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Prolab: Perceptually Uniform Projective Colour Coordinate System

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Prolab: Perceptually Uniform Projective Colour Coordinate System Date of publication xxxx 00, 0000, date of current version December 31, 2020. Digital Object Identifier 10.1109/ACCESS.2017.DOI ProLab: perceptually uniform projective colour coordinate system IVAN A. KONOVALENKO1,2, ANNA A. SMAGINA1, DMITRY P. NIKOLAEV1,2, (Member, IEEE), AND PETR P. NIKOLAEV1,3. 1Institute for Information Transmission Problems of the Russian Academy of Sciences, Bolshoy Karetny per. 19, build.1, Moscow, 127051, Russia 2Smart Engines Service, LLC, Prospect 60-Letiya Oktyabrya, 9, Office 605, Moscow, 117312, Russia 3Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, 141700, Russia Corresponding author: Dmitry P. Nikolaev (e-mail: [email protected]). The work was partially financially supported by RFBR in the context of scientific projects NoNo 17-29-03370 and 19-29-09075. ABSTRACT In this work, we propose proLab: a new colour coordinate system derived as a 3D projective transformation of CIE XYZ. We show that proLab is far ahead of the widely used CIELAB coordinate system (though inferior to the modern CAM16-UCS) according to perceptual uniformity evaluated by the STRESS metric in reference to the CIEDE2000 colour difference formula. At the same time, angular errors of chromaticity estimation that are standard for linear colour spaces can also be used in proLab since projective transformations preserve the linearity of manifolds. Unlike in linear spaces, angular errors for different hues are normalized according to human colour discrimination thresholds within proLab. We also demonstrate that shot noise in proLab is more homoscedastic than in CAM16-UCS or other standard colour spaces. This makes proLab a convenient coordinate system in which to perform linear colour analysis. INDEX TERMS color, mathematical model, linearity, image representation, image color analysis, noise measurement I. INTRODUCTION A. COLOUR SPACES AND COLOUR COORDINATE SYSTEMS The purpose of this work is to develop a new colour co- The human perception of colour is defined by the spa- ordinate system for the analysis of colour images of the tial distribution of retinal irradiance and the internal state visible light spectrum. Most existing algorithms for colour of the visual system. Under photopic conditions, there are analysis operate with distances in a colour space, and some three types of active cones, the reactions of which can be of them also rely on linear properties of colour distributions. considered continuously dependent on the irradiance. Un- Meanwhile, the colour space metric, as a rule, is not derived der the assumption that photoreceptors are negligibly small arXiv:2012.07653v2 [cs.CV] 11 Jan 2021 strictly from physical models, but rather from the properties and uniformly distributed, the response to the irradiance at of human colour perception using psychophysiological ex- each point on the retina can be represented as three scalar perimental data. The colour coordinate system proposed in reactions. All other elements of the visual system employ this work is based on the colour perception model as well not arbitrary parameters of the irrandiance, but these three as on physical models of image formation, so for accurate scalar reactions exclusively. In this model, given a fixed problem statement we have to provide a detailed introduc- internal state, the colour perception caused by a uniformly tion. Sections I-A and I-B are dedicated to colour models in illuminated part of the retina depends on a three-dimensional psychophysics; Sections I-C and I-D to the aspects of colour manifold, regardless of the visual system’s complexity and its image formation and processing. Then, in Section I-E, we internal state. We shall call such a manifold a colour space, formulate the problem to be solved and propose the general and its elements we call colours. idea of the solution. Section I-F addresses the most relevant works in the field. In Section I-G, the desired properties of To perform colour mapping, various colour coordinate the proposed coordinate system are listed. systems are used. In any of them, the coordinates of the colour vector c are related to the spectral irradiance F(λ) through some vector functional Ψ, which can be additionally parameterized with context information and the internal state VOLUME 4, 2016 1 Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS θ of the visual system: B. EVALUATION METRIC AND PERCEPTUAL UNIFORMITY c = Ψθ[F(λ)]; (1) Psychophysical experiments not only reveal the spectral basis where λ is the light wavelength. The colour coordinate space of the colour space perceived by people, but they also help is called linear if Ψ is linear for fixed θ. to determine its metric parameters. This can be done, for In 1853, Grassmann showed in his experiments [1] that instance, by measuring changes in the thresholds in spectral under