Improving the Color Constancy of Prints by Ink Design

A computer simulation was performed to investigate optimal fourand five-ink sets in order to minimize color inconstancy while maintaining a large color gamut. The effect of including a spectrally-flat black ink on color constancy was explored. The relationship between ink shape and the color constancy of prints was also analyzed. The results show that the ink shape in the middle wavelength region is very important for the color constancy of prints. The minimum and maximum color inconstancy indices that the different ink sets can achieve, which were optimized by different objective functions, were explored. The results show that color constancy can be improved by both ink design and lookup table creation. Introduction Color constancy is the general tendency of the color of an object to remain constant when the level and color of the illumination are changed. It is a result of both physiological and psychological compensations. Conversely color inconstancy is the undesirable change in color caused by changes in illumination. There does not yet exist a computational theory sufficient to explain the mechanism of color constancy of human vision. For many applications, it is very important that colored materials exhibit color constancy. However they often deviate significantly in hue when viewed under different light sources. In fact, color inconstancy is unavoidable. That means the perceived color always changes under different illuminants. Color inconstancy is a very important factor to evaluate for the image quality of prints since prints are viewed under many different lighting conditions. The color constancy of neutral has always been a design criterion for photographic dyes. Ohta explored the optimum combination of cyan, magenta, and yellow dyes to create stable grays under different illuminants. Because of the popularity of ink-jet printing technologies, fortuitous color constancy is less commonplace. The practices of only using CMY for pictorial images and using dye-based black inks with long wavelength reflectance “tails” result in appreciable color inconstancy. As a consequence, in addition to color gamut expansion, color constancy should be an ink-design criterion. This paper is divided into two sections. In the first section, the theoretical ink design for color constancy was described. In the second section, color constancy properties of lookup tables combined by the optimized ink sets were evaluated. Optimizing the Ink Spectra while Minimizing Color Inconstancy As mentioned above, color inconstancy depends on the ink spectral properties and can be minimized by optimizing the ink spectral properties while maintaining a relative large color gamut. Method In this section, the different parts of the optimization algorithm are described. These metrics and models include the color inconstancy metric, theoretical ink curve function, printing model, gamut calculation, and optimization objective function. Color Inconstancy Index Generally a color inconstancy index (CII) is used as a metric to evaluate the extent of color inconstancy. The color inconstancy index is the total color difference between a sample’s colorimetric coordinates under reference and test illuminants using a perceptually uniform color-difference equation. The calculation of the color inconstancy index is described in references 1, 7 and 8: Tristimulus values are calculated for illuminants of interest from an object’s spectral reflectance. Using a chromatic-adaptation transform such as CIECAT02, corresponding colors are calculated from each illuminant to D65. The corresponding-color tristimulus values are converted to CIELAB using D65 as the reference white. A weighted CIE94 color difference is calculated with kL = kC = 2 between pairs of corresponding colors. In this manner, hue inconstancy is penalized twice as much as lightness or chroma inconstancy. As a rule-of-thumb, samples with excellent color constancy have CII values below unity. Absorption Bands of Hypothetical Inks by Symmetric Cubic Spline Function In order to perform ink curve optimization, several different functions were considered to model ink spectra, including a Gaussian function, symmetric cubic-spline function, triangular function, line function, etc. All of these functions can model both one and two peak reflectance curves. At first, the Gaussian functions were used to simulate hypothetical ink curves. Six parameters, two sets each of height, width, and peak wavelength, were necessary to simulate one ink curve in reflectance space. The advantage of this model is that it is very flexible when simulating different ink curves. The disadvantage is that the model has so many parameters that convergence and optimization speeds were slow. Moreover, some parameters became useless in some situations. The width and the peak wavelength were not useful when the height is zero, for example. From previous research, we found that these optimized inks were not as complicated as real ink curves, the ink curves are close to 13th Color Imaging Conference Final Program and Proceedings 159 block ink curves so they are able to provide the higher lightness and chroma. The purpose of this theoretical research is to provide direction for ink design rather than producing real inks. Therefore, a simple model can be used to optimize the ink curves. From experience, the symmetric cubic-spline function can provide better convergence properties. Only two parameters, peak wavelength and width, are necessary to model one ink curve. However the symmetric cubic-spline function only can simulate single peak spectra in density space. As a consequence, it cannot directly model the green ink curve. Therefore, in this research the green ink was not considered. The spectral density of a theoretical ink was modeled by a symmetric cubic spline function with the peak wavelength and width, shown in Eq. 1: Dλ,ink = w + 3w w− λ ( )+ 3w w− λ ( )2 −3 w− λ ( )3 { } 6w for...λ < w Dλ,ink = 2w− w ( )3 6w for...w < λ <2w Dλ,ink =0 for...2w < λ Rλ,ink =10 Dλ,ink (1) where Dλ,ink represents the spectral density of the inks, Rλ,ink represents the spectral reflectance of the inks, λ is the peak wavelength of an absorption band, w is the width of an absorption band, and the peak densities are normalized to 2.5, corresponding to the minimum reflectance, 0.0032. Virtual Printing Model and Printing Gamut In order to evaluate different ink sets, the spectra of overprints with different ink amounts should be predicted from the ink and substrate spectra. Therefore a virtual model was developed in our previous research. 5 The virtual model can be divided into two parts. In the first part, Kubelka-Munk turbid media theory was used to predict the spectra of overprints with full area coverage, or Neugebauer primaries, with the assumption that transparent ink-jet inks would penetrate the paper support and yield a homogenous colored layer. In the second part, the Yule-Nielson modified Spectral Neugebauer model (YNSN) was used to predict the spectra of samples from ink amounts of different inks based on the spectra of Neugebauer primaries. The Yule-Nielsen exponent, n value, was chosen as ten based on previous research. The color gamut created by a specific ink set can be calculated with factorial area coverage data. For example, we selected 11 steps from 0% to 100% area coverage in intervals of 10% for each ink. By combining these steps of four colors, there were 11=14,641 samples. According to the area coverage, corresponding spectral reflectances were calculated by the virtual printer model, tristimulus values calculated for illuminant D50 and the 1931 observer. In order to evaluate the effect of printing gamut size, the tristimulus values were transformed to a more uniform color space, CIE94 corrected CIELAB space. The color gamut was assumed as a convex hull and calculated with the MATLAB built-in Quick hull algorithm and notated as V(Rλ,pred). Rλ,pred was calculated from the virtual printing model. Function V(x) was used to calculate the gamut volume produced by the Lab values in CIE94 corrected CIELAB space. Objective Function for Minimizing the Color Inconstancy Index A computer simulation was developed by the authors to investigate the optimum ink combinations for maximizing color gamut. At that time, color gamut was the only optimization objective. In this research, color inconstancy was used as the optimization objective and color gamut was used as a constrained condition. The optimization objective is minimizing the color inconstancy of prints between two illuminations by investigating optimum ink reflectance spectra. The objective function is expressed in Eq. 2. I = CII(Rλ,pred,i) i=1 n Σ