Weighted LabPQR Interim Connection Space Based on Human Color Vision for Spectral Color Reproduction

1 School of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology, Shanghai 200093, China 2 College of Communication and Art Design, University of Shanghai for Science and Technology, Shanghai 200093, China 3Department of Printing and Packaging Engineering, Shanghai Publishing and Printing College, Shanghai 200093, China 4Department of Materials and Chemical Engineering, Henan Institute of Engineering, Zhengzhou 450007, China


Introduction
The spectral reflectance is define as an object "fingerprinting" that accurately carry the fundamental color information, so spectral color reproduction could match originals under arbitrary illuminants and observers [1][2][3].However, a highdimensional spectral data need large storage space and computational complexity.In addition, the raw spectral data is not suited for spectral image processing, gamut boundary description, and spectral gamut mapping in the spectral color reproduction in many applications such as textile color, art reproduction, and printers with inks [3][4][5][6].So it is necessary to construct an interim connection space (ICS) both for spectral color representation and reproduction.
In the quest for an optimal spectral color reproduction, an impressive number of ICSs have been proposed in the literature, and ICSs are classified into two types of categories.The first type is the algorithm that applies multivariate statistical analysis theory to optimize spectral color information.Bakke et al. [7] proposed PCA-based ICS, which applied principal component analysis (PCA) on dimensionality reduction of the spectra and spectral reconstruction.Zhang et al. [8] proposed two ICSs called ICS 2SI and ICS 3SI, which applied PCA on extracting the widely used illuminants and light sources.These ICSs have the higher spectral and colorimetric representing accuracy and no obvious illuminants selective property.However, these ICSs have not effectively gamut boundary, gamut mapping algorithms, and spectral gamut visualization.In addition, ICSs are not compatible with the widely used colorimetric management system and have no actual physical meaning.
The second type called compensation approach reinserts metameric black spectrum to compensate for the loss of spectral details caused by the colorimetric tristimulus values.Derhak and Rosen [9] had proposed an ICS called LabPQR with six dimensions.The first three dimensions were CIELAB values under a specific viewing condition, and the additional dimensions were spectral reconstruction dimensions describing a metameric black (PQR) with PCA.As a matter of fact, the LabPQR ICS could well represent the spectral information with low dimension, and most of the proposed spectral gamuts mapping algorithms in LabPQR were based on the optimal method [8].Several variations of the LabPQR had been proposed recently, such as the LabRGB color space [10] and XYZLMS color space [3].These ICSs could be better to be compatible with the widely used colorimetric 2 Journal of Spectroscopy management system and spectral gamut visualization.In addition, LabPQR have effectively gamut boundary and gamut mapping algorithms [11].It had reported that LabPQR could be used successfully in several applications [12,13].However, these ICSs have obvious illuminants dependence property.In additional, the LabPQR ICS is with high spectral but low colorimetric representing accuracy.The main reason is that PQR dimensions of LabPQR apply PCA to extract the residual error of the spectral reflectance, and are not weighted by the human color vision that usually has different sensitivities over different wavelengths.
This paper presents a weighted LabPQR ICS based on human color vision for the spectral color reproduction, which is the desire to achieve a visually more spectral color mapping between reproductions and originals.To define the most optimal weight function, three different weight functions were tested.These weighted LabPQR (wLabPQR) ICSs were evaluated under different illuminants and light sources compared with the traditional, nonweighted LabPQR ICS.

Mathematical Background
2.1.LabPQR [9,[11][12][13].The LabPQR is a six-dimensional ICS.The first three dimensions are CIELAB values under a specific condition, and the additional dimensions are spectral reconstruction dimensions (PQR).The PQR coordinates represent the spectral difference between original and reconstructed spectra from colorimetric values.
Spectral reconstruction from six-dimensional LabPQR could be determined as where T is a  by 3-transformation matrix, V is a  by 3 matrix describing PQR bases, N  is a tristimulus vector, N  is a vector of PQR values, and  counts wavelength .Note that T is applied to tristimulus values converted from CIELAB values.Using a set of the tristimulus vectors, the transformation matrix T is determined by a matrix calculation using the least square analysis: where R is measured spectral reflectance of training samples; the superscript "" shows the pseudoinverse of the proposed matrix.
The PQR bases V are derived from principal component analysis (PCA) on a set of spectral differences between the original spectra and the reconstructed spectra through the inverse transformation with T from N  .This spectral difference is expressed as Only the first three eigenvectors are preserved as the PQR bases: where k  are eigenvectors in a set of the spectral difference.

