author    = "S. Mann",
  title     = "Comparametric Equations",
  journal   = "{IEEE} Trans. Image Proc.",
  year      = 2000,
  volume    = 9,
  number    = 8,
  note      = "ISSN 1057-7149",
  month     = "August",
  pages     = "1389-1406"}


It is argued that, hidden within the flow of signals from typical cameras, through image processing, to display media, is a homomorphic filter. While homomorphic filtering is often desirable, there are some occasions where it is not. Thus cancellation of this implicit homomorphic filter is proposed, through the introduction of an anti--homomorphic filter. This concept gives rise to the principle of quantigraphic image processing, wherein it is argued that most cameras can be modelled as an array of idealized light meters each linearly responsive to a semi--monotonic function of the quantity of light received, integrated over a fixed spectral response profile. This quantity is neither radiometric nor photometric, but, rather, depends only on the spectral response of the sensor elements in the camera. A particular class of functional equations, called comparametric equations, is introduced as a basis for quantigraphic image processing. Comparametric equations are fundamental to the analysis and processing of multiple images differing only in exposure. The well--known ``gamma correction'' of an image is presented as a simple example of a comparametric equation, for which it is shown that the underlying quantigraphic function does not pass through the origin. For this reason it is argued that exposure adjustment by gamma correction is inherently flawed, and alternatives are provided. These alternatives, when applied to a plurality of images that differ only in exposure, give rise to a new kind of processing in the ``amplitude domain'' (as opposed to the time domain or the frequency domain). While the theoretical framework presented in this paper originated within the field of wearable cybernetics (wearable photographic apparatus) in the 1970s and early 1980s, it is applicable to the processing of images from nearly all types of modern cameras, wearable or otherwise. This paper is a much revised draft of a 1992 peer--reviewed but unpublished report by the author, entitled ``Lightspace and the Wyckoff principle''.

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You can also download this paper in Proprietary Document Format (PDF), as it appeared in IEEE Trans. Image Processing.

If you're familiar with this work and you've read the paper (or the textbook) on Comparametric Equations already, you may just want to take a quick look at Table 1. (Note that Table 1 is not very clear in the PDF of the paper above, so the better scan here might also be of use even if you're reading the whole paper for the first time.)
Fun with comparametric equations: There is also some interesting work on comparametrics happening at the Image Understanding Laboratory (Florida International University) founded and directed by Dr. Frank M. Candocia