/[escript]/trunk/doc/inversion/Regularization.tex
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revision 4121 by gross, Tue Dec 11 04:04:47 2012 UTC revision 4122 by gross, Thu Dec 20 05:42:35 2012 UTC
# Line 33  The coefficients $\omega^{(0)}_k$, $\ome Line 33  The coefficients $\omega^{(0)}_k$, $\ome
33  may depend on their location within the domain. We assume that for given level set function $k$ the  may depend on their location within the domain. We assume that for given level set function $k$ the
34  weighting factors $\omega^{(0)}_k$, $\omega^{(1)}_{ki}$ are scaled such that  weighting factors $\omega^{(0)}_k$, $\omega^{(1)}_{ki}$ are scaled such that
35  \begin{equation}\label{ref:EQU:REG:5}  \begin{equation}\label{ref:EQU:REG:5}
36  \int_{\Omega} ( \omega^{(0)}_k  + \frac{\omega^{(1)}_{ki}}{L_i^2} ) \; dx = \alpha_k \cdot  vol(\Omega)  \int_{\Omega} ( \omega^{(0)}_k  + \frac{\omega^{(1)}_{ki}}{L_i^2} ) \; dx = \alpha_k
37  \end{equation}  \end{equation}
38  where $\alpha_k$ defines the scale which is typically set to one. $L_i$ is the width of the domain in $x_i$ direction.  where $\alpha_k$ defines the scale which is typically set to one. $L_i$ is the width of the domain in $x_i$ direction.
39  Similarly we set for $l<k$ we set  Similarly we set for $l<k$ we set
40  \begin{equation}\label{ref:EQU:REG:6}  \begin{equation}\label{ref:EQU:REG:6}
41  \int_{\Omega} \frac{\omega^{(c)}_{lk}}{L^4} \; dx = \alpha^{(c)}_{lk} \cdot  vol(\Omega)  \int_{\Omega} \frac{\omega^{(c)}_{lk}}{L^4} \; dx = \alpha^{(c)}_{lk}
42  \end{equation}  \end{equation}
43  where $\alpha^{(c)}_{lk}$ defines the scale which is typically set to one and  where $\alpha^{(c)}_{lk}$ defines the scale which is typically set to one and
44  \begin{equation}\label{ref:EQU:REG:6b}  \begin{equation}\label{ref:EQU:REG:6b}
# Line 115  K^{reg}(m) = \frac{1}{2} Line 115  K^{reg}(m) = \frac{1}{2}
115  \sum_{k} \mu_k \cdot ( \omega^{(0)}_k \cdot m_k^2 + \omega^{(1)}_{ki}m_{k,i}^2 )  \sum_{k} \mu_k \cdot ( \omega^{(0)}_k \cdot m_k^2 + \omega^{(1)}_{ki}m_{k,i}^2 )
116  +  \sum_{l<k} \mu^{(c)}_{lk} \cdot \omega^{(c)}_{lk}  \cdot  \chi(m_l,m_k)  +  \sum_{l<k} \mu^{(c)}_{lk} \cdot \omega^{(c)}_{lk}  \cdot  \chi(m_l,m_k)
117  \end{equation}  \end{equation}
   
118  We need to provide the gradient of the cost function $J^{reg}$  with respect to the level set functions $m$.  We need to provide the gradient of the cost function $J^{reg}$  with respect to the level set functions $m$.
119  The gradient is represented by two functions $Y$ and $X$ which define the  The gradient is represented by two functions $Y$ and $X$ which define the
120  derivative of the cost function kernel with respect to $m$ and to the gradient $m_{,i}$, respectively:  derivative of the cost function kernel with respect to $m$ and to the gradient $m_{,i}$, respectively:
121  \begin{equation}\label{ref:EQU:REG:101}  \begin{equation}\label{ref:EQU:REG:101}
122  \begin{array}{rcl}  \begin{array}{rcl}
123    Y_k & = & \displaystyle{\frac{\partial K^{reg}}{\partial m_k}} \\    Y_k & = & \displaystyle{\frac{\partial K^{reg}}{\partial m_k}} \\
124     X_{k,i} & = & \displaystyle{\frac{\partial K^{reg}}{\partial m_{k,i}}}     X_{ki} & = & \displaystyle{\frac{\partial K^{reg}}{\partial m_{k,i}}}
125  \end{array}  \end{array}
126  \end{equation}  \end{equation}
127  For $Y$ we get  For the case of a single valued level set function $m$ we get
128  \begin{equation}\label{ref:EQU:REG:102}  \begin{equation}\label{ref:EQU:REG:202}
129  Y_k = \mu_k \cdot \omega^{(0)}_k \cdot m_k  Y = \mu \cdot \omega^{(0)} \cdot m
130    \end{equation}
131    and
132    \begin{equation}\label{ref:EQU:REG:203}
133     X_{i} = \mu \cdot \omega^{(1)}_{i} \cdot m_{,i}
134    \end{equation}
135    For a two-valued level set function $(m_0,m_1)$ we have
