Comparametric equation explained

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

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]}

Related concepts

• Parametric equation
• Functional equation
• Contraction mapping

Notes and References

1. Comparametric equations with practical applications in quantigraphicimage processing",IEEE Transactions on Image Processing, Volume 9, Issue 8,Issue Date: Aug 2000,pages 1389-1406,,INSPEC Accession Number: 6682161,Digital Object Identifier: 10.1109/83.855434,Date of Current Version: 06 August 2002IEEE Signal Processing Society,download: http://wearcam.org/comparam.htm
2. 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.
3. 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.
4. Mann, S. (2000). Comparametric equations with practical applications in quantigraphic image processing. IEEE transactions on image processing, 9(8), 1389-1406.
5. 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.
6. 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.
7. 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. Chinese. 54. 5. 10.7544/issn1000-1239.2017.20160122. 1000-1239.
8. Web site: Periodic solutions for nonlinear dilation equations. Grindrod. Peter.