Convergence Problems

It is often possible to get into situations where it is impossible to create spectra between zero and one for the various components (usually the red curve). For spectra defined from 400nm to 700nm and an XYZ to RGB matrix based on Sony monitor data with a white point set at (.333,.333)², it is not possible to get the red curve below one. The area of the spectrum contributing solely to red is not large enough. Even though all the matching functions are low past 700nm, the red is still significant and more importantly, the other two matching functions are negative. Two factors are responsible for causing curves to rise above one: the width of the visible spectrum, and the chromaticity coordinates for the primaries used to create the XYZ to RGB matrix. The width of the spectrum can be increased, however for small numbers of spectral samples (9 or less), uniform bin size results in poor sampling of the regions of the spectrum where the matching functions are changing more frequently. As the primaries get more saturated, this problem gets worse. Consider three monochromatic primaries. The only way to (almost) match these is to have a zero for every basis function except the one containing that wavelength. For the sum of the red, green, and blue curves to have the same intensity as a constant unit spectrum, the value of the single non-zero coefficient must be much larger than one.