在《恒星结构与演化》这门课上，曾经用mesa进行过恒星演化的模拟。当时在绘制各种恒星的演化动画时，想到如果能将恒星的有效温度映射到具体对应的颜色上，画出来的动画是不是就更加生动形象了？于是我就到网上四处搜集黑体温度与颜色的关系，结果却什么也没搜到，加上ddl在即，于是只好作罢。

凑巧的是，没过多久，又有一位 ~~老同事~~ 也问了我一个类似的问题：为什么A型恒星的温度和电焊的温度差距如此之大，发出的光却都是蓝白色？当时我的回答是前者可以视为黑体而后者则不然，二者的光谱不同，在某种能够变换到颜色上的积分运算结果不同，从而导致得到了不同的颜色。

那么问题来了，光谱是怎么计算出对应的颜色的？为此我搜集了部分色彩学的文献及网页资料，最后便找到了下述的答案。

光谱到颜色的计算过程

从光谱到颜色的计算过程大致如下：

光谱 & CIE XYZ三色系的CMF $\rightarrow$ XYZ分量 $\rightarrow$ RGB分量 $\rightarrow$ 归一化后的，可以直接使用的RGB颜色

光谱

在光学中已经知道，一个物体在各个波段的辐射强度可以使用光谱 $\phi_\lambda(\lambda)$ 来描述，其中 $\lambda$ 为波长。有了光谱 $\phi_\lambda(\lambda)$ 后，就可以通过积分的方式求出各个指定波长范围 $[\lambda_1,\ \lambda_2]$ 内的辐射强度：
$I_{12}=\int_{\lambda_1}^{\lambda_2}\phi_\lambda(\lambda)d\lambda$
对于黑体而言，辐射为：
$S_\lambda(\lambda,\ T)=\frac{8\pi hc}{\lambda^5}\frac{1}{e^{hc/\lambda kT}-1}$

颜色的定量描述——CIE XYZ三色系

说到颜色的定量描述，最著名的一个色彩坐标系便是CIE XYZ三色系（CIE XYZ Trichromatic System），它将色彩空间里的三个基矢分别定义为X，Y和Z，在这个系统中的所有颜色都由这三个基失的线性叠加来描述。也就是说，只要知道了X，Y和Z的值，就可以确定一个具体的颜色

但如何确定X，Y和Z？这个时候就需要用到类似于滤光片透光率曲线的“人眼透光率曲线”了，在色彩学中称作color matching functions（CMF）。CMF是根据人眼的感光特性，将这个特性分解到X，Y和Z这三个分量上，从而分别得到了三条“透光率曲线” $\bar x(\lambda)$ ， $\bar y(\lambda)$ 和 $\bar z(\lambda)$ ：

在这里插入图片描述

类似于天文观测中的流量计算，对于任一光谱 $\phi_\lambda(\lambda)$ ，这里的三个分量的“流量”计算方式也几乎是一模一样的：
$\begin{aligned} X&=k\int_{380\ \mathrm{nm}}^{780\ \mathrm{nm}}\phi_\lambda(\lambda)\bar x(\lambda)d\lambda\\ Y&=k\int_{380\ \mathrm{nm}}^{780\ \mathrm{nm}}\phi_\lambda(\lambda)\bar y(\lambda)d\lambda\\ Z&=k\int_{380\ \mathrm{nm}}^{780\ \mathrm{nm}}\phi_\lambda(\lambda)\bar z(\lambda)d\lambda \end{aligned}$
计算出了X，Y和Z后，将其归一化：
$\begin{aligned} x&=\frac{X}{X+Y+Z}\\ y&=\frac{Y}{X+Y+Z}\\ z&=\frac{Z}{X+Y+Z} \end{aligned}$
$(x,\ y,\ z)$ 自由度为2，省略 $z$ 坐标，画出的 $(x,\ y)$ 平面即为 $(x,\ y)$ 色度图。至此，任意颜色都可以使用这个平面上的坐标来定量表示

XYZ到RGB的转换

实际上在最早的CIE 1931规范提出时，我们所熟知的RGB系统比XYZ系统要出现的更早。由于它们都是在色彩空间中使用三个不同的基矢，因此在这两个系统之间必然能够进行相应的线性变换