136    \begin{equation}\label{ref:EQU:REG:302}
137    Y_k = \mu_k \cdot \omega^{(0)}_k \cdot m_k \mbox{ for } k=0,1
138  \end{equation}  \end{equation}
139  and for $X$ (and ignoring the cross gradient component):  and for $X$
140  \begin{equation}\label{ref:EQU:REG:103}  \begin{equation}\label{ref:EQU:REG:303}
141   X_{k,i} = \mu_k \cdot \omega^{(1)}_{ki} \cdot m_{k,i}  \begin{array}{rcl}
142     X_{0i} &  = & \mu_0 \cdot \omega^{(1)}_{0i} \cdot m_{0,i} + \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot
143    \left( (m_{1,j}m_{1,j} ) \cdot m_{0,i} - (m_{1,j}m_{0,j} ) \cdot m_{1,i} \right) \\
144     X_{1i} &  = & \mu_1 \cdot \omega^{(1)}_{1i} \cdot m_{1,i} + \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot
145    \left( (m_{0,j}m_{0,j} ) \cdot m_{1,i} - (m_{1,j}m_{0,j} ) \cdot m_{0,i} \right)
146    \\
147    \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
150    of a linear PDE which is solved using \escript \class{LinearPDE} class. In the \escript notation we need to provide
151  We also need to provide an approximation of the inverse of the Hessian operator which provides a  \begin{equation}\label{ref:EQU:REG:600}
152  level set function $h$ for a given value $r$ represented by the pair of values $(Y,X)$. If one ignores the correlation function  \begin{array}{rcl}
153  the inner product defines the Hessian operator of the cost function. In this approach we set   A_{kilj} & = & \displaystyle{\frac{\partial X_{ki}}{\partial m_{l,j}}} \\
154  \begin{equation}\label{EQU:REG:8}  D_{kl} & =  &  \displaystyle{\frac{\partial Y_{k}}{\partial m_{l}}}
155   < p,   h > = [p, r]  \end{array}
156  \end{equation}  \end{equation}
157  for all $p$. This problem can be solved using \escript \class{LinearPDE} class by setting  For the case of a single valued level set function $m$ we get
158  \begin{equation}\label{EQU:REG:8b}  \begin{equation}\label{ref:EQU:REG:601}
159  \begin{array}{rcl}  \begin{array}{rcl}
160   A_{ij} & =&  \mu \cdot \omega^{(1)}_i \cdot \delta_{ij}  \\   A_{ij} & =&  \mu \cdot \omega^{(1)}_i \cdot \delta_{ij}  \\
161  D & = &  \mu \cdot \omega^{(0)}  D & = &  \mu \cdot \omega^{(0)}
162  \end{array}  \end{array}
163    \end{equation}
164    For a two-valued level set function $(m_0,m_1)$ we have
165    \begin{equation}\label{ref:EQU:REG:602}
166    D_{kl}  =   \mu_k \cdot \omega^{(0)}_k \cdot \delta_{kl}
167  \end{equation}  \end{equation}
168  and $X$ and $Y$ as defined by $r$ for the case of a single-valued level set function.  and
169  For a vector-valued level-set function one sets:  \begin{equation}\label{ref:EQU:REG:603}
 \begin{equation}\label{EQU:REG:8c}  
170  \begin{array}{rcl}  \begin{array}{rcl}
171   A_{kilj} & = & \mu_k \cdot \omega^{(1)}_{ki} \cdot \delta_{ij} \cdot \delta_{kl}  \\  A_{0i0j} & = & \mu_0 \cdot \omega^{(1)}_{0i} \cdot \delta_{ij} + \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot
172  D_{kl} & =  &  \mu_k \cdot \omega^{(0)}_k \cdot \delta_{kl}  \left( (m_{1,j'}m_{1,j'} )\cdot \delta_{ij}  -  m_{1,i} \cdot m_{1,j} \right)    \\
173    A_{0i1j} & = & \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot \left( 2 \cdot m_{0,i} \cdot  m_{1,j}
174    - m_{1,i} \cdot  m_{0,j} - ( m_{1,j'} m_{0,j'} ) \cdot  \delta_{ij}
175    \right)  \\
176    A_{1i0j} & = & \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot \left( 2 \cdot m_{1,i} \cdot  m_{0,j}
177    - m_{0,i} \cdot  m_{1,j} - ( m_{1,j'} m_{0,j'} ) \cdot  \delta_{ij} \right)  \\
178    A_{1i1j} & = &  \mu_1 \cdot \omega^{(1)}_{1i} \cdot \delta_{ij} + \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot
179    \left( (m_{0,j'}m_{0,j'} ) \cdot \delta_{ij}  -  m_{0,i} \cdot m_{0,j}  ) \right)
180  \end{array}  \end{array}
181  \end{equation}  \end{equation}
182    
183    

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