Geometric invariants under illuminant transformations

An object colour's Commission Internationale de l'Éclairage XYZ coordinates can change when it is viewed under different illuminants. The set of XYZ coordinates for all object colours, which is called the object‐colour solid, likewise varies under different illuminants. This article shows that, despite these changes, some properties are invariant under illuminant transformations. In particular, as long as the illuminant is nowhere zero in the visible spectrum, optimal colours take the same Schrödinger form, and no two optimal colours are metameric. Furthermore, all object‐colour solids have the same shape at the origin: they all fit perfectly into the convex cone (which we will call the spectrum cone) generated by the spectrum locus. The spectrum cone, itself, does not vary when the illuminant changes. The object‐colour solid for one illuminant can be transformed into the solid for another illuminant, by an easily visualized sequence of expansions and contractions of irregular rings, called zones. © 2012 Wiley Periodicals, Inc. Col Res Appl, 39, 179–187, 2014     

Four‐transition 0‐1 functions for reflectance spectra

It is known that a reflectance spectrum for an optimal colour takes on the value 0 or 1 at every wavelength, with at most two transitions between those values. This article shows that any non‐optimal colour can be produced by a reflectance spectrum that takes on the value 0 or 1 at every wavelength, with at most four transitions. While the two‐transition optimal spectrum is unique, the four‐transition non‐optimal spectrum is not unique.     

Generation of virtual illuminants for a balanced colorimetric match

Colorimetric matching is employed under a series of virtual illuminants to produce more controllable and equalised colour‐difference results under multiple illuminants. A method based on weighted principal component analysis of different artificial lights and the weighted mean is used to create a series of virtual illuminants. The weights were selected in a manner to create the virtual illuminants that benefit from the impact of a given illuminant, while the effects of other light sources on the formation of desired illuminants were also considered. The generated virtual illuminants were implemented in a colorimetric colour‐matching trial. Using this method, by choosing suitable weights for different lighting conditions, more balanced colour‐difference values were presented in the data set than was expected. The advantages of the suggested method were evaluated by the matching of a collection of 135 woollen samples under different virtual illuminants.     

A zonohedral approach to optimal colours

This article demonstrates that the CIE XYZ colour solid is a zonoid. An approximating zonohedral colour solid is constructed explicitly from a set of generating vectors, which are integrals of colour‐matching functions over narrow intervals of the visible spectrum. The zonohedral approach yields an intuitive, constructive proof of the Optimal Colour Theorem: the reflectance function of an optimal colour takes on only the values 0 or 1, with at most two transition wavelengths. In addition, zonohedral techniques can simplify computations: for example, optimal colours can be found without calculating transition wavelengths. Finally, zonohedra provide a simple, unified approach to colour space and eliminate much of the confusion arising from chromaticity diagrams. © 2011 Wiley Periodicals, Inc. Col Res Appl, 2013     

An integrated spectral imaging system for producing color images of static and moving objects

An integrated spectral imaging system constructed by synchronizing a programmable light source, a high‐speed monochrome camera, and a display device is proposed to produce tristimulus images of static and moving objects effectively in real time onto the display. This system is called the CIE‐XYZ display. Active spectral illuminants, containing both the device characteristics of camera and display, are projected onto object surfaces as time sequence. The images are captured synchronously by the camera and quickly transmitted to the display device in the RGB signal form so that the accurate tristimulus images are displayed. First, we describe the principle of the CIE‐XYZ tristimulus display. The theoretically optimal illuminants contain negative parts in the spectral curves. Second, we design practical illuminants with all positive spectral curves. The color images in our system are composed of the time sequence of RGB component images. Then, the synthesized color images on the display contain color artifacts when objects move fast. An image processing algorithm for correcting the motion color artifact is proposed based on optical flow estimation using a graphics processing unit. The comprehensive performance of the proposed system and algorithms is examined using both static and moving objects. © 2014 Wiley Periodicals, Inc. Col Res Appl, 40, 329–340, 2015     

