Robust Estimation

 The images are most often handled in conventional 3-D coordinate systems like RGB, and conversion between the log-spectral space and RGB is necessary.

The conversion from the spectral distribution to the RGB vector is based on the standard CIE-procedure. The computation of a spectrum from a given RGB-vector distribution is ill-defined since every RGB-vector corresponds to a whole subspace of color spectra (metamerism).

For a given RGB vector we compute first the corresponding tristimulus vector (X,Y,Z). This 3-D vector is then converted into an intensity value I and a pair of chromaticity values $(\xi, \eta).$ From all the Munsell/NCS colors the one with the nearest chromaticity values $(\hat{\xi}, \hat{\eta})$ is selected. Let this spectrum be $\widehat{\sigma(\lambda)}$, and the corresponding intensity value $\hat{I}$.Then the spectrum which yields the RGB vector is estimated as $\left(I/\hat{I}\right)\widehat{\sigma(\lambda)}$.

The result of the conversion depends on the definition of $I, \xi, \eta.$ We have experimented with several traditional color spaces, (for example: the CIE (a,b) distribution ) but found that a space derived from the CIE-Lab space by a 45 degree rotation and scaling gives the most uniform coverage of the chromaticity values: the employed color space The distribution of the log-spectra coefficients is very diverse. Some examples are the histogram of the 1. Coefficient , histogram of the 2. Coefficient , histogram of the 3. Coefficient and histogram of the 4. Coefficient . They are not necessarily symmetrical and can have long tails. To characterize such distributions with a single number, a location estimate, robust techniques must be used. The mode, i.e. the most probable value, is such a robust location estimate. In a Bayesian framework, the mode is a MAP estimator which minimizes the uniform error cost function [10, page 210]. We used the least median of squares (LMedS) estimator [11] to compute the modes. They are shown with a dashed line in the previous histograms.