GoPiGo 小汽車︰格點圖像算術《色彩空間》灰階‧丁


Standard illuminant

A standard illuminant is a theoretical source of visible light with a profile (its spectral power distribution) which is published. Standard illuminants provide a basis for comparing images or colors recorded under different lighting.

 Relative spectral power distributions (SPDs) of CIE illuminants A, B, and C from 380 nm to 780 nm.

CIE illuminants

The International Commission on Illumination (usually abbreviated CIE for its French name) is the body responsible for publishing all of the well-known standard illuminants. Each of these is known by a letter or by a letter-number combination.

Illuminants A, B, and C were introduced in 1931, with the intention of respectively representing average incandescent light, direct sunlight, and average daylight. Illuminants D represent phases of daylight, Illuminant E is the equal-energy illuminant, while Illuminants F represent fluorescent lamps of various composition.

There are instructions on how to experimentally produce light sources (“standard sources”) corresponding to the older illuminants. For the relatively newer ones (such as series D), experimenters are left to measure to profiles of their sources and compare them to the published spectra:[1]

At present no artificial source is recommended to realize CIE standard illuminant D65 or any other illuminant D of different CCT. It is hoped that new developments in light sources and filters will eventually offer sufficient basis for a CIE recommendation.

— CIE, Technical Report (2004) Colorimetry, 3rd ed., Publication 15:2004, CIE Central Bureau, Vienna

Nevertheless, they do provide a measure, called the Metamerism Index, to assess the quality of daylight simulators.[2][3] The Metamerism Index tests how well five sets of metameric samples match under the test and reference illuminant. In a manner similar to the color rendering index, the average difference between the metamers is calculated.[4]


下之所見,恐生爭議不知所云也。舉例來說『白物』不在『白光』裡,果真是『白色』的嗎?沒有所謂『物理白』,『白物』實隨『日光白』!且借 ColorPy 之『D65 sRGB』預設的『色彩空間』,看看到底 □□ 『白是不白』?!


【D65 光源】


【CIE A 光源】




【5778 K 黑體輻射光源】


【如果太陽是 6500K 黑體】



※ 註




一個給定目標亮度在不同的場景中可以引起不同的明度感覺;比如White錯覺Wertheimer-Benary錯覺(Wertheimer-Benary effect)。


\mu ={R+G+B \over 3}






 黑色曲線為亮適應光度函數曲線,綠色曲線為暗適應光度函數曲線。實線為CIE 1931標準。斷續線為1978年修正數據。點線為2005年修正數據。橫坐標單位為nm。

人眼能相當精確地判斷兩種顏色的光亮暗感覺是否相同。所以為了確定眼睛的光譜響應,可將各種波長的光引起亮暗感覺所需的輻射通量進行比較。在較明亮環境中人的視覺對波長為555.016nm的綠色光最為敏感。設任意波長為  \lambda 的光和波長為555.016nm的光產生同樣亮暗感覺所需的輻射通量分別為 \Psi _{{555.016}} \Psi _{{\lambda }},把後者和前者之比

  V(\lambda )={\frac {\Psi _{{555.016}}}{\Psi _{{\lambda }}}}

叫做光度函數(luminosity function)或視見函數(visual sensitivity function)。例如,實驗表明,1mW的555.0nm綠光與2.5W的400.0nm紫光引起的亮暗感覺相同。於是在400.0nm的光度函數值為

V(400.0nm)={\frac {10^{{-3}}}{2.5}}=0.0004.

衡量光通量的大小,要以光度函數為權重把輻射通量折合成對人眼的有效數量。對波長為  \lambda 的光,輻射強度為  \psi (\lambda ),光通量為  \Phi _{v},則有

\Phi _{v}=K_{{max}}\int V(\lambda )\psi (\lambda )d\lambda

式中  K_{{max}}是波長為555.016nm的光功當量,也叫做最大光功當量,其值為683 lm/W。




1 \cdot \vec{R} + 1 \cdot \vec{G} + 1 \cdot \vec{B} = \vec{W}

實質確定了『所選擇』最大『強度』之『紅』 \vec{R} 、『綠』 \vec{G} 、『藍』 \vec{B} 的哩◎


r \cdot \vec{R} + g \cdot \vec{G} + b \cdot \vec{B}

定義了一個 (r, g, b) 『色彩空間』乎??這些 r, g, b 之所以在 [ 0, 1 ] 區間內『取值』,表達『相對最大』之『百分比』耶!!

Numeric representations

 A typical RGB color selector in graphic software. Each slider ranges from 0 to 255.

 Hexadecimal 8-bit RGB representations of the main 125 colors

A color in the RGB color model is described by indicating how much of each of the red, green, and blue is included. The color is expressed as an RGB triplet (r,g,b), each component of which can vary from zero to a defined maximum value. If all the components are at zero the result is black; if all are at maximum, the result is the brightest representable white.

These ranges may be quantified in several different ways:

  • From 0 to 1, with any fractional value in between. This representation is used in theoretical analyses, and in systems that use floating point representations.
  • Each color component value can also be written as a percentage, from 0% to 100%.
  • In computers, the component values are often stored as integer numbers in the range 0 to 255, the range that a single 8-bit byte can offer. These are often represented as either decimal or hexadecimal numbers.
  • High-end digital image equipment are often able to deal with larger integer ranges for each primary color, such as 0..1023 (10 bits), 0..65535 (16 bits) or even larger, by extending the 24-bits (three 8-bit values) to 32-bit, 48-bit, or 64-bit units (more or less independent from the particular computer’s word size).

For example, brightest saturated red is written in the different RGB notations as:

Notation RGB triplet
Arithmetic (1.0, 0.0, 0.0)
Percentage (100%, 0%, 0%)
Digital 8-bit per channel (255, 0, 0) or sometimes
#FF0000 (hexadecimal)
Digital 16-bit per channel (65535, 0, 0)

In many environments, the component values within the ranges are not managed as linear (that is, the numbers are nonlinearly related to the intensities that they represent), as in digital cameras and TV broadcasting and receiving due to gamma correction, for example.[15] Linear and nonlinear transformations are often dealt with via digital image processing. Representations with only 8 bits per component are considered sufficient if gamma encoding is used.[16]

Following is the mathematical relationship between RGB space to HSI space (hue, saturation, and intensity: HSI color space):

{\displaystyle {\begin{aligned}I&={\frac {R+G+B}{3}}\\S&=1\,-\,{\frac {3}{(R+G+B)}}\,\min(R,G,B)\\H&=\cos ^{-1}\left({\frac {{\frac {1}{2}}((R-G)+(R-B))}{(R-G)^{2}+(R-B)(G-B)}}\right)^{\frac {1}{2}}\end{aligned}}}

Color depth

The RGB color model is one of the most common ways to encode color in computing, and several different binary digital representations are in use. The main characteristic of all of them is the quantization of the possible values per component (technically a Sample (signal) ) by using only integer numbers within some range, usually from 0 to some power of two minus one (2n – 1) to fit them into some bit groupings. Encodings of 1, 2, 4, 5, 8 and 16 bits per color are commonly found; the total number of bits used for an RGB color is typically called the color depth.

