%% This file is part of Enblend.
%% Licence details can be found in the file COPYING.


\section[Exposure Weighting]{\label{sec:exposure-weighting}%
  \genidx{weighting!exposure}%
  \gensee{exposure weighting}{weighting, exposure}%
  Exposure Weighting}

\optidx{--exposure-optimum}%
Exposure weighting prefers pixels with a luminance~$Y$ close to the user-chosen optimum value
(option~\option{--exposure-optimum}, default: \val{val:default-exposure-optimum}) of the
normalized, real-valued luminance interval~$(0, 1)$.

\genidx{projector!grayscale}%
\gensee{grayscale projector}{projector, grayscale}%
\optidx{--gray-projector}%
\acronym{RGB}-pixels get converted to luminance before using the grayscale projector given by
\sample{--gray-projector}, which defaults to \code{average}.  Grayscale pixels simply are
identified with luminance.

\genidx{luminance interval!normalized}%
\gensee{normalized luminance interval}{luminance interval, normalized}%
In the normalized luminance interval 0.0 represents pure black and 1.0 represents pure white
independently of the data type of the input image.  This is, for a \acronym{JPEG} image the
luminance~255 maps to 1.0 in the normalized interval and for a 32~bit \acronym{TIFF} picture the
highest luminance value~4294967295 also maps to 1.0.  The middle of the luminance interval, 0.5,
is where a neutral gray tone ends up with every camera that had no exposure correction dialed
in, for example the image of any gray-card or white-card.

The exposure weighting algorithm only looks at a single pixel at a time; the pixel's
neighborhood is not taken into account.


\subsection[Built-In Functions]{\label{sec:built-in-functions}%
  \genidx{weighting!exposure!built-in}%
  Built-In Functions}

\genidx{exposure weight function!\code{gauss}}%
\genidx{exposure weight function!\propername{Gaussian}}%
Up to \App{} version~4.1 the only weighting function is the \propername{Gaussian}
\begin{equation*}\refrep{equ:weight:gauss}%
    w_{\mathrm{exp}}(Y) =
    \exp\left(-\frac{1}{2}
              \left( \frac{Y - Y_{\mathrm{opt}}}{\mathit{width}} \right)^2\right),
\end{equation*}
\noindent whose maximum position~$Y_{\mathrm{opt}}$ and $width$ are controlled by the command
line options \option{--exposure-optimum} and~\option{--exposure-width} respectively, where
$Y_{\mathrm{opt}}$ defaults to \val{val:default-exposure-optimum} and $width$ defaults to
\val{val:default-exposure-width}.  \figureName~\ref{fig:gaussian} shows some
\propername{Gaussians}.


\begin{figure}
  \ifreferencemanual\begin{maxipage}\fi
  \centering
  \includeimage{gaussian}
  \ifreferencemanual\end{maxipage}\fi

  \caption[\propername{Gaussian} weight function]{\label{fig:gaussian}%
    \App{}'s \propername{Gaussian} function with the parameters \metavar{optimum} = 0.5 and
    three different \metavar{width}s: 0.1, 0.2, and~0.4.}
\end{figure}


\optidx{--exposure-optimum}%
\optidx{--exposure-width}%
The options \option{--exposure-optimum} and~\option{--exposure-width} serve to fine-tune the
final result without changing the set of input images.  Option~\option{--ex\shyp po\shyp
  sure\hyp op\shyp ti\shyp mum} sets the point of optimum exposure.  Increasing the
\metavar{optimum} makes \App{} prefer lighter pixels, rendering the final image lighter, and
vice versa.  Option~\option{--exposure-width} defines the \metavar{width} of acceptable
exposures.  Small values of \metavar{width} penalize exposures that deviate from
\metavar{optimum} more, and vice versa.

