1. 上海出版印刷高等专科学校, 上海 200093
2. 河南牧业经济学院, 河南 郑州 450011
3. 河南工程学院, 河南 郑州 450007
4. 上海理工大学, 上海 200093

Spectral Evaluation Metric Based on Human Perception
TIAN Quan-hui1, LIU Shi-wei2, ZHU Ming3, LIU Zhen4
1. Lab of Green Platemaking and Standardization for Flexographic Printing, Shanghai Publishing and Printing College, Shanghai 200093, China
2. Department of Printing and Packaging Engineering, Henan University of Animal Husbandry and Economy, Zhengzhou 450011, China
3. Department of Materials and Chemical Engineering, Henan University of Engineering, Zhengzhou 450007, China
4. College of Communication and Art Design, Shanghai University of Science and Technology, Shanghai 200093, China

Biography: TIAN Quan-hui, (1974-), female, PhD., associate professor, Shanghai Publishing and Printing College e-mail: tqh0509@163.com

Abstract

Reasonable evaluation of spectral color reproduction and chromaticity accuracy is fundamental to evaluate the accurate of spectral reconstruction algorithm. A simple and direct evaluation index is necessary to the color reproduction. This index requires to directly characterize the spectral difference and color difference for color reproduction. Based on the study of the reproduction of the accuracy evaluation on spectral matching, this paper proposes three spectral evaluation metrics based on color visual perception. These methods are weighted spectral evaluation metrics based on human visual function. The results of weighted metrics are analyzed with Munsell color system. These metrics get the uniform distribution and stability in there replication experiment, which proves that weighted metrics are accurate metrics for color spectral reconstruction, and take into account the color perception and spectral similarity. The empirical result is that the weighted metrics indicate the actual human color perception and spectral error at same time.The metrics based on human perception solve the problem of quantitative evaluation of spectral estimation accuracy of natural spectrum matching and color spectral reconstruction.The evaluation methods in this paper show simple and intuitive color spectrum reproduction accuracy.

Keyword: Spectral perception evaluation; Spectral error; Spectral estimation metric; Estimation metric; Color perception
Introduction

In recent years, the research of color restoration based on multispectral imaging system has become a hot topic in the field of image science. In the color restoration workflow, the image spectral data is required to match with the actual color spectrum. It is a key point that is necessary for the quantitative evaluation of the spectral matching accuracy between the reconstructed color spectral data and the actual corresponding color spectral data, and the evaluation of spectral estimation performance of multispectral imaging systems[1, 2].

In previous studies, there are two kinds of methods for the evaluation of spectral matching accuracy. The first one is based on constant-chroma matching. The spectral reflectance of color must be approximately similar in order to achieve matching. So the evaluation is using the CIE color difference and the metameric index[3]. This method takes into account the colorimetric properties of the spectral response of the human eye, but it does not really indicate the similarity of the two spectra.

The second one is based on the difference of the two spectral curves mainly used in the field of remote sensing, such as spectral distance, spectral root Mean square error (sRMS), and Goodness-of-fit coefficient (GFC)[4, 5]. Then there are researches to find out the smaller difference of spectral curve does not mean the smaller of difference of color perception[4, 5, 6].

The researchers now use the CIE color difference andsRMS as the evaluation indices to estimate chromaticity and spectral differences in statistical parameters at same time. This method uses a number of indicators to take into account both the differentia of chromaticity and spectral distribution. But these indices are only logical repetition because the chromaticity is the reflection of the spectral parameters, and difficult to draw the constant conclusion definitely. Prof. Li proposed the spectral matching skewness index to evaluate the Spectral matching of two spectral vectors, which solved the problem of quantitative evaluation of spectral estimation accuracy of natural spectrum matching and multispectral camera. But it cannot be considered as the difference of two color spectra based on perception of these colors[7]. However, due to the diversity and complexity of color reproduction, there is no simple and intuitive evaluation method now which could take into account the difference of chroma and spectrum at the same time.

This paper proposes a calculation method to evaluate the spectral similar of the reconstruction spectral color and actual spectral color with one index, which could present both the chromatic differences and spectral differences based on human perception.

1 Theory

According to the characteristics of human visual system, the weighted matrix of human visual color matching function is used to weigh the spectral evaluation metric.

