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Color Algebras

A color algebra refers to a system for computing sums and products of colors, analogous to additive and subtractive color mixtures. The difficulty addressed here is the fact that, because of metamerism, we cannot know with certainty the spectrum that produced a particular color solely on the basis of sensory data. Knowledge of the spectrum is not required to compute additive mixture of colors, but is critical for subtractive (multiplicative) mixture. Therefore, we cannot predict with certainty the multiplicative interactions between colors based solely on sensory data. There are two potential applications of a color algebra: first, to aid modeling phenomena of human visual perception, such as color constancy and transparency; and, second, to provide better models of the interactions of lights and surfaces for computer graphics rendering.

color rendering

Factors affecting the perception of transparent motion

It is possible to create a perception of transparency by combining patterns having different motions. Two particular combination rules, have specific interpretations in terms of physical phenomena: additive (specular reflection) and multiplicative (shadow illumination). Arbitrary combination rules applied to random patterns generate percepts in which the motions of the two patterns are visible, but have super-imposed noise. It is also possible to combine the patterns (using an exclusive-OR rule) so that only noise is visible. Within a one-dimensional family of combination rules which include addition and multiplication, there is a range where smooth motions are seen with no superimposed noise; this range is centered about the additive combination. This result suggests that the motion system deals with a linear representation of luminance, and is consistent with the analysis of motion by linear sensors. This research gives tentative validation the use in beam splitters (which combine images additively) in the construction of heads-up aviation displays. Further work is needed to determine if the superiority of additive combination generalizes to the case of full-color imagery (there are results in the literature suggesting that subtractive color mixture yields the best legibility of overlapping alphanumerics).

Mulligan, Jeffrey B.

Color Algebras

A color algebra refers to a system for computing sums and products of colors, analogous to additive and subtractive color mixtures. We would like it to match the well-defined algebra of spectral functions describing lights and surface reflectances, but an exact correspondence is impossible after the spectra have been projected to a three-dimensional color space, because of metamerism physically different spectra can produce the same color sensation. Metameric spectra are interchangeable for the purposes of addition, but not multiplication, so any color algebra is necessarily an approximation to physical reality. Nevertheless, because the majority of naturally-occurring spectra are well-behaved (e.g., continuous and slowly-varying), color algebras can be formulated that are largely accurate and agree well with human intuition. Here we explore the family of algebras that result from associating each color with a member of a three-dimensional manifold of spectra. This association can be used to construct a color product, defined as the color of the spectrum of the wavelength-wise product of the spectra associated with the two input colors. The choice of the spectral manifold determines the behavior of the resulting system, and certain special subspaces allow computational efficiencies. The resulting systems can be used to improve computer graphic rendering techniques, and to model various perceptual phenomena such as color constancy.

color constancy

The X-Ray Core of the Low-Luminosity Radio Galaxy 3C346 and ASCA Spectroscopy to Test BL LAC/Radio Galaxy Unification

Radio galaxies are relatively faint sources for Advanced Spacecraft for Cosmology Astrophysics (ASCA), and so in order to get the best possible results from the observations two things have been necessary, both of which delayed the fast preparation of papers. Firstly, the best possible data screening and background subtraction were necessary to improve the signal-to-noise, and all our several initial analysis trials were discarded in favor of using FTOOLS versions 4.1 and above. Secondly, we found that the ASCA spectra were statistically too poor to discriminate well between non-thermal and thermal models, never mind the mixture of the two which we expected on the basis of our ROSAT spatial separation of components in radio galaxies. This means that in each case we have needed to combine the ASCA spectroscopy with analysis of data from other X-ray or radio observations in order to exploit the ASCA data to the full. Our analysis for 3C 346 has yielded the cleanest final result. This powerful radio galaxy at a redshift of 0.161, lies in a poor cluster, which we have separated well from the dominant X-ray component of unresolved emission using a spatial analysis of archival ROSAT data. We were then able to fix the thermal component in our ASCA spectral analysis, and have found evidence that the unresolved emission varied by 32 +/- 13% over the 18 months between the ROSAT and ASCA observations. The unresolved X-ray emission does not suffer from intrinsic absorption, and we have related it to radio structures on both milliarcsecond scales and the arcsecond scales which Chandra can resolve. The source is a target of a Chandra AO2 proposal which we have recently submitted to follow up on our ASCA (and ROSAT) work. 3C 346's orientation to the line of sight is uncertain. However, the absence of X-ray absorption, and the radio/optical/X-ray colors, when combined with with previous radio evidence that the source is a foreshortened radio galaxy of the FRII class, suggest that the radio jets are seen at an angle to the line of sight of about 30 deg, intermediate between the radio-galaxy and quasar classes. The relatively hard ASCA response has allowed us to place an upper limit of 5.6 x 10(exp 43) ergs/ s on the 2-10 keV luminosity of any central X-ray component absorbed bN, gas which might be obscuring the broad-line emission region. Attached to this report is an almost final draft of a paper which we have prepared for submission to the Astrophysical Journal. Our combined ASCA and ROSAT results for NGC 6251 rule out our previously preferred flat-spectrum model and inverse-Compton interpretation for the source based on ROSAT data alone, but a softer X-ray spectrum and moderate absorption bring all the available data (including our early VLA HI measurements) into consistency, and we are reasonably confident that we understand the processes responsible for the X-ray emission. We have made some more sensitive HI absorption measurements which are currently being analyzed, and our plans are to publish our ASCA analysis in conjunction with the new HI results. The ASCA data for NGC 4261 have been difficult to interpret. A re-analysis of our ROSAT data with a wider range of physical parameters brings the ROSAT and ASCA results into reasonable agreement only if the emission from hot gas dominates more than suggested by our earlier work, which is itself unexpected since the radio core is bright and a large jet-related X-ray component would bring the source into agreement with results for others of its type. However, we have recently received our Chandra A01 data for this source, with the spatial resolution which allows us to separate thermal and non-thermal emission components. Our ASCA results will be re-interpreted once the analysis of our Chandra data is complete. The interpretation of the ASCA data for BL Lac object 3C 371 is ongoing, in conjunction with analysis of archival multifrequency data. Radio galaxies are complex in their X-ray properties, and hindsight has shown that the spatial resolution of ASCA is too poor for a reliable interpretation of the data without drawing on other observations. However, the ASCA spectra have made a useful contribution to the interpretation of these sources, and the groundwork is now there for more sensitive work using Chandra and XMM-Newton.

Worrall, Diana