Bart, it doesn't have anything to do with integer or float; if you want to compute the volume of any shape its an integral over all three dimensions. To go mathematical, a triple integral of the constant function over its volume. For a cube, that simplifies to multiplying the lengths of the sides. What it is NOT is the triple integral of the delta E function, which is what is being claimed above. That makes no sense - it's a mathematical nonsense.

Sandy, this calculation makes sense on a mathematical level, and, although I could quibble with the terminology, I don't find it too much of a stretch. Whether CIEL*a*b* is the right metric is also open to debate, but it has the advantage of familiarity. If you know the joke about the drunk looking for his keys, Lab is our lamp-post. If you don't, I need little prodding to tell it.

The details:

Lab is a 3D Cartesian space (there is also an associated cylindrical space, but that needn't concern us here). In any such space, one can compute a the volume enclosed by a closed surface, either, and you point out, through integration, or, more likely these days, through summations of the volumes of small elementary 3D solids (tetrahedra, for example) in a digital computer. The units of the volume are the product of the units of each of the three axes. If all three axes have the same unit, say, meters, we use a kind of shorthand to define the unit of the volume: rather than say meter-meter-meter, we say cubic meter.

The units of the axes in CIEL*a*b* happen to be one DeltaE apart, since sqrt(0^2+1^2+0^2) = 1, and similarly for the other two axes. Therefore, it is convenient to refer to the unit of volumes in CIEL*a*b* as cubic DeltaE.

Now, let us turn to the right metric question. CIEL*a*b* was created to describe color in a way in which perceptual differences between closely-spaced colors could be characterized as a scalar, DeltaE. There are other measures for describing other color differences, but that's not part of this discussion. If Lab were perfect for its intended purpose, discrimination ellipsoids (plots of Just Noticeable Differences, or JNDs) produced by psychologists would map to spheres in Lab. In general, they do not. There have been attempts to device color spaces that more nearly meet this criterion, and they have met with some success, but none has caught the attention of photographers like CIEL*a*b*. As an aside, Lab traditionally has been favored by color scientists working with paper and ink. Lab has a cousin, CIEL*u*v* which has been the color space of choice for color scientists working with emissive displays. The two spaces share some genetic material, notably the luminance axis. Unfortunately, Luv is not more perceptually uniform than Lab (or worse, either, although the worst errors occur in different places).

There's another problem with Lab as a metric for gamut volumes: it wasn't created to be perceptually uniform over immense color differences. There is an argument that large color differences are merely the sum of many small color differences, but I think that's dangerous thinking.

Bruce Lindbloom has produced a modification of Lab that is more perceptually uniform, and in addition, does not exhibit the perceived hue shifts along constant hue angles that Lab (and Luv) possess. There are other spaces with similar objectives. From a technical point of view, they might be better choices for gamut volume calculations, but Lab is the devil we know.

And, I think importantly, we have to keep our eyes on the prize here. Our objective in describing gamut volumes is to come up with a scalar to characterize a color space's gamut. But we need to keep in mind that that's a very crude measure of a gamut, and is not useful in most circumstances. If I'm printing an image, I want to know something about the colors in my image that the printer can't print. I don't care about the colors that

**aren't **in my image that the printer

**can **print. Knowing the volume of the gamut of my image and the volume of the gamut of the printer doesn't help me at all. So why obsess on making that calculation more accurate?