Geometric representation


The RGB color model mapped to a cube. The horizontal x-axis as red values increasing to the left, y-axis as blue increasing to the lower right and the vertical z-axis as green increasing towards the top. The origin, black is the vertex hidden from view.

See also RGB color space

Since colors are usually defined by three components, not only in the RGB model, but also in other color models such as CIELAB and Y’UV, among others, then a three-dimensional volume is described by treating the component values as ordinary cartesian coordinates in a euclidean space. For the RGB model, this is represented by a cube using non-negative values within a 0–1 range, assigning black to the origin at the vertex (0, 0, 0), and with increasing intensity values running along the three axes up to white at the vertex (1, 1, 1), diagonally opposite black.

An RGB triplet (r,g,b) represents the three-dimensional coordinate of the point of the given color within the cube or its faces or along its edges. This approach allows computations of the color similarity of two given RGB colors by simply calculating the distance between them: the shorter the distance, the higher the similarity. Out-of-gamut computations can also be performed this way.









GoPiGo 小汽車︰格點圖像算術《色彩空間》灰階‧丙

『眼睛』並非單純的『物理量』度量儀表,『感官知覺』雖有相當之『共性』,『差異』還是存在的。因此明白『完美的黑』 ─ 入眼光譜為零 ─ 與『理想的白』─ 所有可見光輻射強度均等 ─ 之所以為重要『色覺參考』思過半矣!或可『讀』『解』公孫龍之『白馬非馬論』乎?






』色是各種叫白色東西,所共有顏色。有這種顏色嗎?然而『白馬』的白、『白羊』的白和『白狐』的白,雖然在其類中都是稱之為者,彼此類間確又有所不同,當然也與『共相抽象的『』色有所不同。比方說把『白羊』之『白』用之於馬,『那馬』──  羊白之馬 ──叫做『黃馬』,為什麼呢?因為『白羊』之白近之於馬類』之然它羊類』之







馬『』是看起來『』馬的東西,『』字的構創之來歷。但是『』能回答『馬是什麼嗎?』,也許只因人們心中有著的『概念』,它不是的『概念』,也不同於的『概念』,由於馬牛羊外『不同 ,正以『區別』這些不同的動物。想當日創生『』『』字之時,卻遇著了『牛頭馬面』來訪,這兩個字會『寫的』和今天不一樣嗎?




── 弗雷格的理念就是想讓『表達』和『論證清晰明白 ──

─── 摘自《白馬非馬論








格拉斯曼定律 (色彩)



若兩單色光組合成一測試色光,則觀測者感知到的三原色數值為兩單色光分別被單獨觀測的三原色數值之和。換句話說,如果光束一及光束二為單色光,而  {\displaystyle (R_{1},G_{1},B_{1})}  {\displaystyle (R_{2},G_{2},B_{2})}分別為觀測者對光束一及光束二的感知三原色數值,當此二光束合併時,觀測者感知的三原色數值為  {\displaystyle (R,G,B)},其中:

{\displaystyle R=R_{1}+R_{2}\,}
  {\displaystyle G=G_{1}+G_{2}\,}
  {\displaystyle B=B_{1}+B_{2}\,}


{\displaystyle R=\int _{0}^{\infty }I(\lambda )\,{\bar {r}}(\lambda )\,d\lambda }
  {\displaystyle G=\int _{0}^{\infty }I(\lambda )\,{\bar {g}}(\lambda )\,d\lambda }
  {\displaystyle B=\int _{0}^{\infty }I(\lambda )\,{\bar {b}}(\lambda )\,d\lambda }

  I(\lambda)為該光束對波長的強度分布;  {\displaystyle {\bar {r}}(\lambda )}  {\displaystyle {\bar {g}}(\lambda )}  {\displaystyle {\bar {b}}(\lambda )}則分別為人眼中三種錐狀細胞對不同波長的反應強度。


自能曉得任選『自然』或『科技』中『可調變』之『紅』、『綠』 、『藍』,比方講用







Cathode ray tubes

 Spectra of constituent blue, green and red phosphors in a common cathode ray tube.

Cathode ray tubes produce signal-generated light patterns in a (typically) round or rectangular format. Bulky CRTs were used in the black-and-white household television (“TV”) sets that became popular in the 1950s, as well as first-generation, tube-based color TVs, and most earlier computer monitors. CRTs have also been widely used in scientific and engineering instrumentation, such as oscilloscopes, usually with a single phosphor color, typically green. Phosphors for such applications may have long afterglow, for increased image persistence.

The phosphors can be deposited as either thin film, or as discrete particles, a powder bound to the surface. Thin films have better lifetime and better resolution, but provide less bright and less efficient image than powder ones. This is caused by multiple internal reflections in the thin film, scattering the emitted light.

White (in black-and-white): The mix of zinc cadmium sulfide and zinc sulfide silver, the ZnS:Ag+(Zn,Cd)S:Ag is the white P4 phosphor used in black and white television CRTs. Mixes of yellow and blue phosphors are usual. Mixes of red, green and blue, or a single white phosphor, can also be encountered.

Red: Yttrium oxidesulfide activated with europium is used as the red phosphor in color CRTs. The development of color TV took a long time due to the search for a red phosphor. The first red emitting rare earth phosphor, YVO4:Eu3+, was introduced by Levine and Palilla as a primary color in television in 1964.[21] In single crystal form, it was used as an excellent polarizer and laser material.[22]

Yellow: When mixed with cadmium sulfide, the resulting zinc cadmium sulfide (Zn,Cd)S:Ag, provides strong yellow light.

Green: Combination of zinc sulfide with copper, the P31 phosphor or ZnS:Cu, provides green light peaking at 531 nm, with long glow.

Blue: Combination of zinc sulfide with few ppm of silver, the ZnS:Ag, when excited by electrons, provides strong blue glow with maximum at 450 nm, with short afterglow with 200 nanosecond duration. It is known as the P22B phosphor. This material, zinc sulfide silver, is still one of the most efficient phosphors in cathode ray tubes. It is used as a blue phosphor in color CRTs.

The phosphors are usually poor electrical conductors. This may lead to deposition of residual charge on the screen, effectively decreasing the energy of the impacting electrons due to electrostatic repulsion (an effect known as “sticking”). To eliminate this, a thin layer of aluminium (about 100 nm) is deposited over the phosphors, usually by vacuum evaporation, and connected to the conductive layer inside the tube. This layer also reflects the phosphor light to the desired direction, and protects the phosphor from ion bombardment resulting from an imperfect vacuum.

