Geodesic Based Ink Separation for Spectral Printing

An ink separation algorithm is introduced for printing with 6 to 9 inks. A spectral gamut mapping algorithm is also introduced that projects an input reflectance onto the manifold of the printer spectral gamut space The ink separation, which is finding the best ink combination to reproduce a given reflectance, is done by applying an interpolation between printer gamut points neighboring a projected point point’s geodesic location. The technique finds the best manifold projection using ISOMAP. The algorithm searches for the lowest dimensionality that holds the spectral information accurately. Using this method we were able to find a good ink combination given an input reflectance for both a 6-ink and 9-ink printer model. Introduction In comparison to standard color printing, spectral printing aims to reproduce a given reflectance spectrum rather than produce a metameric reflectance spectrum that simply matches a given color. Spectral printing aims to reduce a problem that can arise in metameric color printing which is that the reproduced color may match under one illuminant, but not match well under some other illuminant. Clearly, if the printed output reflectance matches the input reflectance, the printed color will match the input color under all illuminants. Spectral printing requires a significantly larger number of inks than the standard CMYK ones, but this increases the computational complexity of printing algorithms in terms of both time and space. In particular, standard gamut-mapping algorithms map colors within a 3-dimensional space. Generally, their computational complexity increases rapidly with dimension, so that they become intractable for the gamut-mapping of spectra represented in, say, 11 dimensions. For example, a gamut-mapping algorithm that relies on the computation of the convex hull of the measured gamut will not work, since computing a d-dimensional convex hull of n points requires order O(n**floor(d/2)+1) operations. Bakke et al. [7] address this problem by reducing the dimensionality via principal components analysis and then computing the convex hull in up to 8 dimensions. The first part of this paper introduces an ink separation method based on spectral data. This method uses interpolation in geodesic locations to find the best ink combination to match a spectral reflectance. McIntosh et al. [12] have previously applied the idea of interpolating over geodesic distances rather than Euclidean distances to improve an image segmentation algorithm. The second part of the paper a spectral gamut mapping algorithm is introduced based on manifold projection. The last part of the paper evaluates performance of the two models against some of the existing approaches. The result is presented in both Root Mean Squared of the spectral reproduction and ∆E94 under 11 different illuminations. ISOMAP and Multidimensional Scaling ISOMAP [2] is a nonlinear generalization of classical Multidimensional Scaling (MDS) [1]. MDS maps the input data to a lower dimensional space, subject to the constraint that pairwise distances between data points are preserved as much as possible. The main idea of ISOMAP is to perform MDS, not on the input space distances, but on the geodesic distances between points on the data manifold. The geodesic distances represent the shortest paths along the curved surface of the manifold. This can be approximated by a sequence of short steps between neighboring sample points. ISOMAP then applies MDS to the geodesic, rather than straight line, distances to find a low-dimensional mapping that preserves these pairwise distances. Thin Plate Spline Interpolation As is typical of interpolation methods, thin-plate spline (TPS) interpolation [10] constructs a function f that matches a given set of data values yi, corresponding to a given set of data vectors Xi = [Xi,1, Xi,2, ... Xi,D] in the sense that yi = f(Xi). Xiong et al. extended the TPS model to N-dimensions and applied it to illumination estimation successfully [9]. For the spectral printing process, in this paper TPS is used to find a continuous function that maps between the set of inks and each of the output dimensions. For instance if the output spectral reflectance of an 8-ink printer is measured from 380nm to 730nm with a 10 nm sampling, TPS is used to create 36 separate functions mapping from the 8 input dimensions to each reflectance wavelength 380nm, 390nm, .... to 730nm individually Geodesic Location and Ink Separation Interpolation is a common approach to ink separation, and we use interpolation here. In general, an ink combination is interpolated as a weighted combination of ink formulas from nearby experimentally measured data points. The weights typically are derived from the distance of the point to be interpolated from its neighbors. The distance metric can be defined in many different ways. For instance, the distance between two spectral reflectances can be measured as the Euclidean distance between them. We propose interpolation based on geodesic distances on the gamut manifold. Many spaces appear to have a high dimensionality in a linear space, but actually have lower intrinsic dimensionality. A good example is the Swiss Roll example of Tenenbaum et al. shown in Figure 1, along with its projection into 2 dimensions shown in Figure 2. Clearly, the geodesic distances on the 2D manifold are not the same as the direct Euclidean distances in 3-space. 16th Color Imaging Conference Final Program and Proceedings 67 Figure 1: Swiss Roll representation in 3 dimensions Figure 2: Un-folded Swiss Roll data into 2 dimensions using ISOMAP The proposed ink separation method uses the geodesic distances between data points. The algorithm is as follows: 1. Given a training set of points (reflectances of print samples), the geodesic distances between the input reflectance (the reflectance to be printed) and all training points in the gamut are calculated 2. The geodesic distances are used in MDS to calculate the point locations in a space of lower dimension. 3. Thin Plate Spline interpolation is used based on the data point locations in the new space Steps 1 and 2 are part of the standard ISOMAP algorithm [2]. ISOMAP makes the assumption that the Euclidean distances to points within the local neighborhood of a given point P approximate the corresponding geodesic distances. The geodesic distances to a point Q outside the local neighborhood is calculated as the sum the distances between neighboring points along the shortest path from P to Q. Spectral Gamut Mapping based on Manifold Projection Bastani et al. [6] showed that almost 99% of the scene reflectances in spectral space fall outside of the printer gamut even if the printer ink sensitivity is optimized and we use as many as 12 inks. This means spectral gamut mapping becomes an important part of spectral reproduction work. In this section we present a spectral gamut mapping algorithm based on manifold projection. The steps are as follows: 1. Given a printer gamut space, calculate the data point geodesic locations using ISOMAP 2. Transfer the input spectral reflectance using the same transformation 3. After the transformation, gamut mapping is then applied in the transformed space, which tends to have a lower dimensionality.