Editing Comparametric equation

{{Short description|Mathematical concept}}
A '''comparametric equation''' is an [[equation]] that describes a [[parameter|parametric]] relationship between a [[function (mathematics)|function]] and a [[Scaling (geometry)|dilated]] version of the same function, where the equation does not involve the [[parameter]].  For example, ''ƒ''(2''t'') = 4''ƒ''(''t'') is a comparametric equation, when we define ''g''(''t'') = ''ƒ''(2''t''), so that we have ''g'' = 4''ƒ'' no longer contains the parameter, ''t''.  The comparametric equation ''g'' = 4''ƒ'' has a family of solutions, one of which is ''ƒ'' = ''t''<sup>2</sup>.
<ref>
Comparametric equations with practical applications in quantigraphic
image processing",
IEEE Transactions on Image Processing, Volume 9, Issue 8,
Issue Date: Aug 2000,
pages 1389–1406,
{{ISSN|1057-7149}},
INSPEC Accession Number: 6682161,
Digital Object Identifier: 10.1109/83.855434,
Date of Current Version: 06 August 2002
IEEE Signal Processing Society,
download: http://wearcam.org/comparam.htm
</ref>

To see that ''ƒ'' = ''t''<sup>2</sup> is a solution, we merely substitute back in: ''g'' = ''ƒ''(2''t'') = (2''t'')<sup>2</sup> = 4''t''<sup>2</sup> = 4''ƒ'', so that ''g'' = 4''ƒ''.

Comparametric equations arise naturally in [[signal processing]] when we have multiple measurements of the same phenomenon, in which each of the measurements was acquired using a different sensitivity.  For example, two or more differently exposed pictures of the same subject matter give rise to a comparametric relationship, the solution of which is the response function of the camera, image sensor, or imaging system.  In this sense, comparametric equations are the fundamental mathematical basis for [[high-dynamic-range imaging|HDR (high dynamic range) imaging]],<ref>Ali, M. A., & Mann, S. (2012, March). Comparametric image compositing: Computationally efficient high dynamic range imaging. In 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (pp. 913–916). IEEE.</ref><ref>Ai, T., Ali, M. A., Steffan, G., Ovtcharov, K., Zulfiqar, S., & Mann, S. (2014, May). Real-time HDR video imaging on FPGA with compressed comparametric lookup tables. In 2014 IEEE 27th Canadian Conference on Electrical and Computer Engineering (CCECE) (pp. 1–6). IEEE.</ref><ref>Mann, S. (2000). Comparametric equations with practical applications in quantigraphic image processing. IEEE transactions on image processing, 9(8), 1389–1406.</ref> as well as HDR audio.<ref>Janzen, R., & Mann, S. (2012, April). High dynamic range simultaneous signal compositing, applied to audio. In 2012 25th IEEE Canadian Conference on Electrical and Computer Engineering (CCECE) (pp. 1–6). IEEE.</ref><ref>Janzen, R., & Mann, S. (2016, December). Feedback control system for exposure optimization in high-dynamic-range multimedia sensing. In 2016 IEEE International Symposium on Multimedia (ISM) (pp. 119–125). IEEE.</ref>

Comparametric equations have been used in many areas of research, and have many practical applications to the real world. They are used in [[radar]], [[microphone array]]s, and have been used in processing crime scene video in [[homicide]] trials in which the only evidence against the accused was video recordings of the murder.

== Solution ==
An existing solution is comparametric camera response function (CCRF) for real-time comparametric analysis. It has applications in the analysis of multiple images.<ref>{{Cite journal|others=吴安, 金西, 杜学亮, 张克宁, 姚春赫, 马淑芬|title=HDR视频算法优化及硬件实现|url=http://crad.ict.ac.cn/EN/abstract/abstract3440.shtml|journal=计算机研究与发展|language=Chinese|volume=54|issue=5|doi=10.7544/issn1000-1239.2017.20160122|issn=1000-1239}}</ref><ref>{{Cite web|url=http://www.reading.ac.uk/web/files/maths/MPS_2008_03.pdf|title=Periodic solutions for nonlinear dilation equations|last=Grindrod|first=Peter}}</ref>

==References==
{{reflist}}

==Related concepts==
*[[Parametric equation]]
*[[Functional equation]]
*[[Contraction mapping]]

[[Category:Multivariable calculus]]
[[Category:Equations]]