Positive Bradford Curves through Sharpening

The Bradford curves are used in the CIECAM97 colour appearance model, and are ‘sharpened’ in the sense that they have narrower support than the cone fundamentals. Spectral sharpening is a method which finds the linear combination of a set of sensors that is most sensitive to a given interval of the visible spectrum. Here we investigate the relationship between the apparently sharp Bradford curves and the spectral sharpening method (since spectral sharpening was not used to derive the Bradford curves). We find that Bradford curves can be derived using spectral sharpening but the sharpening intervals are not the ones we would might have expected or wished for. The Bradford intervals are far from the ‘prime wavelength’ intervals: those parts of the visible spectrum where there is maximal visual sensitivity. Also, independent of any sharpening argument, the Bradford curves are unexpected in the sense that they have some negative sensitivity. Here we address both these concerns and produce sharpened versions of the Bradford curves that are both all-positive and also sharpened within wavelength intervals around the prime wavelengths. In a sense, we are continuing the work of MacAdam, and Pearson and Yule, in forming positive combinations of the colour-matching functions. However, the advantage of the spectral sharpening approach is that not only can we produce positive curves, but the process is ‘steerable’ in that we can produce positive curves with as good or better properties for sharpening within a given set of sharpening intervals. At base, however, it is positive colours in the transformed space that are the prime objective. Therefore we also carry out sharpening of sensor curves governed not by positivity of the curves themselves but of colours resulting from them. Curves that result have negative lobes, but generate positive colours. We find that this type of constrained sharpening generates the best results, almost as good as for unconstrained sharpening but without the penalty of negative colours.