/[escript]/release/3.4.2/doc/inversion/Regularization.tex
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revision 4122 by gross, Thu Dec 20 05:42:35 2012 UTC revision 4268 by gross, Mon Mar 4 00:41:08 2013 UTC
# Line 25  $\chi$ is a given symmetric, non-negativ Line 25  $\chi$ is a given symmetric, non-negativ
25  \begin{equation}\label{EQU:REG:4}  \begin{equation}\label{EQU:REG:4}
26   \chi(a,b) =  ( a_{,i} a_{,i}) \cdot ( b_{,j} b_{,j}) -   ( a_{,i} b_{,i})^2   \chi(a,b) =  ( a_{,i} a_{,i}) \cdot ( b_{,j} b_{,j}) -   ( a_{,i} b_{,i})^2
27  \end{equation}  \end{equation}
28  where summations over $i$ and $j$  are performed. Notice that cross-gradient function  where summations over $i$ and $j$  are performed, see~\cite{GALLARDO2005a}. Notice that cross-gradient function
29  is measuring the angle between the surface normals of contours of level set functions. So  is measuring the angle between the surface normals of contours of level set functions. So
30  minimizing the cost function will align the surface normals of the contours.  minimizing the cost function will align the surface normals of the contours.
31    
# Line 146  and for $X$ Line 146  and for $X$
146  \\  \\
147  \end{array}  \end{array}
148  \end{equation}    \end{equation}  
149  We also need to provide an approximation of the inverse of the Hessian operator. The operator evaluation is executes as a solution  We also need to provide an approximation of the inverse of the Hessian operator as discussed in section~\ref{chapter:ref:inversion cost function:gradient}.
 of a linear PDE which is solved using \escript \class{LinearPDE} class. In the \escript notation we need to provide  
 \begin{equation}\label{ref:EQU:REG:600}  
 \begin{array}{rcl}  
  A_{kilj} & = & \displaystyle{\frac{\partial X_{ki}}{\partial m_{l,j}}} \\  
 D_{kl} & =  &  \displaystyle{\frac{\partial Y_{k}}{\partial m_{l}}}  
 \end{array}  
 \end{equation}  
150  For the case of a single valued level set function $m$ we get  For the case of a single valued level set function $m$ we get
151  \begin{equation}\label{ref:EQU:REG:601}  \begin{equation}\label{ref:EQU:REG:601}
152  \begin{array}{rcl}  \begin{array}{rcl}

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