Open Access
Add to BibSonomy
Abstract
We have developed a new method for selecting the test color sample set (TCSS) used to calculate CIE 2017 color fidelity index (CIE-Rf). Taking a Large Set as a starting point, a new optimized color sample set (OCSS) is obtained by clustering analysis. Taking metamerism phenomenon into account, spectra clustering is performed within the class obtained from color appearance attributes clustering. The CIE-Rf of 1202 light sources are calculated and analyzed by taking the Large Set, OCSS and CIE color evaluation sample set (CIE CESS-99) as TCSS. Through analyzing CIE-Rf, the performance of the OCSS is further investigated. The results show that the clustering analysis method developed in this paper can be well used in selecting test color samples, and the obtained OCSS can represent Large Set well and be better used for color fidelity metrics of light sources.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The CIE general color rendering index (CIE-Ra) is used to assess the ability of white light sources to render colors. To reduce misinterpretation by users and make a clear distinction with the other aspects of color quality, the color rendering index (CRI) is renamed color fidelity index (CFI) [1]. CFI can quantitatively evaluate the color fidelity quality of a light source by calculating the color difference of test color samples [2]. Therefore, test color sample set (TCSS) is undoubtedly considered as a very important dimension when calculating CFI. The number and color of sample set will affect the calculation speed and accuracy, respectively [3]. In the existing calculation models of CFI, the TCSS is selected according to one or two of the principles of the color appearance distribution uniformity, spectral uniformity, mathematical simulation efficiency, color constancy and metamerism [4–6].
The 14 color samples of CIE-Ra are commonly used for assessing the color rendering capabilities of artificial light sources [7] and they are named as CRI-14 in this paper. Limitations of CRI-14 have been extensively documented, as a result, there are many past efforts to develop sample set for evaluating light sources’ color rendition [6–10]. Another color rendering evaluation method CRI2012 uses two TCSS, named HL17 and Real set, to calculate Ra,2012 [1]. The HL17 is a mathematically derived set containing 17 samples. And the Real set consists of 210 real reflectance functions. For the color fidelity index IES Rf, the color evaluation sample set (IES CESS-99) contains of 99 color samples which are derived from a Large Set [1]. The Large Set, a large collection of reflectance data, contains about 105,862 reflectance spectra from various types of objects: flowers and other natural objects, skin tones, textiles, paints, plastics, printed materials, and color systems. These objects are representatives of materials found in interiors and nature; they also possess a variety of surface finishes (some are glossy, purely diffuse). After that, CIE 2017 color fidelity index (CIE-Rf) optimizes the partial wavelengths’ spectrum data of the 99 color samples (CIE CESS-99) [11].
Using a very large number of test samples has the drawback of increasing the calculation time of the index. Although such calculations can be performed in a matter of seconds in the modern computer area, nonetheless this might present a problem for many lamp manufacturers who need to do such calculations for every white light source they produce. It is therefore desirable to reduce the number of samples to a minimum. In addition, even if the calculation time would not be an issue, a large number of samples does not necessarily imply that the calculation result of CFIs are accurate, as many reflectance samples are not independent due to the limited number of common dyes used to produce them. Thus, asn ideal test sample set would therefore contain only a limited number of samples, thereby reducing computation time [2].
In this paper, we have developed a new method in selecting test color samples for color fidelity metric of light sources. Firstly, we review the previous relevant research, especially the selection of the TCSS. CIE-Rf and its color sample set CIE CESS-99 are relatively advanced research at present. Therefore, we decided to optimize the color samples starting from a set of 105862 measured real spectra, that is, the Large Set used for deriving CIE CESS-99. Secondly, considering the metamerism, Q clustering is first performed on the color appearance attribute values to obtain the initial clustering categories. AP clustering is then performed on the reduced-dimensional spectrum of the color samples in each initial clustering category. The clustering centers of AP make up OCSS. Thirdly, by taking Large Set, OCSS and CIE CESS-99 as TCSS, CIE-Rf of 1202 light sources is calculated. Finally, the performance of OCSS is investigated by analyzing CIE-Rf calculated by the three color sample sets.
2. Review of the previous work
2.1 CIE CRI-14
The CIE-Ra is an official evaluation index of light source color rendering in the technical documentation of CIE in 1974. As extensively reported, such a measure has many limitations including the shortage of the color samples [7]. CIE-Ra is calculated by 14 specific colors which are from the Munsell color system, and induce more sensitivity to some wavelengths than others. The first 8 specified color samples are used to calculate the general CIE-Ra. They have the same brightness and different tones in the CIE1964U* V* W* uniform color space. The last 6 samples, which contain of the saturated color, green leaves and skin color, are used to calculate the special CRI Ri.
