
This appendix is a formal mathematical view of \emph{inverse colorimetry}.
Let $Q^\infty$ be the space of all (measurable) functions from the interval 
$[\lambda_1,\lambda_2]$ to $[0,1]$.
Each function in $Q^\infty$ corresponds to the reflectance function of a material.
A ``material responder" is a linear operator 
$\Lambda : Q^\infty \to \mathbb{R}^3$,
where points in $\mathbb{R}^3$ are tristimulus responses.
Define $Z := \Lambda(Q^\infty)$; 
$Z$ is a compact convex body - the \emph{object color solid} 
(\emph{R{\"o}sch Farbk{\"o}rper}).
Points in the boundary $\partial Z$ correspond to \emph{optimal colors}
(\emph{Optimalfarben}).
Non-optimal colors correspond to points in the interior $\operatorname{int}(Z)$.
The centroid method for spectral reflectance estimation is a
continuous function
$\gamma : \operatorname{int}(Z) \to \operatorname{int}(Q^\infty)$ so that
$$\text{the composition } \hspace{30pt} \operatorname{int}(Z) ~~
{\overset{\gamma}\rightarrow} ~~
\operatorname{int}(Q^\infty) ~~
{\overset{\Lambda}\twoheadrightarrow}~~
\operatorname{int}(Z) \hspace{30pt}  \text{ is the identity on } \operatorname{int}(Z).$$
Whenever $\Lambda \circ \gamma = \operatorname{id}_{\operatorname{int}(Z)}$,
$\gamma$ is said to be a \emph{right-inverse for} $\Lambda$
(or a \emph{section of} $\Lambda$).
Note that $\gamma$ is not defined for optimal colors in $\partial Z$.
For more details on $\gamma$, see Davis \cite{Davis2018}.
There are many other possible right-inverses.
Assuming that $\Lambda$ has the property that
every optimal color comes from a spectrum with $\le 2$ transitions (jumps),
Alexander Logvinenko has constructed a right-inverse 
that is defined on all of $Z$ (including $\partial Z$)
and whose image is a 3D space of step functions 
(the \emph{rectangle color atlas})
that is the same for every such $\Lambda$, see \cite{Logvinenko2009}.

There is a similar situation for a ``light responder" except that
the domain $Q^\infty$ is replaced by the \emph{non-negative orthant} $L_+^1$,
which is all integrable functions from the interval $[\lambda_1,\lambda_2]$ to $[0,\infty)$.
Each function in $L_+^1$ represents the spectrum of a source of light.
The image $C := \Lambda(L_+^1)$ is now an unbounded convex cone,
see Logvinenko \cite{Logvinenko2015}.
But in this unbounded case the centroid method right-inverse cannot 
be defined on all of $C$, but only on a proper subset of $C$.
For more about this situation see Davis \cite{Davis2018} Section 13.

