sectionOptimalColors: compute sections of an optimal color surface by hyperplanes

In colorSpec: Color Calculations with Emphasis on Spectral Data

compute sections of an optimal color surface by hyperplanes

Description

Consider a colorSpec object x with type equal to 'responsivity.material'. The set of all possible material reflectance functions (or transmittance functions) is convex, closed, and bounded, and this implies that the set of all possible output responses from x is also convex, closed, and bounded. The latter set is called the object-color solid or Rösch Farbkörper for x. If the dimension of the response of x is 2, this solid is a convex polygon that is centrally symmetric - a zonogon. If the dimension of the response of x is 3 (e.g. RGB or XYZ), this solid is a special type of centrally symmetric convex polyhedron called a zonohedron, see Centore. This function only supports dimensions 2 and 3. Denote this object-color solid by Z.

A color on the boundary of Z is called an optimal color. Let the equation of a hyperplane be given by:

<v,normal> = \beta

where normal is orthogonal to the hyperplane, and \beta is the plane constant, and v is a variable. The purpose of the function sectionOptimalColors() is to compute the intersection of the hyperplane and \partialZ.

In dimension 3 this hyperplane is a 2D plane, and the intersection is generically a convex polygon. If the plane is a supporting plane, then the intersection is a face (a vertex, an edge, or a facet) of Z; in this case the function computes the center of the face.

In dimension 2 this hyperplane is a line, and the intersection is generically 2 points. If the line is supporing line, then the intersection is a vertex or an edge of Z; in the case of an edge the function computes the center of the edge.

Of course, the intersection can also be empty.

The function is essentially a wrapper around zonohedra::section.zonohedron() and zonohedra::section.zonogon().

Usage

## S3 method for class 'colorSpec'
sectionOptimalColors( x, normal, beta )

Arguments

x	a colorSpec object with type equal to 'responsivity.material' and M spectra, where M=2 or 3.
normal	a nonzero vector of dimension M, that is the normal to the family of parallel hyperplanes
beta	a vector of numbers of positive length. The number beta[k] defines the k'th plane <v,normal> = beta[k].

.

Details

In the case that the dimension of x is 3, then Z is a zonohedron. For processing details see: zonohedra::section.zonohedron().

In the case that the dimension of x is 2, then Z is a zonogon. For processing details see: zonohedra::section.zonogon().

Value

The function returns a list with an item for each value in vector beta. Each item in the output is a list with these items:

beta	the value of the plane constant \beta
section	an NxM matrix, where N is the number of points in the section, and M is the dimension of normal. The points of the section are the rows of the matrix. If the intersection is empty, then N=0.

In case of global error, the function returns NULL.

References

Centore, Paul. A Zonohedral Approach to Optimal Colours. Color Research & Application. Vol. 38. No. 2. pp. 110-119. April 2013.

Logvinenko, A. D. An object-color space. Journal of Vision. 9(11):5, 1-23, (2009).
https://jov.arvojournals.org/article.aspx?articleid=2203976. doi:10.1167/9.11.5.

See Also

vignette Plotting Chromaticity Loci of Optimal Colors, probeOptimalColors(), zonohedra::section.zonohedron(), zonohedra::section.zonogon()

Examples

wave = seq(420,680,by=5)
Flea2.scanner = product( A.1nm, "material", Flea2.RGB, wavelength=wave )
seclist = sectionOptimalColors( Flea2.scanner, normal=c(0,1,0), beta=10 )
nrow( seclist[[1]]$section )
##  [1] 89 


seclist[[1]]$section[ 1:5, ]
##  the polygon has 89 vertices, and the first 5 are:
##           Red Green      Blue
##  233 109.2756    10 3.5391342
##  185 109.5729    10 2.5403628
##  136 109.8078    10 1.7020526
##  86  109.9942    10 1.0111585
##  35  110.1428    10 0.4513051

colorSpec documentation built on April 4, 2025, 1:59 a.m.