To reduce the image degradation by reflection of ambient light, contrast can be increased by several methods. In addition to black masking of unused areas of screen, the phosphor particles in color screens are coated with pigments of matching color. For example, the red phosphors are coated with ferric oxide (replacing earlier Cd(S,Se) due to cadmium toxicity), blue phosphors can be coated with marine blue (CoO·nAl2O3) or ultramarine (Na8Al6Si6O24S2). Green phosphors based on ZnS:Cu do not have to be coated due to their own yellowish color.[2]















GoPiGo 小汽車︰格點圖像算術《色彩空間》灰階‧乙



Grey (British English) or gray (American English;[citation needed] see spelling differences) is an intermediate color between black and white. It is a neutral or achromatic color, meaning literally that it is a color “without color.”[2] This means that there are equal components of red, green, and blue. The variations in intensity of these colors uniformly produce different shades of grey. It is the color of a cloud-covered sky, of ash and of lead.[3]

The first recorded use of grey as a color name in the English language was in AD 700.[4] Grey is the dominant spelling in European and Commonwealth English, although gray remained in common usage in the UK until the second half of the 20th century.[5] Gray has been the preferred American spelling since approximately 1825,[6] although grey is an accepted variant.[7][8]

In Europe and the United States, surveys show that grey is the color most commonly associated with neutrality, conformity, boredom, uncertainty, old age, indifference, and modesty. Only one percent of respondents chose it as their favorite color.[9]



因為『純灰』指稱沒有『色相』 HUE 之『顏色』也◎




顏色名稱 紅綠藍含量 角度 代表物體
紅色 R255,G0,B0 血液草莓
橙色 R255,G128,B0 30° 橙子
黃色 R255,G255,B0 60° 香蕉杧果
黃綠 R128,G255,B0 90° 檸檬
綠色 R0,G255,B0 120° 樹葉
青綠 R0,G255,B128 150° 軍裝
青色 R0,G255,B255 180° 水面天空
靛藍 R0,G128,B255 210° 水面天空
藍色 R0,G0,B255 240° 樹根墨水
紫色 R128,G0,B255 270° 葡萄茄子
品紅 R255,G0,B255 300° 桃子
紫紅 R255,G0,B128 330° 墨水




Black is the darkest color, resulting from the absence or complete absorption of light. Like white and grey, it is an achromatic color, literally a color without hue.[1] It is one of the four primary colors in the CMYK color model, along with cyan, yellow, and magenta, used in color printing to produce all the other colors. Black is often used to represent darkness; it is the symbolic opposite of white (or brightness).

Black was one of the first colors used by artists in neolithic cave paintings. In the 14th century, it began to be worn by royalty, the clergy, judges and government officials in much of Europe. It became the color worn by English romantic poets, businessmen and statesmen in the 19th century, and a high fashion color in the 20th century.[2]

In the Roman Empire, it became the color of mourning, and over the centuries it was frequently associated with death, evil, witches and magic. According to surveys in Europe and North America, it is the color most commonly associated with mourning, the end, secrets, magic, force, violence, evil, and elegance.[3]



In the visible spectrum, black is the absorption of all colors.

Black can be defined as the visual impression experienced when no visible light reaches the eye. Pigments or dyes that absorb light rather than reflect it back to the eye “look black”. A black pigment can, however, result from a combination of several pigments that collectively absorb all colors. If appropriate proportions of three primary pigments are mixed, the result reflects so little light as to be called “black”.

This provides two superficially opposite but actually complementary descriptions of black. Black is the absorption of all colors of light, or an exhaustive combination of multiple colors of pigment. See also primary colors.

In physics, a black body is a perfect absorber of light, but, by a thermodynamic rule, it is also the best emitter. Thus, the best radiative cooling, out of sunlight, is by using black paint, though it is important that it be black (a nearly perfect absorber) in the infrared as well.

In elementary science, far ultraviolet light is called “black light” because, while itself unseen, it causes many minerals and other substances to fluoresce.

On January 16, 2008, researchers from Troy, New York‘s Rensselaer Polytechnic Institute announced the creation of the then darkest material on the planet. The material, which reflected only 0.045 percent of light, was created from carbon nanotubes stood on end. This is 1/30 of the light reflected by the current standard for blackness, and one third the light reflected by the previous record holder for darkest substance.[30] As of February 2016, the current darkest material known is claimed to be Vantablack.[31][32]

A material is said to be black if most incoming light is absorbed equally in the material. Light (electromagnetic radiation in the visible spectrum) interacts with the atoms and molecules, which causes the energy of the light to be converted into other forms of energy, usually heat. This means that black surfaces can act as thermal collectors, absorbing light and generating heat (see Solar thermal collector).

Absorption of light is contrasted by transmission, reflection and diffusion, where the light is only redirected, causing objects to appear transparent, reflective or white respectively.


Spectral color

A spectral color is a color that is evoked by a single wavelength of light in the visible spectrum, or by a relatively narrow band of wavelengths, also known as monochromatic light. Every wavelength of visible light is perceived as a spectral color, in a continuous spectrum; the colors of sufficiently close wavelengths are indistinguishable.

The spectrum is often divided into named colors, though any division is somewhat arbitrary: the spectrum is continuous. Traditional colors include: red, orange, yellow, green, blue, and violet.

The division used by Isaac Newton, in his color wheel, was: red, orange, yellow, green, blue, indigo and violet; a mnemonic for this order is “Roy G. Biv“. In modern divisions of the spectrum, indigo is often omitted.

One needs at least trichromatic color vision for there to be a distinction between spectral and non-spectral colours[dubious ]: trichromacy gives a possibility to perceive both hue and saturation in the chroma. In color models capable of representing spectral colors,[1] such as CIELUV, a spectral color has the maximal saturation.

The CIE xy chromaticity diagram. The spectrum colors are the colors on the horseshoe-shaped curve on the outside of the diagram. All other colors are not spectral: the bottom straight line is the line of purples, whilst within the interior of the diagram are unsaturated colors that are various mixtures of a spectral color or a purple color with white, a grayscale color. White is in the central part of the interior of the diagram, since when all colors of light are mixed together, they produce white.

In color spaces

 This metrically accurate diagram shows that the spectral locus is almost flat on the red – bright green segment, is strongly curved around green, and becomes less curved between green/cyan and blue

In color spaces which include all, or most spectral colors, they form a part of boundary of the set of all real colors. If luminance is counted, then spectral colors form a surface, otherwise their locus is a curve in a two-dimensional chromaticity space.

Theoretically, only RGB-implemented colors which might be really spectral are its primaries: red, green, and blue, whereas any other (mixed) color is inherently non-spectral. But due to different chromaticity properties of different spectral segments, and also due to practical limitations of light sources, the actual distance between RGB pure color wheel colors and spectral colors shows a complicated dependence on the hue. Due to location of R and G primaries near the almost “flat” spectral segment, RGB color space is reasonably good with approximating spectral orange, yellow, and bright (yellowish) green, but is especially poor in reaching a visual appearance of spectral colors between green and blue, as well as extreme spectral colors. The sRGB standard has an additional problem with its “red” primary which is shifted to orange due to a trade-off between purity of red and its reasonable luminance, so that the red spectral became unreachable. Some samples in the table below provide only rough approximations of spectral and near-spectral colors.

CMYK is usually even poorer than RGB in its reach of spectral colors, with notable exception of process yellow, which is rather close to spectral colors due to aforementioned flatness of the spectral locus in the red–green segment.

Note that spectral color are universally included to scientific color models such as CIE 1931, but industrial and consumer color spaces such as sRGB, CMYK, and Pantone, do not include any of spectral colors.






