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The Radiance HDR (RGBE, XYZE) color spaces
The RGBE color space In this color space, the underlying coordinates are the linear sRGB values (what I sometime call r, g, and b); that is, the sRGB coordinate values before the application of "gamma precompensation". Each coordinate is recorded in floating point form with an 8bit fractional significand [more about that shortly] and an 8 bit exponent. However, the same exponent applies to all three coordinates. (It is the one suited to the largest coordinate value.) Thus there are a total of 32 bits stored per pixel, 8 each for R, G, B, and the common exponent, E. If compression is used, it is of the runlength encoded (RLE) form. The fractional significand In some types of binary floating point representation (such as IEEE754), the significand (sometimes called the mantissa) has a value (S) in this range: 1≤S<2 That is, in binary, the range is: 1.0000... through 1.1111... [yes, that is a binary point] This being the case, the leading (units place) "1" need not be recorded, since its value is always known ("1"). Thus we can get, for example, 9 bits of precision by only recording 8 bits. Of course, this only works if the exponent is always chosen such that the significand will be in the range mentioned. (The number represented is the significand times 2 to the power of the exponent.) These "4 significant digit" "floating point" expressions would all represent the same value (the significands are all in binary, the rest in decimal): 10.01 x 2^7 1.001 x 2^8 0.1001 x 2^9 but we only use the second representation for that particular number so as to maintain the range of the significand that allows us to "know" its units digit and not have to record it. But if we are going to have three coordinates, with values perhaps quite different, and force them to share a common exponent, this can't happen. Imagine these three values, with separate "optimum" exponents, following the rule above: R: 1.001 x 2^9 G: 1.010 x 2^8 B: 1.110 x 2^7 If we wish them to share a common exponent, we might think to do that this way: R: 1.0010 x 2^9 G: 0.1010 x 2^9 B: 0.0111 x 2^9 But now we would have to store all the digits of each significand  we can no longer say that we "know" the value of the units digit. So we would have to store 5 bits for each value. So in the RGBE system, we do just that. Well, not quite. Actually, by convention, those three values would be stored as: R: 0.10010 x 2^9 G: 0.01010 x 2^9 B: 0.00111 x 2^9 And of course now we no longer need to store the units digit, which is always "0". But we have to store the same number of digits as before: 5. Actually, of course there is no "binary point" stored; it occurs by implication. Thus those three significands would be stored as:This outlook follows a different norm for the range of the significand. If we don't saddle a scheme with the need to use the same exponent for several values, but just chose the "optimum" exponent for each value, then under this convention, the range of the significand, S will always be: 0.5≤S<1 That is, in binary, the range is: 0.1000... through 0.1111... But in RGBE, where we must choose an exponent that suits the largest value and then make the other values use it, only for the largest value will the significand lie in the range just stated. We see that in the earlier example for R, G, and B, where only for R (the largest value) does its significand lie in that range. The exponent As in many floating point schemes, the exponent is recorded with a bias (here +128) to allow representation of a range of negative and positive values. I don't have the detailed specification, but from various hints, I suspect that the range of the exponent itself is ±126. Black When all coordinate values are below the range of the floating point scheme (near black), the significands are all recorded as 0 and the exponent is 128 (recorded, with its offset, as 0). The "precision" of the scheme In evaluating various color spaces, there is often concern about the "precision" of the encoding,. which we can think of as the ratio of size of an encoding step (the difference between adjacent values that have legitimate representations) and the average value of the adjacent "steps". If we only think of the "largest" coordinate value (or others that are able to share the same exponent), that ratio depends on where in the range of the significant we lie. Here are two pairs of adjacent "steps" we could consider with the 8bit significand of RGBE: These are at the top of the range: 0.11111110 0.11111111 The difference between those is 0.00000001. The average of them is 0.111111101. The ratio is roughly 0.004, or 0.4%. (That is considered "not really a problem" with respect to banding.) These are at the bottom of the range: 0.10000000 0.10000001 The difference between those is 0.00000001. The average of them is 0.100000001. The ratio is roughly 0.008, or 0.8%. (That, though coarser, is still considered "not really a problem" with respect to banding.) Now, what about values that have to share an exponent that is "too big" for them? These are like voltages we measure with the voltmeter set on too large a scale (so it will fit all the voltages we are measuring). Their "precision" might be quite low. Here would be a legitimate pair of adjacent significands (the worst case in the situation mentioned): 0.00000000 0.00000001 The difference between those is 0.00000001. The average of them is 0.000000001. The ratio is 2, or 200%! The gamut The gamut of the RGBE color space is the same as that of the sRGB color space. The EYZE color space In this color space, the underlying coordinates are the coordinates X, Y, and Z of the CIE XYZ color space. The rest of the scheme works just exactly as we saw for the RGBE color space. The gamut The gamut of the XYZE color space is the same as that of the CIE XYZ color space. In chromaticity, it embraces the entire visual gamut (and then some). The price we pay for that is that the precision, in terms of color perception, is less than for the RGBE color space, but still quite good. Best regards, Doug 
#2




In connection with these color spaces, attention is often given to their precision (I discussed that above with regard to the RGBE color space) and also their "dynamic range". How might we reckon that for the RGBE (and XYZE) color spaces.