2.2 HL17 and Real set
The CRI2012 metric is proposed by Smet et al. in 2012 [1]. Some improvements have been made to CIE-Ra including the improved sample set. These improvements are designed to make experimental data more consistent with human color perception. The TCSS of CRI2012 includes HL17 and Real set [12,13].
The HL17 is obtained based on mathematical functions and has a minimum change of spectral sensitivity across spectrum. The Real set provides additional information on the expected color shifts when changing illumination. Both HL17 and Real set use about 100000 reflectance samples, accumulated by the University of Leeds, as a starting point.
The ideal 1000 wavelength-shifted spectrum set with fine uniform spectral sensitivity is created by using a Monte Carlo approach. It is used for generating simulated spectra and adapting the Leeds spectra when generating HL17 set. Then CRI2012 designs a set of reflectance spectra with a fairly smooth spectral feature that shifts through the spectrum. By using a series of mathematical methods, the optimal sample set HL17 that emulates the 1000 wavelength-shifted sample set is obtained.
As for the Real set, the 100000 samples are firstly sampled in CAM02-UCS space according to the lightness, chroma and hue. The color difference is set by comparing the color constancy of the metameric samples under different light sources, and then 180 samples are obtained by sampling and filling. Five representative skin tones of major ethnic groups are selected four times. In addition, ten reflectance spectra representing typical fine art paints are added as well, bringing the total number of samples to 210 [2].
2.3 CIE CESS-99
The CIE CESS-99 is consistent with IES CESS-99 [10]. IES CESS-99 is selected from Large Set which is introduced in section one. Firstly, considering the validity range of color error formulas, IES conservatively chooses the samples in the NCS gamut. Then IES partitions the (J’, a’, b’) color space into cubic pixels and keeps only one of the metameric samples in each pixel. The selection procedure yields a set of real samples that uniformly distribute in color space. Later, IES introduces a flatness figure of merit (F) to achieve the spectral uniformity. The Final set containing 99 color samples is built by iterative processing [14].
In this paper, as mentioned in section one, the OCSS is selected from a set of measured real spectra, that is, the Large Set used for deriving CIE CESS-99. Accordingly, we choose CIE-Rf to evaluate the color fidelity of the light sources and the performance of OCSS.
3. The optimizing selection of OCSS based on clustering analysis
3.1 The optimization selection process of OCSS
According to the following postulations and principles, the typical color samples are selected from Large Set and they make up the OCSS.
Firstly, for the selected typical color samples, the uniformity of their distribution in color space should be considered. The sample set should yield sufficient accuracy in condition of uniform coverage to all color dimensions (hue, chroma, lightness).
Secondly, the index performance could be improved by adopting highly-saturated samples. A light source may exhibit good performance for non-saturated samples while perform poorly with saturated samples, especially for RGB (red-green-blue) white LEDs with strong peaks in their spectra. However, the reverse is found not to be the case. Therefore, we separately select highly-saturated samples at the color gamut boundary of the Large Set [1,15].
Thirdly, metameris, color appearance attributes and the spectra information of the TCSS will affect the evaluation results of color fidelity properties. To obtain OCSS, Q and AP clustering analysis are used for the color appearance attributes and spectral characteristics of Large Set’s samples, respectively.
Besides, in order to avoid the influence of non-principal component spectra on spectral difference calculation and reduce the computational complexity, we firstly reduce the dimension of color samples’ spectra.
Based on the above principles, the overall logical structure and technical route of the workflow of the OCSS generation are determined.
The workflow of OCSS generation is given in Fig. 1. It includes three computational steps:
- 1) the calculation of color appearance attributes and classification of color gamut boundary of Large Set's samples. The Large Set are classified as set Ωg and set Ωf, and in order to retain highly saturated samples in OCSS, the same processing is performed on the two set. If sets “g” and “f” are clustered together, the cluster centers may not be the highly saturated sample at the boundary, but mostly from set “f”.
- 2) spectral dimension reduction of the samples.
- 3) clustering analysis of samples’ color appearance attributes and low-dimensional spectra.
- 4) the two typical color sample sets Og and Of make up OCSS directly.
From the spectral data of Large Set's color samples, the color appearance attributes J'a'b’ are calculated. Currently, there are many color appearance models [16–18]. Considering the applicability for this article, we choose CAM16 for which the luminance adaptations to the illuminant are completed in the same space rather than in two different spaces, as in the original CIE CAM02 model. When predicting the color appearance datasets, the CAM16 and CIECAM02 models performed equally well in predicting the lightness data, but the CAM16 model performed better than the CIECAM02 model in predicting the colorfulness and hue composition data. In addition, the CAM16-UCS space predicted these color-difference datasets results at least equal to or better than the CAM02-UCS space [16]. The distribution of Large Set's color samples is shown in Fig. 2.
