metric that appears to meet these principles is proposed, and so to be a satisfactory metric of authenticaiton.
Abstract: Authentication using a path of trusted intermediaries, each able to authenicate the next in the path, is a well-known technique for authenicating entities in a large-scale system. Recent work has extended this technique to include multiple paths in an effort to bolster authentication, but the success of this approach may be unclear in the face of intersecting paths, ambiguities in the meaning of certificates, and interdependencies in the use of different keys. Thus, several authors have proposed metrics to evaluate the confidence afforded by a set of paths. In this paper we develop a set of guiding principles for the design of such metrics. We motivate our principles by showing how previous approaches failed with respect to these principles and what the consequences to authentication might be. We then propose a new metric that appears to meet our principles, and so to be a satisfactory metric of authenticaiton.

167 citations

Journal ArticleDOI
TL;DR: This paper provides a rigorous and general construction of this infinite dimensional "shape manifold" on which a Riemannian metric is placed and uses this to provide a geometrically founded linear approximation of the deformations of shapes in the neighborhood of a given template.
Abstract: In this paper, we discuss a geometrical model of a space of deformable images or shapes, in which infinitesimal variations are combinations of elastic deformations (warping) and of photometric variations. Geodesics in this space are related to velocity-based image warping methods, which have proved to yield efficient and robust estimations of diffeomorphisms in the case of large deformation. Here, we provide a rigorous and general construction of this infinite dimensional "shape manifold" on which we place a Riemannian metric. We then obtain the geodesic equations, for which we show the existence and uniqueness of solutions for all times. We finally use this to provide a geometrically founded linear approximation of the deformations of shapes in the neighborhood of a given template.

167 citations

Book
01 Jan 1959
TL;DR: In this article, it was shown that the assumption that parsimony is equivalent to smallness of number of commonfactors has been shown to be unfounded, and that even after eliminating unique factors, one can leave as many common factors as observed variables or even more common factors than observed variables, as in equation (3)-(6) and equations (7)-(10).
Abstract: from a facet design based on social psychological consideration. "Blind" analyses are now known to be essentially incapable of revealing ordered structures, such as the simplex, even when the order is strikingly apparent to the eye and is the most parsimonious way of viewing the data. (b) At least two different rotations of axes are meaningful for our case: equations (3)-(6) and equations (7)-(10). In each case, our factors are "named" in advance. A third rotation-to principal axes-may also prove meaningful eventually, when the appropriate psychology is worked out, as discussed in the references of footnotes 4 and 10. But our structural conclusions depend on none of these: the additivity of distance functions is all we need and this requires no particular location of reference axes. (c) Even after eliminating unique factors, we have left as many common-factors as observed variables, as in equations (7)-(10), or even more common-factors than observed variables, as in equation (3)-(6). New light is cast by radex theory on the community problem of factor analysis: the assumption that parsimony is equivalent to smallness of number of commonfactors has been shown to be unfounded.13 (d) Using variances as distance functions, instead of standard deviations, implies using a non-Euclidean metric for the variables. The variables lie in one dimension-along a straight line, and in the semantic scale order-when our non-Euclidean metric is used; but they define an n-dimensional space-where n is the number of common-factors-when the Euclidean metric is used. The orthogonality conditions between a, and Y lay no restrictions on their covariances with t1 in pattern (7)-(10); t1 may be oblique or orthogonal to each of these latter factors, without affecting the additivity of distances.

167 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202253
20213,191
20203,141
20192,843
20182,731
20172,341