\optidx{--exposure-weight-function}%
In \App{} version~4.2 several new exposure weight functions have been added.  Select them with
option~\option{--exposure-weight-function}.  For the following presentation we refer to the
linear luminance transform
\begin{equation*}\refrep{equ:linear-luminance-transform}
  z = \frac{Y - Y_{\mathrm{opt}}}{\mathit{width}}.
\end{equation*}
as introduced in \equationabbr~\fullref{equ:linear-luminance-transform}.

\begin{table}
  \ifreferencemanual\begin{maxipage}\fi
  \centering
  \begin{tabular}{p{.28\textwidth}lcc}
    \hline
    \multicolumn{1}{c|}{Exposure Weight} &
    \multicolumn{1}{c|}{\metavar{WEIGHT-FUNC.}} &
    \multicolumn{1}{c|}{Equ.} &
    \multicolumn{1}{c}{Chart} \\
    \hline\extraheadingsep
    \propername{Gaussian} curve (default)%
    \genidx{exposure weight function!\code{gauss}}%
    \genidx{exposure weight function!\propername{Gaussian}}%
    & \code{gauss}, \code{gaussian} &
    \fullref*{equ:weight:gauss} &
    \fullref*{fig:gaussian} \\
    \propername{Lorentz} curve%
    \genidx{exposure weight function!\code{lorentz}}%
    \genidx{exposure weight function!\propername{Lorentz} curve}%
    \genidx{exposure weight function!\propername{Lorentzian}}%
    & \code{lorentz}, \code{lorentzian} &
    \fullref*{equ:weight:lorentz} &
    \fullref*{fig:lorentzian} \\
    Upper half-wave of a sine%
    \genidx{exposure weight function!\code{halfsine}}%
    \genidx{exposure weight function!\code{half-sine}}%
    & \code{halfsine}, \code{half-sine} &
    \fullref*{equ:weight:halfsine} &
    \fullref*{fig:halfsine} \\
    Full sine-wave shifted upwards by one%
    \genidx{exposure weight function!\code{fullsine}}%
    \genidx{exposure weight function!\code{full-sine}}%
    & \code{fullsine}, \code{full-sine} &
    \fullref*{equ:weight:fullsine} &
    \fullref*{fig:fullsine} \\
    Quartic, or bi-square function%
    \genidx{exposure weight function!\code{bisquare}}%
    \genidx{exposure weight function!\code{bi-square}}%
    & \code{bisquare}, \code{bi-square} &
    \fullref*{equ:weight:bisquare} &
    \fullref*{fig:power}
  \end{tabular}
  \ifreferencemanual\end{maxipage}\fi

  \caption[Exposure weight functions]{\label{tab:weight-functions}%
    \genidx{exposure weight functions}%
    Available, compiled-in exposure weight functions.}
\end{table}


Functions \propername{Gaussian}
\begin{equation*}\refrep{equ:weight:gauss}%
    w_{\mathrm{exp}}(z) = \exp\left(-z^2 / 2\right)
\end{equation*}
and \propername{Lorentzian}
\begin{equation*}\refrep{equ:weight:lorentz}
  w_{\mathrm{exp}}(z) = \frac{1}{1 + z^2 / 2}
\end{equation*}
\noindent behave like $1 - z^2$ around the optimum.  However for large $|z|$ the
\propername{Gaussian} weight rolls off like $\exp(-z^2/2)$ and the \propername{Lorentzian} only
as $z^{-2}$.


\begin{figure}
  \ifreferencemanual\begin{maxipage}\fi
  \centering
  \includeimage{lorentzian}
  \ifreferencemanual\end{maxipage}\fi

  \caption[\propername{Lorentzian} function]{\label{fig:lorentzian}%
    \App{}'s \propername{Lorentzian} function with the parameters $\metavar{optimum} = 0.5$ and
    three different \metavar{width}s: 0.1, 0.2, and~0.4.}
\end{figure}


Both, the \propername{Gaussian} and the \propername{Lorentzian} are easy to use, because they do
not go exactly to zero.  Thus, \App{} can select ``better'' pixels even far away from the chosen
optimum.


\begin{figure}
  \ifreferencemanual\begin{maxipage}\fi
  \centering
  \includeimage{halfsine}
  \ifreferencemanual\end{maxipage}\fi

  \caption[Half-Sine function]{\label{fig:halfsine}%
    \App{}'s Half-Sine function with the parameters $\metavar{optimum} = 0.5$ and three
    different \metavar{width}s: 0.1, 0.2, and~0.4.}
\end{figure}


\begin{figure}
  \ifreferencemanual\begin{maxipage}\fi
  \centering
  \includeimage{fullsine}
  \ifreferencemanual\end{maxipage}\fi