以前許多科學家認為,白光是最基本的光,其他顏色的光是在白光上添加了某些元素。但英國科學家艾薩克·牛頓的研究,揭露了白光是由光譜中各種顏色的光組成的。現在光學中,稱黑體在加熱到不同溫度釋放出的輻射光都叫做「白光」,最低發光溫度為2848 K,相當於白熾燈泡的溫度;劇場中白光等溫度達到3200K;白天天光的溫度相當於5400K,但是由多種顏色的光組成的,從最低溫度的紅光到將近25000K的紫光都包括在內。但並不是所有黑體輻射都是白光,宇宙背景輻射也是黑體輻射,只有3K。




White is an achromatic color, a color without hue.[1]

Light with a spectral composition that stimulates all three types of the color sensitive cone cells of the human eye in nearly equal amounts appears white. White is one of the most common colors in nature, the color of sunlight, and the color of sunlight reflected by snow, milk, chalk, limestone and other common minerals. In many cultures white represents or signifies purity, innocence, and light, and is the symbolic opposite of black, or darkness. According to surveys in Europe and the United States, white is the color most often associated with perfection, the good, honesty, cleanliness, the beginning, the new, neutrality, and exactitude.[2]

In ancient Egypt and ancient Rome, priestesses wore white as a symbol of purity, and Romans wore a white toga as a symbol of citizenship. In the Middle Ages and Renaissance a white unicorn symbolized chastity, and a white lamb sacrifice and purity; the widows of kings dressed in white rather than black as the color of mourning. It sometimes symbolizes royalty; it was the color of the French kings (black being the color of the queens) and of the monarchist movement after the French Revolution as well as of the movement called the White Russians (not to be confounded with Belarus, literally “White Russia”) who fought the Bolsheviks during the Russian Civil War (1917–1922). Greek and Roman temples were faced with white marble, and beginning in the 18th century, with the advent of neoclassical architecture, white became the most common color of new churches, capitols and other government buildings, especially in the United States. It was also widely used in 20th century modern architecture as a symbol of modernity, simplicity and strength.

White is an important color for almost all world religions. The Pope, the head of the Roman Catholic Church, has worn white since 1566, as a symbol of purity and sacrifice. In Islam, and in the Shinto religion of Japan, it is worn by pilgrims; and by the Brahmins in India. In Western cultures and in Japan, white is the most common color for wedding dresses, symbolizing purity and virginity. In many Asian cultures, white is also the color of mourning.[3]

The white color on television screens and computer monitors is created with the RGB color model by mixing red, green and blue light at equal intensities.[citation needed]



White is the color the human visual system senses when the incoming light to the eye stimulates all three types of color sensitive cone cells in the eye in nearly equal amounts.[24] Materials that do not emit light themselves appear white if their surfaces reflect back most of the light that strikes them in a diffuse way.

In 1666, Isaac Newton demonstrated that white light could be broken up into its composite colors by passing it through a prism, then using a second prism to reassemble them. Before Newton, most scientists believed that white was the fundamental color of light.

White light can be generated by the sun, by stars, or by earthbound sources such as fluorescent lamps, white LEDs and incandescent bulbs. On the screen of a color television or computer, white is produced by mixing the primary colors of light: red, green and blue (RGB) at full intensity, a process called additive mixing (see image below). White light can be fabricated using light with only two wavelengths, for instance by mixing light from a red and cyan laser or yellow and blue lasers. This light will however have very few practical applications since color rendering of objects will be greatly distorted.

The fact that light sources with vastly different spectral power distributions can result in a similar sensory experience is due to the way the light is processed by the visual system. One color that arise from two different spectral power distributions is called a metamerism.

The International Commission on Illumination defines white (adapted) as “a color stimulus that an observer who is adapted to the viewing environment would judge to be perfectly achromatic and to have a luminance factor of unity. The color stimulus that is considered to be the adapted white may be different at different locations within a scene.[25]

The adaptation mentioned in the CIE definition above is the chromatic adaptation by which the same colored object in a scene experienced under very different illuminations will be perceived as having nearly the same color. The same principle is used in photography and cinematography where the choice of white point determines a transformation of all other color stimuli. Changes in or manipulation of the white point can be used to explain some optical illusions such as The dress.

Many of the light sources that emit white light emit light at almost all visible wavelengths (sun light, incandescent lamps of various Color temperatures). This has led to the notion that white light can be defined as a mixture of “all colors” or “all visible wavelengths”. This misconception is widespread[26][27] and might originally stem from the fact that Newton discovered that sunlight is composed of light with wavelengths across the visible spectrum. Concluding that since “all colors” produce white light then white must be made up of “all colors” is a common logical error called affirming the consequent, which might be the cause of the misunderstanding.

Why snow, clouds and beaches are white

Snow is a mixture of air and tiny ice crystals. When white sunlight enters snow, very little of the spectrum is absorbed; almost all of the light is reflected or scattered by the air and water molecules, so the snow appears to be the color of sunlight, white. Sometimes the light bounces around inside the ice crystals before being scattered, making the snow seem to sparkle.[28]

In the case of glaciers, the ice is more tightly pressed together and contains little air. As sunlight enters the ice, more light of the red spectrum is absorbed, so the light scattered will be bluish.[29]

Clouds are white for the same reason as ice. They are composed of water droplets or ice crystals mixed with air, very little light that strikes them is absorbed, and most of the light is scattered, appearing to the eye as white. Shadows of other clouds above can make clouds look gray, and some clouds have their own shadow on the bottom of the cloud.[30]

Many mountains with winter or year-round snow cover are named accordingly: Mauna Kea means white mountain in Hawaiian, Mont Blanc means white mountain in French. Changbai Mountains literally meaning perpetually white mountains, marks the border between China and Korea.

Beaches with sand containing high amounts of quartz or eroded limestone also appear white, since quartz and limestone reflect or scatter sunlight, rather than absorbing it. Tropical white sand beaches may also have a high quantity of white calcium carbonate from tiny bits of seashells ground to fine sand by the action of the waves.[31]



白點white point),在技術文檔中常被稱作參考白色(reference white)或目標白色(target white),是一組三色視覺值(tristimulus values)或色度值(Chromaticity)。它被用來在圖像捕獲、編碼和再現時定義白色[1]白點的數值不是恆定的,需要按照使用場合的不同對它的數值做出相應的更改。例如在室內攝影時常用白熾燈來照明,而白熾燈的光線色溫相對日光較低,因而呈現出偏橙的顏色。倘若在這種情況下依然使用以日光為準的白點數值來定義白色,攝影的結果就會出現偏色的問題。

White point

A white point (often referred to as reference white or target white in technical documents) is a set of tristimulus values or chromaticity coordinates that serve to define the color “white” in image capture, encoding, or reproduction.[1] Depending on the application, different definitions of white are needed to give acceptable results. For example, photographs taken indoors may be lit by incandescent lights, which are relatively orange compared to daylight. Defining “white” as daylight will give unacceptable results when attempting to color-correct a photograph taken with incandescent lighting.


An illuminant is characterized by its relative spectral power distribution. The white point of an illuminant is the chromaticity of a white object under the illuminant, and can be specified by chromaticity coordinates, such as the x, y coordinates on the CIE 1931 chromaticity diagram (hence the use of the relative SPD and not the absolute SPD, because the white point is only related to color and unaffected by intensity).[2]

Illuminant and white point are separate concepts. For a given illuminant, its white point is uniquely defined. A given white point, on the other hand, generally does not uniquely correspond to only one illuminant. From the commonly used CIE 1931 chromaticity diagram, it can be seen that almost all non-spectral colors (all except those on the line of purples), including colors described as white, can be produced by infinitely many combinations of spectral colors, and therefore by infinitely many different illuminant spectra.