Highly-saturated color samples have more stringent requirements for the color rendering of the light sources [15]. Therefore, special consideration needs to be given to the effect of highly-saturated samples on the model. The Large Set are classified as highly-saturated sample set Ωg at the color gamut boundary and the set Ωf within the color gamut. We first divide Large Set into different lightness layers according to lightness, and then find the most saturated color sample at different hue angles on each lightness layer, so as to obtain the sample set Ωg at the color gamut boundary. And by removing the samples of Ωg from Large Set, we get sample set Ωf. The detail steps are as follows:
- 1) Divide J’ at equal lightness interval ΔJ’ = n to obtain 100/ n = m different lightness layers. The number of color samples in each lightness layer is different.
- 2) In each lightness layer, the hue angle h (0 °∼360 °) is divided at equal interval Δh = k, where h = arctan (b’/ a’), and 360/ k = l hue regions are obtained.
- 3) In each hue region, the most vivid color samples are selected according to the saturation Eqs. C= (a’2+ b’2)1/2.
In this study, J’a’b’ of Large Set’s 105862 samples are calculated under illumination D65 and CIE 1964 standard observer function. And we consider four cases of combinations of n and k values: n=2, k=3; n=2, k=1; n=1, k=1 and n=1, k=0.5. By the overall process described later in the paper, it is found that as the ratio of the number of saturated samples to the Large set increases, the final optimization advantage increases first and then decreases. And the optimization effect is best when the boundary saturation sample accounts for 23.8%, that is, n and k are both set to 1, and set Ωg containing 24843 boundary highly saturated color samples and Ωf containing 81019 color samples within the color gamut are obtained. In fact, when the hue interval k is small, not all hue regions have color samples.
Since the spectral accuracy of principal component analysis (PCA) is better [19,20], this paper uses this mainstream dimension reduction method PCA. By using PCA, the 81-dimensional spectral data ranging from 380 nm to 780nm at 5nm interval is directly subjected to the first six principal components. For all the samples of Large Set, the contribution rate of each of the first six principal components can be obtained and the cumulative contribution rate exceeds 99.98%.
Clustering analysis is carried out for Ωf and Ωg color sample sets. The clustering analysis of J'a'b’ effectively removed color selection bias in Large Set. Indeed, if many nearly-identical color appearance attributes samples are included in the Large Set, they all contribute to the same color difference and their outsized representation in the sample set does not dominate the final results [14]. So, Q clustering of the Large Set samples’ color appearance attribute J'a'b’ is needed. Considering the metamerism, AP clustering analysis is performed on the dimensionality reduction spectra of the samples within each class of Q cluster results. Their AP clustering centers are selected as the typical color sample sets Of and Og. And they make up the OCSS.
3.2 The clustering analysis of Large Set
One of the commonly used methods of statistical analysis is clustering analysis [21–23]. In statistics and data mining, AP clustering and Q clustering are the algorithms based on the concept of “message passing” or “similarity” between data points [24–26]. Unlike clustering algorithms such as k-means and k-medoids [27–29], AP clustering and Q clustering don’t require clusters number to be determined or estimated before running the algorithm, which is one of their major advantages in clustering analysis. Therefore, this paper mainly uses these two methods.
3.2.1 The Q clustering analysis of color appearance attributes J'a'b’
Q clustering is a kind of clustering mode without the preset of clusters number. There are two parameters to be set in this hierarchical clustering algorithms. One is the measurement mode of the distance between objects. Another is the connection mode between classes [22,30]. We take the Euclidean distance as the similarity matrix and the Ward connection method as the connection rule to measure the connection strength in our algorithm. The steps in detail are as follows:
- 1) Initialize. Set each sample as an independent object.
- 2) Search the two objects with the nearest distance in the data set. Select the top two biggest values in the similarity matrix, until only one object left. In this step, it’s needed to calculate the distance between objects as similarity. For a data set made up of n objects, there are n*(n – 1)/2 pairs in the data set. The result of this computation is commonly known as a distance matrix. As shown in Eq. (1), we calculate the Euclidean distance between objects in this study. Given an n-by-m data matrix X, which is treated as n (1-by-m) row vectors x1, x2, …, xn, the various distances d(i, j) between the vector xi and xj are defined as follows: where the subscript v refers to the vth attribute of the sample.
- 3) Merge. Merge the nearest two objects by the Ward method into a new one and update the similarity matrix at the same time. Ward's linkage uses the incremental sum of squares, that is, the increase in the total within-cluster sum of squares as a result of joining two clusters. The within-cluster sum of squares is defined as the sum of the squares of the distances between all objects in the cluster and the centroid of the cluster. The sum of squares metric is equivalent to the following distance metric d(r, s), which is the Eqs. linkage uses:$$d({r,s} )= \sqrt {\frac{{2{n_s}{n_r}}}{{({{n_s} + {n_r}} )}}} \; \parallel \overline {{x_r}} - {\overline {{x_s}}\parallel_2}$$where || ||2 is the Euclidean distance, $\overline {{x_r}} $and $\overline {{x_s}} $ are the centroids of clusters r and s, nr and ns are the number of the samples in clusters r and s.