colorimetric conditions, a linear 3D colour coordinate stimuli for a human eye at different points of the colour space. system of the human vision system can be constructed, while The colour coordinate space is called perceptually uniform the internal state of the latter can be neglected due to these (hereinafter – uniform) if the Euclidean distances between conditions. This reduced the problem of determining the colours in it correspond to the differences perceived by a colour space for the human eye to a linear colour coordinate- human eye. Liminal difference vector lengths are uniform based construction in the spectral irradiance space F(λ). In across all of the points and in any direction in such a space. this system, the relationship between the retinal spectral irra- The CIE XYZ linear coordinate system provides a colour diance and the colour coordinates is constructed analytically: space with a natural Euclidean representation, but it is sig- nificantly non-uniform in this regard. Z 1 def There have been many attempts to create a uniform colour cx = F(λ)X(λ)dλ, (2) 0 coordinate space. In 1948, Richard Hunter proposed the first uniform space, Hunter Lab [5]. Later, David MacAdam where X(λ) are the colour matching functions of a standard proposed a space based on a research by Dean Judd [6]. This observer, and c are the colour coordinates in CIE XYZ. x space was standardized by CIE in 1960 as a uniform chro- Thus far, various colour coordinate systems have been maticity space (CIE 1960 UCS). As its name implies, this proposed for the standard observer. These systems vary in coordinate system does not include any brightness compo- terms of their suitability for the certain applications [2]. Some nent. Soon after that, Gunter Wyszecki proposed a space [7] of them imply colorimetric conditions. Others are related to based on the latter one, that was adopted as the CIE 1964 the various colour perception models, which parametrize θ (U*, V*, W*) Color Space (or CIEUVW). It allowed for the in one way or another. Assuming that the visual context and calculation of the colour differences even with mismatched the internal state do not affect the spectral sensitivity, the brightness. In 1976, CIELAB space was developed based on coordinate vectors in any of such coordinate systems can be Hunter Lab [8]. It is still the most used uniform colour co- expressed via CIE XYZ coordinates independently of F(λ): ordinate system when it comes to complex stimuli (images) analysis. cΦ = ΨΦ;θ[F(λ)] = Φθ(cx); (3) However, the CIELAB space is only approximately uni- where Φθ is the transformation from CIE XYZ to the given form, so from the moment of its inception, there have been coordinate system under known internal state θ. continuous attempts to create more uniform coordinate sys- Usually, colour perception models consider at least the tems (e.g. [9]). At the same time, non-Euclidean colour adaptation of the visual system to the dominant illumi- difference formulas were being developed in order to provide nance [3]. In von Kries’ model [4], this is expressed within more precise correlation with experiment regarding human the transformation (3) as a componentwise division of the in- perception. The successful outcome of these efforts was the put coordinate vector cx by the light source colour coordinate CIEDE2000 formula [10], [11], which is still considered > vector cx (θ): the most accurate one available [12]. Nevertheless, for some applications it is preferable to operate in colour coordinate −1 > Φθ(cx) = Φ0 diag cx (θ) cx ; (4) spaces with uniform Euclidean distance (e.g. for the devel- opment of effective search structures). Therefore, uniform where Φ0 is a transformation that is independent of the spaces are still being actively researched and developed. illumination. Currently, the CAM16-UCS space [13], developed in 2016, In systems with the same adaptation model, the coordinate is considered to be the accuracy standard among these. transformation could be performed bypassing CIE XYZ and, obviously, such transformation does not require information C. COLOUR SPACES OF VISIBLE SPECTRUM on the illumination: CAMERAS 8 def −1 With certain reservations, all of the aforesaid can also be <ca = Aθ(cx) = A0 diag c> (θ) cx x attributed to technical visual systems. Of course, we are not def −1 > (5) :cb = Bθ(cx) = B0 diag cx (θ) cx considering the perception of technical systems. In technical vision systems, the term ‘colour’ usually means a three- =) c = B A −1(c ) ; b 0 0 a dimensional vector that is passed for further processing. where ca and cb are the colour coordinate vectors in the two Also, the internal state θ of such systems, as a rule, could be coordinate systems, defined by the transformations Aθ and neglected or considered to be known. But the most important Bθ, respectively. thing is that the spectral basis of camera space is significantly 2 VOLUME 4, 2016 Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS different from the standard observer ones.
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