\title{Color-based per-pixel blending of detail textures}
\author{Papavasiliou Dimitris \\
\small\texttt {dpapavas@gmail.com}}
\date{\today}

\documentclass[12pt]{article}
\usepackage{hyperref}

\begin{document}
\maketitle

\section{Introduction}
Detail texture blending is a technique that can be used to add high
frequency detail to a base texture when directly using higher
resolution versions of this base texture is either impractical or not
worth the cost in terms of added texture memory usage and bandwidth
\cite{detail}.  In many cases as for example when texturing terrain,
more than one detail maps may be necessary to represent a usually
small number of discrete categories of small-scale detail, such as
grass, soil and rock.  The solution to this problem, also known as
splatting \cite{splatting}, has traditionally been to use a grayscale
mask texture for each detail map to control the way it is blended over
the base texture to produce the final result.  These masks can either
be automatically calculated based on analysis of the surface geometry
(for example the height, slope or aspect of the terrain) or manually
created by an artist.

The technique to be described in the next section is similar to the
latter approach but instead of using additional mask textures for each
detail map the base map itself is used.  This eliminates the problem
of creating separate masks as well as the added cost in terms of the
needed memory bandwidth and GPU texture units.  Perhaps the biggest
gain though is that the blending function is evaluated per-pixel and
the resolution of the transition between different detail maps is not
bound by the resolution of the used mask textures.

This presentation does not imply that the technique is novel, it may
have been employed by others.  In fact the notion of using the base
texture color to select detail maps seemed to the author so
straightforward that it was taken for granted that a quick research on
the internet would yield the details needed to carry it out.  The fact
that this did not in fact prove true was the main incentive for
publishing this work in the hope that it will be useful to others.

\section{Blending based on color}

The basic idea of the method is the following.  Instead of using masks
to define how much of each texture to blend into a given fragment of
the terrain the color of the base texture is used. A color is assigned
to each detail map used, for example some shade of green for grass, a
shade of brown for soil, gray for rock and so on.  Then, when
rendering the terrain, at each output fragment each detail map is
blended with a contribution inversely proportional to the 'distance'
of the base texture color from the color assigned to the detail map in
question.  Conceptually this would mean that where the color of the
base map is green the prevailing detail texture would be that of grass
but as the base texture fades towards brown say so would the detail
map fade towards a mixture of grass and soil and finally towards
mostly soil.  Smooth transitions in the base map ensure smooth
transitions between detail textures and one can utilize as many detail
textures as the number of texturing units available.

In terms of artistic control the basic advantage of this method is
that it is very easy to use and extremely versatile: the base map
allows the definition of the general layout of the terrain texture and
can either be created by hand or sourced from aerial and satellite
imagery.  Once this is defined any number of detail maps can be used,
provided that the hardware has enough texturing units of course, but
older hardware can be supported by leaving out detail maps based on
the number of units available.  For example in addition to the basic
set of soil, grass and rock detail maps one could use a separate map
of sparse grass and assign it to a green-brown shade halfway between
the colors assigned to grass and soil.  By varying the color as well
as the power parameter that will be explained in the next section, an
infinity of variations can be achieved.  Using the same set of maps
the terrain can be made to look more grassy or more barren by changing
the colors assigned to each detail map (also making it possible to
provide multiple 'skins' of the same terrain by specifying different
color configurations).  By changing these colors in real-time it would
be possible to animate the terrain shading.  The result is almost
guaranteed to be aesthetically pleasing (provided suitable base and
detail maps are used) since at every point all maps are mixed in a
smooth and controllable fashion.  The only downside is that you lose
some control over the way the textures are applied since you do not
specify it directly.

\section{Detail texture synthesis through multivariate interpolation}

As it turns out the method is in fact a straight application of an
existing mathematical method in the field of multivariate
interpolation named inverse distance weighting \cite{idw}.  Let us
therefore cast the problem into a more mathematical formulation.

Let $f$ be a function $f: R^3 \rightarrow R^N$ mapping three
dimensional points representing colors to N-dimensional points
representing weights for $N$ discrete detail maps.  That is the value
of $f$ is a vector with each component lying in $[0, 1]$ and defining
how much one of the detail maps contributes to the final detail
texture.  Given then the color $\mathbf{x_b} \in R^3$ of the base map
at some point and the colors $D_i,\; i = 1,...,N$ of each detail map at
this point it is straightforward to calculate the final blended color
$C$ as

\begin{equation}
C = \sum_{i = 1}^{N} {f_i(\mathbf{x_b}) D_i}
\end{equation}

In the equation above $f_i$ is the i-th component of the value of $f$,
which is the weight of the i-th detail map.

