A comparametric equation is an equation that describes a parametric relationship between a function and a dilated version of the same function, where the equation does not involve the parameter. For example, ƒ(2t) = 4ƒ(t) is a comparametric equation, when we define g(t) = ƒ(2t), so that we have g = 4ƒ no longer contains the parameter, t. The comparametric equation g = 4ƒ has a family of solutions, one of which is ƒ = t2. [1]
To see that ƒ = t2 is a solution, we merely substitute back in: g = ƒ(2t) = (2t)2 = 4t2 = 4ƒ, so that g = 4ƒ.
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 HDR (high dynamic range) imaging,[2][3][4] as well as HDR audio.[5][6]
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 arrays, 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.
SolutionEdit
An existing solution is comparametric camera response function (CCRF) for real-time comparametric analysis. It has applications in the analysis of multiple images. [7][8]
ReferencesEdit
^ 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
^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.
^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.
^Mann, S. (2000). Comparametric equations with practical applications in quantigraphic image processing. IEEE transactions on image processing, 9(8), 1389-1406.
^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.
^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.
comparametric, equation, comparametric, equation, equation, that, describes, parametric, relationship, between, function, dilated, version, same, function, where, equation, does, involve, parameter, example, comparametric, equation, when, define, that, have, l. A comparametric equation is an equation that describes a parametric relationship between a function and a dilated version of the same function where the equation does not involve the parameter For example ƒ 2t 4ƒ t is a comparametric equation when we define g t ƒ 2t so that we have g 4ƒ no longer contains the parameter t The comparametric equation g 4ƒ has a family of solutions one of which is ƒ t2 1 To see that ƒ t2 is a solution we merely substitute back in g ƒ 2t 2t 2 4t2 4ƒ so that g 4ƒ 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 HDR high dynamic range imaging 2 3 4 as well as HDR audio 5 6 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 arrays 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 EditAn existing solution is comparametric camera response function CCRF for real time comparametric analysis It has applications in the analysis of multiple images 7 8 References Edit 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 Ali M A amp 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 Ai T Ali M A Steffan G Ovtcharov K Zulfiqar S amp 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 Mann S 2000 Comparametric equations with practical applications in quantigraphic image processing IEEE transactions on image processing 9 8 1389 1406 Janzen R amp 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 Janzen R amp 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 吴安 金西 杜学亮 张克宁 姚春赫 马淑芬 HDR视频算法优化及硬件实现 计算机研究与发展 in Chinese 54 5 doi 10 7544 issn1000 1239 2017 20160122 ISSN 1000 1239 a href Template Cite journal html title Template Cite journal cite journal a CS1 maint others link Grindrod Peter Periodic solutions for nonlinear dilation equations PDF Related concepts EditParametric equation Functional equation Contraction mapping Retrieved from https en wikipedia org w index php title Comparametric equation amp oldid 1027481975, wikipedia, wiki, book, books, library,
