Preface

The trigger for writing this book was, for me, the realisation that Grassmann's laws?-?the cornerstone of colorimetry (which is the basis of colour science)?-?seemed not to be quite correct. True, it has been reported from time to time before that Grassmann's laws get violated under some special circumstances such as high light intensity and/or at the far periphery in the retina. However, I realised that one of the most important aspects of Grassmann's laws?-?the transitivity law?-?just could not be possibly true for colour matching even under normal condition of viewing. Intransitivity of colour matching undermines the major achievement of colorimetry: the definition of colour as a class of equivalence (i.e., a class of visually indiscriminable lights) since in this case colour matching cannot be conceptualised as an equivalence relation. The problem, as I saw it, was not only that the whole grand construction of colorimetry, it turned out, was built on sand, but that, despite this, colour television, the very idea of which would have been impossible without colorimetry, still successfully worked.

In the end, resolving this apparent paradox proved to be rather easy (Logvinenko, 2006). However, in doing this I had been becoming more and more confident in the opinion that the very foundations of colour science should be reconsidered. Furthermore, it should be done on quite different grounds, involving rather different mathematics.

It so happened that thus far the basic mathematical tool of colour science has been linear algebra. There is good reason for this. 'Many modern ideas in linear algebra?-?especially the concept of linear functional and the resulting distinction between a linear space and its dual space?-?seem to be specially designed to clarify thinking about colour theory' (Krantz, 1975a, p. 283).

Really, lights can be mixed together. Formally, this translates into the statement that there is an algebraic operation on the set of lights. Although colour is a mental, not physical attribute of light, a similar algebraic operation can be defined on the set of colours, namely, colour mixing. As known, the colour of the mixture of two lights depends on the colour of each light rather than their individual spectral compositions. Grassmann was, perhaps, the first to realise that this implies there being an algebraic structure of colour, which is relatively independent of that of light (Grassmann, 1853).

It is the algebraic structure of colours that enables us to represent colours as vectors in the linear space. Also it allows us to use matrix algebra to perform colour computation, that has been widely used in colour science and its applications (e.g., colorimetry, colour imaging, computer graphics). Formally, the algebraic structure of colour is defined as that of the classes of visually equivalent (i.e., subjectively indistinguishable) lights. The vector representation of colours results from mapping the lights onto the classes of visual equivalence (the colour stimulus map). The vector representation of colour, empirically based on colour matching, puts colorimetry and its methods on par with the physical sciences, which has been acknowledged by the inclusion of colorimetry by Richard Feynman in his celebrated lectures on physics (Feynman et al., 1963, Ch. 35).

It should be said, however, that there are natural limitations for the application of linear algebra even in colorimetry, not to mention colour science in general. The finite dimensionality of linear spaces which linear algebra deals with requires the discretisation of continuous functions that naturally arise in colorimetry (e.g., the light energy distributions over the visible spectrum and colour matching functions), which is a problem in itself. Being free of this limitation, functional analysis (dealing with infinite-dimensional linear spaces) seems more appropriate for colour science purposes. We found that many problems in colorimetry are much easier to tackle from the infinite-dimensional standpoint.

The linear structure is not all that colour inherits from light. There is a natural physical measure of proximity between different lights. The fact that close lights have colours which look subjectively close to each other testifies that the light proximity is somehow retained in colour. It means that the colour stimulus map is, in a sense, continuous. While Grassmann stated the law of continuity for colour mixing more than 150 years ago, it has not been elaborated on to the same extent as his other laws of colour mixing. Moreover, the topological concepts underlying the notion of continuity still remain alien to colour science despite the fact that, as shown in this book, they prove to be rather effective for solving a number of problems in colour science.

Topological methods are particularly pertinent to colour science because the colour manifold itself, as well as some other important objects (e.g., the object-colour solid), is convex. All these objects inherit convexity from their physical prototypes. For example, all lights, as a whole, can be represented by a convex cone (in the infinite-dimensional vector space). As the colour stimulus map preserves convexity, the full gamut of colours can also be represented as a convex cone in the vector space, but with finite dimensions. Using the methods of convex analysis for infinite-dimensional topological vector spaces, we have solved some long-standing problems in colour science.

It should be mentioned that the achievements of colorimetry in colour measurement have been made at a cost of neglecting colour perception (i.e., the structure of colour appearance). Really, colorimetry can predict whether two lights will appear of same colour or different, but not what exactly they will look like. Although a number of colour appearance models have been put forward (Wyszecki, 1986; Fairchild, 2005), they are quite far from meeting the colorimetric standards of precision and accuracy. It is hardly surprising since colorimetry has proven to be, in fact, the science about photopigments in the human eye. The participation of a human observer in a colorimetric experiment is intentionally reduced to the role of a comparator, with only two outputs: same or different. The success of a colorimetric study depends on how well a human turns into, virtually, a technical device. Ideally, the influence of all factors (including perceptual factors), with the exception of photopigments, is supposed to be eliminated. The effectiveness of colour television suggests that we are pretty close to this ideal. One has, however, to admit that colorimetry has not quite reached the level of exact sciences, like physics (as evidenced by the aforementioned violation of Grassmann's laws). The reason is that, surprisingly, a human, alas, is unable to entirely simulate such a simple device as a comparator. The fact is that human responses, even such simple ones as binary 'yes/no' s, are inherently fuzzy, whereas colorimetry implies crisp responses, simply ignoring this fuzziness. It was interesting for us to see what would become of colorimetry after the revision of Grassmann's laws, taking into account the fuzziness of human responses.

In order to look into colour perception, we chose to completely break with colorimetry and start from scratch. Guided by Hering's idea of component hues, we attempted a new approach to colour perception based on the same theoretical framework as that used by us to expose the main results obtained in colorimetry. It allowed us to develop techniques to measure colour appearance with an accuracy meeting, we believe, the standards adopted in colorimetric measurements.

Although, human colour vision was our main concern, we hope many results presented in the book can be applicable to animals' colour vision as well as various technical devices equipped with artificial photosensors (e.g., cameras). Our intention was to outline mathematical foundations?-?the most general principles?-?which can be used when dealing with a particular system of colour vision based on the same principles as human colour vision. For example, a notion of metamerism is applicable to any live or artificial system of photosensors with different spectral characteristics. The mathematical problem of describing metameric lights (i.e., indistinguishable for these photosensors) naturally arises. It turns out that there is a general solution to this problem (an algorithm is presented in Chapter 6). In fact, much of the book is devoted to various mathematical problems arising in colour science, and their solving.

Going beyond the usual repertoire of mathematical methods used by colour scientists would not have been possible without the participation of a professional mathematician, such as was Vladimir Levin. He made proofs of almost all theorems, lemmas, and propositions in the book. Unfortunately, his untimely death in 2012 prevented him from taking part in the final stage of work on this book. In particular, the Outline (Chapter 1), Chapters 9, 13, 17, 19-21, and the Epilogue were written after he passed away.

It is also worth mentioning the following context, which played a not insignificant part in the genesis of the premise for this book. Since colour is the subject of various disciplines, and given the rapid progress in almost every one of them in recent years, there is an urgent need for a guide that presents the subject from a...