In order to calculate the needed mapping $f$ we specify an arbitrary
number of known points for it.  In general one can specify values for
$f$ with arbitrary weights for each detail map but in the current
implementation it was assumed that each point specified would have one
component equal to one and all others equal to zero.  In other words
the initial values map colors to pure detail maps instead of mapping
them to mixtures.  This choice was made for the sake of simplicity but
it can readily avoided should more control over the blending be
required.

According to the inverse distance weighting method, given a set of
initial values $u_i$ for the mapping $f$ on some points $\mathbf{x_i}$ the
values on any other point can be interpolated through the equation

\begin {equation}
    u(\mathbf{x}) = \sum_{i = 1}^{N}{ \frac{ w_i(\mathbf{x}) u_i } {
    \sum_{j = 1}^{N}{ w_j(\mathbf{x}) } } }
\end {equation}

where

\begin {equation}
    w_i(\mathbf{x}) = \frac{1}{d(\mathbf{x},\mathbf{x}_i)^p}
\end {equation}

Here $\mathbf{x}$ denotes an interpolated (arbitrary) point,
$\mathbf{x_i}$ is an interpolating (known) point, $d$ is a given
distance metric from the known point $\mathbf{x_i}$ to the unknown
point $\mathbf{x}$, $N$ is the total number of known points and $p$
is a positive real number, called the power parameter.

In our case, where $f$ maps colors to detail map mixing weights, what
this equation means intuitively is that for some point on the terrain
the weight of each detail map is the inverse of the distance between
the color assigned to the detail map and the color of the base map at
this point so that a detail map contributes more to the final result
over the areas of the terrain where the base color is closer to the
color assigned to it. The normalization of each weight through
division by the sum of all weights ensures that the final blended
detail map will have the same mean brightness as the input detail
maps.

The power parameter is particularly interesting as greater values of p
assign greater influence to values closest to the interpolated point
resulting in sharper separation of the detail maps and more
well-defined map transitions.  Furthermore as the weighting function
is evaluated per-pixel the definition of the transitions is not bound
by the resolution of the base map so the latter needn't be overly
large (assuming it is sampled using bi-linear interpolation).  A base
map at a 1 m resolution for example only allows you to directly
specify the mixture of detail maps that is to be applied every 1 m but
since the mapping will be evaluated per pixel yielding a different set
of detail map weights based on the interpolated base map values one
can get pleasantly sharp detail map transitions by making $p$
sufficiently high.

The distance function used was the usual Euclidean distance metric but
instead of specifying the colors and calculating their distance in the
RGB color space the HSL color space \cite{color} was chosen.  This
choice was made on the very scientific basis that `it seemed
reasonable' given that the HSL color space is more intuitive in terms
of what a distance along each axis means.  It might be possible to use
the RGB color space instead which would incur a minor saving on the
calculations needed for the conversion but the author didn't have much
luck with a quick implementation.\footnote{It seemed to work in
general but it proved difficult to map gray to a detail map such as
rock or asphalt.  The grass kept creeping into the road.}

Apart from the fact that the colors assigned to the detail maps have
to be specified in the HSL space and that the color sampled from the
base map has to be transformed into that space in order to calculate
the distance there's an additional implication of this choice that
should be noted.  The HSL space is cylindrical since the hue
represents an angle.  When calculating the distance this should be
taken into account as the difference between a hue of $350^{\circ}$ and
a hue of $20^{\circ}$ is $30^{\circ}$ and not $330^{\circ}$.  Using
the Euclidean metric the distance between two colors $(h_a, s_a, l_a)$
and $(h_b, s_b, l_b)$ can be calculated as

\begin {equation}
       d = \sqrt{\left[\min\left(h_a - h_b, 1 - h_a +
       h_b\right)\right]^2 + \left(s_a - s_b\right) +
       \left(l_a - l_b\right)}
\end {equation}

\section {Implementation details}

Once the detail map for the given terrain fragment has been calculated
there are a number of possibilities for blending them onto the terrain
such as using additive blending as in