关于线性变换的参数，在很多文献中也都能找到。例如，RGB变换到XYZ的方式如下：
$\left(\begin{matrix} X\\Y\\Z \end{matrix}\right)= \left(\begin{matrix} 2.768892 & 1.751748 & 1.130160\\ 1.000000 & 4.590700 & 0.060100\\ 0 & 0.056508 & 5.594292 \end{matrix}\right) \left(\begin{matrix} R\\G\\B \end{matrix}\right)$
将变换矩阵求逆即可用于XYZ到RGB的变换，变换后再根据需求进行相应的归一化，就能在采用RGB规范的显示设备上将这个颜色显示出来了

算法运行结果展示

不同温度下的黑体颜色

根据前面提到的黑体谱以及CMF，分别对不同温度的黑体进行计算得到XYZ，随后再通过线性变换得到RGB并进行归一化，作图得到下述的结果：

在这里插入图片描述

这个图和各种色温参考图相比，还是有比较大的偏差的，可能是因为CMF曲线的使用有所差异（这里的CMF曲线是从python的colour-science包中相应的数据库里掏出来的）

赫罗图改进

现在已经知道了黑体温度与颜色的具体对应关系，因此就尝试了一下对几个月前画的一张赫罗图进行了相应的改进，现在各点的颜色都能够与有效（黑体）温度相对应，显得更加科学了：

最开始的问题：不同温度，辐射颜色却相同的原因？

答案自然是光谱的不同所致了。光谱不同，得到的X，Y，Z分量不同，表现出来的颜色自然也就不同了

也有文献（McCamy, C. , 1992）研究了CIE中的等温线——它们是近似地以某个点为中心，向外辐射出各个不同斜率的直线，在同一条直线上，物体对应的温度是相等的。也就是说，温度相同的物体，的确会随着光谱的不同而呈现出不同的颜色

参考文献

[1] JÁNOS SCHANDA (2007). UNDERSTANDING THE CIE SYSTEM, 29-34.

[2] McCamy, C. (1992). Correlated color temperature as an explicit function of chromaticity coordinates . Color Research & Application, 17(2), 142–144.

相关代码

黑体谱与RGB颜色的转换

import numpy as np
import matplotlib.pyplot as plt
import json
from colour.plotting import plot_chromaticity_diagram_CIE1931def load_cmf():"""获取XYZ坐标系的cmfReturns-------[xbar, ybar, zbar] : [float[], float[], float[]]三个坐标变量的cmf"""xbar = []ybar = []zbar = []with open("data_cmfs_standard_observer.json", "r") as f:D = json.load(f)for each_waveLength in range(380, 781):each_waveLength_str = str(each_waveLength)[each_xbar, each_ybar, each_zbar] = D[each_waveLength_str]xbar.append(each_xbar)ybar.append(each_ybar)zbar.append(each_zbar)return [xbar, ybar, zbar]def get_bb_spectrum(T):"""计算某一温度下对应的黑体谱Parameters----------T : float温度(K)Returns-------waveLength : float[]波长(nm)bb_spectrum : float[]计算得到的归一化黑体谱"""h = 6.626e-34c = 3e8k = 1.381e-23waveLength = np.arange(380, 781, 1) # unit : nmwaveLength_metric = waveLength / 1e9 # unit : mbb_spectrum = 8 * np.pi * h * c / waveLength_metric**5 / (np.exp(h * c / (waveLength_metric * k * T)) - 1)bb_spectrum = bb_spectrum / max(bb_spectrum) # normalizationreturn [waveLength, bb_spectrum]def plotCMF(waveLength, xbar, ybar, zbar):"""绘制CMF"""plt.plot(waveLength, xbar, 'r-', label=r'$\bar