Influences of lighting time course and background on categorical colour constancy with RGB‐LED light sources

Some previous studies have investigated the influence of the lighting time course and viewing background on the colour constancy using two‐dimensional flat stimuli simulated on a monitor. In the present study, we investigated the categorical colour constancy in real scenes by manipulating (a) the lighting time course, that is, adaptation period to the illuminant (brief adaptation or complete adaptation) and (b) the background structure of a stimulus (a uniform gray background with an approximately 25% spectral reflectance or a multicolour background consisting of the Macbeth ColorChecker and some fruit models). The neutral (u′ = 0.1994, v′ = 0.4671), red (u′ = 0.2433, v′ = 0.4622), green (u′ = 0.1525, v′ = 0.4697), blue (u′ = 0.2049, v′ = 0.4198), and yellow (u′ = 0.1892, v′ = 0.5112) illuminants were produced by an RGB‐LED lamp. For each chromatic illumination condition, subjects categorized 240 surfaces with Munsell Value 5/ in four viewing conditions with different combinations of the lighting time course and the background structure. A total of seven subjects participated in experiments with red and green illuminants and five subjects with blue and yellow illuminants. The results showed that the constancy index was the lowest (0.66) in the brief adaptation and gray background condition and the highest (0.74) in the complete adaptation and multicolour background condition. The results suggest that increasing the adaptation period alone or adding chromatic cues in the background with a brief adaptation can help to improve the colour constancy, and a time‐taking reference to surrounding coloured objects with the long presentation of the illuminant may also contribute to obtaining colour constancy.     

Recovery of reflectance spectra from CIE tristimulus values using a progressive database selection technique

A procedure for creating efficient reflectance spectra from CIE tristimulus colour values is described using a modified linear model. By fixing certain criteria based on colour difference values, the proposed technique preliminarily selects a series of suitable samples from a main data set containing the reflectance values of a large number of different coloured samples, based on the colour specifications of a given sample. In this way, a series of different databases containing the reflectance values of confirmed samples relating to the particular samples are formed. Then, a well-known principal components linear model is applied using three basis functions. This operation yields higher cumulative variances in comparison with the original database, having the same number of basis vectors. The performance of the proposed method is evaluated using a different collection of samples and the resulting spectra show considerable improvements in terms of root mean square error as well as colour difference values under different illuminants.     

Colour management in digital textile printing

On the basis of simulated prints, a customer can decide on the colour appropriateness for a selected pattern. This article presents the possibilities and correct procedure for colour management application in the field of digital printing onto a textile substrate. The aim of the research was to establish whether long‐term and expensive pre‐ and after‐treatments can be avoided with the help of an appropriate development of printer profiles. The latter would be conducted through a print simulation on paper printed with a Canon laser or ink‐jet printer. The results have confirmed that a print simulation on paper can be conducted with both printers, while a Canon ink‐jet printer offers better results. In addition, by calculating the colour inconstancy index CMCCON02, the illuminant influence on the colour change of substrates under five different illuminations was determined. The results showed that simulated colours should be compared with original colours under daylight illuminants (D50, D55 and D75), while indoor illuminants A and F2 are not appropriate for a comparison, especially in the case of simulation with a Canon ink‐jet printer (CMCCON02 index >5).     

An efficient method for solid‐colour and multicolour region segmentation in real yarn‐dyed fabric images