After the setting the above appropriate parameter, the data results can be regarded as classified by the distance, that is, the color difference. That’s the reason we use Q clustering analysis to process J'a'b’.
The color samples have been classified into two sets Ωg and Ωf in section 3.1. As shown in Fig. 3, the Q clustering algorithm is performed directly on the color appearance attributes J’a’b’ calculated in color space CAM16-UCS, and we get the initial categories. For the Q clustering, we choose Square Euclidean distance as the distance criteria and Ward's method as the linkage criteria, so that the initial category obtained by Q clustering is actually divided according to the color difference. At the same time, the “evalclusters” function is introduced for the determination of the appropriate clusters number [33]. Finally, Ωg and Ωf are grouped into 12 and 29 categories respectively. They are marked as Θgi (i=1, 2, …, 12) and Θfi (j=1, 2, …, 29) and used as initial sample set for subsequent processing. Taking the saturated sample set Ωg as an example, Fig. 4 shows its results of Q clustering.
As shown in Fig. 4, one color represents a color sample set Θgi whose samples have similar color difference with their cluster center.
3.2.2 The AP clustering analysis of the samples’ spectra of Θgi and Θfi
In Section 3.2.1, all samples of Large Set were classified into 41 categories (Θgi: 12 +Θfi: 29) based on the color appearance attributes J’a’b’. However, the samples with same J’a’b’ may have different spectra, that is, the metamerism phenomenon and this phenomenon needs to be taken into account. Therefore, we perform spectra clustering on each of 41 categories. Because the high-dimensional spectral information is cumbersome and computationally difficult, we performed spectral dimension reduction before spectra clustering analysis.
The AP clustering algorithm proposed by Frey and Dueck is based on neighbor information propagation and it automatically locates all the available cluster centers [23]. The core feature of AP clustering is its sole use of responsibility and availability indicators to decide the probability of a point becoming a cluster center without prior knowledge. The purpose of AP clustering is to produce a similarity clustering model between N samples, where in this paper the negative Euclidean distance squared is selected as the similarity measure function for any two samples,
where S (i, j) is the similarity between point xi and xj. When carrying out AP clustering for spectral principal components, we added contribution rate as weight to calculate S, which is introduced in Eq. (10).The AP clustering uses the responsibility R (i, k) and availability A (i, k) to generate candidate cluster center points. Each iterations of the AP clustering algorithm is the process of alternately updating information between the two parameters R (i, k) and A (i, k). Here R (i, k) is the likelihood of kth point xk to be the cluster center of ith point xi, A (i, k) denotes the suitability of xi, and xk is its cluster center. The detailed calculation steps of the AP clustering algorithm are as follows:
- 1) Initialize the similarity matrix S by the similarity between any two samples. Set up the largest number of iterations tmax.
- 4) Update R (i, k) and A (i, k) of each sample.
Steps (3-4) are utilized to compute the R (i, k) and A (i, k) for each sample. Here lam in (8-9) denotes the damping factor. It is just there for numerical stabilization and can be regarded as a slowly converging learning rate. When updating the messages, it is important to avoid numerical oscillations in some cases. It is advised to choose a damping factor within the range of 0.5 to 1.
- 5) If t is greater than the maximum number of iterations tmax or the model reaches the termination condition, terminate the process. Otherwise, go back to step 2 [34].
As shown in Fig. 5, the same processing is performed on sets Θgi and Θfi. In this section, the spectra dimension of color samples of Θgi and Θfj is reduced from 81 to 6. Each principal component is then multiplied by the corresponding contribution rate. For each initial category Θgi or Θfj, AP clustering analysis is performed on the 6-dimensional spectra and contribution rates. In other words, a total of 41 (12 + 29) AP clustering analysis are performed. Calculating the Euclidean distance when using the contribution rate as the weight can improve the accuracy of the similarity result. The weighted Euclidean distance equation is shown in (10).
After AP clustering analysis is carried out, the AP cluster centers are merged into the typical color sample set Og and Of. As shown in Fig. 6, three initial classes are randomly selected from Θf to show the results of spectra clustering within each class. A solid point represents a typical color sample. They are the cluster centers obtained by performing AP clustering on the spectra within Θf5, Θf9 and Θf11. As can be seen from Fig. 6, when intra-class spectra clustering is performed on Θf5, Θf9 and Θf11, they are divided into three, two and two categories known by the number of solid points. These 8 solid points are part of the optimization color samples. The AP cluster centers of the other 38 initial classes are directly shown in Fig. 7.
Og contains 15 highly-saturated samples and Of contains 80 samples within the boundaries of the three-dimensional color gamut. The three-dimensional distribution of OCSS, which consists of 95 samples, is shown in Fig. 7.
