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\begin{document}
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\title{ {\Huge Convexity and Transitions} \\ {\Large a strict examination of the 1931 CIE inverted-U}}
\author{Glenn Davis     \url{    <gdavis@gluonics.com>}}
\maketitle
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% Setup stuff.
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% ----------------------------------------------------------------------------
\section{Introduction}

In the fundamental paper \cite{Logvinenko2009},
Logvinenko investigates the statement that 
a color is optimal iff it comes from a (reflectance or transmittance) spectrum
that only takes the values 0 and 1, and has 0 or 2 transitions.
He calls this the \emph{two-transition assumption}.
He plots chromaticity diagrams for the cone fundamentals of Govardovskii et al. (Figure 4)
and of Stockman et Sharpe (Figure 7) and remarks that they are not convex.
By a theorem in \cite{West}, there are optimal colors for these two sets of
cone fundamentals whose transmittance spectra have more than 2 transitions.
So the two-transition assumption is false in these two cases.
He also plots the standard 1931 CIE chromaticity diagram (Figure 5) and remarks:
\begin{quote}
However, the completed spectral contour (in the unit plane) derived from
the color matching functions adopted by the CIE as the
standard colorimetric observer (Figure 5) is convex. 
This indicates that for this observer the two-transition assumption holds true.  \emph{[page 5]}
\end{quote}
The goal of this vignette is to show that, from an extremely strict viewpoint,
the standard 1931 CIE inverted-U is not convex either,
and the two-transition assumption does not hold.

To state this all precisely requires a lot of tedious mathematics,
which is then followed by an analysis at both 5nm and 1nm.

The featured functions from \textbf{colorSpec} used in this vignette are
\code{responsivityMetrics()},
\code{canonicalOptimalColors()}, and 
\code{bandRepresentation()}.


<<packs, echo=TRUE, message=FALSE>>=
library( colorSpec )
@




\section{Wavelengths and Subintervals}

Suppose we are given $N$ wavelengths:
$\lambda_1 < \lambda_2 < \ldots < \lambda_N$.
Define $N$ intervals $I_i := [\beta_{i-1},\beta_i]$ 
where
\begin{equation}
\beta_0 := \tfrac{3}{2}\lambda_1 - \tfrac{1}{2}\lambda_2 ~~~~~
\beta_i := (\lambda_i + \lambda_{i+1})/2, ~ i{=}1,\ldots,N-1  ~~~~~
\beta_N := \tfrac{3}{2}\lambda_N - \tfrac{1}{2}\lambda_{N-1}
\end{equation}
The intervals $I_i$ are a partition of $[\beta_0,\beta_N]$.
Note that $[\beta_0,\beta_N]$ is slightly bigger than $[\lambda_1,\lambda_N]$ because the endpoints are extended.
Define the $i'th$ step $\mu_i := \operatorname{length}(I_i), ~ i{=}1,\ldots,N$.
If the sequence $\{\lambda_i\}$ is \emph{regular} ($\mu_i$ is constant), 
then  $\{\beta_i\}$ is regular with the same step,
and each $\lambda_i$ is the center of $I_i$.

\section{Band Functions}

Let $B$ be the set of all functions on $[\beta_0,\beta_N]$ that take the values 0 or 1
and have finitely many transitions (jumps).
As in \cite{Centore2013}, we identify the endpoints $\beta_0$ and $\beta_N$ to form a circle,
so if the values at $\beta_0$ and $\beta_N$ are different, then this is considered to be a transition.
Equivalently $B$ is the set of all indicator functions $\mathbf{1}_S$
where $S$ is a disjoint unit of finitely many arcs in the circle.
We call these arcs \emph{bands}.
For a given function $f \in B$, twice the number of the bands is the number of transitions,
unless $S$ is the entire circle when there is 1 band and 0 transitions.
In any case the number of transitions is even.
We think of such an $f(\lambda)$ as a transmittance function of a filter,
and a superposition of bandpass and bandstop filters.
If the endpoints are in the interior of a band, then the band corresponds to a bandstop filter,
and otherwise it corresponds to a bandpass filter.
It is clear that a given $f$ has either 0 or 1 bandstop filters.