This paper presents an efficient approach to solid‐colour and multicolour region segmentation in real yarn‐dyed fabric images. The approach is based on a novel model describing the spectral response of a multispectral imaging system to yarn‐dyed fabrics. The model indicates that solid‐colour regions cannot be distinguished from multicolour regions in terms of reflectance, tristimulus, or CIELAB values owing to a geometric term representing the influence of fabric surface condition on measured colours. The geometric term makes it difficult to determine the segmentation thresholds of CIEXYZ and CIELAB colour histograms. However, solid‐colour and multicolour regions can be detected in CIExyY space because chromaticity coordinates are impervious to the geometric term. The CIExyY histograms of a solid‐colour region accord with one Gaussian distribution, but those of a multicolour region accord with a combination of two Gaussian distributions. The CIEXYZ, CIELAB, and CIExyY colour distributions of both solid‐colour and multicolour yarn‐dyed fabrics were analysed in detail in simulation and real experiments. Experimental results show that solid‐colour yarn‐dyed regions can be distinguished from multicolour yarn‐dyed fabric regions by the shapes of CIExyY histograms, but cannot be distinguished by the shapes of CIEXYZ or CIELAB histograms.     

Designing a Jet Mill Nozzle of Maximum Efficiency

A method of calculation for an accelerating jet mill nozzle profile which provides a maximum kinetic energy for solid particles at the nozzle outlet, is developed. The gas‐solid flow in the nozzle is described by a well‐known one‐dimensional model. The effect of both solid loading and particle size on the optimal nozzle profile, as well as on the main flow parameters is studied. Distribution of particle velocities in the computed optimal nozzle is compared to that in a conical convergent‐divergent nozzle of the same length. Computations demonstrated that optimized nozzles are profitable for fairly low solid loadings. The kinetic energy of solids at the optimized nozzle outlet is about 50% higher than that in a conical nozzle if the solid/gas mass flow ratio equals unity.     

Spectral dependence of colorimetric characterisation of scanners

The spectral dependence of the colorimetric characterisation of a typical scanner was investigated. Different colour sets, including ColorChecker SG, Kodak Q‐60 colour input target, a set of plain woven coloured fabrics with a large colour gamut, and randomly selected samples of the Munsell Book of Color were used as training and testing sets in the colorimetric characterisation of a scanner by employing a non‐linear regression method. The coefficient matrices were optimised for each particular media in the training stage and used to predict the device‐independent colorimetric data, i.e. CIELab values of other media from their corresponding RGB values measured by the scanner. In order to extract the differences between the applied sets and determine the actual dimensions of their reflectance spectra, the principal component analysis technique was employed. As expected, it was observed that the different sets benefit from diverse dimensional properties and, in some cases, the spectral behaviours of the first few eigenvectors were apparently different. It was demonstrated that scanner colorimetric characterisation depended on the spectral properties of the applied colour set in the training stage and, consequently, the testing errors increased with increasing the spectral dissimilarity between the sets that were used in training and testing sequences. It was concluded that, to achieve better colour reproduction results, the scanner should be characterised for each media with specific spectral properties.     

Mathematical approach for predicting non‐negative tristimulus values using the CAT02 chromatic adaptation transform

It has been reported that for certain colour samples, the chromatic adaptation transform CAT02 imbedded in the CIECAM02 colour appearance model predicts corresponding colours with negative tristimulus values (TSVs), which can cause problems in certain applications. To overcome this problem, a mathematical approach is proposed for modifying CAT02. This approach combines a non‐negativity constraint for the TSVs of corresponding colours with the minimization of the colour differences between those values for the corresponding colours obtained by visual observations and the TSVs of the corresponding colours predicted by the model, which is a constrained non‐linear optimization problem. By solving the non‐linear optimization problem, a new matrix is found. The performance of the CAT02 transform with various matrices including the original CAT02 matrix, and the new matrix are tested using visual datasets and the optimum colours. Test results show that the CAT02 with the new matrix predicted corresponding colours without negative TSVs for all optimum colours and the colour matching functions of the two CIE standard observers under the test illuminants considered. However, the accuracy with the new matrix for predicting the visual data is approximately 1 CIELAB colour difference unit worse compared with the original CAT02. This indicates that accuracy has to be sacrificed to achieve the non‐negativity constraint for the TSVs of the corresponding colours. © 2011 Wiley Periodicals, Inc. Col Res Appl, 2011     