Fig. 5. Workflow of AP clustering of the products of 6-dimensional spectra and contribution coverage.
4. The calculation of CIE-Rf and performance analysis of OCSS
In this section, the Large Set, CIE CESS-99 and OCSS are taken as TCSS, the CIE-Rf of 1202 light sources are calculated. The performance of our proposed color samples and CIE CESS-99 can be compared by calculating the values Absolute Difference (AD), Mean Absolute Difference (MAD), and correlation coefficient of CIE-Rf, as well as doing regression analysis. This paper uses 1202 light sources of various occasions including clothing sales environment, home environment, museum booth, restaurant lighting environment, office environment, backlit billboard and 401 light sources from Ref. [1]. Light source types cover CIE standard illuminant, fluorescent lamps [35], LED lamps [36–40], incandescent lamps, tungsten halogen lamps, and high pressure mercury lamps. Their color temperature ranges from 1634k to 10462 K [41] as shown in Table 1. They are numbered as light source 1, light source 2, light source 3…light source 1202, respectively.
Table 1. Statistics of light source types.
4.1 The calculation of analysis index
A total of 10818 color sample data sets are calculated: 3 kinds of analysis index (CIE-Rf, AD, and residual) *3 kinds of TCSS * 1202 light sources = 10818.
The Large Set is used as reference color sample set. If the indexes of one sample set is closer to Large Set, this set is more representative of Large Set, and it has better performance.
Compared with AD, MAD is an absolutely value and there is no positive or negative phase cancellation. Therefore, MAD can better reflect the actual situation when predicting difference between the data. The calculation equations are shown in (11) and (12),
where n is the number of the light sources; CFIkt is the CFI of the kth light source calculated by taking the OCSS or CIE CESS-99 as the TCSS; CFIkLS is the CFI of the kth light source calculated by taking Large Set as the TCSS.The calculation process is as follows:
- 1) Using Large Set as the TCSS, CIE-Rf of 1202 light sources is calculated and recorded as Rf -LS;
- 2) Using OCSS and CIE CESS-99 as the TCSS, CIE-Rf is calculated and recorded as Rf -95 and Rf -99, respectively;
- 3) Using Rf -LS as reference data, AD and MAD between Rf -95, Rf -99 and Rf -LS are calculated respectively;
- 4) The performance of the color sample set with smaller MAD is better in evaluating the color fidelity of light sources.
4.2 The calculation results and analysis
The calculation results of the AD between Rf-95, Rf -99 and the reference data set Rf -LS are illustrated in Fig. 8. The MAD is shown in the Table 2.
Table 2. The MAD, ADmax, ADmin and Variance of CIE -Rf.
Seen from the scatter plot Fig. 8, the AD between Rf -95 and Rf -LS is smaller than that between Rf -99 and Rf -LS. Considering 1202 light sources as a whole, we can see in Table 2, MAD calculated by OCSS is an order of magnitude smaller than CIE CESS-99 (0.32 VS 3.74). The maximum difference between Rf -95 and Rf -LS is only 3.40, which is about a quarter of 12.83. The smaller the variance of AD, the lower the deviation of the data from the MAD. It can be found that Rf -95’s stability is better. Regardless of which set of data in Table 2, Rf -95 is closer to Rf -LS. That is, OCSS is more representative of the Large Set when evaluating the fidelity of the light sources.
The two sets of data Rf -95 and Rf -99 are also linearly regressed. As an illustration of CIE-Rf calculations, Fig. 9 shows the correlation between Rf -95/ Rf -99 and Rf -LS for the 1202 Spectral Power Distributions (SPDs). Ideally, all data points are spread over a straight line that passes the origin and has a slope of 1. That is, the value of the ordinate is equal to the abscissa, indicating that CIE-Rf calculated by the sample set represented by the ordinate is equivalent to the one by Large Set.
From the point of dispersion, the closeness of the linear regression line and the solid line can be used to visually characterize the correlation between Rf -95/ Rf -99 and Rf -LS in Fig. 9. We can see that the regression line obtained by Rf -95 is closer to the solid line and Rf-95’s intercept of the regression line is smaller. When Rf-LS is in the range of 80∼90 or about 40 and 60, Rf -99 has a larger difference with Rf-LS, and the value is lower. For a value of Rf -LS = 80, Rf -99 spans the range 71∼78 with 8 numerical changes, Rf -95 spans the range 78∼81 with only 4 numerical changes. Overall, the smaller the Rf -LS, the greater the dispersion of the scatter points.
Considering that the light sources with a terrible CFI has a low application rate in various occasions, we select Rf -LS greater than 70 and make a linear regression again as shown by the red dotted line in Fig. 10. We find that the regression line of Rf -99 shifts downward, but Rf -95’s almost coincides with the straight lines passing through the origin and having a slope of 1. As can be seen from the distribution of data points, Rf -99 performed poorly in the 80∼90 range because of its large discrete distribution. Rf -95 has the best performance in the 90∼100 range for it is almost equal to Rf -LS. All of the above shows that CIE-Rf calculated by OCSS is closer to the one calculated by Large Set. The OCSS obtained in this paper has better performance.
