Spectral sharpening with positivity
Spectral sharpening is a method for developing camera or other optical-device sensor functions that are more narrowband than those in hardware, by means of a linear transform of sensor functions. The utility of such a transform is that many computer vision and color-correction algorithms perform better in a sharpened space, and thus such a space can be used as an intermediate representation for carrying out calculations. In this paper we consider how one may sharpen sensor functions such that the transformed sensors are all positive. We show that constrained optimization can be used to produce positive sensors in two fundamentally different ways: by constraining the coefficients in the transform or by constraining the functions directly. In the former method, we prove that convexity can be used to constrain the solution exactly. In a sense, we are continuing the work of MacAdam and of Pearson and Yule, who formed positive combinations of the color-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 colors 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 colors resulting from them. Curves that result have negative lobes but generate positive colors. We find that this type of constrained sharpening generates the best results, which are almost as good as for unconstrained sharpening but without the penalty of negative colors. All methods discussed may be used with any number of sensors.