Use of a predictive colour model for managing the colour appearance of two‐colour woven fabrics

The aim of this study was to implement a two‐dimensional colour appearance model for prediction of the colour values of weft threads when the optical mixing of a two‐colour woven structure had to match the colour appearance of a single‐colour reference woven fabric. Five single‐colour woven fabrics were woven from five threads of similar hue. One of the samples was chosen as a reference, for which the colour appearance was the goal to be achieved in the two‐colour woven fabrics prepared with the other available warp threads and newly dyed weft threads. The colour values of dyed weft threads were predicted by a two‐dimensional colour appearance model. With dyed weft threads, managing the colour appearance of the two‐colour woven fabric was enabled to achieve the colour values of the reference. In the results, colour deviations between the predicted and measured colour values of weft threads revealed some limitations to the colour appearance model and performance of the dyeing process. After the production of the two‐colour woven fabric, the colour appearance matched the appearance of the reference, resulting in deviations of ΔE_CMC(2:1) = 1.2‐7.8. Moreover, the differences between theoretically predicted and measured colour values of the two‐colour woven fabric were evaluated as small, ranging from ΔE_CMC(2:1) = 1.5‐1.9. The results demonstrated the efficiency of implementing the colour appearance model and the dyeing process of weft threads as an approach to achieve the defined colour appearance of two‐colour woven fabrics, which with small colour deviations matches the colour of a single‐colour reference.     

Illuminant metamerism potentiality of metameric pairs

Indices for describing the degree of metamerism are based on either the deviation of the spectra of a metameric pair or the colour difference of the pair under test conditions. The magnitude of illuminant metamerism is commonly evaluated by measuring the colour difference under the test illuminant. The calculated colour differences absolutely vary with the selected test illuminants, so the measured (special) index of metamerism could be considered as a test‐illuminant‐dependent value. The spectral‐based indices of metamerism act as a single‐number value, but most of them are criticised for their poor correlation with visual assessment. In this paper, a general metric is developed for evaluating the upper limit of the degree of illuminant metamerism. The suggested approach combines the advantages of general and special indices, avoiding their drawbacks at the same time. The performance of the formula is analysed in a number of numerical experiments, as well as by practical testing.     

Model for Predicting Solids Velocity Fluctuations in Sedimenting Suspensions

Computational fluid dynamics (CFD) simulations of sedimenting suspensions in two‐dimensional periodic domains using a Eulerian two‐fluid model were performed with the commercial software ANSYS Fluent®. Three particle systems belonging to Geldart groups A and B were selected to gather data on the solids velocity fluctuations. It is proposed that solids velocity fluctuations are composed of local and global parts. Therefore, the solids velocity signal was segregated into low‐frequency and high‐frequency fluctuations using the FFT solver in MATLAB®. A model is proposed to predict the velocity fluctuations of a solid phase at low volume fractions and the model results are compared with the CFD results. The model is capable of capturing the solid particles fluctuation in the dilute limit.     

Multiphase CFD Simulation of a Solid Bowl Centrifuge

This study presents some results from the numerical simulation of the flow in an industrial solid bowl centrifuge used for particle separation in industrial fluid processing. The computational fluid dynamics (CFD) software Fluent was used to simulate this multiphase flow. Simplified two‐dimensional and three‐dimensional geometries were built and meshed from the real centrifuge geometry. The CFD results show a boundary layer of axially fast moving fluid at the gas‐liquid interface. Below this layer there is a thin recirculation. The obtained tangential velocity values are lower than the ones for the rigid‐body motion. Also, the trajectories of the solid particles are evaluated.     