1.1 Weighted Matrix

In order to quantify the color perception, W. D. Wright and J. Guild did a large number of color vision experiments, and summarized a set of RGB data to matchequal energy visible-spectrum, which is on behalf of the average human color visual characters, and is the CIE1931 standard color observer CIE1931 color matching function[9, 10, 11]. The CIE 1931 standard color observer matching function is applied to 2-degreecentral visual observation conditions (field of view in the range of 1-degree to 4-degree), which is the main function of cone cell of fovea, and presents the precise color perception. Then we constructed a weighted matrix according to the color matching function[13].

If there is a spectral color matrix r, which describes m pieces of color samples and spectral n-dimensional, is expressed as r1, r2, …, rm. Therefore, the tristimulus of color (Γ ) can be expressed as a vector in Eq.(1)[12], where E is lighting source vector shown in Eq.(2), A is color matching function vector shown in Eq.(3), k is the coefficient of lightness normalization.

$Γ=kATEri(1)E=e1…0︙⋱︙0…en(2)A=x̅1y̅1z̅1………x̅ny̅nz̅n(3)$

Defined a matrix (Λ ) as Λ T=ATE, which defined as the weighted color matching function (A) with lighting source (E). The matrix Λ is a 3× n matric and presents the adjusting of human perception in given light source.

Then Eq. (1) could be changed as Eq. (4).

$Γ=ΛTsi(4)$

Multiplying Λ (Λ TΛ )-1 to Eq. (4), Vector Γ could be get as Eq.(5).

$Λ(ΛTΛ)-1Γ=Λ(ΛTΛ)-1ΛTsi(5)$

Suppose

$R=Λ(ΛTΛ)-1ΛT(6)$

Then the color spectral data could be divided to basic spectral reflectance (eλ ) and metamerism stimulation black (Bλ ), shown in Eq.(7) and Eq.(8). That means the spectral energy distribution of any color stimulus is composed of a basic color stimulus and a metamerism spectrum black stimulation.

$eλ=Rri(7)Bλ=(I-R)ri(8)$

The tristimulus value of the color is determined by the basic color stimulus because the tristimulation of the metamerismblack spectrum is zero. So we define the diagonal matrix of R-matric as weighted matrix w(λ ), shown in Eq. (9). The color matching function and weighted matrix are shown in Figure 1.

$w(λ)=diag(R)=diagΛ(ΛTΛ)-1ΛT=diag(EA(ATEEA)-1ATE)(9)$

 Figure Option Fig.1 Color matching function (x, y, z) and weighted matrix (w)

1.2 Weighted Spectral Root Mean Square Error (WRMS)

WRMS is defined as Eq.(10).

$WRMS=∑λ=1n(w(λ))|rλi-r^λi|2n(10)$

n is spectral dimension. r is spectral color. The hat symbol denotes ( $r^$) reconstruction, and | | is the vector norm. w(λ ) is weighted matrix.

1.3 Weighted Goodness-of-fit coefficient (WGFC)

We define WGFC as Eq. (11).

$WGFC=w(λ)rTr^rTrr^Tr^(11)$

r is spectral color. The hat symbol denotes ( $r^$) reconstruction, and T is the vector transposition. w(λ ) is weighted matrix.

1.4 Weighted Spectral Similarity (WSS)

WSS is defined by Eq.(12).

$WSS(r, r^)=w(λ)(r-r̅)T(r^-r^¯)[(r-r̅)T(r-r̅)(r^-r^¯)T(r^-r^¯)]12(12)$

Where, r and $r^$are the sample spectrum and the reconstructed spectrum. T is the vector transposition. w(λ ) is weighted matrix.

2 Experiment

The Munsell Book of Coloris a collection of color patches that uniformly occupy a large gamut in perceptual space. The color chips were mounted uniformly different hue, chroma and lightness in Munsell. Therefore, we used 1269 spectral reflectance $r^λi$by each swatch of Munsell chips from Munsell colors matt (from http://www.uef.fi/en/web/spectral), which were measured with Perkin-Elmer lambda 9 UV/VIS/NIR spectrophotometer of 1 nm wavelength resolution and 3 80 to 800 nm wavelength interval by Jouni Hiltunen[13].

Then we cleaned the spectral data of Munsell of 10 nm wavelength resolution and 400 nm to 700 nm wavelength interval for the human visual system is most sensitive to the wavelengths in the middle of the visual part of the spectrum.

Different spectral metrics were computed and compared with adjacent color samples in the Munsell Book of Coloris. The experiment procedure is shown as Figure 2.

 Figure Option Fig.2 The procedure of experiment

3 Results and discussion
3.1 Variability of different spectral evaluation metrics

Comparing the different spectral evaluation metrics to color-difference of color samples which are the adjacent color samples in the Munsell Book of Coloris, the spectral evaluation metrics vary differently as is shown in Figure 3.