In order to further quantify the results of the optimization, we also calculate some indexes of the regression line and Spearman correlation coefficient (SCC) between Rf-95/ Rf -99 and Rf -LS [42,43]. The intercept of the regression line on the vertical axis can be used to judge the accuracy of Rf -95 and Rf -99 simulation Rf -LS [42]. The degree of dispersion of the points can also determine the representativeness of the screening color sample to Large Set. The degree of dispersion of the entire equation can usually be measured by the Standard Error of the Regression (SER) of the residual. Around the regression line, the more the sample points disperse, the larger the value of SER presents [30].
The calculation results are shown in Table 3. The intercept and SER of regression line made from data points of Rf -95 and Rf -LS on the vertical axis are smaller than the one made form Rf -99 and Rf -LS, especially when Rf -LS is greater than 70. It means that Rf -95 is better at simulating Rf-LS and data points are less discrete. The SCC shows a strong correlation between Rf -95/ Rf -99 and Rf-LS. And OCSS still better than CIE CESS-99 for the SCC between Rf -95 and Rf -LS is about 0.025 higher than Rf -99.
Table 3. The correlation index between Rf -95/ Rf -99 and Rf –LS.
4.3 Deep discussion
It can be known from section 4.2 that there were a large difference between the performance of Rf-95 and Rf -99. We think there are two reasons for such a big performance difference. One is the differences between the selection method of 95 color samples and the CESS-99. CESS-99 removed a large part of the color samples outside the NCS color gamut from the beginning, and we filtered the samples based on all color samples in Large Set. This may lead to the result calculated by optimized color sample in this paper to be closer to Large Set. But there is no relevant data (more than 60,000 data restricted by NCS), this hypothesis needs to be verified by related research in the future. In our opinion, another is the boundary highly saturated color samples.
In order to further verify the influence of the highly saturated sample set Og on the calculation results, we added the 15 highly saturated samples of set Og to CESS-99 and recorded the calculation result as Rf -114; removed 15 highly saturated samples in OCSS-95 and recorded the calculation result as Rf -80. These results are shown in Table 4, Fig. 11 and Fig. 12.
Table 4. The correlation index between Rf -114/ Rf -80 and Rf –LS.
It is found that the correlation coefficient has not changed much. But from the perspective of MAD, intercept and SER, we can see that Rf -114 has been improved, and optimization advantage of Rf -80 has been reduced the after removing 15 saturated samples.
These trends can also be seen in the linear regression figure. Comparing left and right pictures, we find that saturated samples can increase CIE Rf value: the scatter points of Rf -114 and the regression line move up as a whole compared with Rf -99, but the scatter points of Rf -80 move down as a whole compared with Rf -95. And these movements are both closer to Rf-LS, which indicates that the highly saturated samples are significant in terms of accuracy of calculations.
5. Conclusion
Based on clustering analysis, this paper developed a method to select the test color sample set and the set is used for evaluating light sources’ fidelity for CIE-Rf. It is found that the OCSS outperforms the existing CIE CESS-99, since it is more representative of Large Set when calculating CIE-Rf. The reason for the difference between OCSS and CESS-99 is further analyzed, which is the classification of color samples inside and at the color gamut, which contributes the better performance of OCSS in this paper.
Q clustering is first performed on the color appearance attribute values to obtain the initial clustering categories. AP clustering is then performed on the reduced-dimensional spectrum of the color samples in each initial clustering category. The clustering centers of AP make up OCSS. Taking OCSS, CIE CESS-99 and Large Set as TCSS to calculate CIE-Rf of 1202 light sources. The calculation results are marked as Rf-95, Rf-99 and Rf-LS. The AD, MAD, SCC and linearly regression analysis between the paired data Rf-95/Rf-99 and Rf-LS are used to measure the representativeness of the color samples. The correlation analysis results show that Rf-95 is very close to Rf-LS when evaluating the fidelity of light sources of various types and occasions. Compared with CIE CESS-99, OCSS with less samples is more representative of the Large Set. Therefore, the method of clustering analysis can be well used for the selection of test color samples. Besides, when both chromaticity values and spectral information are taken into account, the OCSS obtained in this paper can be better used as test color sample set to evaluate the fidelity of light sources of various types and occasions.
Funding
National Natural Science Foundation of China (61605012).
Acknowledgments
Kevin Smet who kindly shared the data of the spectra of Large Set and 401 light sources is greatly acknowledged.
Disclosures
The authors declare no conflict of interest.
