Step 3
sRGB is scaled based on human perception, while XYZ is scaled based on physical luminance.
The human perception of brightness depends on the physical luminance in a complex way. At low levels of light the relation is approximately linear, so for low brightness (< 0.04045) the luminance is the brightness divided by 12.92. At levels of luminance above low the relation is exponential, and that is the equation for levels of brightness which are not low (> 0.04045).
Step 5
The axes of the XYZ and sRGB colour spaces are different, and neither correspond to the cone responses of the human eye.
The sRGB axes are based on three actual colours which can be used in combination to produce a large range of colours.
The axes of XYZ were chosen so that Y represented the luminance, and X and Z the range of colours at that luminance.
(There is a different colour space called LMS where the axes do correspond to the cone responses.)
This means that a matrix transformation is needed to convert the values based on sRGB to XYZ values.
Step 6
First the XYZ values are scaled based on a typical colour value for sunlight, which is (0.95047, 1.0, 1.08883), called illuminant D65.
This is relatively close to the ideal white of (1.0 ,1.0, 1.0), but the actual distribution of wavelengths in sunlight varies depending on various factors, and this is a typically used value.
LAB is again based on human perception, so step 6 is effectively reversing the process from Step 3, but LAB uses a different approximation function of the luminance to brightness relation compared to sRGB.
I am not sure I understand exactly what you are asking here, but I'll try to explain.
There is no single correct way to specify colours, the XYZ colour space is the result of experiments which provided a method of describing all colours which humans can perceive based on combinations of wavelengths of light. The XYZ colour space is however not easy to understand and use practically, so other colour spaces were created based on the XYZ colour space but which are easier to understand and use. LAB colour space aims to have a space where the distance between two colours in the colour space is more similar to the difference that humans perceive between those colours, and is therefore a better representation of human colour vision. LAB also takes into account the light which is illuminating a coloured object. There is no single RGB colour space, but when people refer to RGB they usually mean the sRGB colour space. This cannot actually represent the full range of colours in the XYZ colour space, but is used in practice for the range of colours which e.g. colour monitors can display, based on combining specific red, green and blue lights.
Bruce Lindbloom has a good webpage which gives the specific formulas for conversion.
If you convert sRGB to XYZ you are using the specific values which the red, green and blue of sRGB have in the XYZ colour space to calculate where that sRGB colour would lie in the XYZ colour space.
Timothy James Maguire Thanks for your answer. It's helpful. What I meant was I would like to know the physical interpretation of the equations used:
a* = 500(f(X/Xn) -f(Y/Yn))
b* = 200(f(Y/Yn-f(Z/Zn))
L = 116f(Y/Yn) -16
Where do these specific numbers come from? What do the quantities f(X/Xn), f(Y/Yn) and f(Z/Zn) signify?
(It's possible that the answer is obvious to others and that's why I'm unable to find an explanation. Apologies if it's dumb question).
I'm new to this field but I have to explain this to people from a non-scientific background.
The Xn, Yn and Zn are the XYZ values of the reference white. These are to allow e.g. for coloured surfaces to look different depending on the light which is shining on them. If you use the equal-energy radiator which is the standard "illuminant E", then these values are all equal to 1 and for explanation purposes you could take those quantities out of the equation as they would have no effect.
The values L, a and b don't represent something physical, they are attempts to model the human perception of colour. This is part of the field of psychophysics.
If the physical amount of light is doubled, this doubles the physical luminance, but humans do not perceive the light as being twice as bright, but only around 37% brighter.
The function "L = 116f(Y/Yn) -16" is a model of the human perception of the brightness of the light. There have been different functions used to model the perception of lightness, but they are similar in the final result and provide a good match for the human perception of brightness.
Producing numeric values which represent the human perception of the difference of colours of the same perceived brightness is not as clear as the numeric value for perceived brightness. The a and b in the LAB space are an attempt to do so, but there are problems with the results.
In sRGB the function for perceived brightness is applied separately to the red, green and blue components. This is useful for practical reproduction of a colour, and is not primarily aimed at producing numeric values which are otherwise meaningful.
Yes, Professor Albert Henry Munsell did experiments around 1905 to determine the relation between the physical luminance and the perceived brightness (also called lightness, or in his own system "value") of colours. Various functions were then tried during the 20th century before the truncated cube root function used in LAB was chosen as the best fit for his experimental observations.