  \caption[Full-Sine function]{\label{fig:fullsine}%
    \App{}'s Full-Sine function with the parameters $\metavar{optimum} = 0.5$ and three
    different \metavar{width}s: 0.1, 0.2, and~0.4.}
\end{figure}


Again, Half-Sine
\begin{equation}\refrep{equ:weight:halfsine}
  w_{\mathrm{exp}}(z) =
  \left\{\begin{array}{cl}
  \cos(z) & \mbox{if } |z| \leq \pi/2 \\
  0       & \mbox{otherwise.}
  \end{array}\right.
\end{equation}
and Full-Sine
\begin{equation}\refrep{equ:weight:fullsine}
  w_{\mathrm{exp}}(z) =
  \left\{\begin{array}{cl}
  (1 + \cos(z)) / 2 & \mbox{if } |z| \leq \pi \\
  0                 & \mbox{otherwise.}
  \end{array} \right.
\end{equation}
\noindent behave like $1 - z^2$ around the optimum, like \propername{Gaussian} and
\propername{Lorentzian}.  However for large $|z|$ they both are exactly zero.  The difference is
how they decrease just before they reach zero.  Half-Sine behaves like $z - z'$ and Full-Sine
like $(z - z'')^2$, where $z'$ and $z''$ are the respective zeros.


\begin{figure}
  \ifreferencemanual\begin{maxipage}\fi
  \centering
  \includeimage{power}
  \ifreferencemanual\end{maxipage}\fi

  \caption[Bi-Square function]{\label{fig:power}%
    \App{}'s Bi-Square function with the parameters $\metavar{optimum} = 0.5$ and three
    different \metavar{width}s: 0.1, 0.2, and~0.4.}
\end{figure}


Bi-Square
\begin{equation}\refrep{equ:weight:bisquare}
  w_{\mathrm{exp}}(z) =
  \left\{
  \begin{array}{cl}
    1 - z^4 & \mbox{if } |z| \leq 1 \\
    0       & \mbox{otherwise.}
  \end{array}
  \right.
\end{equation}
\noindent is the only predefined function that behaves like $1 - z^4$ around the optimum.

The weight functions Half-Sine, Full-Sine, and Bi-Square are more difficult to use, because they
yield exactly zero if the normalized luminance of a pixel is far enough away from the optimum.
This can lead to pathologies if the luminances of the same pixel position in all $N$ input
images are assigned a weight of zero.  For all-zero weights \App{} falls back to weighing
equally.  This is, each pixel gets a weight of $\slfrac{1}{N}$, which may or may not be the
desired result.  However, if the \metavar{width} is chosen so large that the weight never
vanishes or the input images span a large enough range of luminances for each and every pixel,
the problem is circumnavigated.

\optidx{--exposure-cutoff}%
Another way of cutting off underexposed or overexposed pixels is to use
option~\option{--exposure-cutoff}, which has the additional benefit of allowing to choose upper
and lower cutoff separately.


\begin{figure}
  \ifreferencemanual\begin{maxipage}\fi
  \centering
  \includeimage{exposure-weights}
  \ifreferencemanual\end{maxipage}\fi

  \caption[Comparison of exposure weight functions]{\label{fig:exposure-weights}%
    Comparison of all of \App{}'s built-in exposure weight functions~$w(Y)$ for the default
    values of $\metavar{optimum} = 0.5$ and $\metavar{width} = 0.2$.  Note that all functions
    intersect at $w = \slfrac{1}{2}$, this is, they share the same \acronym{FWHM}.}
\end{figure}


\figureName~\ref{fig:exposure-weights} compares all available exposure weight functions for the
same parameters, namely their defaults.  They all intersect at $w = \slfrac{1}{2}$ independently
of \metavar{optimum} or \metavar{width}, making it simple to just try them out without
fundamentally changing brightness.


\subsection[User-Defined Functions]{\label{sec:user-defined-functions}%
  \genidx{weighting!exposure!user-defined}%
  User-Defined Exposure Weighting Functions}

%% -- Commented out for Stable Branch 4.2
%% Depending on how \App{} was compiled it may support dynamically-linked exposure weighting
%% functions, \acronym{OpenCL} exposure weighting functions, or both.

\input{enfuse-exposure-weighting-user-dynamic}
%% -- Commented out for Stable Branch 4.2
%%\input{enfuse-exposure-weighting-user-opencl}


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