References
1. S. Jost, C. Cauwerts, and P. Avouac, “CIE 2017 color fidelity index Rf: a better index to predict perceived color difference,” J. Opt. Soc. Am. A 35(4), B202–B213 (2018). [CrossRef]
2. S. J. Schanda, L. Whitehead, and M. R. Luo, “CRI2012: A proposal for updating the CIE colour rendering index,” Light. Res. Technol. 45(6), 689–709 (2013). [CrossRef]
3. A. David, “Color fidelity of light sources evaluated over large sets of reflectance samples,” Leukos 10(2), 59–75 (2014). [CrossRef]
4. K. R. Gegenfurtner, “Color metamerism and the structure of illuminant space,” J. Opt. Soc. Am. A 35(4), B231–B238 (2018). [CrossRef]
5. K. A. G. Smet, A. David, and L. Whitehead, “Why Color Space Uniformity and Sample Set Spectral Uniformity Are Essential for Color Rendering Measures,” Leukos 12(1-2), 39–50 (2016). [CrossRef]
6. P. Khanh, Q. Bodrogi, D. Vinh, and Stojanovic, “Colour preference, naturalness, vividness and colour quality metrics, Part 1: Experiments in a room,” Light. Res. Technol. 49(6), 697–713 (2017). [CrossRef]
7. L. Cheng, M. R. Luo, and G. H. Cui, “Evaluation of the CIE Color Rendering Index,” Color. Technol. 127(2), 129–135 (2011). [CrossRef]
8. K. W. Houser, M. Wei, A. David, M. R. Krames, and X. S. Shen, “Review of measures for light-source color rendition and considerations for a two-measure system for characterizing color rendition,” Opt. Express 21(8), 10393–10411 (2013). [CrossRef]
9. K. Smet, W. R. Ryckaert, M. R. Pointer, G. Deconinck, and P. Hanselaer, “Correlation between color quality metric predictions and visual appreciation of light sources,” Opt. Express 19(9), 8151–8166 (2011). [CrossRef]
10. P. Khanh, Q. Bodrogi, D. Vinh, and Stojanovic, “Colour preference, naturalness, vividness and colour quality metrics, Part 2: Experiments in a viewing booth and analysis of the combined dataset,” Light. Res. Technol. 49(6), 714–726 (2017). [CrossRef]
11. H. Yaguchi and A. David, “CIE 2017 COLOUR FIDELITY INDEX FOR ACCURATE SCIENTIFIC USE,”,” Color Res. Appl. 42(5), 590 (2017). [CrossRef]
12. M. R. Luo, G. Cui, and C. Li, “Uniform colour spaces based on CIECAM02 colour appearance model,” Color Res. Appl. 31(4), 320–330 (2006). [CrossRef]
13. C. Li, M. Ronnier Luo, C. Li, and G. Cui, “The CRI-CAM02UCS colour rendering index,” Color Res. Appl. 37(3), 160–167 (2012). [CrossRef]
14. A. David, P. T. Fini, K. W. Houser, Y. Ohno, M. P. Royer, K. A. G. Smet, M. Wei, and L. Whitehead, “Development of the IES method for evaluating the color rendition of light sources,” Opt. Express 23(12), 15888–15906 (2015). [CrossRef]
15. Q. Liu, Z. Huang, K. D. Xiao, M. R. Pointer, S. Westland, and M. R. Luo, “Gamut Volume Index: a color preference metric based on meta-analysis and optimized colour samples,” Opt. Express 25(14), 16378–16391 (2017). [CrossRef]
16. C. J. Li, Z. Q. Li, and Z. F. Wang, “Comprehensive color solutions: CAM16, CAT16, and CAM16-UCS,” Color Res. Appl. 42(6), 703–718 (2017). [CrossRef]
17. S. Hermans, K. A. G. Smet, and P. Hanselaer, “Exploring the applicability of the CAM18sl brightness prediction,” Opt. Express 27(10), 14423–14436 (2019). [CrossRef]
18. S. Hermans, K. A. G. Smet, and P. Hanselaer, “Color appearance model for self-luminous stimuli,” J. Opt. Soc. Am. A 35(12), 2000–2009 (2018). [CrossRef]
19. J. Zabalza, J. Ren, J. Ren, Z. Liu, and S. Marshall, “Structured covariance principal component analysis for real-time onsite feature extraction and dimensionality reduction in hyperspectral imaging,” Appl. Opt. 53(20), 4440–4449 (2014). [CrossRef]
20. Y. Wang, “A Study of Key Technologies in Multispectral Image Color Reproduction,” Xi An University. (2010).
21. M. Hoon, S. De, S. Imoto, and Miyano, “The C Clustering Library,” The University of Tokyo, Institute of Medical Science, Human Genome Center, Tokyo, Japan (2017).
22. L. Anselin, “Classic Clustering Methods,” at https://geodacenter.github.io/workbook/7b_clusters_2/lab7b.html (Accessed 2019).