Although there is generally no one-to-one correspondence between illuminants and white points, in the case of the CIE D-series standard illuminants, the spectral power distributions are mathematically derivable from the chromaticity coordinates of the corresponding white points.[3]

Knowing the illuminant’s spectral power distribution, the reflectance spectrum of the specified white object (often taken as unity), and the numerical definition of the observer allows the coordinates of the white point in any color space to be defined. For example, one of the simplest illuminants is the “E” or “Equal Energy” spectrum. Its spectral power distribution is flat, giving the same power per unit wavelength at any wavelength. In terms of both the 1931 and 1964 CIE XYZ color spaces, its color coordinates are [k, k, k], where k is a constant, and its chromaticity coordinates are [x, y] = [1/3, 1/3].

Diagram of the CIE 1931 color space that shows the Rec. 2020 (UHDTV) color space in the outer triangle and Rec. 709 (HDTV) color space in the inner triangle. Both Rec. 2020 and Rec. 709 use Illuminant D65 for the white point.











GoPiGo 小汽車︰格點圖像算術《色彩空間》灰階‧甲



Radiometry is a set of techniques for measuring electromagnetic radiation, including visible light. Radiometric techniques in optics characterize the distribution of the radiation’s power in space, as opposed to photometric techniques, which characterize the light’s interaction with the human eye. Radiometry is distinct from quantum techniques such as photon counting.

The use of radiometers to determine the temperature of objects and gasses by measuring radiation flux is called pyrometry. Handheld pyrometer devices are often marketed as infrared thermometers.

Radiometry is important in astronomy, especially radio astronomy, and plays a significant role in Earth remote sensing. The measurement techniques categorized as radiometry in optics are called photometry in some astronomical applications, contrary to the optics usage of the term.

Spectroradiometry is the measurement of absolute radiometric quantities in narrow bands of wavelength.[1]



Photometry (optics)

Photometry is the science of the measurement of light, in terms of its perceived brightness to the human eye.[1] It is distinct from radiometry, which is the science of measurement of radiant energy (including light) in terms of absolute power. In modern photometry, the radiant power at each wavelength is weighted by a luminosity function that models human brightness sensitivity. Typically, this weighting function is the photopic sensitivity function, although the scotopic function or other functions may also be applied in the same way.

Photopic (daytime-adapted, black curve) and scotopic [1] (darkness-adapted, green curve) luminosity functions. The photopic includes the CIE 1931 standard [2] (solid), the Judd-Vos 1978 modified data [3] (dashed), and the Sharpe, Stockman, Jagla & Jägle 2005 data [4] (dotted). The horizontal axis is wavelength in nm.




色度學(Colorimetry),又名比色法,是量化和物理上描述人們顏色知覺的科學和技術。 色度學同光譜學(spectrophotometry)相近 ,但是色度學更關心的是於人們顏色知覺物理相關的光譜。最常用的是CIE1931色彩空間和相關的數值。



In photography and computing, a grayscale or greyscale digital image is an image in which the value of each pixel is a single sample, that is, it carries only intensity information. Images of this sort, also known as black-and-white, are composed exclusively of shades of gray, varying from black at the weakest intensity to white at the strongest.[1]

Grayscale images are distinct from one-bit bi-tonal black-and-white images, which in the context of computer imaging are images with only two colors, black and white (also called bilevel or binary images). Grayscale images have many shades of gray in between.

Grayscale images are often the result of measuring the intensity of light at each pixel in a single band of the electromagnetic spectrum (e.g. infrared, visible light, ultraviolet, etc.), and in such cases they are monochromatic proper when only a given frequency is captured. But also they can be synthesized from a full color image; see the section about converting to grayscale.

Numerical representations


A sample grayscale image

The intensity of a pixel is expressed within a given range between a minimum and a maximum, inclusive. This range is represented in an abstract way as a range from 0 (total absence, black) and 1 (total presence, white), with any fractional values in between. This notation is used in academic papers, but this does not define what “black” or “white” is in terms of colorimetry.

Another convention is to employ percentages, so the scale is then from 0% to 100%. This is used for a more intuitive approach, but if only integer values are used, the range encompasses a total of only 101 intensities, which are insufficient to represent a broad gradient of grays. Also, the percentile notation is used in printing to denote how much ink is employed in halftoning, but then the scale is reversed, being 0% the paper white (no ink) and 100% a solid black (full ink).

In computing, although the grayscale can be computed through rational numbers, image pixels are stored in binary, quantized form. Some early grayscale monitors can only show up to sixteen (4-bit) different shades, but today grayscale images (as photographs) intended for visual display (both on screen and printed) are commonly stored with 8 bits per sampled pixel, which allows 256 different intensities (i.e., shades of gray) to be recorded, typically on a non-linear scale. The precision provided by this format is barely sufficient to avoid visible banding artifacts, but very convenient for programming because a single pixel then occupies a single byte.

Technical uses (e.g. in medical imaging or remote sensing applications) often require more levels, to make full use of the sensor accuracy (typically 10 or 12 bits per sample) and to guard against roundoff errors in computations. Sixteen bits per sample (65,536 levels) is a convenient choice for such uses, as computers manage 16-bit words efficiently. The TIFF and the PNG (among other) image file formats support 16-bit grayscale natively, although browsers and many imaging programs tend to ignore the low order 8 bits of each pixel.

No matter what pixel depth is used, the binary representations assume that 0 is black and the maximum value (255 at 8 bpp, 65,535 at 16 bpp, etc.) is white, if not otherwise noted.



Converting color to grayscale

Conversion of a color image to grayscale is not unique; different weighting of the color channels effectively represent the effect of shooting black-and-white film with different-colored photographic filters on the cameras.

Colorimetric (luminance-preserving) conversion to grayscale

A common strategy is to use the principles of photometry or, more broadly, colorimetry to match the luminance of the grayscale image to the luminance of the original color image.[2][3] This also ensures that both images will have the same absolute luminance, as can be measured in its SI units of candelas per square meter, in any given area of the image, given equal whitepoints. In addition, matching luminance provides matching perceptual lightness measures, such as L* (as in the 1976 CIE Lab color space) which is determined by the linear luminance Y (as in the CIE 1931 XYZ color space) which we will refer to here as Ylinear to avoid any ambiguity.