Let $[0,1]^N$ denote the $N$-cube and define a function $p()$
\begin{equation}\label{eqnp}
p : B \twoheadrightarrow [0,1]^N  ~~~  \text{by} ~~~  p(f) := \mathbf{y} \equiv (y_1,\ldots,y_N)  ~~~ \text{where}  ~~~   y_i = {\mu_i}^{-1} \int_{I_i} f(\lambda) \, d\lambda
\end{equation}
Note that $y_i$ is the mean of $f$ on $I_i$.
It is straightforward to show that $p()$ is surjective
and it follows that $p()$ has a right-inverse (or \emph{section}),
i.e. a function $p^+ : [0,1]^N \to B$ so that $p \circ p^+$ is the identity on $[0,1]^N$.
Such a section is fairly easy to construct, but $p^+(\mathbf{y})$ is certainly not unique, except in special cases.
If $\mathbf{v} \in [0,1]^N$ is a vertex of the cube, then $p^+(\mathbf{v})$ \emph{is} unique.
Another important case is $\mathbf{y}_{ij}=(0,\ldots,0,y_i,1,\ldots,1,y_j,0,\ldots,0)$ and $y_i,y_j \in (0,1)$.
There is a unique $f \in p^{-1}(\mathbf{y}_{ij})$ with 2 transitions (1 passband),
but an arbitrarily large number of bands of $f$ in the intervals $I_i$ and $I_j$ can be created without changing
the value of $p()$.
In the extreme case where $\mathbf{y}$ is in the interior of the cube (all $y_i \in (0,1)$),
there is a band function $f \in p^{-1}(\mathbf{y})$ with $\lceil N/2 \rceil$ bands.

In \textbf{colorSpec} software, the function $p()$ is implemented as \code{bandMaterial()},
and $p^+()$ is implemented as \code{bandRepresentation()}.
In the latter case, the function tries to find a function with the minimum number of bands;
see the corresponding man page for details.


\section{Responsivity Function}

Let $\mathbf{w} : [\beta_0,\beta_N] \to \mathbb{R}^3$ be a step function that take the constant value $\mathbf{w}_i$ on $I_i$.
Define a function
\begin{equation}
\Gamma : B \to \mathbb{R}^3  ~~~  \text{by} ~~~ \Gamma(f) := \int_{\beta_0}^{\beta_N} f(\lambda) \mathbf{w}(\lambda) \, d\lambda
~~=~~ \sum_i^N \left( \int_{I_i} f(\lambda)\, d\lambda \right) \mathbf{w}_i
\end{equation}
And define a similar function
\begin{equation}\label{eqnGammaN}
\Gamma^N : [0,1]^N \to \mathbb{R}^3  ~~~  \text{by} ~~~ \Gamma^N(y) ~=~ \Gamma^N(y_1,\ldots,y_N)  ~~:=~~  \sum_i^N y_i \mu_i \mathbf{w}_i
\end{equation}
By \ref{eqnp} it follows that $\Gamma^N(p(f)) = \Gamma(f)$.
Define $Z := \Gamma^N([0,1]^N)$;
since $Z$ is the linear image of a cube, $Z$ is a \emph{zonohedron}, see \cite{Centore2013}.
We now have a commutative diagram in which all 3 maps are surjective:
\begin{center}
\begin{tikzpicture}[scale=2]
\node (C) at (2,0.75) {$B$};
\node (F) at (2,0) {$[0,1]^N$};
\node (J) at (3.4,0) {$Z$};
\draw [->>] (C) to  node[left]{$p$} (F);
\draw [->>] (C) to  node[above]{$\Gamma$} (J);
\draw [->>] (F) to  node[below]{$\Gamma^N$} (J);
\end{tikzpicture}
\end{center}
If $f \in B$ has 0 or 2 transitions, then $\Gamma(f)$ is called a \emph{Schr{\"o}dinger color}, see \cite{West}.

In \textbf{colorSpec} software, the function $\Gamma^N()$ is implemented in \code{product()},
and is a simple matrix multiplication,
see the corresponding man page for details.