 Figure Option Fig.3 Comparing Δ E2 000 to the spectral distance, sRMS, GFC, WRMS, WGFC, and WSS of measured spectrum ( $rλi$) and reconstructed spectrum ( $r^λi$) (L is the lighterness of color samples)

The CIE 2000 color-difference index (Δ E2 000) is concerned as the chromatic aberration of human perception[14]. If the variability of spectral evaluation metric corresponded with Δ E2 000, the spectral evaluation metric would be considered to predict the difference of human color perception as Δ E2 000. All of weighted spectral evaluation metrics are better than unweighted metrics from the distribution of data in Figure 3.

3.2 Distribution of spectral evaluation metrics

The sRMS, GFC, WRMS, WGFC and WSS of the adjacent spectral reflectance of Munsell are computed with the given formulas in reference 3-5, normalized with their maximum, and got the results of different spectral evaluation metrics in Figure 4, in which the size of bubble presents the results of spectral evaluation metric. The bigger the bubble is, the larger the result of spectral evaluation metric is. The locations of bubble are a-b values of swatch of the Munsell Book of Coloris.

 Figure Option Fig.4 The distribution of sRMS, GFC, WRMS, WGFC, and WSS of the adjacent spectral reflectance

Shown in Figure 4, all of distribution of the spectral evaluation metrics are not uniformity. Because the size of bubble indicates the value of the spectral evaluation metric, and the adjacent color swatches of Munsell are equal perception of uniformity hue, lightness and chroma, the bubbles of Figure 4 should be looked similarly if the spectral metrics are corresponding with the color visual perception. The bubbles of sRMS and WRMS look more no uniformity than the others’ from Figure 4. That means the sRMS and WRMS are not good at performance the same different color perception responds to the same spectral error.

3.3 Histogram of spectral evaluation metrics

There are 5 graphs for the histogram of sRMS, GFC, WRMS, WGFC and WSS in Figure 5. The distribution frequencies of different spectral metrics of adjacent patches in Munsell color system are shown in Figure 5. The distribution frequencies of GFC concentrate on about 1, which makes the GFC does not vary with the change of spectral error or color perception. But the distribution frequencies of WGFC and WSS conform to Gaussian distribution approximately, that means WGFC and WSS are stable and that have probability density functions that can be expressed analytically.

 Figure Option Fig.5 Histogram of different spectral evaluation metrics with Munsell

3.4 Relationship of spectral evaluation metric and Color perception

We compared all of spectral evaluation metrics and analysed the relationship of spectral evaluation metrics and color perception with the spectral reflectance of Munsell. Figure 6 is the relationship of hue angle of color sample and spectral reflectance metrics of adjacent patches in Munsell color.

 Figure Option Fig.6 Relationship of hue angle of Munsell and spectral reflectance metrics

From Figure 6, WGFC and WSS are stablizing. Comparatively, the diagrams of WRMS and sRMS are similar, and the distributions do not vary with hue of color perception clearly. Refering to Figure 5, the histogram of different matics, sRMS, WRMS and GFC of Munsell are relative stable. WGFC of Munsell is distributed at the range of 0.09 to 0.11, and is scattered at 0 and 360 of the hue value. The similar distribution is shown at WSS. That means WGFC and WSS of Munsell are changed with the changing of color hue and the largest changing at red color.

Figure 7 is the relationship of lightness of Munsell and spectral reflectance metrics. WGFC and WSS are distribted uniformly with lightness. That means that matrics could indicate the spectral difference and lightness difference at same time.

 Figure Option Fig.7 Relationship of lightness of Munsell and spectral reflectance metrics

4 Conclusion

The spectral estimation metric is important for analyzing the precise of color spectral reconstruction and color reproduction. The spectral estimation metric presents not only the spectral error, but also the color perception when it is used for color reproduction. We proposed weighted metric that could indicate the actual human color perception and spectral error at the same time. The empirical evidence obtained shows that the values of the metrics range from the given range, and closing to the middle of the range are the conditions for good and excellent spectral matches. The human-vision weighted metrics are better than sRMS and color difference for the distinguishing the difference of human color perception and spectral error of based spectral-based color reproduction. The human-vision weighted evaluation metric solves the problem of quantitative evaluation of spectral estimation accuracy of natural spectrum matching and color spectral reconstruction.

The authors have declared that no competing interests exist.

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