Different line scan prism manufacturers might compare two possible curves in their advertising and argue that the steeper flanks should enable better color reproduction with greater selectivity.
The argumentation seems conclusive, but is it correct? In order to be able to judge this statement, it is necessary to take a look at the basics.
The colorimetry most commonly used today is based on the so-called CIE standard observer of 1931. At that time a spectral sensitivity curve for the color-sensitive receptors of the eye was determined by a comparative evaluation of color areas with three spectral lines. These curves were then transferred to the CIE normal-valence system. The color spaces, such as sRGB, AdobeRGB, eciRGB, and CIELAB are all still defined using this CIE standard valence system.
Each RGB camera supplies a color triple (R, G, B) for a given color. This RGB value is a device-specific description for the submitted sample and cannot be interpreted without reference to exactly this camera. Both colors and color spacings are always defined in relation to the CIE standard valence system. It follows that it is necessary to transfer the device-specific RGB values of the camera into that system.
In 1927, Robert Luther first formulated the necessary conditions for a technical recording device to be able to reproduce the visual behavior of the eye. This is the case when the spectral sensitivity curve consisting of filter and spectral characteristics of the receiver differs from the standard spectral value curves by only one factor k. However, this condition is not met by any standard RGB camera. The reason is that filters are optimized with regard to the output color space.
For example, the widely used sRGB standard is based on the primary colors typical for monitors at the time of definition. This ensures that a camera output image produces a usable color impression on a monitor without conversion.
A good filter characteristic in an industrial environment should meet both requirements as far as possible. This could be achieved if the spectral curves are fundamental metamers of the normal spectral value curves. This condition is fulfilled if the filter function can be mapped to the standard spectral value function by a non-singular 3×3 matrix.
The transformation of the standard spectral value curves into filter curves for common output color spaces generally leads to curves with negative components that cannot be realized physically.
Actual filter functions always represent a compromise between theoretical considerations and practical feasibility.
Three Examples for Comparison
The curves in Figures 3-5 differ essentially in the slope steepness. Due to the dichroic filters used, the curves in Figure 5 are much steeper than in Figures 3 and 4. The camera manufacturer with the Figure 5 curves argues that the steepness of the flanks allows good color separation.
Neither of the two curves meets the Luther criterion mentioned above, which would allow a correct transformation of the RGB values into the standard color space. For which of the two curves does this succeed with fewer deviations?
Behind this question is the issue of which curve comes closest to a fundamental metamer of the normal spectral value curves. It can be answered with the help of the Cohen matrix.
This can be understood as a projector that projects the filter function of the sensors onto the fundamental metamers of the standard spectral value functions. These functions are divided into parts that lie in the space spanned by the metamers and the part perpendicular to it. The vertical part is called the black metamer. The name derives from the fact that in the investigation of metameric behaviour of color spectra, this is the part of the spectrum that does not contribute to color discrimination by an observer.
In the technical processing of color information, the black metamer of the sensor spectrum is the portion of the signal that “disturbs” the mapping to the standard spectral values. The lower this proportion, the better the transformation.
Figures 6, 7, and 8 are the spectral curves of the black metamer for the two color filter functions
Spectral sensitivity curves 1:
Spectral sensitivity curves 2:
Spectral sensitivity curves 3:
The signal component of the disturbing black metamer results from the inner product of the color spectrum and the spectral black metamer curves. In curve 3, this is significantly larger than in curves 1 and 2, especially in the critical green component. This interference component cannot be completely corrected without knowledge of the color spectrum.
Simulation with ColorChecker
In addition, in a reproduction of the scanning process from the spectra β of the n=24 color fields, the spectral sensor sensitivities S, of the associated RGB values are calculated. An Illumination with a D50 characteristic is assumed.
The RGB values obtained in this way are mapped to the XYZ values with a 3×3 matrix, M. The statistics of the differences in the color distance measure ∆E are determined according to CIE2000. Matrix M is determined using the least squares method.
These error statistics belong to the two sensor variants:
As expected, the RGB values obtained with filter curves 1 can be transferred to the standard color space with the smallest deviations. While curves 2 are still acceptable, the high maximum error of curve 3 is critical.
The ∆E deviations can be significantly reduced with nonlinear methods. However, these always also result in a greater sensitivity of the transformation to small deviations of the process parameters.
However, it should also be mentioned that this transformability is only one aspect of the filter design. When comparing the curves, it is obvious that the areas under curves 2 and 3 are significantly larger. This means that the available light is better utilized, which can result in a better signal-to-noise ratio.