Consider a colorSpec object
type equal to
The set of all possible material reflectance functions (or transmittance functions)
is convex, closed, and bounded (in any reasonable function space),
and this implies that the set of all possible output responses
x is also convex, closed, and bounded.
The latter set is called the object-color solid or Rösch Farbkörper for
A color on the boundary of the object-color solid is called an optimal color.
The special points W (the response to the perfect reflecting diffuser)
and 0 are on the boundary of this set.
The interior of the line segment of neutrals joining 0 to W is in the interior of the
It is natural to parameterize this segment from 0 to 1 (from 0 to W).
The solid is symmetrical about the neutral gray midpoint G=W/2.
Now suppose that
x has 3 spectra (3 responses)
and consider a color response R not equal to G.
There is a ray based at G and passing through R
that intersects the boundary of the
object-color solid at an optimal color B on the boundary
with Logvinenko coordinates (δ,ω).
If these 2 coordinates are combined with α, where
R = (1-α)G + αB,
it yields the Logvinenko coordinates
(α,δ,ω) of R.
These coordinates are also denoted by ADL; see References.
A response is in the object-color solid iff α ≤ 1.
A response is optimal iff α=1.
The coordinates of 0 are (α,δ,ω)=(1,0,0). The coordinates of W are (α,δ,ω)=(1,1,0). The coordinates of G are undefined.
## S3 method for class 'colorSpec' computeADL( x, response )
a colorSpec object with
a numeric Nx3 matrix with responses in the rows, or a numeric vector that can be converted to such a matrix, by row.
For each response, a ray is computed and the ray tracing is
computeADL() returns a
data.frame with a row for each response.
The columns in the data frame are:
the input response vector
the computed ADL coordinates of the response vector
the reparameterized λ in the interval [0,1]; see References
lambda.1 and lambda.2 at the 2 transitions, in nm. lambda.1 < lambda.2 => bandpass, and lambda.1 > lambda.2 => bandstop.
If an individual ray could not be traced, the row contains
NA in appropriate columns.
In case of global error, the function returns
The optimal color boundary is not differentiable at 0 and W. There may be numerical iteration failures if the response is near the neutral axis.
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.
Godau, Christoph and Brian Funt. XYZ to ADL: Calculating Logvinenko's Object Color Coordinates. Proceedings Eighteenth IS&T Color Imaging Conference. San Antonio. Nov 2009.
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D50.eye = product( D50.5nm, 'varmat', xyz1931.1nm, wave='auto' ) computeADL( D50.eye, c(30,50,70) ) # response.x response.y response.z ADL.alpha ADL.delta ADL.lambda omega lambda.1 lambda.2 # 30 50 70 0.7364348 0.5384243 473.3909184 0.3008561 427.1431 555.5176 #since alpha < 1, this response is *inside* the object-color solid
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