To convert a color from a colorspace based on an RGB color model to a grayscale representation of its luminance, weighted sums must be calculated in a linear RGB space, that is, after the gamma compression function has been removed first via gamma expansion.[4]

For the sRGB color space, gamma expansion is defined as

C_{\mathrm {linear} }={\begin{cases}{\frac {C_{\mathrm {srgb} }}{12.92}},&C_{\mathrm {srgb} }\leq 0.04045\\\left({\frac {C_{\mathrm {srgb} }+0.055}{1.055}}\right)^{2.4},&C_{\mathrm {srgb} }>0.04045\end{cases}}

where Csrgb represents any of the three gamma-compressed sRGB primaries (Rsrgb, Gsrgb, and Bsrgb, each in range [0,1]) and Clinear is the corresponding linear-intensity value (Rlinear, Glinear, and Blinear, also in range [0,1]). Then, linear luminance is calculated as a weighted sum of the three linear-intensity values. The sRGB color space is defined in terms of the CIE 1931 linear luminance Ylinear, which is given by

{\displaystyle Y_{\mathrm {linear} }=0.2126R_{\mathrm {linear} }+0.7152G_{\mathrm {linear} }+0.0722B_{\mathrm {linear} }}.[5]

The coefficients represent the measured intensity perception of typical trichromat humans, depending on the primaries being used; in particular, human vision is most sensitive to green and least sensitive to blue. To encode grayscale intensity in linear RGB, each of the three primaries can be set to equal the calculated linear luminance Y (replacing R,G,B by Y,Y,Y to get this linear grayscale). Linear luminance typically needs to be gamma compressed to get back to a conventional non-linear representation. For sRGB, each of its three primaries is then set to the same gamma-compressed Ysrgb given by the inverse of the gamma expansion above as

{\displaystyle Y_{\mathrm {srgb} }={\begin{cases}12.92\ Y_{\mathrm {linear} },&Y_{\mathrm {linear} }\leq 0.0031308\\1.055\ Y_{\mathrm {linear} }^{1/2.4}-0.055,&Y_{\mathrm {linear} }>0.0031308.\end{cases}}}

In practice, because the three sRGB components are then equal, it is only necessary to store these values once in sRGB-compatible image formats that support a single-channel representation. Web browsers and other software that recognizes sRGB images will typically produce the same rendering for such a grayscale image as it would for an sRGB image having the same values in all three color channels.



ColorPy/colorpy/ – Conversions between color models
Defines several color models, and conversions between them.
The models are:

xyz – CIE XYZ color space, based on the 1931 matching functions for a 2 degree field of view.

Spectra are converted to xyz color values by integrating with the matching functions in

xyz colors are often handled as absolute values, conventionally written with uppercase letters XYZ, or as scaled values (so that X+Y+Z = 1.0), conventionally written with lowercase letters xyz.
This is the fundamental color model around which all others are based.

rgb – Colors expressed as red, green and blue values, in the nominal range 0.0 – 1.0. These are linear color values, meaning that doubling the number implies a doubling of the light intensity. rgb color values may be out of range (greater than 1.0, or negative), and do not account for gamma correction. They should not be drawn directly.

irgb – Displayable color values expressed as red, green and blue values, in the range 0 – 255. These have been adjusted for gamma correction, and have been clipped into the displayable range 0 – 255.
These color values can be drawn directly.

Luv – A nearly perceptually uniform color space.

Lab – Another nearly perceptually uniform color space.

As far as I know, the Luv and Lab spaces are of similar quality.
Neither is perfect, so perhaps try each, and see what works best for your application.

The models store color values as 3-element NumPy vectors.
The values are stored as floats, except for irgb, which are stored as integers.





Power law for video display

A gamma characteristic is a power-law relationship that approximates the relationship between the encoded luma in a television system and the actual desired image luminance.

With this nonlinear relationship, equal steps in encoded luminance correspond roughly to subjectively equal steps in brightness. Ebner and Fairchild[9] used an exponent of 0.43 to convert linear intensity into lightness (luma) for neutrals; the reciprocal, approximately 2.33 (quite close to the 2.2 figure cited for a typical display subsystem), was found to provide approximately optimal perceptual encoding of grays.

The following illustration shows the difference between a scale with linearly-increasing encoded luminance signal (linear gamma-compressed luma input) and a scale with linearly-increasing intensity scale (linear luminance output).

Linear encoding VS = 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Linear intensity  I = 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0




【 Linear encoding 】


【 Linear intensity  】


On most displays (those with gamma of about 2.2), one can observe that the linear-intensity scale has a large jump in perceived brightness between the intensity values 0.0 and 0.1, while the steps at the higher end of the scale are hardly perceptible. The gamma-encoded scale, which has a nonlinearly-increasing intensity, will show much more even steps in perceived brightness.

A cathode ray tube (CRT), for example, converts a video signal to light in a nonlinear way, because the electron gun’s intensity (brightness) as a function of applied video voltage is nonlinear. The light intensity I is related to the source voltage Vs according to

{\displaystyle I\propto V_{\rm {s}}^{\gamma }}

where γ is the Greek letter gamma. For a CRT, the gamma that relates brightness to voltage is usually in the range 2.35 to 2.55; video look-up tables in computers usually adjust the system gamma to the range 1.8 to 2.2,[1] which is in the region that makes a uniform encoding difference give approximately uniform perceptual brightness difference, as illustrated in the diagram at the top of this section.

For simplicity, consider the example of a monochrome CRT. In this case, when a video signal of 0.5 (representing mid-gray) is fed to the display, the intensity or brightness is about 0.22 (resulting in a dark gray). Pure black (0.0) and pure white (1.0) are the only shades that are unaffected by gamma.

To compensate for this effect, the inverse transfer function (gamma correction) is sometimes applied to the video signal so that the end-to-end response is linear. In other words, the transmitted signal is deliberately distorted so that, after it has been distorted again by the display device, the viewer sees the correct brightness. The inverse of the function above is:

  {\displaystyle V_{\rm {c}}\propto V_{\rm {s}}^{1/\gamma }}

where Vc is the corrected voltage and Vs is the source voltage, for example from an image sensor that converts photocharge linearly to a voltage. In our CRT example 1/γ is 1/2.2 or 0.45.

A color CRT receives three video signals (red, green and blue) and in general each color has its own value of gamma, denoted γR, γG or γB. However, in simple display systems, a single value of γ is used for all three colors.

Other display devices have different values of gamma: for example, a Game Boy Advance display has a gamma between 3 and 4 depending on lighting conditions. In LCDs such as those on laptop computers, the relation between the signal voltage Vs and the intensity I is very nonlinear and cannot be described with gamma value. However, such displays apply a correction onto the signal voltage in order to approximately get a standard γ = 2.5 behavior. In NTSC television recording, γ = 2.2.

The power-law function, or its inverse, has a slope of infinity at zero. This leads to problems in converting from and to a gamma colorspace. For this reason most formally defined colorspaces such as sRGB will define a straight-line segment near zero and add raising x + K (where K is a constant) to a power so the curve has continuous slope. This straight line does not represent what the CRT does, but does make the rest of the curve more closely match the effect of ambient light on the CRT. In such expressions the exponent is not the gamma; for instance, the sRGB function uses a power of 2.4 in it, but more closely resembles a power-law function with an exponent of 2.2, without a linear portion.



Simple monitor tests


To see whether one’s computer monitor is properly hardware adjusted and can display shadow detail in sRGB images properly, they should see the left half of the circle in the large black square very faintly but the right half should be clearly visible. If not, one can adjust their monitor’s contrast and/or brightness setting. This alters the monitor’s perceived gamma. The image is best viewed against a black background.

This procedure is not suitable for calibrating or print-proofing a monitor. It can be useful for making a monitor display sRGB images approximately correctly, on systems in which profiles are not used (for example, the Firefox browser prior to version 3.0 and many others) or in systems that assume untagged source images are in the sRGB colorspace.

