Numerical methods for smoothest reflectance reconstruction

Abstract

Three numerical methods are presented for finding the smoothest reflectance curve associated with a given triplet of tristimulus values. The methods differ in how “smooth” is defined, and also differ in the domain of colors over which they are applicable. The first method is very quick and applies to any tristimulus values, but sometimes can yield reflectance curves with portions that fall outside the range 0 to 1. The second method applies to colors within the spectral locus (real colors) and guarantees that the reflectances produced are positive. The third method applies to colors within the object color solid (object colors) and guarantees that the reflectances fall within the range 0 to 1. The methods are shown to create reflectances that closely resemble those of real colors (natural and synthetic). Focus is given to implementing the numerical methods in very short MATLAB/Octave functions and to understanding the numerical behavior of the methods near the limits of their respective domains of applicability in terms of matrix conditioning and discretization artifacts.

Adaptive and non-adaptive convergent evolution in feather reflectance of California Channel Islands songbirds

“…To reconstruct soil reflectance spectra for visual modelling, we first downloaded standard red, green, blue (sRGB) values of the nine most common Santa Cruz Island and California mainland surface soils from a soil colour dataset of the contiguous United States with a spatial resolution of 200 m [45]. These values were converted to reflectance curves using the ‘real colours’ approach of Burns [46]. Briefly, this method seeks the smoothest reflectance function corresponding to a point in sRGB colour space by solving the system of linear equations that minimize the square of the slope of the reflectance curve, summed over the entire curve ([46]; see the electronic supplementary material, appendix S1 Modelling camouflage, and figure S6 for reconstructed curves and comparison to empirical soil reflectance curves from other regions).…”

The location of optimal object colors with more than two transitions

“…Here, we examine how other illuminants affect the results. Five high‐chroma Munsell colors were selected, 5R 5/14, 5Y 8/16, 5G 7/10, 5B 6/10, and 5P 4/12, and an illuminant was created with the same chromaticity, using a “smoothest” reflectance reconstruction technique 25‐26 . They are all shown at the top of Figure 9, scaled to have a maximum value of 1.…”

The Use of Colorimeters to Support Remote Sensing Techniques on Asphalt Pavements

“…The color space to reflectance conversion was performed as per the conversion methods of Burns (2020) [33]. Numerical methods can be used to reconstruct a reflectance distribution from a set of tristimulus values, XYZ.…”

“…The study used the method that is intended to be used on object colors. Burns (2020) [33] claimed that this method provides the best match with minimum root mean square errors compared to other numerical methods and guarantees a reflectance value between 0 and 1. This is achieved by using a hyperbolic tangent change of variables.…”

“…A is an n × 3 array of color matching functions (CMFs), A = [x, y, z], A w is the illuminant (W) color matching functions (CMFs) computed as WA, W is an n × n matrix with W on the main diagonal and zeros elsewhere, is the transpose of the matrix An assumption taken into consideration is that the scalar product y × W = 1, due to illuminant normalization. This a common practice used in previous studies as well [33][34][35]. The derivative (tanh(z) + 1) is (sec h 2 (z)) and λ is a (3 × 1) vector of Lagrande multipliers, thus by applying the Lagrange multipliers function, the stationary conditions would be:…”