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Comparametric equation
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{{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, ''Ζ''(2''t'') = 4''Ζ''(''t'') is a comparametric equation, when we define ''g''(''t'') = ''Ζ''(2''t''), so that we have ''g'' = 4''Ζ'' no longer contains the parameter, ''t''. The comparametric equation ''g'' = 4''Ζ'' has a family of solutions, one of which is ''Ζ'' = ''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 1389β1406, {{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 ''Ζ'' = ''t''<sup>2</sup> is a solution, we merely substitute back in: ''g'' = ''Ζ''(2''t'') = (2''t'')<sup>2</sup> = 4''t''<sup>2</sup> = 4''Ζ'', so that ''g'' = 4''Ζ''. 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. 913β916). 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. 1β6). IEEE.</ref><ref>Mann, S. (2000). Comparametric equations with practical applications in quantigraphic image processing. IEEE transactions on image processing, 9(8), 1389β1406.</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. 1β6). 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. 119β125). 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.
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