On some operating systems running the X Window System, one can set the gamma correction factor (applied to the existing gamma value) by issuing the command xgamma -gamma 0.9 for setting gamma correction factor to 0.9, and xgamma for querying current value of that factor (the default is 1.0). In OS X systems, the gamma and other related screen calibrations are made through the System Preferences. Microsoft Windows versions before Windows Vista lack a first-party developed calibration tool.


In the test pattern to the right, the linear intensity of each solid bar is the average of the linear intensities in the surrounding striped dither; therefore, ideally, the solid squares and the dithers should appear equally bright in a properly adjusted sRGB system.













GoPiGo 小汽車︰格點圖像算術《色彩空間》故事


一六一四年 John Napier 約翰‧納皮爾在一本名為《 Mirifici Logarithmorum Canonis Descriptio  》── 奇妙的對數規律的描述 ── 的書中,用了三十七頁解釋『對數log ,以及給了長達九十頁的對數表。這有什麼重要的嗎?想一想即使在今天用『鉛筆』和『紙』做大位數的加減乘除,尚且困難也很容易算錯,就可以知道對數的發明,對計算一事貢獻之大的了。如果用一對一對應的觀點來看,對數把『乘除』運算『變換加減』運算

\log {a * b} = \log{a} + \log{b}

\log {a / b} = \log{a} - \log{b}


\log {a^n} = n * \log{a}

傳聞納皮爾還發明了的『骨頭計算器』,他的書對於之後的天文學、力學、物理學、占星學的發展都有很大的影響。他的運算變換 Transform 的想法,開啟了『換個空間解決數學問題』的大門,比方『常微分方程式的  Laplace Transform』與『頻譜分析的傅立葉變換』等等。


Rendered by

不只如此這個對數關係竟然還跟人類之『五官』── 眼耳鼻舌身 ── 受到『刺激』── 色聲香味觸 ── 的『感覺』強弱大小有關。一七九五年出生的 Ernst Heinrich Weber 韋伯,一位德國物理學家,是一位心理物理學的先驅,他提出感覺之『方可分辨』JND just-noticeable difference 的特性。比方說你提了五公斤的水,再加上半公斤,可能感覺差不了多少,要是你沒提水,說不定會覺的突然拿著半公斤的水很重。也就是說在『既定的刺激』下, 感覺的方可分辨性大小並不相同。韋伯實驗後歸結成一個關係式︰

ΔR/R = K

R:  既有刺激之物理量數值
ΔR:  方可分辨 JND 所需增加的刺激之物理量數值
K: 特定感官之常數,不同的感官不同

。之後  Gustav Theodor Fechner  費希納,一位韋伯派的學者,提出『知覺』perception 『連續性假設,將韋伯關係式改寫為︰

dP = k  \frac {dS}{S}


P = k \ln S + C

假如刺激之物理量數值小於 S_0 時,人感覺不到 P = 0,就可將上式寫成︰

P = k \ln \frac {S}{S_0}

這就是知名的韋伯-費希納定律,它講著:在絕對閾限 S_0 之上,主觀知覺之強度的變化與刺激之物理量大小的改變呈現自然對數的關係,也可以說,如果刺激大小按著幾何級數倍增,所引起的感覺強度卻只依造算術級數累加。

─── 摘自《千江有水千江月





P = k \ln \frac {S}{S_0} 之『無因次』 \frac{S}{S_0} 『表述』實優於

P = k \ln S + C 也。

追求五官之『絕對 □ 感』者,或當知『等距量表』哩︰



一九四七年,匈牙利之美籍猶太人數學家,現代電腦創始人之一。約翰‧馮‧諾伊曼 Jhon Von Neumann 和德國-美國經濟學家奧斯卡‧摩根斯特恩 Oskar Morgenstern 提出只要『個體』的『喜好性』之『度量』滿足『四條公設』,那麼『個體』之『效用函數』就『存在』,而且除了『零點』的『規定』,以及『等距長度』之『定義』之外,這個『效用函數』還可以說是『唯一』的。就像是『個體』隨身攜帶的『理性』之『溫度計』一樣,能在任何『選擇』下,告知最大『滿意度』與『期望值』。現今這稱之為『期望效用函數理論』 Expected Utility Theory。

由於每個人的『冷熱感受』不同,所以『溫度計』上的『刻度』並不是代表數學上的一般『數字』,通常這一種比較『尺度』只有『差距值』有相對『強弱』意義,『數值比值』並不代表什麼意義,就像說,攝氏二十度不是攝氏十度的兩倍熱。這一類『尺度』在度量中叫做『等距量表』 Interval scale 。

溫度計』量測『溫度』的『高低』,『理性』之『溫度計』度量『選擇』的『優劣』。通常在『實驗經濟學』裡最廣泛採取的是『彩票選擇實驗』 lottery- choice experiments,也就是講,請你在『眾多彩票』中選擇一個你『喜好』 的『彩票』。

這樣就可以將一個有多種『機率p_i,能產生互斥『結果A_i 的『彩票L 表示成︰

L = \sum \limits_{i=1}^{N} p_i A_i ,  \  \sum \limits_{i=1}^{N} p_i  =1,  \ i=1 \cdots N



L\prec MM\prec L,或 L \sim M

任意的兩張『彩票』都可以比較『喜好度』 ,它的結果只能是上述三種關係之一,『偏好 ML\prec M,『偏好 LM\prec L,『無差異L \sim M

遞移性公設】 Transitivity

如果 L \preceq M,而且 M \preceq N,那麼 L \preceq N

連續性公設】 Continuity

如果 L \preceq M\preceq N , 那麼存在一個『機率p\in[0,1] ,使得 pL + (1-p)N = M

獨立性公設】 Independence

如果 L\prec M, 那麼對任意的『彩票N 與『機率p\in(0,1],滿足 pL+(1-p)N \prec pM+(1-p)N

對於任何一個滿足上述公設的『理性經紀人』 rational agent ,必然可以『建構』一個『效用函數u,使得 A_i \rightarrow u(A_i),而且對任意兩張『彩票』,如果 L\prec M \Longleftrightarrow \  E(u(L)) < E(u(M))。此處 E(u(L)) 代表對 L彩票』的『效用期望值』,簡記作 Eu(L),符合

Eu(p_1 A_1 + \ldots + p_n A_n) = p_1 u(A_1) + \cdots + p_n u(A_n)

它在『微觀經濟學』、『博弈論』與『決策論』中,今天稱之為『預期效用假說』 Expected utility hypothesis,指在有『風險』的情況下,任何『個體』所應該作出的『理性選擇』就是追求『效用期望值』的『最大化』。假使人生中的『抉擇』真實能夠如是的『簡化』,也許想得到『快樂』與『幸福』的辦法,就清楚明白的多了。然而有人認為這個『假說』不合邏輯。一九五二年,法國總體經濟學家莫里斯‧菲力‧夏爾‧阿萊斯 Maurice Félix Charles Allais ── 一九八八年,諾貝爾經濟學獎的得主 ── 作了一個著名的實驗,看看實際上人到底是怎麼『做選擇』的,這個『阿萊斯』發明的『彩票選擇實驗』就是大名鼎鼎的『阿萊斯悖論』 Allais paradox 。