Weighted LabPQR (wLabPQR).
It is clear that the PCA is the well-known linear model that equally treats spectral reflectance over the whole wavelength, which could not well represent the characteristic of color information [14,15].The purpose of wPCA is to improve the color reproduction accuracy at the cost of the accuracy of spectral reconstruction in color technology and science [16,17].
In the wPCA, it is noted that, before calculating the correlation matrix C  , each sample point requires being multiplied or weighted with proper coefficients or a weight function (), resulting in weighted data E  , where the matrix W is a diagonal matrix with main diagonal of the values of the weigh function ().The superscript "+" is the matrix transpose.After reproduction, this same weight function can be separated from the spectral data to achieve representatives of the original spectral curves [17].Consider 2.3.Weight Function.The main goal of the wPCA algorithm is to minimize the weighted squared reconstruction error [18]: To evaluate the performance of the weight functions, three weight functions were selected.The weight functions are not limited, but the color-matching functions well reflect human vision characteristics.Three weight functions plotted in Figure 1 are connected with CIE1931 (), (), and () color-matching function that involves brightness information and chromatic information.
The first weight function  1 () (WF1) was introduced by Tian and Tang [16], which is generated by adding the three matching functions and normalizing the maximum of value to be 1: The second weight function  2 () (WF2) is generated by adding the three matching functions to the constant functions, which normalize the maximum of value to be 1.The WF2 was introduced by Laamanen et al. [17].Consider because values of CIE1931 (), (), and () colormatching function are not less than zero.According to (7), we proposed the third weight function  3 () (WF3) generated by calculating the square root of adding the three matching functions, which normalize the maximum of value to be 1:

Experiments and Procedure
To evaluate the performance of nonweighted LabPQR (ICS NW) and wLabPQR with three different weight functions (ICS WF1, ICS WF2, and ICS WF3), all the four ICS were constructed, respectively.The CIE1931 2 ∘ standard observer was adopted for all four ICSs.The CIE standard illuminant D65 was employed to construct the first three dimensions of the four ICSs.The spectral reflectance of Munsell (1269 chips) and spectral image (fruits and flowers) [19] were adopted to construct and assess the performance of the ICSs.For most applications, these continuous functions could be sampled at 10 nm intervals without a signification loss of accuracy [20].So, all the spectrums, light sources, and illuminants were sampled at 10 nm intervals between 400 and 700 nm.After that, all the parameters of the four ICSs were determined; the spectrums of the testing samples were converted to the four ICSs and then transformed back to spectral reflectance.
The color difference is a more powerful tool to access the performance of ICS as the eventual criterion for evaluating the spectral color reproduction accuracy is the human color vision under various illuminating environments [8].Therefore, the CIELAB color differences between the original and reconstructed spectrum of Munsell and spectral image testing samples are calculated under the CIE standard illuminants, light emitting diode (LED) light sources, and tungsten halogen (TH) light source.The dominant spectral power distributions (SPDs) of the selected illuminants and light sources were relatively smooth and with various distributions, which did not contain much spiky radiance.[21].The root mean square error (RMSE) and goodness of fit coefficient (GFC) were employed to evaluate the spectral difference between the original and reconstructed testing samples [22].The smaller the RMSE the closer the original spectrum, while the GFC is just the reverse.