\section{Chromaticity Polygons}

From this point on, we require that all $\mathbf{w}_i$, $i=1,\ldots,N$
lie in some linear open halfspace in $\mathbb{R}^3$, except if $\mathbf{w}_i{=}0$.
This means that there is a vector $\mathbf{u}$ so that all $\langle \mathbf{u}, \mathbf{w}_i \rangle > 0$,
except if $\mathbf{w}_i{=}0$.
If all responsivities are non-negative, which is the usual case, then we can take $\mathbf{u}{=}(1,1,1)$.
We now define the vertices $\mathbf{v}_i := \mathbf{w}_i / \langle \mathbf{u}, \mathbf{w}_i \rangle$
which are in the plane
$\{ \mathbf{v} | \langle \mathbf{v},\mathbf{u} \rangle = 1\}$.
These are the vertices of what we call the \emph{chromaticity polygon} $P$ in the previously mentioned plane.
The CIE inverted-U is the classical example;
where $\mathbf{w}_i$ is $(\bar{x},\bar{y},\bar{z})$ at $\lambda_i$,
and $\mathbf{v}_i$ is the CIE chromaticity $(x,y)$ at $\lambda_i$
(after the final coordinate $z$ of $\mathbf{v}_i$ is dropped).

We also consider the central projection of $P$ onto the unit sphere $S^2$,
and call this the \emph{spherical chromaticity polygon} $P_S$.
It is clearly contained in the hemisphere centered at $u/\left| u \right|$.
The internal angles of $P$ and $P_S$ may be different,
but whether an internal angle $\theta$ is convex ($\theta < \pi$),
straight ($\theta{=}\pi$),
or concave|reflex ($\theta > \pi$),
is the same in $P$ and $P_S$.

If for all distinct indexes $i,j,k$, the vectors $\mathbf{w}_i,\mathbf{w}_j,\mathbf{w}_k$ are linearly independent
we say that the responsivities are in \emph{general position}.
If they are \emph{not} in general position, 
then $\mathbf{w}_i,\mathbf{w}_j,\mathbf{w}_k$ are linearly dependent for some distinct $i,j,k$,
which means that one of these 3 is a linear combination of the other 2.
By re-indexing assume the one is $\mathbf{w}_i$ and the others are $\mathbf{w}_j$ and $\mathbf{w}_k$.
There are 3 ways such a degeneracy can happen:

\begin{enumerate}
\item $\mathbf{w}_i = 0$  
\item $\mathbf{w}_i = \alpha \mathbf{w}_j$, where $\alpha \ne 0$ and $\mathbf{w}_j \ne 0$  
\item $\mathbf{w}_i = \alpha \mathbf{w}_j + \beta \mathbf{w}_k$, where $\alpha \ne 0$, $\beta \ne 0$, and $\mathbf{w}_j,\mathbf{w}_k$
are linearly independent
\end{enumerate}

For the chromaticity polygon $P$, with 2D vertices $\mathbf{v}_i$, these translate to 3 polygon degeneracies:
\begin{enumerate}[label=\arabic*$'$.]
\item vertex $\mathbf{v}_i$ is undefined
\item vertices $\mathbf{v}_i$ and $\mathbf{v}_j$ are identical
\item vertices $\mathbf{v}_i$, $\mathbf{v}_j$, and $\mathbf{v}_k$ are distinct but collinear,
with $\mathbf{v}_i$ between $\mathbf{v}_j$ and $\mathbf{v}_k$
\end{enumerate}

The chromaticity polygon $P$ is not simple in general; it is just a closed polygonal path.
In the next section we discuss the case where $P$ is \emph{convex}, 
which means that all internal angles are $\le \pi$.
For convex $P$ we allow all 3 of these degeneracies.
However, each group of identical vertices
and each group of distinct collinear vertices must have contiguous indexes.
A subset of $\{1,\ldots,N\}$ is \emph{contiguous} iff
the indexes are consecutive, with wraparound from $N$ to $1$ allowed.
So for this vignette, a convex $P$ is simple, except possibly for contiguous identical vertices.



% ----------------------------------------------------------------------------

\section{The Optimal Color Theorem}

The preliminaries are done and we can finally state the main result from \cite{West}:

\begin{theorem}\label{thm:main}
With $\Gamma$,  $Z$,  and $P$ as defined above, the following are equivalent:
\begin{enumerate}
\item  for any $z \in Z$, $z \in \partial Z$ iff there is an $f \in \Gamma^{-1}(z)$ with 0 or 2 transitions 
\item the chromaticity polygon $P$ is convex
\end{enumerate}
Moreover, in part 1, $p(f)$ is unique for all $z$ iff all vertices of $P$ are defined.
\end{theorem}
A point $z \in \partial Z$ is called an \emph{optimal color}.
A corollary of the theorem is that if $P$ is not convex,
then there are optimal colors that are not {Schr{\"o}dinger colors}.
We explore examples of this in the next two sections.


