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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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Proceedings ArticleDOI
23 Jun 1998
TL;DR: The novel contributions are that in a stratified context the various forms of providing metric information can be represented as circular constraints on the parameters of an affine transformation of the plane, providing a simple and uniform framework for integrating constraints.
Abstract: We describe the geometry constraints and algorithmic implementation for metric rectification of planes. The rectification allows metric properties, such as angles and length ratios, to be measured on the world plane from a perspective image. The novel contributions are: first, that in a stratified context the various forms of providing metric information, which include a known angle, two equal though unknown angles, and a known length ratio; can all be represented as circular constraints on the parameters of an affine transformation of the plane-this provides a simple and uniform framework for integrating constraints; second, direct rectification from right angles in the plane; third, it is shown that metric rectification enables calibration of the internal camera parameters; fourth, vanishing points are estimated using a Maximum Likelihood estimator; fifth, an algorithm for automatic rectification. Examples are given for a number of images, and applications demonstrated for texture map acquisition and metric measurements.

414 citations

Journal ArticleDOI
TL;DR: The theory of metric regularity is an extension of two classical results: the Lyusternik tangent space theorem and the Graves surjection theorem as mentioned in this paper, and it was also well understood that the phenomena behind the results are of metric origin, not connected with any linear structure.
Abstract: The theory of metric regularity is an extension of two classical results: the Lyusternik tangent space theorem and the Graves surjection theorem. Developments in non-smooth analysis in the 1980s and 1990s paved the way for a number of far-reaching extensions of these results. It was also well understood that the phenomena behind the results are of metric origin, not connected with any linear structure. At the same time it became clear that some basic hypotheses of the subdifferential calculus are closely connected with the metric regularity of certain set-valued maps. The survey is devoted to the metric theory of metric regularity and its connection with subdifferential calculus in Banach spaces.

414 citations

Journal ArticleDOI
TL;DR: A new evolutionary approach, particle swarm optimization, is adapted for single-slice 3D-to-3D biomedical image registration and a new hybrid particle swarm technique is proposed that incorporates initial user guidance.
Abstract: Biomedical image registration, or geometric alignment of two-dimensional and/or three-dimensional (3D) image data, is becoming increasingly important in diagnosis, treatment planning, functional studies, computer-guided therapies, and in biomedical research. Registration based on intensity values usually requires optimization of some similarity metric between the images. Local optimization techniques frequently fail because functions of these metrics with respect to transformation parameters are generally nonconvex and irregular and, therefore, global methods are often required. In this paper, a new evolutionary approach, particle swarm optimization, is adapted for single-slice 3D-to-3D biomedical image registration. A new hybrid particle swarm technique is proposed that incorporates initial user guidance. Multimodal registrations with initial orientations far from the ground truth were performed on three volumes from different modalities. Results of optimizing the normalized mutual information similarity metric were compared with various evolutionary strategies. The hybrid particle swarm technique produced more accurate registrations than the evolutionary strategies in many cases, with comparable convergence. These results demonstrate that particle swarm approaches, along with evolutionary techniques and local methods, are useful in image registration, and emphasize the need for hybrid approaches for difficult registration problems.

413 citations

Journal ArticleDOI
TL;DR: The optimal correspondence is found by an efficient dynamic-programming method both for aligning pairs of curve segments and pairs of closed curves, and is effective in the presence of a variety of transformations of the curve.
Abstract: We present a novel approach to finding a correspondence (alignment) between two curves. The correspondence is based on a notion of an alignment curve which treats both curves symmetrically. We then define a similarity metric based on the alignment curve using two intrinsic properties of the curve, namely, length and curvature. The optimal correspondence is found by an efficient dynamic-programming method both for aligning pairs of curve segments and pairs of closed curves, and is effective in the presence of a variety of transformations of the curve. Finally, the correspondence is shown in application to handwritten character recognition, prototype formation, and object recognition, and is potentially useful in other applications such as registration and tracking.

412 citations

Journal ArticleDOI
TL;DR: In this paper, it was shown that the topological properties of the symplectic diffeomorphism groups are related to the topology of the phase space geometry of the manifold.
Abstract: In this paper we show that symplectic maps have surprising topological properties. In particular, we construct an interesting metric for the symplectic diffeomorphism groups, which is related, but not obviously, to the topological properties of symplectic maps and phase space geometry. We also prove a certain number of generalised symplectic fixed point theorems and give an application to a Hamiltonian system.

412 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