Results and Discussion
The performance and feasibility of wLabPQR ICSs were tested by odd and even chips of Munsell as training and testing samples, respectively, comparing the results with the nonweighted LabPQR.The CIELAB color differences between the original and reconstructed spectrum of Munsell testing samples were calculated under various illuminants and light sources.The statistical results are shown in Table 1.It illustrated that wLabPQR ICSs results outperform nonweighted LabPQR ICS and ICS WF3 performs best according to the experimental result.The mean color difference, the maximum color difference, and the percentage of testing samples with color differences greater than 3 of ICS WF1 and ICS WF2 are not much numerical difference, but these are a much smaller than that of ICS NW.It indicated that the ICS NW has obviously illuminant selective phenomenon.The last row of Table 1 shows the variance of the color difference statistics under all the illuminants and light sources of the four ICSs.The variances could represent the robustness of the four ICSs: the smaller the variance, the more robust the performance of the ICS under various illuminants and light sources [8].The variance results indicate that ICS WF3 is with the highest robustness and then are ICS WF2 and ICS WF1.The LabPQR is with relatively low robustness.
The RMSE and GFC statistics between the original and reconstructed reflectance of Munsell testing samples are shown in Table 2.It is shown that ICS NW performed best and next is ICS WF2 and ICS WF3; the ICS WF1 performs worst according to the experiment result.Similar results were shown by Laamanen et al. [17]: weighting clearly improves the reproduction of color information, but at the cost of the reproduction of spectral reflectance curve.The reason is that the higher the weight function numerical value, the lower the spectral reproduction error.Figure 2 shows an example of spectral reconstructions of one Munsell spectrum formed with four ICSs.From the results it can be concluded that the middle part of the weighted spectrum is reconstructed more accurately than the nonweighted spectrum, but both ends of the spectrum are quite the contrary.The reason can be found from the curve shape of the weight function shown in Figure 1.
To access the medium dependence of the ICS, the colorimetric and spectral representing accuracy of the four ICSs were also evaluated with spectral image as testing samples.All the spectral reflectance of spectral image was transformed into the four ICSs and then transformed back to spectral reflectance with the Munsell chips as training   samples.The statistics of color and spectral difference under all the illuminants and light sources between the original and reconstructed spectrum are illustrated in Tables 3 and  4, respectively.It indicated that the colorimetric and the spectral representing accuracy of spectral image are lower than Munsell testing samples as a whole for all four ICSs.As results presented, the comparative results in the spectral and colorimetric representing accuracy of the four ICSs for the spectral image have similar trends with the Munsell testing samples.The wLabPQR clearly improved the color reproduction accuracy comparing to the nonweighted LabPQR, even though it decreased slightly the spectral reproduction accuracy.In addition, the robust performance of the ICS WF1 is relatively higher than that of ICS WF2 under various illuminants and light sources.
The overall performances of the four ICSs are shown Table 5.It illustrated that the performance orders of the four ICSs are ICS WF3, ICS WF1, ICS WF2, and ICS NW according to the experimental results.The ICS NW is with high spectral but low colorimetric representing accuracy and robustness, and it is not suitable for spectral color reproduction, while the wLabPQR is with low spectral but high colorimetric representing accuracy and robustness; ICS WF3 is especially most suitable for spectral color reproduction at low dimension.

Conclusion
In this paper, the weighted LabPQR method was presented for spectral color reproduction.The method is largely based on nonweighted LabPQR, but it differs from the normal LabPQR in that PQR dimensions of LabPQR are weighted by the human color vision.This is done to retain more color information rather than spectral information in spectral color reproduction.
To evaluate the performance and feasibility of the weighed LabPRQ, PQR dimensions are weighted by three different weight functions based on human color vision composed of the three ICSs (ICS WF1, ICS WF2, and ICS WF3).The reflectance of the color chips of Munsell and spectral image was employed as the samples in this study.As results presented, weighting obviously improves the colorimetric representing accuracy and robustness, but at the cost of the spectral representing accuracy.However, the weighted LabPQR ICSs achieve more accurate reconstruction at higher human eye sensitive wavelengths that retain an amount of human color vision information.These results further imply that weight function WF3 proposed in this study outperformed the other two weight functions, and ICS WF3 is most suitable for spectral color reproduction.

Figure 2 :
Figure 2: Example of the spectral reconstructions of one Munsell spectrum.

Table 1 :
Colorimetric representing accuracy comparison of the four ICSs with odd and even chips of Munsell as training and testing sample, respectively.>3 means the percentage of testing samples with color differences greater than 3 CIELAB units.
a Mean means the mean value of testing samples with color differences.b Max means the maximal value of testing samples with color differences.c %

Table 2 :
Spectral representing accuracy comparison of the four ICSs with odd and even chips of Munsell as training and testing samples, respectively.

Table 3 :
Colorimetric representing accuracy comparison of the four ICSs with chips of Munsell and spectral image as training and testing sample, respectively.

Table 4 :
Spectral representing accuracy comparison of the four ICSs with chips of Munsell and spectral image as training and testing sample, respectively.

Table 5 :
Overall performance of the four ICSs.