Evaluation of wall boundary condition parameters for gas–solids fluidized bed simulations

Wall boundary conditions for the solids phase have significant effects on numerical predictions of various gas–solids fluidized beds. Several models for the granular flow wall boundary condition are available in the open literature for numerical modeling of gas–solids flow. A model for specularity coefficient used in Johnson and Jackson boundary conditions by Li and Benyahia (Li and Benyahia, AIChE J. 2012;58:2058–2068) is implemented in the open‐source CFD code‐MFIX. The variable specularity coefficient model provides a physical way to calculate the specularity coefficient needed by the partial‐slip boundary conditions for the solids phase. Through a series of two‐dimensional numerical simulations of bubbling fluidized bed and circulating fluidized bed riser, the model predicts qualitatively consistent trends to the previous studies. Furthermore, a quantitative comparison is conducted between numerical results of variable and constant specularity coefficients to investigate the effect of spatial and temporal variations in specularity coefficient. Published 2013 American Institute of Chemical Engineers AIChE J, 59: 3624–3632, 2013     

The minimum number of measurements for colour,sparkle, and graininess characterisation in gonio‐apparent panels

Materials with new visual appearances have emerged over the last few years. In the automotive industry in particular there is a growing interest in materials with new effect finishes, such as metallic, pearlescent, sparkle, and graininess effects. Typically, for solid colours the mean of three measurements with repetitions is sufficient to obtain a representative measurement for colour characterisation. However, gonio‐apparent panels have non‐homogeneous colours, and there are no studies that recommend the minimum number of repetitions for colour, sparkle, and graininess characterisation of this type of panel. We assume that colour panels incorporating special‐effect pigments in their colour recipes will require a higher minimum number of measurements than solid colour panels. Therefore, the purpose of this study is to verify this assumption by using a multiangle BYK‐mac spectrophotometer, given that it is currently the only commercial device that can measure colour, sparkle, and graininess values simultaneously. In addition, a possible methodology is given for establishing the minimum number of measurements when characterising gonio‐apparent materials using a specific instrument, able to be implemented in future instruments when determining multiple appearance attributes (colour, gloss, sparkle, etc.) for many coloration technologies. Thus, we studied the minimum number of measurements needed to characterise the colour, sparkle, and graininess of three types of sample with solid, metallic, and pearlescent coatings respectively. Twenty measurements were made at twenty random positions (different target areas) of 90 samples. The minimum number of measurements for all these variables was determined on the basis of the point at which the cumulative mean value became constant. Thus, applying new statistical tools, it is clearly shown that metallic and pearlescent panels require more colour measurements than solid panels, in particular when geometries are being measured in a specular direction. As regards texture (sparkle and graininess), more measurements are needed for graininess than for sparkle, and more for metallic panels than for pearlescent panels.     

Design of vertical air classifiers for municipal solid waste

A previously published diffusion-based mathematical model is used to simulate data from two vertical air classifiers operating on shredded municipal solid waste streams. Good agreement with the model is achieved at high values of the air/solids ratio, that is, the region where interaction between solid particles would be expected to be insignificant. At lower values of the air/solids ratio the concept of a recovery efficiency is introduced to take account of interaction effects. This efficiency appears to show the expected dependence on air/solids ratio. A design method is proposed, based on feed particle size distribution and relationships between recoveries and product qualities.     

Individual differences in human colour vision as derived from stiles & Burch 10° colour matching functions

Individual differences between the 49 Stiles & Burch observers have been analyzed using the object‐colour space put forth recently (J of Vision 2009;9:1–23). A set of rectangular reflectance spectra has been used as a common frame of reference for representing object colours for all the observers. Being metameric to one of these rectangular reflectance spectra, every reflectance spectrum can be geometrically represented as a point in the three‐dimentional space. The interindividual differences reveal themselves in that, for various observers, the same reflectance spectrum maps to different points in this space. It has been found that on average such differences do not exceed the differences in object‐colour appearance induced by an illumination shift from daylight to the fluorescent daylight simulator F1. Such small individual variations have been accounted for by the fact that the cone spectral tuning curves have a special form that mitigates the individual differences in cone spectral positioning. © 2012 Wiley Periodicals, Inc. Col Res Appl, 2013