As almost always, the issue is what should be consider to be the "smallest nonzero luminance that is "handled" by the system. I'll play with that in a moment with respect to the RGBE color space. First, lets do the easy part: what is the greatest luminance the system can represent. I will persist in my suspicion that the range of the exponent in its floating point representation is ±126 (although in fact something funny happens at the low end  more on that presently). The largest value that can be represented is (written in my usual hybrid of binary and decimal): 0.11111111 x 2^126 That is close to 8.5 x 10^+37, often considered to be about 10^+38. Now, for the low end. If we did not arbitrarily cut off the working of the floating point scheme, the smallest nonzero value would be represented as: 0.00000001 x 2^126. That is close to 4.6 x 10^41. But recall that here we are in the realm of what would be called in some floating point systems "denormal numbers". That is, they do not have the full precision implied by the size of our significand field. For example, in this example, the significand has only one significant (binary) digit. Perhaps a more appropriate reckoning of the "lowest"value would be the lowest one that had the full number of significant digits in its significand. That would be: 0.10000000 x 2^126 That is close to 5.9 x 10^39. But perhaps we should instead consider this one (which has a nice parallelism with the maximum I spoke of before): 0.11111111 x 2^126 That is about 1.2 x 10^38. In fact, the ratio of the maximum to this minimum is exactly 2^252. That is very nearly 7.2 x 10^75, or roughly 10^76. So perhaps it is from this train of thought that it is often said that the dynamic range of the color space is 10^76, or "76 decades", or "76 orders of magnitude". Now, according to one author, values less than 10^38 are arbitrarily recorded as zero. That is essentially equivalent to my final proposal above. That is, in the terminology of other floating point schemes, we do not use any "denormal" numbers. On the other hand, the C++ code (by Bruce Walter) that supposedly follows the code of the original developer of the RGBE color space (Greg Ward) says that this "lower cutoff" occurs at a value of 10^32 (a value 1 million times higher). (That could be a transcription error.) If that is authentic, then the dynamic range of the scheme is about 70 decades. Whatever it is, it's a bunch. ********** By the way, Bart suggested in a recent post that (from a rigorous outlook) it is somewhat inappropriate to refer to the "dynamic range" of a coding system; he suggests that "dynamic range" is a property of a scene, and when we "rate" an encoding system, we are speaking of not its dynamic range but rather its capability to handle a certain dynamic range in a scene (where that term is essentially synonymous with the "contrast ratio" of the scene). [We might wonder why we need a "new" name for that.] At the time I concurred. But upon reflection, I think not. As I examine the tradition of the use of "dynamic range" in, for example, electrical systems (including recording systems), I find that it is normally used for a capability of the signal handling system or recording system, not for a property of a signal that it can handle. Of course the two are closely linked. But in fact I cannot endorse taking a term that has traditionally referred to a capability of a "recording system" and redefine it as a property of a scene (one for which there is already a recognized name). Of course, we today plaster the moniker HDR on everything associated with this (including, presumably, on the shoulder straps of HDR cameras, and on the albums in which we place prints that have successfully been derived from high contrast ratio scenes in various ways). So just remember: if it says "duck" on it, it is something on which it says "duck". Maybe to distinguish a duck call from a turkey call in our field kit. Best regards, Doug 
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