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Object-Colour Manifold

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Abstract

Colorimetry can predict which lights will look alike. Such lights are called metameric. Two lights are metameric if they have the same tri-stimulus values. Using the tri-stimulus values as Cartesian coordinates one can represent light colours as points in a 3D space (referred to as the colorimetric space). All the light colours make a tri-dimensional manifold which can be represented as a circular cone in the colorimetric space. Furthermore, colorimetry also claims that reflecting objects illuminated by the same light will look alike as soon as they reflect metameric lights. All the object colours are then represented as a closed solid inscribed in the light colour cone provided the illumination is fixed. However, as argued in this article, the reflected light metamerism does not guarantee that the reflecting objects will look identical (referred to as colour equivalence), especially when there are multiple illuminants. Moreover, colour equivalence cannot be derived from metamerism. The colour of a reflecting object under various illuminations is shown to be specified by six numbers (referred to as its six-stimulus values) that can be established by experiment. Using the six-stimulus values one can represent the colours of all the reflecting objects illuminated by various illuminants as a cone (without a vertex) through a 5D ball.

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Notes

  1. Strictly speaking, light cannot be black. The absence of light is experienced as darkness which is different from the experience of black (Volbrecht and Kliegl 1998).

  2. Furthermore, as shown elsewhere there simply cannot be, out of principle, a human observer with the CIE 1931 colour matching functions (Logvinenko 2009b).

  3. It should be noted that there is a natural limit for the spectral bandwidth of an illuminant exceeding which object-colour perception as such no longer occurs. For example, when a multi-coloured paper design (a so-called Mondrian) is illuminated by a narrow-band light-emitting diode, the paper is perceived as achromatic (of different shades of grey) through a “wash” of colour that can be described as the media (atmosphere) colour rather than surface colour (Schirillo, personal communication). Therefore, a monochromatic light cannot be considered as an illuminant for real objects not only for theoretical reasons, but simply because the monochromatic light would not secure any object-colour perception.

  4. In contrast to an asymmetric colour matching which involves only manipulating the object’s reflectance, in an object-colour matching experiment both the object’s reflectance and the object’s illumination are adjusted by observers to obtain colour equivalence.

  5. Of course, it is not easy to manipulate the object reflectance systematically. Furthermore, the rectangle object colour atlas \(\mathcal{A}_{x}\) cannot be implemented as real reflecting objects (e.g., papers). Still, one can use papers that approximate the rectangle reflectance functions.

  6. As only positive spectral power distributions are taken into consideration, the weights q n and q cannot be both zero in (4).

  7. The visible spectrum circle results from identifying the ends of the visible spectrum interval.

  8. Experimental evidence supporting the distinction between material and lighting hues can be found in Tokunaga and Logvinenko (2010a).

  9. A solution might exist if one imposes some additional constraints on the object set. Specifically, there is a solution if consideration is restricted to object sets that are three-dimensional. Such a constraint is explicitly made in the linear models (Maloney and Wandell 1986; Maloney 2003). Sometimes, this constraint is implicitly included. For example, Forsyth assumes no mismatching of metamers (Forsyth 1990, p. 7), however, this can be the case only when the dimensionality of the object set is not more than the number of photoreceptors (or sensors).

  10. If one counts each material (respectively, lighting component) as three unknowns, then the system (32) comprises 3N equations with 6N unknowns.

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Acknowledgements

This work was supported by a research grant (EP/C010353/1) from EPSRC. I wish to thank Brian Funt for fruitful discussions and encouragement while working on the article. I am also very grateful to Michael H. Brill for valuable criticism of the early draft.

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  1. Department of Vision Sciences, Glasgow Caledonian University, City Campus, Cowcaddens Road, Glasgow, G4 0BA, UK

    Alexander D. Logvinenko

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Logvinenko, A.D. Object-Colour Manifold. Int J Comput Vis 101, 143–160 (2013). https://doi.org/10.1007/s11263-012-0555-2

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