<?xml version="1.0" encoding="ISO-8859-1"?>
<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" version="5.0-subset Scilab" xml:lang="en" xml:id="pl_ph_direct">
  <info>
    <pubdate>19-May-2010</pubdate>
  </info>

  <refnamediv>
    <refname>pl_ph_direct</refname>
    <refpurpose>Discrete Pasch-Hausdorff envelope</refpurpose>
  </refnamediv>

   <refsynopsisdiv>
     <title>Calling Sequence</title>
     <synopsis>ph = pl_ph_direct(X, Y, S, lambda)</synopsis>
   </refsynopsisdiv>

   <refsection>
     <title>Parameters</title>
     <variablelist>

       <varlistentry>
         <term>X</term>
         <listitem>
           <para>column vector of size m.</para>
         </listitem>
       </varlistentry>

       <varlistentry>
         <term>Y</term>
         <listitem>
           <para>column vector of size m.</para>
         </listitem>
       </varlistentry>

       <varlistentry>
         <term>S</term>
         <listitem>
           <para>column vector of size n.</para>
         </listitem>
       </varlistentry>

       <varlistentry>
         <term>lambda</term>
         <listitem>
           <para>constant.  Scaling parameter for the norm function.</para>
         </listitem>
       </varlistentry>

       <varlistentry>
         <term>ph</term>
         <listitem>
           <para>The Pasch-Hausdorff envelope evaluated at S.</para>
         </listitem>
       </varlistentry>

     </variablelist>
   </refsection>

   <refsection>
     <title>Description</title>

     <para>
       Computes the Pasch-Hausdorff envelope of a function described
       by the vectors X and Y, where y(i)=f(x(i)).  Returns the
       envelope evaluated at the points in the vector S.  The
       Pasch-Hausdorff envelope is the inf-convolution of the function
       with a scaled norm term,
       <!--<programlisting><![CDATA[
    (f ic lambda||.||)(s) = inf [f(x) + lambda||x-s||].
                             x
]]></programlisting>-->
			<para>
				<inlinemediaobject>
					<imageobject>
						<imagedata fileref="pl_ImagesOfEquationsByLatex/pl_ph_direct/pl_ph_direct_1.png"/>
					</imageobject>
				</inlinemediaobject>
			</para>
       <para>The discrete Pasch-Hausdorff envelope is computed using the
       formula</para>
       <!--<programlisting><![CDATA[
    S = min [y_i + lambda |x_i - s_i|].
         i
]]></programlisting>-->
			<para>
				<inlinemediaobject>
					<imageobject>
						<imagedata fileref="pl_ImagesOfEquationsByLatex/pl_ph_direct/pl_ph_direct_2.png"/>
					</imageobject>
				</inlinemediaobject>
			</para>
     

     <para>
       This function runs in O(mn) time.
     </para>
	 </para>
   </refsection>

   <refsection>
     <title>Examples</title>
     <programlisting role="example"><![CDATA[
lambda = 0.5;
X = [-2;-1;0;1;2];
Y = abs(abs(X)-1);
S = X;
ph = pl_ph_direct(X,Y,S,lambda),
]]></programlisting>
   </refsection>

   <refsection>
     <title>Authors</title>
     <para>
       <literal>Bryan Gardiner</literal>, University of British Columbia, BC, Canada
     </para>
   </refsection>

</refentry>
