• Corpus ID: 119134743

# A New Weighted Metric: the Relative Metric I

@inproceedings{Hasto2001ANW,
title={A New Weighted Metric: the Relative Metric I},
author={Peter A. Hasto},
year={2001}
}
• P. Hasto
• Published 3 August 2001
• Mathematics
The M –relative distance , denoted by ρ M is a generalization of the p –relative distance, which was introduced in [10]. We establish necessary and suﬃcient conditions under which ρ M is a metric. In two special cases we derive complete characterizations of the metric. We also present a way of extending the results to metrics sensitive to the domain in which they are deﬁned, thus ﬁnding some connections to previously studied metrics. An auxiliary result of independent interest is an inequality…
7 Citations

## Figures from this paper

• Computer Science
• 2019
This work proves that another function, studied by O. Dovgoshey, P. Hariri, and M. Vuorinen is a metric and gives upper and lower bounds for it.
• Computer Science
Complex Variables and Elliptic Equations
• 2020
This work studies this Barrlund metric and gives sharp bounds for it in terms of other metrics of current interest and proves sharp distortion results for the Barrlund metrics under quasiconformal maps.
• Computer Science
IEEE Transactions on Image Processing
• 2012
A series of normalized and generalized metrics based on the important ingredients of SSIM are constructed and it is shown that such modified measures are valid distance metrics and have many useful properties, among which the most significant ones include quasi-convexity, a region of convexity around the minimizer, and distance preservation under orthogonal or unitary transformations.
• Mathematics
• 2005
The Apollonian metric is a generalization of the hyperbolic metric, defined in a much larger class of open sets. Beardon introduced the metric in 1998, and asked whether its isometries are just the
• Computer Science
Multimedia Tools and Applications
• 2016
This paper deduces the metric of the structural similarity constraint, and further it proves it does’t hold non-crossing-edges property, and constructs the rate-distortion function of optimal structural similarity constraints, which is equivalent to minimize the average distortion for a given embedding rate.

## References

SHOWING 1-10 OF 19 REFERENCES

• P. Seittenranta
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 1999
A new Möbius-invariant metric δG defined on an open set G⊂[Rmacron]n with at least two boundary points is introduced. This metric coincides with the hyperbolic metric if G is the unit ball. Some
We study a certain monotonicity property of ratios of means, which we call a strong inequality. These strong inequalities were recently shown to be related to the so-called relative metric. We also
In a very interesting and recent note, Tung-Po Lin [I] obtained the least value q and the greatest value p such that M <L<M P q is valid for all distinct positive numbers x and y where M (x +.y.)s
• A. Barrlund
• Computer Science
SIAM J. Matrix Anal. Appl.
• 2000
The conjecture that the p-relative distance, $\varrho_p(\alpha,\tilde{\alpha})=|\alpha-\tilde{\alpha}| /{\sqrt[p]{|\alpha|^p+|\tilde{\alpha}|^p}}$, is a metric is proved.
• Mathematics
• 1994
Abstract A monotone form of L′Hospital′s rule is obtained and applied to derive inequalities between the arithmetic-geometric mean of Gauss, the logarithmic mean, and Stolarsky′s identric mean. Some
2-Dimensional Characterization.- The Parallelogram Equality and Derived Equalities.- Norm Derivatives Characterizations.- James' Isoceles Orthogonality (Midpoints of Chords).- Birkhoff Orthogolaity.-
The classical perturbation theory for Hermitian matrix eigenvalue and singular value problems provides bounds on the absolute differences between approximate eigenvalues (singular values) and the
• Mathematics
Nature
• 1927
Prof. FORSYTH'S latest work appears opportunely at a time when there is quite a notable revival of interest in the calculus of variations. To those who desire an account of the subject which, while