\begin {equation}
     f = b + d - 0.5
\end {equation}

where $f$ is the final pixel color, $b$ the base color and $d$ the
detail map color.  That would work with both color and grayscale
detail maps.  Assuming monochrome detail maps one could simply use
some form of multiplicative blending such as

\begin {equation}
     f = b d
\end {equation}

In the case of color detail maps another possibility (which is the
one chosen by the author) is to use the luminance of the base color to
provide intensity variation to the detail maps as well as baked in
lighting if the base map is lit:

\begin {equation}
     f = \frac{b_r + b_g + b_b}{3} d
\end {equation}

There are some possible modifications to the scheme outlined so far.
For one, one could modify the distance metric by weighting each of the
axes, which correspond to hue, saturation and lightness, with an
arbitrary weight.  This would allow one more control since it would be
possible to specify how important each component of the input colors
is.  For example when mapping a shade of green to a map of grass
specifying a higher weight for hue would make it more significant by
assigning larger distances to points that differ by some distance in
hue than to points that differ by the same distance in saturation or
lightness.  This allows more fine-tuning based on intuitive color
qualities and was one of the reasons the HSL color space was chosen.
Although initially implemented the author chose to disallow
user-setting of weights for the sake of simplicity after observing
that you can do quite well without them.

Another area that deserves more exploration is the weighting function
used.  Instead of using the inverse Euclidean distance one could, for
example, assign a weight of one to the detail map with the least
distance and zero to all others.  This would yield a sharp mixing of
detail maps suitable for cel-like shading.  Of course a function with
more quantization steps would be preferable.

Finally it should be noted that this method could scale well to more
levels of hierarchy.  For example for a flight simulator it should be
possible to use two (or more) palettes of detail maps one with
geotypical detail that is visible from a relatively low height such as
fields, cities and the like and one with the usual close-range detail,
that is grass, rock, etc.  One could then use the base map to blend
the first palette and the result to blend the second and blend these
yet again based on the height to get the final fragment color.  Of
course when the viewing point is high enough only the first palette
needs to be used.

An implementation of the algorithm using GLSL is given in the appendix.

\section{Results}

The web page cited in \cite{home} contains images as well as a few
videos captured from a motorcycle simulation where this method is
employed.  Although the base texture and detail maps employed are
quite rudimentary and hardly any time has been taken experimenting
with the colors assigned to the detail maps which guide the process of
mixing they still allow the basic qualities of the method to be
assessed.  There is probably a lot of room for improvement though,
through proper fine-tuning of the various parameters as well as the
base map.

\begin{thebibliography}{9}


\bibitem{detail} Detail textures\\ \scriptsize\url{http://www.opengl.org/resources/code/samples/sig99/advanced99/notes/node88.html}\normalsize
\bibitem{splatting} Terrain Texture Compositing by Blending in the
Frame-Buffer \\ \scriptsize\url{http://www.cbloom.com/3d/techdocs/splatting.txt}\normalsize
\bibitem{idw} Inverse distance weighting \\ \scriptsize\url{http://en.wikipedia.org/wiki/Inverse_distance_weighting}\normalsize
\bibitem{color} HSL and HSV color spaces \\ \scriptsize\url{http://en.wikipedia.org/wiki/HSL_and_HSV}\normalsize
\bibitem{home} Color-based per-pixel blending of detail textures
sample page \\ \scriptsize\url{http://www.nongnu.org/techne/research/blending}\normalsize
\end{thebibliography}

\appendix
\section{GLSL source code}
The implementation is quite straightforward and simple, although
there are a few details such as converting the base texture texel to
the HSL color space and computing a distance in this color space which
is cylindrical.  The following listing contains the code used by the
author to implement this shader in GLSL.  The hue in this
implementation has been remapped to lie in the range [0,1) although it
is customarily given in degrees. Also in the code below \texttt{N} is
an integer constant holding the number of detail maps
used, \texttt{pigments} is a uniform array of N 3-vectors holding the
colors assigned by the user to each map and \texttt{detailScale} is a
uniform array of N 2-vectors specified by the user to control the
scale of each detail map.