x(\lambda)$')plt.plot(waveLength, ybar, 'g-', label=r'$\bar y(\lambda)$')plt.plot(waveLength, zbar, 'b-', label=r'$\bar z(\lambda)$')plt.xlabel('Wavelength (nm)')plt.ylabel('CMF')plt.legend()plt.savefig("CMF.png", dpi=600)def convolution(fx1, fx2):"""计算两个函数相乘的积分，已满足x轴坐标对应的条件"""fx1 = np.array(fx1)fx2 = np.array(fx2)F = np.sum(fx1 * fx2)return Fdef calculate_CIE_pos(bb_spectrum, xbar, ybar, zbar):"""根据黑体谱和CMF，计算CIE坐标"""X = convolution(bb_spectrum, xbar)Y = convolution(bb_spectrum, ybar)Z = convolution(bb_spectrum, zbar)x = X / (X + Y + Z)y = Y / (X + Y + Z)z = Z / (X + Y + Z)return [x, y, z]def show_bb_color_in_CIE(T):"""在CIE上显示温度为T的黑体对应颜色"""[xbar, ybar, zbar] = load_cmf()plot_chromaticity_diagram_CIE1931(standalone=False)[waveLength, bb_spectrum] = get_bb_spectrum(T=T)[x, y, z] = calculate_CIE_pos(bb_spectrum, xbar, ybar, zbar)plt.plot(x, y, 'k.')plt.show()def from_xyz_to_RGB(x, y, z):"""将CIE上的坐标转换为RGB坐标"""M = np.matrix([[0.41845516, -0.15865714, -0.08283193],[-0.09116486, 0.25242564, 0.01570531],[0.00092086, -0.00254975, 0.17859499]])xyz = np.matrix([[x], [y], [z]])RGB = M * xyzRGB = RGB / max(RGB)RGB[np.where(RGB < 0)] = 0RGB = np.array(RGB)return [RGB[0][0], RGB[1][0], RGB[2][0]]def plot_bb_color_temperature_relation():"""画出黑体的颜色-温度关系图"""[xbar, ybar, zbar] = load_cmf()deltaT = 50plt.figure(figsize=(10,1))for T in range(1000, 12000, deltaT):[waveLength, bb_spectrum] = get_bb_spectrum(T=T)[x, y, z] = calculate_CIE_pos(bb_spectrum, xbar, ybar, zbar)RGB = from_xyz_to_RGB(x, y, z)plt.fill([T, T+deltaT, T+deltaT, T, T], [0, 0, 1, 1, 0], c=RGB)plt.yticks([])plt.xlabel("Blackbody Temperature (K)")plt.gca().spines['left'].set_visible(False)plt.gca().spines['right'].set_visible(False)plt.gca().spines['bottom'].set_visible(False)plt.gca().spines['top'].set_visible(False)plt.tight_layout()plt.savefig("bb_color_temperature_relation.png", dpi=600)def bb_temperature_to_RGB(T):"""将黑体的温度所对应的颜色转换为RGB值Returns-------RGB : float[3]RGB值，已进行归一化，可以直接在pyplot中使用"""[xbar, ybar, zbar] = load_cmf()[waveLength, bb_spectrum] = get_bb_spectrum(T=T)[x, y, z] = calculate_CIE_pos(bb_spectrum, xbar, ybar, zbar)RGB = from_xyz_to_RGB(x, y, z)return RGBif __name__ == "__main__":RGB = bb_temperature_to_RGB(T=5000)print(RGB)