彩票甲:百分之百的機會得到一百萬元。【期望值 100 萬】

彩票乙:百分之十的機會得到五百萬元,百分之八十九的機會得到一百萬元,百分之一的機會什麼也得不到。【期望值 139 萬】



彩票丙:百分之十一的機會得到一百萬元,百分之八十九的機會什麼也得不到。【期望值 11 萬】

彩票丁:百分之十的機會得到五百萬元,百分之九十的機會什麼也得不到。【期望值 50 萬】


那麼這又是為什麼呢?也許說設想『人只是理性的』的這種想法,並不符合『合理性』,畢竟『人的心理』是『複雜的』,而且『人類行為』也是『多樣的』。於是自一九七九年起,以色列裔美國心理學家丹尼爾‧卡內曼 Daniel Kahneman 和以色列著名認知心理學者阿摩司‧特沃斯基 Amos Tversky 系統的研究『行為經濟學』 behavioral economic theory 這一領域,開創了現今稱為的『展望理論』prospect theory,試圖回答『為什麼』人是這麼『做選擇』的,此『前景理論』這麼講︰

People make decisions based on the potential value of losses and gains rather than the final outcome, and that people evaluate these losses and gains using certain heuristics.


U = \sum \limits_{i=1}^N w(p_i)v(A_i)

,此處 A_i 是各個可能結果,而 v 是『價值函數』 value function ,表示不同可能結果,在決策者心中的『相對價值』。而 w 是『機會權重函數』 probability weighting function ,藉此表現通常人對於『極不可能』發生的事,往往會『過度反應』 over-react,而對『高度可能』出現的事,常常又會『反應不及』 under-react。從而形成一條穿過『參考點』的『S 型曲線』。那個 U 就是一個人在作『得失決策』時的『總體評估』,或者說『預期效用』。





價值函數 value function


這條『S 型曲線』的不對稱性呈現出,當人們面對一個『損失』的『結果』,所產生之『厭惡感』或者說『傷感情』,比『獲益』之『情況』下所生的『滿意度』也許講『感覺好』,更為『強烈』。這使『展望理論』基本上不同於『期望效用函數理論』。有人將此理論的引申結論,整理成︰


── 『人的行為』應當用著『純理性』來『定義』嗎?


─── 摘自《物理哲學·下中…



Stevens’s power law

Stevens’s power law is a proposed relationship between the magnitude of a physical stimulus and its perceived intensity or strength. It is often considered to supersede the Weber–Fechner law on the basis that it describes a wider range of sensations, although critics argue that the validity of the law is contingent on the virtue of approaches to the measurement of perceived intensity that are employed in relevant experiments. In addition, a distinction has been made between local psychophysics, where stimuli are discriminated only with a certain probability, and global psychophysics, where the stimuli would be discriminated correctly with near certainty (Luce & Krumhansl, 1988). The Weber–Fechner law and methods described by L. L. Thurstone are generally applied in local psychophysics, whereas Stevens’s methods are usually applied in global psychophysics.

The theory is named after psychophysicist Stanley Smith Stevens (1906–1973). Although the idea of a power law had been suggested by 19th-century researchers, Stevens is credited with reviving the law and publishing a body of psychophysical data to support it in 1957.

The general form of the law is

  {\displaystyle \psi (I)=kI^{a},}

where I is the magnitude of the physical stimulus, ψ(I) is the subjective magnitude of the sensation evoked by the stimulus, a is an exponent that depends on the type of stimulation, and k is a proportionality constant that depends on the units used.

The table to the bottom lists the exponents reported by Stevens.


Continuum Exponent  a Stimulus condition
Loudness 0.67 Sound pressure of 3000 Hz tone
Vibration 0.95 Amplitude of 60 Hz on finger
Vibration 0.6 Amplitude of 250 Hz on finger
Brightness 0.33 5° target in dark
Brightness 0.5 Point source
Brightness 0.5 Brief flash
Brightness 1 Point source briefly flashed
Lightness 1.2 Reflectance of gray papers
Visual length 1 Projected line
Visual area 0.7 Projected square
Redness (saturation) 1.7 Red–gray mixture
Taste 1.3 Sucrose
Taste 1.4 Salt
Taste 0.8 Saccharin
Smell 0.6 Heptane
Cold 1 Metal contact on arm
Warmth 1.6 Metal contact on arm
Warmth 1.3 Irradiation of skin, small area
Warmth 0.7 Irradiation of skin, large area
Discomfort, cold 1.7 Whole-body irradiation
Discomfort, warm 0.7 Whole-body irradiation
Thermal pain 1 Radiant heat on skin
Tactual roughness 1.5 Rubbing emery cloths
Tactual hardness 0.8 Squeezing rubber
Finger span 1.3 Thickness of blocks
Pressure on palm 1.1 Static force on skin
Muscle force 1.7 Static contractions
Heaviness 1.45 Lifted weights
Viscosity 0.42 Stirring silicone fluids
Electric shock 3.5 Current through fingers
Vocal effort 1.1 Vocal sound pressure
Angular acceleration 1.4 5 s rotation
Duration 1.1 White-noise stimuli




Stevens generally collected magnitude estimation data from multiple observers, averaged the data across subjects, and then fitted a power function to the data. Because the fit was generally reasonable, he concluded the power law was correct. This approach ignores any individual differences that may obtain and indeed it has been reported that the power relationship does not always hold as well when data are considered separately for individual respondents (Green & Luce 1974).

Another issue is that the approach provides neither a direct test of the power law itself nor the underlying assumptions of the magnitude estimation/production method.

Stevens’s main assertion was that using magnitude estimations/productions respondents were able to make judgements on a ratio scale (i.e., if x and y are values on a given ratio scale, then there exists a constant k such that x = ky). In the context of axiomatic psychophysics, (Narens 1996) formulated a testable property capturing the implicit underlying assumption this assertion entailed. Specifically, for two proportions p and q, and three stimuli, x, y, z, if y is judged p times x, z is judged q times y, then t = pq times x should be equal to z. This amounts to assuming that respondents interpret numbers in a veridical way. This property was unambiguously rejected (Ellermeier & Faulhammer 2000, Zimmer 2005). Without assuming veridical interpretation of numbers, (Narens 1996) formulated another property that, if sustained, meant that respondents could make ratio scaled judgments, namely, if y is judged p times x, z is judged q times y, and if y is judged q times x, z is judged p times y, then z should equal z. This property has been sustained in a variety of situations (Ellermeier & Faulhammer 2000, Zimmer 2005).

Because Stevens fit power functions to data, his method did not provide a direct test of the power law itself. (Luce 2002), under the condition that respondents’ numerical distortion function and the psychophysical functions could be separated, formulated a behavioral condition equivalent to the psychophysical function being a power function. This condition was confirmed for just over half the respondents, and the power form was found to be a reasonable approximation for the rest (Steingrimsson & Luce 2006).

It has also been questioned, particularly in terms of signal detection theory, whether any given stimulus is actually associated with a particular and absolute perceived intensity; i.e. one that is independent of contextual factors and conditions. Consistent with this, Luce (1990, p. 73) observed that “by introducing contexts such as background noise in loudness judgements, the shape of the magnitude estimation functions certainly deviates sharply from a power function”.


仍在,『理論』尚未『 ○ 滿』耶??