The values 200 and 500 were determined by an attempt to make the resulting colour space as perceptually even as possible.
Work is still being done on producing a more perceptually even colour space, CIECAM02 was published in 2002, but CIELAB is a lot simpler.
Step 3
sRGB is scaled based on human perception, while XYZ is scaled based on physical luminance.
The human perception of brightness depends on the physical luminance in a complex way. At low levels of light the relation is approximately linear, so for low brightness (< 0.04045) the luminance is the brightness divided by 12.92. At levels of luminance above low the relation is exponential, and that is the equation for levels of brightness which are not low (> 0.04045).
Step 5
The axes of the XYZ and sRGB colour spaces are different, and neither correspond to the cone responses of the human eye.
The sRGB axes are based on three actual colours which can be used in combination to produce a large range of colours.
The axes of XYZ were chosen so that Y represented the luminance, and X and Z the range of colours at that luminance.
(There is a different colour space called LMS where the axes do correspond to the cone responses.)
This means that a matrix transformation is needed to convert the values based on sRGB to XYZ values.
Step 6
First the XYZ values are scaled based on a typical colour value for sunlight, which is (0.95047, 1.0, 1.08883), called illuminant D65.
This is relatively close to the ideal white of (1.0 ,1.0, 1.0), but the actual distribution of wavelengths in sunlight varies depending on various factors, and this is a typically used value.
LAB is again based on human perception, so step 6 is effectively reversing the process from Step 3, but LAB uses a different approximation function of the luminance to brightness relation compared to sRGB.
Okay. You've been really helpful. Thank you. Just one more thing: what about the weighting factors used in step 5? have they been experimentally determined as well? I don't understand how multiplying them with the R G B and summing results in decoupling of colour and luminance.
I am not sure if it is best to say they were determined experimentally.
The red, green and blue in sRGB were defined by HP and Microsoft in 1996.
The values in step 5 are the values which convert sRGB co-ordinates of (1,0,0), (0,1,0) and (0,0,1) to the XYZ co-ordinates corresponding to those chosen colours. Mathematically speaking this is a linear transformation matrix.
Other RGB spaces definen by e.g. Adobe or Apple define different colours for red, green and blue. Conversion from these colour spaces to XYZ would require different values in step 5.
The red, green and blue defined for RGB colour spaces do not necessarily correspond with the colour people imagine when they think of red, green or blue.
The red is fairly close, maybe slightly orange-red, but the blue is a very deep, dark blue-violet, which can appear slightly purplish. The green is a very bright vivid green.
The luminance of the final colour depends mostly on the amount of sRGB green in the colour, so the luminance value Y mostly depends on the sRGB green value. Human eyes perceive medium wavelengths as brighter than short or long wavelengths.
I hope this makes it clearer.
Unlike sRGB, the colour spaces XYZ and LAB do not have an easily specified range of meaningful values. They have maximum ranges, where some combinations of values within the range are not real colours. From my work, if you convert colours in the sRGB range into LAB the possible range of values is from -64 a to +62 a, and from -50 b to +110 b. The sRGB range is not the full range of possible colours, so there are real colours with values outside of this range.
If you get values around 150,000 then there is probably an error in the calculation. The easiest way to check this might be to check the range of values at each step. The values sR, sG and sB should be in the range 0 to 255. In steps 2 and 3 the var_R, var_G and var_B should be in the range 0 to 1. Step 4 should be taking the values in the 0 to 1 range and converting them to values in the 0 to 100 range. In step 5 the X, Y and Z values should still be approximately in the 0 to 100 range. In step 6 the var_X, var_Y and var_Z should be in the 0 to 1 range.
I assume by intensity you mean the property of the colour which is 0 for all colours on the grayscale from black to white, but high for primary colours such as red, yellow and blue. The intensity of the colour can be calculated from the values of a colour in all three systems, but none of the systems have a single value which corresponds to intensity. In the LAB colour space the greyscale from white to black is represented by all the colours where a=0 and b=0, and the intensity of the colour is referred to as the "chroma" and is calculated as the square root of the sum of the squares of a and b. The calculated intensity value of the same colour will vary depending on which colour space it is represnted in.
Yes, plotting L along its length would be a representation of the perceived brightness of colours in the image. I do not think there is a simple way to correlate the result with individual parts of the image.
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