CIE 1931规范下的CMF（从colour-science包中抠出，可直接复制使用）

json格式的，key为波长，单位是纳米，value分别为 $\bar x(\lambda)$ ， $\bar y(\lambda)$ 和 $\bar z(\lambda)$ 的值

{"360": [0.0001299, 3.917e-06, 0.0006061], "361": [0.000145847, 4.393581e-06, 0.0006808792], "362": [0.0001638021, 4.929604e-06, 0.0007651456], "363": [0.0001840037, 5.532136e-06, 0.0008600124], "364": [0.0002066902, 6.208245e-06, 0.0009665928], "365": [0.0002321, 6.965e-06, 0.001086], "366": [0.000260728, 7.813219e-06, 0.001220586], "367": [0.000293075, 8.767336e-06, 0.001372729], "368": [0.000329388, 9.839844e-06, 0.001543579], "369": [0.000369914, 1.104323e-05, 0.001734286], "370": [0.0004149, 1.239e-05, 0.001946], "371": [0.0004641587, 1.388641e-05, 0.002177777], "372": [0.000518986, 1.555728e-05, 0.002435809], "373": [0.000581854, 1.744296e-05, 0.002731953], "374": [0.0006552347, 1.958375e-05, 0.003078064], "375": [0.0007416, 2.202e-05, 0.003486], "376": [0.0008450296, 2.483965e-05, 0.003975227], "377": [0.0009645268, 2.804126e-05, 0.00454088], "378": [0.001094949, 3.153104e-05, 0.00515832], "379": [0.001231154, 3.521521e-05, 0.005802907], "380": [0.001368, 3.9e-05, 0.006450001], "381": [0.00150205, 4.28264e-05, 0.007083216], "382": [0.001642328, 4.69146e-05, 0.007745488], "383": [0.001802382, 5.15896e-05, 0.008501152], "384": [0.001995757, 5.71764e-05, 0.009414544], "385": [0.002236, 6.4e-05, 0.01054999], "386": [0.002535385, 7.234421e-05, 0.0119658], "387": [0.002892603, 8.221224e-05, 0.01365587], "388": [0.003300829, 9.350816e-05, 0.01558805], "389": [0.003753236, 0.0001061361, 0.01773015], "390": [0.004243, 0.00012, 0.02005001], "391": [0.004762389, 0.000134984, 0.02251136], "392": [0.005330048, 0.000151492, 0.02520288], "393": [0.005978712, 0.000170208, 0.02827972], "394": [0.006741117, 0.000191816, 0.03189704], "395": [0.00765, 0.000217, 0.03621], "396": [0.008751373, 0.0002469067, 0.04143771], "397": [0.01002888, 0.00028124, 0.04750372], "398": [0.0114217, 0.00031852, 0.05411988], "399": [0.01286901, 0.0003572667, 0.06099803], "400": [0.01431, 0.000396, 0.06785001], "401": [0.01570443, 0.0004337147, 0.07448632], "402": [0.01714744, 0.000473024, 0.08136156], "403": [0.01878122, 0.000517876, 0.08915364], "404": [0.02074801, 0.0005722187, 0.09854048], "405": [0.02319, 0.00064, 0.1102], "406": [0.02620736, 0.00072456, 0.1246133], "407": [0.02978248, 0.0008255, 0.1417017], "408": [0.03388092, 0.00094116, 0.1613035], "409": [0.03846824, 0.00106988, 0.1832568], "410": [0.04351, 0.00121, 0.2074], "411": [0.0489956, 0.001362091, 0.2336921], "412": [0.0550226, 0.001530752, 0.2626114], "413": [0.0617188, 0.001720368, 0.2947746], "414": [0.069212, 0.001935323, 0.3307985], "415": [0.07763, 0.00218, 0.3713], "416": [0.08695811, 0.0024548, 0.4162091], "417": [0.09717672, 0.002764, 0.4654642], "418": [0.1084063, 0.0031178, 0.5196948], "419": [0.1207672, 0.0035264, 0.5795303], "420": [0.13438, 0.004, 0.6456], "421": [0.1493582, 0.00454624, 0.7184838], "422": [0.1653957, 0.00515932, 0.7967133], "423": [0.1819831, 0.00582928, 0.8778459], "424": [0.198611, 0.00654616, 0.959439], "425": [0.21477, 0.0073, 1.0390501], "426": [0.2301868, 0.008086507, 1.1153673], "427": [0.2448797, 0.00890872, 1.1884971], "428": [0.2587773, 0.00976768, 1.2581233], "429": [0.2718079, 0.01066443, 1.3239296], "430": [0.2839, 0.0116, 1.3856], "431": [0.2949438, 0.01257317, 1.4426352], "432": [0.3048965, 0.01358272, 1.4948035], "433": [0.3137873, 0.01462968, 1.5421903], "434": [0.3216454, 0.01571509, 1.5848807], "435": [0.3285, 0.01684, 1.62296], "436": [0.3343513, 0.01800736, 1.6564048], "437": [0.3392101, 0.01921448, 1.6852959], "438": [0.3431213, 0.02045392, 1.7098745], "439": [0.3461296, 0.02171824, 1.7303821], "440": [0.34828, 0.023, 1.74706], "441": [0.3495999, 0.02429461, 1.7600446], "442": [0.3501474, 0.02561024, 1.7696233], "443": [0.350013, 0.02695857, 1.7762637], "444": [0.349287, 0.02835125, 