Spectral Reproduction using LabPQR : Inverting the Fractional-Area-Coverage-to-Spectra Relationship

The tuning of parameters used in inverting the spectral characterization of a six-color inkjet printer was performed. This approach was necessary for building lookup tables for use in spectral color management. First, spectra were converted to a low-dimensional analog known as an Interim Connection Space (ICS). LabPQR was used as the ICS. LabPQR was defined with three colorimetric dimensions (L*a*b*) plus three dimensions describing a metameric black (PQR). Once converted to ICS units, the spectral characterization related printed fractional area coverages to LabPQR. The inversion process minimized the weighted sum of CIEDE2000 and a Euclidian distance in PQR coordinates. A weight series was performed to find the optimal trade-off between the colorimetric and spectral error. A 1:50 weighting ratio, CIEDE2000 to PQR difference, was deemed best. Introduction An important goal of spectral color management is to reproduce images that match originals under arbitrary illuminants. Spectral reproduction requires new approaches including spectral profiling of devices, Spectral Profile Connection Spaces, spectral image processing and new quality metrics. Spectral color management will take advantage of all these concepts and require transformation chains that deliver high-quality results quickly. In previous work [1]-[3], a spectral reproduction workflow from scene to hardcopy was proposed. One of the difficulties associated with spectral reproduction is its high dimensionality since more information is necessary for reproducing samples with illuminant-independence than needed for more traditional colorimetric reproduction. The proposed workflow included a step where spectra of high dimensionality were converted to a lowerdimensional encoding known as an Interim Connection Space (ICS) [3][4]. Recently, Derhak and Rosen proposed an ICS called LabPQR [5]-[7]. LabPQR is a six-dimensional ICS that has three colorimetric axes (L*a*b*) plus three spectral reconstruction axes (PQR). PQR describes a stimulus’ metameric black [8]. The spectral characterization of a printer [9] yields the forward relationship from fractional area coverage to spectra. Unfortunately, spectra are typically 31 or more dimensional values. For the purposes of spectral color management, the spectra are then converted to the lower-dimensional ICS, in this case LabPQR. An inversion of the printer characterization is necessary so fractional area coverages can be chosen for a requested spectrum. Spectral gamut mapping [5]-[7] is necessary when considering the problem of spectral color management because an answer must be delivered for any arbitrary spectral request. How to choose appropriate printer values for an out-of-spectral-gamut request is considered in this paper. Theory LabPQR LabPQR [5]-[7] is a six-dimensional ICS. The first three dimensions are CIELAB values under a particular viewing condition, and the last three are spectral reconstruction dimensions describing a metameric black (PQR). The reconstituted spectra from LabPQR is expressed as: p c VN TN R + = ˆ , (1) where T is a n by 3 transformation matrix, V is a n by 3 matrix describing PQR bases, c N is a 3 by 1 tristimulus vector, and p N is a 3 by 1 vector of PQR values (n counts wavelengths). Note that T is applied to tristimulus values converted from CIELAB values. Using a set of the tristimulus vectors, T is determined by a matrix calculation using least square analysis: ( ) 1 , , , − = T m c m c T m c N N RN T , (2) where R is a n by m matrix of spectra of m training samples. The PQR bases V is derived from Principal Component Analysis (PCA) on a set of spectral differences between the original spectra and the reconstructed spectra through an inverse transformation with T from m c , N . This spectral difference is expressed as: m c, TN R E − = . (3) Only the first three eigenvectors are preserved as the PRQ bases: ) , , ( 3 2 1 v v v V = , (4) where i v are eigenvectors in a set of the spectral difference. Spectral Gamut Mapping Spectral gamut mapping has two aspects to it: colorimetric and spectral. In the current approach, the two are combined and considered simultaneously. Fractional area coverages of an inkjet printer for arbitrary requested spectra are computed by minimizing a single objective function: the weighted sum of CIEDE2000 color difference and normalized Euclidian distance in PQR, defined as: ) IEDE2000 Minimize(C ObjFunc1 n PQR k ∆ + = , (5) where n is the number of samples in a spectrum (i.e., wavelengths) and k is a weighting that may be empirically fitted. Eq. (5) can be globally utilized regardless of whether the requested stimuli are within the colorimetric or spectral response gamuts. Tuning the magnitude of k allows one to choose an optimal trade-off between the colorimetric and spectral error. Choosing smaller values of k increases the relative importance of the colorimetric matching. Eq. (5) is equivalent to minimizing spectral RMS error if the requested stimuli are within the colorimetric response gamut, because the Euclidian distance in PQR between a metameric pair is proportional to spectral RMS error. To show this, let 1 R̂ and 2 R̂ denote reconstructed spectra of a metameric pair with the identical tristimulus values *c N . From Eq.