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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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Journal ArticleDOI
TL;DR: Divide-and-conquer search strategies are described for satisfying proximity queries involving arbitrary distance metrics involving arbitrarydistance metrics.

740 citations

Proceedings ArticleDOI
01 Sep 2009
TL;DR: This work proposes TagProp, a discriminatively trained nearest neighbor model that allows the integration of metric learning by directly maximizing the log-likelihood of the tag predictions in the training set, and introduces a word specific sigmoidal modulation of the weighted neighbor tag predictions to boost the recall of rare words.
Abstract: Image auto-annotation is an important open problem in computer vision. For this task we propose TagProp, a discriminatively trained nearest neighbor model. Tags of test images are predicted using a weighted nearest-neighbor model to exploit labeled training images. Neighbor weights are based on neighbor rank or distance. TagProp allows the integration of metric learning by directly maximizing the log-likelihood of the tag predictions in the training set. In this manner, we can optimally combine a collection of image similarity metrics that cover different aspects of image content, such as local shape descriptors, or global color histograms. We also introduce a word specific sigmoidal modulation of the weighted neighbor tag predictions to boost the recall of rare words. We investigate the performance of different variants of our model and compare to existing work. We present experimental results for three challenging data sets. On all three, TagProp makes a marked improvement as compared to the current state-of-the-art.

739 citations

Journal ArticleDOI
TL;DR: The distance between two points in a fuzzy metric space is a non-negative, upper semicontinuous, normal and convex fuzzy number.

738 citations

Journal ArticleDOI
TL;DR: A new boundary detection approach for shape modeling that detects the global minimum of an active contour model’s energy between two end points and explores the relation between the maximum curvature along the resulting contour and the potential generated from the image.
Abstract: A new boundary detection approach for shape modeling is presented. It detects the global minimum of an active contour model‘s energy between two end points. Initialization is made easier and the curve is not trapped at a local minimum by spurious edges. We modify the “snake” energy by including the internal regularization term in the external potential term. Our method is based on finding a path of minimal length in a Riemannian metric. We then make use of a new efficient numerical method to find this shortest path. It is shown that the proposed energy, though based only on a potential integrated along the curve, imposes a regularization effect like snakes. We explore the relation between the maximum curvature along the resulting contour and the potential generated from the image. The method is capable to close contours, given only one point on the objects‘ boundary by using a topology-based saddle search routine. We show examples of our method applied to real aerial and medical images.

736 citations

Journal ArticleDOI
TL;DR: In this article, the authors establish a general criterion for a set of non-Hermitian operators to constitute a consistent quantum mechanical system, which allows for the normal quantum-mechanical interpretation.

734 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