Hornaes Hans Petter, Wold Jan Henrik, Farup Ivar
Gjøvik University College, P.O.Box 191, 2802, Gjøvik, Norway.
J Math Biol. 2005 Aug;51(2):144-56. doi: 10.1007/s00285-004-0306-2. Epub 2005 Jul 13.
Human colour vision is the result of a complex process involving topics ranging from physics of light to perception. Whereas the diversity of light entering the eye in principle span an infinite-dimensional vector space in terms of the spectral power distributions, the space of human colour perceptions is three dimensional. One important consequence of this is that a variety of colours can be visually matched by a mixture of only three adequately chosen reference lights. It has been observed that there exists one particular set of monochromatic reference lights that, according to a certain definition, is optimal for producing colour matches. These reference lights are commonly denoted prime colours. In the present paper, we intend to rigorously show that the existence of prime colours is not particular to the human visual system as sometimes stated, but rather an algebraic consequence of the manner in which a kind of colorimetric functions called colour-matching functions are defined and transformed. The solution is based on maximisation of a determinant determining the gamut size of the colour space spanned by the prime colours. Cramer's rule for solving a set of linear equations is an essential part of the proof. By means of examples, it is shown that mathematically the optimal set of reference lights is not unique in general, and that the existence of a maximum determinant is not a necessary condition for the existence of prime colours.
人类的色彩视觉是一个复杂的过程，涉及从光的物理学到感知等诸多主题。虽然进入眼睛的光的多样性原则上在光谱功率分布方面跨越一个无限维向量空间，但人类色彩感知的空间是三维的。这一情况的一个重要结果是，仅通过混合三种适当选择的参考光就可以在视觉上匹配多种颜色。据观察，存在一组特定的单色参考光，根据某种定义，它们对于产生颜色匹配是最优的。这些参考光通常被称为原色。在本文中，我们打算严格证明，原色的存在并非如有时所述是人类视觉系统所特有的，而是一种称为颜色匹配函数的比色函数的定义和变换方式的代数结果。该解决方案基于最大化一个决定原色所跨越的颜色空间色域大小的行列式。求解一组线性方程的克莱姆法则是证明的一个重要部分。通过示例表明，在数学上，参考光的最优集通常不是唯一的，并且最大行列式的存在不是原色存在的必要条件。
