Sparse Cellular Neugebauer Model for N-ink Printers

The number of colorants used in printers is growing to improve quality of the prints in terms of gamut size, grain and metamerism. In this paper we demonstrate a modification to the existing models (Cellular Neugebauer Model, with Yule Nielson Correction) to simulate spectral reflectance of an N-ink printer. This is achieved by imposing a constraint for the media ink limit, the modified model is shown to reduce the required number of training points by as much as 97 percent. Imposing this constraint leaves us with sparse training data set and the modified Yule-Nielson Neugebauer Spectral model is shown to handle the sparse data well with average error of 1.48 ΔE94. In this paper, two standard dot placement models (Dot-on-dot v.s. Demichel) are implemented and compared. Introduction Color printers with low number of inks (4 or 6) can be metameric and have limited color gamut. Previous work done by Kohler and Berns 0 and Tzeng [2] and Tzeng and Berns [1] used a larger number of inks to minimize metamerism. In these papers it was shown that reflectance of an ink combination could be predicted accurately using a modified Yule-Nielsen Spectral Cellular Neugebauer (YNSCN) model. One of the main challenges with the Yule-Nielsen Neugebauer model is that the computational complexity grows exponentially with number of inks. One approach by Taplin and Berns [3] was to developed a color reproduction model for 6 color inkjet by combining 10 4-color Neugebauer models. In this paper we introduce a modified YNSCN model developed to model an N-color printer. Its computational complexity is compared on an 8 ink printer. By imposing a media ink limit constraint, the modified YNSCN model is shown to reduce the required number of training points by as much as 97 percent. In this paper, two standard dot placement models (Dot-on-dot and Demichel) are implemented and compared. Also the effect of linearizing the sampling space based on CIELAB color variation and spectral variation is studied. When applying the proposed ink limit constraint to the printer model it will result in the reduction of the number of training points, this reduction leaves the model with a sparse matrix of training data. The modified YNSCN model is shown to handle the sparse data well with average error of 1.48 ∆E Yule-Nielsen Spectral Modified Neugebauer Equation The Cellular Neugebauer Model is essentially an interpolation within a set of test patches that cover the printable colorant space. The interpolation is based on the neighboring points of the hypercube surrounding any requested index, and is generally based on one of two standard dot placement models (Demichel and Doton-Dot), both of which have equations that can be extended to ndimensions. In this paper, the YNSCN model is used and modified to handle large data of ink combinations. YNSN model is defined below for an N-color: n i n iR W R ] [ ~ , / 1 λ λ Σ = (1) Where the weights Wi are computed using the Demichel or dot-ondot equations, n is the empirical value for Yule-Nielsen model to correct for physical and optical dot gain, and Rλ,I is the spectral reflectance of the ith primary. For more information we would like to refer the reader to reference [13]. It has been shown that in order to improve Neugebauer model performance, one can provide more primaries within printer gamut. This approach is called Cellular Neugebauer, which was shown by Heuberger [8] and others [4],[9]. Reducing Complexity of Cellular Neugebauer Model The Cellular Neugebauer model is based on sampling a subdivision of the Neugebauer primaries. For instance, if we have N inks with K samples along each of primary axis, we must print and measure K training patches to use in the model. This means the number of patches grows exponentially with the number of inks used in the printer. For instance for an 8 ink printer, to sample only 4 samples along primaries, would require 4 = 65,536 patches to print and measure for use in the interpolation. To reduce the complexity of the model, we use two concepts: First, the printed patches are subject to ink limiting so only a small number of the potential patches actually need to be printed. Second, linearization of each ink before printing the test patches to keep the Neugebauer cells of a uniformly spaced and further reduce the necessary number of steps per colorant. Linearizing Training Patches In essence the Cellular Neugebauer model is a piecewise linear model, and the Yule-Nielsen correction reduces nonlinearity related to dot gain, but does not capture all of the possible curvature caused by ink interactions, etc. Increasing the sampling size (printing more patches in each dimension) reduces the need for the YN correction. If cells are not printed using uniform perceptual steps with each ink, then the perceived cell spacing at one end of the space can be much larger than the other – and lead to a large color error. By printing patches that are linearly spaced in a perceptual color space such as CIELAB or CIECAM02, one can keep the perceptual spacing of interpolation cells roughly equal. 220 Copyright 2006 Society for Imaging Science and Technology Ink Limiting Printing substrates are commonly faced with a certain ink limit beyond which the page is too saturated to print – in the inkjet realm this leads to issues such as cockle, bleed, dry time and gloss uniformity. It is not reasonable to print and measure patches that violate the ink limit of the substrate media. This observation significantly reduces the complexity of the problem. By imposing the constraint that we do not need to print or measure patches that violate the ink limit, the number of data points to measure for the model can be reduced by up to of 97%. In an inkjet printer, the maximum dispensable weight-per-unit-area W for each colorant i is defined by factors such as drop size, nozzles per inch, and number of passes. This value varies for each ink, and is generally between 50% and 100% of the overall media ink limit. Define the "percent under limit" for each ink i as Ui = Wi/InkLimit) for each ink. Given the number of colorants k, the number of steps per colorant n, and the percent under limit Ui we can compute the complexity subject to the ink-limit constraint. For simplicity, assume that U = min(Ui) for all inks, which will error on the side of over-estimating the complexity. First, consider the case where U = 1. This is the case where the printer is capable of delivering exactly the media ink limit with each ink individually. In two dimensions, the valid sample space is a triangle defined by (0,0), (n,0), (0,n) as shown in Figure 1. In general, the space of printable patches can be represented by a k-simplex defined by the origin and the points along each colorant axis at a distance of n. The area of such a region is: Figure 1: Valid Patches for 2 inks with U=1