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The sigmoid-of-linear activation model

The usual way to compute the response of a sensor to a certain stimulus of known spectral composition is by means of equations like (p. ) from Section :

where is the response of the sensor, is the power spectrum of the stimulus, is the sensor spectral sensitivity, and is wavelength. Since this depends crucially on the linearity of , we cannot apply this equation to the basis functions we derived in Section (p. ) because they are essentially non-linear. The solution I have adopted for this problem is what I will call the sigmoid-of-linear (SL) model. This model rests on the hypothesis that the basis functions of Section can be modeled as the composition of a linear function and a sigmoid function:

where is the linear function, is the usual sigmoid function, and is the spectral sensitivity of . I believe this is a reasonable hypothesis, since saturation (which is modeled by the sigmoid function) is a function of the biochemical machinery of cells, and is determined by total absorbed energy (total ``quantum catch'') rather than by the spectral composition of the stimulus. In Section (p. ), I treated each constant-wavelength data set as independent from the others, and hence applied the sigmoid model at this level, but in the SL model I separate the wavelength dependent component of the response from the radiance dependent component. For , we can use the activation functions from Section (p. ) at relative radiance 1. That gives us the following equations for the SL basis functions:

where , , are the SL basis functions derived from the previous set , , and ; is relative radiance in ; is wavelength; and and , etc., are the sigmoid parameters for , etc. The sign function is as defined before, returning --1, 0, or 1, depending on the sign of its argument. If were linear in , then , and likewise for the other two. The function is a straightforward relative of the usual sigmoid function, meant to deal with both positive and negative values, and subtracting the zero-offset (the value of the function at ) of the sigmoid with the given parameters. The latter is a technicality introduced to prevent discontinuities around the zero-crossings of opponent functions.

All that is left to do now is to determine values for the constants , , , , , . To do this, I fitted the SL functions , , and to their non-linear counterparts , , and , minimizing the mean square error of fit as a function of and , over a regular grid of 441 points (21 coordinates in the radiance and 21 in the wavelength domain), using the steepest gradient descent algorithm of Mathematica's FindMinimum function. The results are shown in Table and Figures to .

With an RMS error of fit of 2-5% over the data set, the model fits rather well. Some of the error is no doubt attributable to the higher intensity ranges of the functions (as evident from the plots of the difference functions), which is the most extrapolated in the original non-linear functions, relative to the data sets they themselves were based on. I will therefore use the SL functions as the basis functions for the NPP color space, because they allow us to continue our exploration of the color-space properties in an analytical way.