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<HTML>
<HEAD>
<TITLE>Diffusion</TITLE>
</HEAD>
<BODY BGCOLOR="#FFFFFF" TEXT="#000000">
<P><font size="+3" color="green"><B>Diffusion</B></font></P>
<TABLE border="1" cols="2" frame="box" rules="all" width="572">
<TR>
<TD width="15%" valign="top"><B>Syntax</B>:</TD>
<TD width="85%" valign="top"><CODE>
DENSITY\DIFFUSION { x y } z { p1 p2 }</CODE>
</TD></TR>
<TR>
<TD width="15%" valign="top"><B>Qualifiers</B>:</TD>
<TD width="85%" valign="top"><CODE>
\PARTIAL, \XPROFILE, \YPROFILE, \BORDER, \AXES, \RESET</CODE>
</TD></TR>
<TR>
<TD width="15%" valign="top"><B>Defaults</B>:</TD>
<TD width="85%" valign="top"><CODE>
if <CODE>z</CODE> is a matrix:
<CODE>x=[1;2;3;...], y=[1;2;3;...],</CODE>
<CODE>p1=0, p2=1,</CODE>
\-PARTIAL, \-XPROFILE, \-YPROFILE, \BORDER, \AXES, \RESET</CODE>
</TD></TR>
</TABLE>
<P>
Diffusion is a form of digital halftoning. A threshold is fixed at
½. Data values are interpolated at each pixel location, and then
normalized to be between <CODE>0</CODE> (white) and <CODE>1</CODE> (black). The
resulting binary output value is compared with the original grey level value. The
difference is called the error for that location. The signal consisting of past error
values is passed through an error filter to produce a correction factor to be added to
future input values. Thus, errors are diffused over a weighted neighborhood.</p>
<P>
<font size="+1" color="green">Input variables</font></P>
<P>
If <CODE>z</CODE> is a vector, the parameters <CODE>x</CODE> and <CODE>y</CODE>
are expected and must be vectors. <CODE>x</CODE> and <CODE>y</CODE> are assumed to represent a scattered set
of points, where <CODE>z[i]</CODE> is the altitude corresponding to the location
<CODE>(x[i],y[i])</CODE>. A matrix is interpolated on these scattered points by means of a Thiessen
triangulation of the plane. The three vectors must be the same length.</p>
<p>
If <CODE>z</CODE> is a matrix, the parameters <CODE>x</CODE> and <CODE>y</CODE> default to
<CODE>[1;2;3;...]</CODE>, but if entered they must be vectors. Each matrix element,
<CODE>z[i,j]</CODE>, is associated with the coordinates <CODE>(x[j],y[i])</CODE>. The
length of <CODE>x</CODE> must be equal to the number of columns of <CODE>z</CODE> and the length of
<CODE>y</CODE> must be equal to the number of rows. The vectors <CODE>x</CODE> and
<CODE>y</CODE> are used for scaling the axes.</p>
<p>
<font size="+1" color="green">Changing the range of values</font></p>
<p>
The optional parameters <CODE>p1</CODE> and <CODE>p2</CODE> can be used to broaden or shrink
the range of data values. If <CODE>z<sub>max</sub></CODE> is the maximum value of the data and
<CODE>z<sub>min</sub></CODE> is the minimum value of the data, the full colour range will be from
a minimum of <CODE>min = p1*(z<sub>max</sub>-z<sub>min</sub>)+z<sub>min</sub></CODE>
to a maximum of <CODE>max = p2*(z<sub>max</sub>-z<sub>min</sub>)+z<sub>min</sub></CODE>.
If <CODE>z</CODE> is a data value and if
<CODE>z < p1*(z<sub>max</sub>-z<sub>min</sub>)+z<sub>min</sub></CODE>,
that data value is treated as <CODE>z<sub>min</sub></CODE>.
If <CODE>z > p1*(z<sub>max</sub>-z<sub>min</sub>)+z<sub>min</sub></CODE>,
that data value is treated as <CODE>z<sub>max</sub></CODE>.
The default values are: <CODE>p1=0</CODE> and <CODE>p2=1</CODE>.</p>
<p>
<font size="+1" color="green">Example</font></P>
<p>
These examples use the following vectors.</p>
<pre>
X=[ 1; 0; 1; 0; .2; .3; .5; .8]
Y=[ 5; 5; 0; 0; 1;1.5; 2.5; 4]
Z=[ 10; 10; 10; 10; -100; 10; -100; 500]
</PRE>
<table>
<tr>
<td><font color="blue"><CODE>DENSITY\DIFFUSION X Y Z</CODE></font></td>
<td><img src="diffusion1.png"></td>
</tr><tr>
<td><font color="blue"><CODE>DENSITY\DIFF\XPRO\YPRO X Y Z</CODE></font></td>
<td><img src="diffusion2.png"></td>
</tr></table>
<p>
<a href="dithering.htm"><img src="../shadow_left.gif">
<font size="+1" color="olive">DITHERING</font></a><br />
<a href="boxes.htm"><img src="../shadow_right.gif">
<font size="+1" color="olive">BOXES</font></a></P>
</BODY>
</HTML>
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