\section{The CIE xyz Responsivities with 5nm step}

In \textbf{colorSpec} software, the CIE responsivities with 5nm step
are stored in the object \code{xyz1931.5nm};
whose values are taken from Table 1 in \cite{ASTM308}.
The wavelengths range from 380 to 780 nm.

Analyze the responsivities, and print the degeneracies.
<<mets5, echo=TRUE, message=FALSE>>=
mets = responsivityMetrics( xyz1931.5nm )
mets$zeros
@
So the responsivity at $\lambda$=780 nm is 0.
This is not a violation of the convexity of $P$.
<<mets6, echo=TRUE, message=FALSE>>=
mets$multiples
@
There are 3 groups of multiples:
735 745 nm (not contiguous),
755 760 nm (contiguous), and
765 770 775 nm (contiguous).
The non-contiguous group is a violation of the convexity of $P$.
Now print the actual concavities in $P$.
<<mets7, echo=TRUE, message=FALSE>>=
mets$concavities
@
These are all violations too.  
The column $\mathtt{extangle}$ is the external angle at the vertex (in radians)
of the spherical chromaticity polygon $P_S$.
The sum of internal and external angles is $\pi$,
so when the external angle is negative, as these are,
the internal angle is greater than $\pi$.
In the vicinity of these wavelengths, we can find optimal colors with more than 2 transitions.
As an example, we choose the canonical optimal color with wavelengths 580 and 585 nm.
<<mets8, echo=TRUE, message=FALSE>>=
wave  = wavelength(xyz1931.5nm)
E.eye = product( illuminantE(1,wave=wave), '*', xyz1931.5nm )
spec = canonicalOptimalColors( E.eye, c(580,585), spectral=TRUE ) 
bandRepresentation( spec )[[1]]
@
So this spectrum is a superposition of 3 bandpass filters, and has 6 transitions.
<<fig1, echo=TRUE, fig.pos="H", fig.height=3.5, out.width='1.0\\linewidth', fig.cap='An example of a transmittance spectrum that is optimal, but has more than 2 transitions' >>=
par( omi=c(0,0,0,0), mai=c(0.5,0.6,0.2,0) )
plot( spec, main=FALSE, legend=FALSE, type='step', lwd=c(3,0.25) )
@




\section{The CIE xyz Responsivities with 1nm step}

In \textbf{colorSpec} software, the CIE responsivities with 1nm step
are stored in the object \code{xyz1931.1nm};
whose values are taken from Table 1 in \cite{wyszecki2000color}.
The wavelengths range from 360 to 830 nm.

Analyze the responsivities, and print the degeneracies.
<<mets10, echo=TRUE, message=FALSE>>=
mets = responsivityMetrics( xyz1931.1nm )
mets$zeros
mets$multiples
@
So there are no wavelengths where the responsivity is 0.
But all responsivities from 699 to 830 are multiples of each other 
(with angular tolerance of about $10^{-6}$ radian).
It is fairly obvious that they were extrapolated in this way intentionally.
Since these wavelengths are contiguous, there are no convexity violations so far.
Now examine the concavities in $P$.
<<mets11, echo=TRUE, message=FALSE>>=
nrow( mets$concavities )
@
This is too many concave vertices to print, so look at the first quartile of external angles instead.
<<mets12, echo=TRUE, message=FALSE>>=
fivenum( mets$concavities$extangle )
mets$concavities[ mets$concavities$extangle <= -0.01606, ]
@

<<mets13, echo=TRUE, message=FALSE>>=
wave  = wavelength(xyz1931.1nm)
E.eye = product( illuminantE(1,wave=wave), '*', xyz1931.1nm )
spec = canonicalOptimalColors( E.eye, c(407,409), spectral=TRUE ) 
bandRepresentation( spec )[[1]]
@
So this spectrum is a superposition of 2 bandpass filters, and has 4 transitions.
<<fig2, echo=TRUE, fig.pos="H", fig.height=3.5, out.width='1.0\\linewidth', fig.cap='An example of a transmittance spectrum that is optimal, but has more than 2 transitions' >>=
par( omi=c(0,0,0,0), mai=c(0.5,0.6,0.2,0) )
plot( spec, main=FALSE, legend=FALSE, type='step', lwd=c(3,0.25) )
@





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\section*{Session Information}
This document was prepared \today \quad with the following configuration:
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knit_hooks$set(output = function(x, options) { x })
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\end{document}