\ttfamily\small
{{\def\dash{\raise2.1pt\hbox{\rule{5pt}{0.3pt}}\hspace{1pt}}\begin{tabbing}
{\texttt{/$\ast$\ Sample\ the\ base\ map.\ $\ast$/}}\\
\\
texel\ =\ vec3(texture2D(sampler,\ gl\_TexCoord{[}0{]}.st));\\
\\
{\texttt{/$\ast$\ Convert\ texel\ to\ HSL.\ $\ast$/}}\\
\\
M\ =\ max(max\ (texel.r,\ texel.g),\ texel.b);\\
m\ =\ min(min\ (texel.r,\ texel.g),\ texel.b);\\
C\ =\ M\ \dash{}\ m;\\
\\
{\texttt{/$\ast$\ If\ C\ is\ zero\ the\ color\ is\ fully\ desaturated\ so}}\\
{\texttt{\ \ \ its\ hue\ is\ undefined.\ \ We\ just\ set\ it\ to\ zero.\ $\ast$/}}\\
\\
{\textbf{if}}\ (C\ $>$\ 0.0)\ \{\\
\ \ \ \ {\textbf{if}}\ (texel.r\ ==\ M)\ \{\\
\ \ \ \ \ \ \ \ H\ =\ mod((texel.g\ \dash{}\ texel.b)\ /\ C,\ 6.0)\ /\ 6.0;\\
\ \ \ \ \}\ {\textbf{else}}\ {\textbf{if}}\ (texel.g\ ==\ M)\ \{\\
\ \ \ \ \ \ \ \ H\ =\ ((texel.b\ \dash{}\ texel.r)\ /\ C\ +\ 2.0)\ /\ 6.0;\\
\ \ \ \ \}\ {\textbf{else}}\ \{\\
\ \ \ \ \ \ \ \ H\ =\ ((texel.r\ \dash{}\ texel.g)\ /\ C\ +\ 4.0)\ /\ 6.0;\\
\ \ \ \ \}\\
\}\ {\textbf{else}}\ \{\\
\ \ \ \ H\ =\ 0.0;\\
\}\\
\\
hsv\ =\ vec3(H,\ C\ /\ M,\ M);\\
\\
{\texttt{/$\ast$\ Now\ compute\ the\ distance\ between\ the\ base\ texture}}\\
{\texttt{color\ and\ the\ color\ assigned\ to\ each\ detail\ map.\ Also}}\\
{\texttt{comput\ the\ sum\ of\ the\ distances\ and\ store\ it\ in\ C.\ $\ast$/}}\\
\\
{\textbf{for}}\ (i\ =\ 0,\ C\ =\ 0.0\ ;\ i\ $<$\ N\ ;\ i\ +=\ 1)\ \{\\
\ \ \ \ vec3\ v;\\
\\
\ \ \ \ v\ =\ hsv\ \dash{}\ pigments{[}i{]};\\
\\
\ \ \ \ {\texttt{/$\ast$\ Take\ into\ account\ that\ v.x\ is\ the\ hue}}\\
{\texttt{\ \ \ \ and\ hence\ an\ angle\ so\ we\ want\ the\ modular}}\\
{\texttt{\ \ \ \ distance\ between\ the\ two\ numbers.\ $\ast$/}}\\
\\
\ \ \ \ v.x\ =\ min(v.x,\ 1.0\ \dash{}\ v.x);\\
\ \ \ \ distances{[}i{]}\ =\ 1.0\ /\ pow(dot(v,\ v),\ power);\\
\ \ \ \ C\ +=\ distances{[}i{]};\\
\}\\
\\
{\texttt{/$\ast$\ Weight\ each\ detail\ map\ based\ on\ its\ normalized}}\\
{\texttt{distance\ and\ sum\ to\ get\ the\ final\ detail\ texel.\ $\ast$/}}\\
\\
{\textbf{for}}\ (i\ =\ 0,\ detail\ =\ vec3(0.0)\ ;\ i\ $<$\ N\ ;\ i\ +=\ 1)\ \{\\
\ \ \ \ detail\ +=\ distances{[}i{]}\ /\ C\ $\ast$\\
\makebox[48pt][l]{}\ \ \ \ \ \ texture2D(detailSampler{[}i{]},\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ detailScale{[}i{]}\ $\ast$\ gl\_TexCoord{[}0{]}.st).rgb;\\
\}\\
\\
{\texttt{/$\ast$\ Modulate\ the\ detail\ texel\ by\ the\ luminance\ of}}\\
{\texttt{the\ base\ map\ to\ get\ the\ final\ value.\ $\ast$/}}\\
\\
result\ =\ (texel.r\ +\ texel.g\ +\ texel.b)\ /\ 3.0\ $\ast$\ detail;
\end{tabbing}}}

\end{document}
