### Color Temperature and Continuous Spectra

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I am trying to understand color temperature, and have gotten tired of looking for references and not finding anything that explains it to my satisfaction. I guess most descriptions are kept necessarily simple, but as a Ph.D. in electromagnetics, I am looking for a more detailed answer.

I understand measuring temperature in Kelvin. A blackbody radiator heated to temperature x Kelvin radiates light at such and such a color. I assume that the spectrum is pretty much a dirac function at a certain frequency.

However, most light we encounter is a continuous spectrum - not a single source at one discrete frequency. So my question is, how does a continuous spectrum of light get mapped into a single color temperature number?

Is it done by taking the frequency for which the spectrum has the highest intensity peak?

Do you integrate the spectrum intensity over the region of visible frequencies and find some sort of weighted average?

Do you find an equivalent RGB mix and somehow map the coefficients (trimistus values?) into a single temperature number?

-- John H. Henderson (jhende03@harris.com), October 05, 2000

Very complicated question that is answered only with formulas. Look at: Stefan-Boltzmann law and the extension of that by Wien known as Wien's displacement law.

Basically, the wavelength for which the spectral emittance is maximum is inversely proportional to the absolute temperature of the complete radiator.

The Wien law is temperature-varied. The corollary is stated as, the maximum spectral emittance of a complete radiator is proportional to the fifth power of its absolute temperature (expressed in Kelvins).

You may also reference Planck's formula for spectral emmittance.

In true definition, color temperature should only be applied to complete radiators, and not to selective radiators and non-selective radiators.

Color temperature is a specification of chromaticity, applicable to only a limited class of chromaticities, and is not a specification of spectral distribution. The color temperature of a nonselective radiator is its true temperature. The color temperature of thermal radiators may be higher or lower than their true temperatures depending upon their emissivities in the spectrum. For example, the color temperature of tungsten is higher than the true temperature of the metal when it is heated.

I hope this may give you some ideas to further explore color temperature to your satisfaction.

-- steve (s.swinehart@worldnet.att.net), October 05, 2000.

On a more practical note: Most colour temperature meters simply have two fairly broadband filtered sensors in them. One weighted towards the red end of the spectrum, and the other towards the blue. The meter reads the relative outputs of the two sensors, and gives an equivalent colour temperature indication on its scale.
RGB tristimulus, or any other RGB reading does not give a scientifically accurate identity to a colour. A single Sodium D line will give the same RGB reading as two single line emmissions at some frequency of Red, and some other frequency of Green, depending on their relative power. It will also give exactly the same reading for a broadband emmission centred about the D line frequency. Fuji's film technologists should think about this.

-- Pete Andrews (p.l.andrews@bham.ac.uk), October 05, 2000.

Pete, could you expand on your last statement, the one regarding Fuji?

Also unless I mistaken the Minolta Color Temp. Meter III is a true three color meter. And likeall color temp. meters it has trouble with like sources that have a very strong single spike.

-- Ellis Vener (evphoto@heartstone.com), October 05, 2000.

"RGB tristimulus, or any other RGB reading does not give a scientifically accurate identity to a colour."

Exactly. Tri-stimulus values do not give a description of the spectral emmittance curve. This is because most radiators of energy are not complete Planckian radiators - as demonstrated in the example of the sodium vapor light, or even with sunlight.

Further, whether the meter is a two color or three color meter doesn't matter, the operating principle is the same. The meter acts to balance the stimulus through two or three filters giving a meter reading (digital or needle deflection) that shows the imbalance between the two or three colors. Originally, all color meters used only red and blue filters as the color temperature was judged as being biased towards red or blue. Very large scale integrated circuit technology, that includes computing power, now gives the ability to compute color temperature from the electrical current generated through 2 or three sensors. Is it all of this any better? The Gossen Sixticolor is still a very good, very accurate meter that uses only two sensors.

-- steve (s.swinehart@worldnet.att.net), October 05, 2000.

"...gives the ability to compute color temperature from the electrical current generated through 2 or three sensors. Is it all of this any better? The Gossen Sixticolor is still a very good, very accurate meter that uses only two sensors."

My apologies, I was distracted answering another question. The statement should say "generated through two or three filters...."

And the reference to the Sixticolor should read "...only uses only two filters."

-- (s.swinehart@worldnet.att.net), October 05, 2000.

John, I think your last question came closest to the answer. The IS&T "Handbook of Photographic Science and Engineering", 1997 has a couple of paragraphs. In section 19.6.1, "Correlated color temperature is defined as the temperature of a blackbody that produces a chromaticity most similar to that of the selective radiator in question." A bit further on, "The uv chromaticity diagram, once called the CIE 1960 UCS diagram, is now obsolete and is only used for defining the color temperature and for computing the color rendering index."

They also defined color temperature as above, except the chromaticity "is identical" rather then "most similar to".

So essentially, color temperature is based on colorimetric matches as Steve indicated in his first response (it sounds like he knows something about this stuff).

If your interest is actually photography related, there is another important parameter called "color rendering index"; defined by the CIE. This is also covered somewhat in the book referenced above. Both subjects are also covered in Hunt's "The Reproduction of Colour" (section 10.11, etc); Hunt is much more likely to be found in a library.

-- Bill C (bcarriel@cpicorp.com), October 05, 2000.

Hi Ellis. If you look at Fuji' published responsivity curves, especially for Velvia, you'll see that they claim almost perfect bandpass filter response for the Red, Green, and Blue sensitive layers. There is almost no overlap shewn between the bands.
Apart from being highly suspect, since dye filters just ain't that good, it's also absolutely useless when it comes to differentiating subtleties of colour.
Imagine a natural spectrum, such as a rainbow, passing through such a collection of filters. Everything within the bandpass of the red filter comes out as red (whatever flavour of red the Magenta and Yellow dye layers produce). Then there is a tiny yellow peak at the minimal crossover of the green and red passbands, and then everything of shorter wavelength is rendered as one uniform green, until the green passband finishes and blue takes over.
You see the absurdity of it? Great for saturation, and if you like tricolour rainbows, but as for realistic colour......

-- Pete Andrews (p.l.andrews@bham.ac.uk), October 06, 2000.

John,

I had the similar question some time before: from point of view of the math it is an obvious nonsense that a whole function (the spectrum) can be described with a single number. But the similar degree of nonsense can be fount in the statement (and in the common practice) that all the variety of colors (the shapes of spectrum, as a physicist may initially think) can be represented by 3D discrete scheme (RGB representation).

The answer is that our eye (and therefore all the RGB-technologies) asks for much less than the true spectral matching. A lot of different spectra shapes can correspond to a single "color" as the eye it perceives. So the spectra are mapped to a "colors", and this already mapped (=simplified) input can be described in a lot simpler way than the initial (unmapped) input. In particular the color shift (bluish <--> reddish) can be satisfactorily described with a singe scalar value: the color temperature.

A good introduction in the subject can be found in the "R.Feunman' Lecture on the Physics", the chapter about color vision (vol. 3 or 4  I don't remember exactly).

I very sympathize with your question: having a not bad education in physics and mathematics I often could not see answers for my obvious doubts in some photo concepts which other people took for granted.

Sorry for my English.

Andrey. St-Petersburg, Russia

-- Andrey Vorobyov (AndreyVorobyov@mail.ru), October 13, 2000.