1.7804334], "445": [0.34806, 0.0298, 1.7826], "446": [0.3463733, 0.03131083, 1.7829682], "447": [0.3442624, 0.03288368, 1.7816998], "448": [0.3418088, 0.03452112, 1.7791982], "449": [0.3390941, 0.03622571, 1.7758671], "450": [0.3362, 0.038, 1.77211], "451": [0.3331977, 0.03984667, 1.7682589], "452": [0.3300411, 0.041768, 1.764039], "453": [0.3266357, 0.043766, 1.7589438], "454": [0.3228868, 0.04584267, 1.7524663], "455": [0.3187, 0.048, 1.7441], "456": [0.3140251, 0.05024368, 1.7335595], "457": [0.308884, 0.05257304, 1.7208581], "458": [0.3032904, 0.05498056, 1.7059369], "459": [0.2972579, 0.05745872, 1.6887372], "460": [0.2908, 0.06, 1.6692], "461": [0.2839701, 0.06260197, 1.6475287], "462": [0.2767214, 0.06527752, 1.6234127], "463": [0.2689178, 0.06804208, 1.5960223], "464": [0.2604227, 0.07091109, 1.564528], "465": [0.2511, 0.0739, 1.5281], "466": [0.2408475, 0.077016, 1.4861114], "467": [0.2298512, 0.0802664, 1.4395215], "468": [0.2184072, 0.0836668, 1.3898799], "469": [0.2068115, 0.0872328, 1.3387362], "470": [0.19536, 0.09098, 1.28764], "471": [0.1842136, 0.09491755, 1.2374223], "472": [0.1733273, 0.09904584, 1.1878243], "473": [0.1626881, 0.1033674, 1.1387611], "474": [0.1522833, 0.1078846, 1.090148], "475": [0.1421, 0.1126, 1.0419], "476": [0.1321786, 0.117532, 0.9941976], "477": [0.1225696, 0.1226744, 0.9473473], "478": [0.1132752, 0.1279928, 0.9014531], "479": [0.1042979, 0.1334528, 0.8566193], "480": [0.09564, 0.13902, 0.8129501], "481": [0.08729955, 0.1446764, 0.7705173], "482": [0.07930804, 0.1504693, 0.7294448], "483": [0.07171776, 0.1564619, 0.6899136], "484": [0.06458099, 0.1627177, 0.6521049], "485": [0.05795001, 0.1693, 0.6162], "486": [0.05186211, 0.1762431, 0.5823286], "487": [0.04628152, 0.1835581, 0.5504162], "488": [0.04115088, 0.1912735, 0.5203376], "489": [0.03641283, 0.199418, 0.4919673], "490": [0.03201, 0.20802, 0.46518], "491": [0.0279172, 0.2171199, 0.4399246], "492": [0.0241444, 0.2267345, 0.4161836], "493": [0.020687, 0.2368571, 0.3938822], "494": [0.0175404, 0.2474812, 0.3729459], "495": [0.0147, 0.2586, 0.3533], "496": [0.01216179, 0.2701849, 0.3348578], "497": [0.00991996, 0.2822939, 0.3175521], "498": [0.00796724, 0.2950505, 0.3013375], "499": [0.006296346, 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[5.870652e-05, 2.12e-05, 0.0], "776": [5.477028e-05, 1.977855e-05, 0.0], "777": [5.109918e-05, 1.845285e-05, 0.0], "778": [4.767654e-05, 1.721687e-05, 0.0], "779": [4.448567e-05, 1.606459e-05, 0.0], "780": [4.150994e-05, 1.499e-05, 0.0], "781": [3.873324e-05, 1.398728e-05, 0.0], "782": [3.614203e-05, 1.305155e-05, 0.0], "783": [3.372352e-05, 1.217818e-05, 0.0], "784": [3.146487e-05, 1.136254e-05, 0.0], "785": [2.935326e-05, 1.06e-05, 0.0], "786": [2.737573e-05, 9.885877e-06, 0.0], "787": [2.552433e-05, 9.217304e-06, 0.0], "788": [2.379376e-05, 8.592362e-06, 0.0], "789": [2.21787e-05, 8.009133e-06, 0.0], "790": [2.067383e-05, 7.4657e-06, 0.0], "791": [1.927226e-05, 6.959567e-06, 0.0], "792": [1.79664e-05, 6.487995e-06, 0.0], "793": [1.674991e-05, 6.048699e-06, 0.0], "794": [1.561648e-05, 5.639396e-06, 0.0], "795": [1.455977e-05, 5.2578e-06, 0.0], "796": [1.357387e-05, 4.901771e-06, 0.0], "797": [1.265436e-05, 4.56972e-06, 0.0], "798": [1.179723e-05, 4.260194e-06, 0.0], "799": 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用于生成黑体温度与RGB值对应关系表的代码

生成的csv文件中，第一列为黑体温度，第2/3/4列分别为r/g/b值，范围[0, 1]

# 建立黑体温度与RGB颜色的映射表from calculateColour import bb_temperature_to_RGB
import numpy as np
import csvT_list = np.arange(200, 100001, 10)
r_list = []
g_list = []
b_list = []
for T in T_list:if(T % 2000 == 0):print(T)RGB = bb_temperature_to_RGB(T)r = RGB[0]g = RGB[1]b = RGB[2]r_list.append(r)g_list.append(g)b_list.append(b)
with open("TCTable.csv", "w") as f:spamwriter = csv.writer(f, delimiter=",")for idx in range(len(T_list)):spamwriter.writerow([T_list[idx], r_list[idx], g_list[idx], b_list[idx]])