Topic
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.
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TL;DR: A twin-comparison approach has been developed to solve the problem of detecting transitions implemented by special effects, and a motion analysis algorithm is applied to determine whether an actual transition has occurred.
Abstract: Partitioning a video source into meaningful segments is an important step for video indexing. We present a comprehensive study of a partitioning system that detects segment boundaries. The system is based on a set of difference metrics and it measures the content changes between video frames. A twin-comparison approach has been developed to solve the problem of detecting transitions implemented by special effects. To eliminate the false interpretation of camera movements as transitions, a motion analysis algorithm is applied to determine whether an actual transition has occurred. A technique for determining the threshold for a difference metric and a multi-pass approach to improve the computation speed and accuracy have also been developed.
1,360 citations
07 May 2006
TL;DR: A fast method for computation of covariances based on integral images, and the performance of the covariance features is superior to other methods, as it is shown, and large rotations and illumination changes are also absorbed by the covariances matrix.
Abstract: We describe a new region descriptor and apply it to two problems, object detection and texture classification. The covariance of d-features, e.g., the three-dimensional color vector, the norm of first and second derivatives of intensity with respect to x and y, etc., characterizes a region of interest. We describe a fast method for computation of covariances based on integral images. The idea presented here is more general than the image sums or histograms, which were already published before, and with a series of integral images the covariances are obtained by a few arithmetic operations. Covariance matrices do not lie on Euclidean space, therefore we use a distance metric involving generalized eigenvalues which also follows from the Lie group structure of positive definite matrices. Feature matching is a simple nearest neighbor search under the distance metric and performed extremely rapidly using the integral images. The performance of the covariance features is superior to other methods, as it is shown, and large rotations and illumination changes are also absorbed by the covariance matrix.
1,338 citations
TL;DR: In this paper, a Hausdorff topology on a fuzzy metric space was defined, and Baire's theorem for fuzzy metric spaces was proved for the first time in the fuzzy metric domain.
1,325 citations
TL;DR: In this article, the RI-MP2 approach is applied to first derivatives of the MP2 correlation energy expression while the (restricted) Hartree-Fock reference is treated in the usual way.
Abstract: The evaluation of RI-MP2 first derivatives with respect to nuclear coordinates or with respect to an external electric field is described. The prefix RI indicates the use of an approximate resolution of identity in the Hilbert space of interacting charge distributions (Coulomb metric), i.e., the use of an auxiliary basis set to approximate charge distributions. The RI technique is applied to first derivatives of the MP2 correlation energy expression while the (restricted) Hartree-Fock reference is treated in the usual way. Computational savings by a factor of 10 over conventional approaches are demonstrated in an application to porphyrin. It is shown that the RI approximation to MP2 derivatives does not entail any significant loss in accuracy. Finally, the relative energetic stabilities of a representative sample of closed-shell molecules built from first and second row elements have been investigated by the RI-MP2 approach, and thus it is tested whether such properties that refer to potential energy hypersurfaces in a more global way can be described with similar consistency to the more locally defined derivatives.
1,310 citations
TL;DR: In this article, a general coefficient of monotonicity, whose maximization is equivalent to optimal satisfaction of the Monotonicity condition, is defined, and which allows various options both for treatment of ties and for weighting error-of-fit.
Abstract: LetA
1,A
2, ...,A
n be anyn objects, such as variables, categories, people, social groups, ideas, physical objects, or any other. The empirical data to be analyzed are coefficients of similarity or distance within pairs (A
i,A
i
), such as correlation coefficients, conditional probabilities or likelihoods, psychological choice or confusion, etc. It is desired to represent these data parsimoniously in a coordinate space, by calculatingm coordinates {x
ia
} for eachA
i
for a semi-metricd of preassigned formd
ij
=d(|x
i1 -x
j1
|, |x
i2 -x
j2|, ..., |x
im
-x
jm
|). The dimensionalitym is sought to be as small as possible, yet satisfy the monotonicity condition thatd
ij
1,310 citations