23. J. Zhang, X. Tuo, Z. Yuan, W. Liao, and H. Chen, “Analysis of fMRI Data Using an Integrated Principal Component Analysis and Supervised Affinity Propagation Clustering Approach,” IEEE Trans. Biomed. Eng. 58(11), 3184–3196 (2011). [CrossRef]
24. T. Bock, “What is Hierarchical Clustering,” at https://www.displayr.com/what-is-hierarchical-clustering/ (Accessed 2019).
25. Kulkarni, Sushant, J. Senthilnath, J. Benediktsson, and Atli, “Classification of Multi-Spectral Satellite Image Using Hierarchical Clustering Algorithms,” in 8th IEEE Symposium Series on Computational Intelligence (IEEE, 2018), pp. 1664–1669.
26. Tsumoto, Shusaku, Hirano, Shoji, Kimura, and Tomohiro, “Empirical Comparison of Similarities for Agglomerative Hierarchical Clustering,” in IEEE International Conference on Systems, Man, and Cybernetics (IEEE, 2018), pp. 3405–3410.
27. R. M. Esteves, T. Hacker, and C. Rong, “Competitive K-Means, a New Accurate and Distributed K-Means Algorithm for Large Datasets,” 2013 IEEE 5th International Conference on Cloud Computing Technology and Science, 17–24, (2013).
28. Xinquan Chen, Hong Peng, and Jingsong Hu, “k-Medoids Substitution Clustering Method and a New Clustering Validity Index Method,” 2006 6th World Congress on Intelligent Control and Automation, 5896–5900, (2006).
29. Löster and Tomá, “Modification of the algorithm for determining the optimal number of clusters in cluster analysis,” presented at 15thConference on Applied Mathematics, (2016).
30. X. L. Ma and Z. M. Yao, Application and method of data analysis: based on SPSS and EXCEl (Posts & Telecom University, 2015).
31. J. Gareth, D. Witten, T. Hastie, and R. Tibshirani, An Introduction to Statistical Learning: with Applications (Springer, 2013).
32. “linkage” at https://ww2.mathworks.cn/help/stats/pdist.html.
33. “Evaluate clustering solutions” at https://ww2.mathworks.cn/help/stats/evalclusters.html.
34. F. Xu, X. Song, K. Tsui, F. Yang, and Z. Huang, “Bearing Performance Degradation Assessment Based on Ensemble Empirical Mode Decomposition and Affinity Propagation Clustering,” IEEE Access 7, 54623–54637 (2019). [CrossRef]
35. Z. Q. Guo, H. L. Lu, T. Shih, Y. Lin, Y. J. Lu, and Z. Chen, “Spectral Optimization of Candle-Like White Light-Emitting Diodes with High Color Rendering Index and Luminous Efficacy,” J. Disp. Technol. 12(11), 1393–1397 (2016). [CrossRef]
36. H. Daicho, K. Enomoto, H. Sawa, S. Matsuishi, and H. Hosono, “Improved color uniformity in white light-emitting diodes using newly developed phosphors,” Opt. Express 26(19), 24784–24791 (2018). [CrossRef]
37. M. Zhao, H. X. Liao, M. S. Molokeev, Y. Y. Zhou, Q. Y. Zhang, Q. L. Liu, and Z. G. Xia, “Emerging ultra-narrow-band cyan-emitting phosphor for white LEDs with enhanced colorrendition,” Light: Sci. Appl. 8(1), 1–5 (2019). [CrossRef]
38. F. Simonetta, C. Bonanomi, and R. Alessandro, “An experiment on the color rendering of different light sources,” Proc. SPIE 8652, 86520F (2013). [CrossRef]
39. G. X. He and H. F. Yan, “Optimal spectra of the phosphor-coated white LEDs with excellent color rendering property and high luminous efficacy of radiation,” Opt. Express 19(3), 2519–2529 (2011). [CrossRef]
40. G. Z. Zhang, K. Ding, G. X. He, and P. Zhong, “Spectral optimization of color temperature tunable white LEDs with red LEDs instead of phosphor for an excellent IES color fidelity index,” OSA Continuum 2(4), 1056–1064 (2019). [CrossRef]
41. P. J. Pardo, E. M. Cordero, M. I. Suero, and ÁL Pérez, “Influence of the correlated color temperature of a light source on the color discrimination capacity of the observer,” J. Opt. Soc. Am. A 29(2), A209–A215 (2012). [CrossRef]
42. Y.S. Lian, “Research on Color Appearance Models Using Chinese Color System Dat,” PLA Information Engineering University. (2007).
43. K. A. G. Smet, W. R. Ryckaert, M. R. Pointer, G. Deconinck, and P. Hanselaer, “Memory colours and colour quality evaluation of conventional and solid-state lamps,” Opt. Express 18(25), 26229–26244 (2010). [CrossRef]
