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1 \chapter{Regularization}\label{Chp:ref:regularization}
2
3 The general cost function $J^{total}$ to be minimized has some of the cost
4 function $J^f$ measuring the defect of the result from the
5 forward model with the data, and the cost function $J^{reg}$ introducing the
6 regularization into the problem and makes sure that a unique answer exists.
7 The regularization term is a function of, possibly vector-valued, level set
8 function $m$ which represents the physical properties to be represented and is,
9 from a mathematical point of view, the unknown of the inversion problem.
10 It is the intention that the values of $m$ are between zero and one and that
11 actual physical values are created from a mapping before being fed into a
12 forward model. In general the cost function $J^{reg}$ is defined as
13 \begin{equation}\label{EQU:REG:1}
14 J^{reg}(m) = \frac{1}{2} \int_{\Omega} \left(
15 \sum_{k} \mu_k \cdot ( \omega^{(0)}_k \cdot m_k^2 + \omega^{(1)}_{ki}m_{k,i}^2 )
16 + \sum_{l<k} \mu^{(c)}_{lk} \cdot \omega^{(c)}_{lk} \cdot \chi(m_l,m_k) \right) \; dx
17 \end{equation}
18 where summation over $i$ is performed. The additional trade--off factors
19 $\mu_k$ and $\mu^{(c)}_{lk}$ ($l<k$) are between zero and one and constant across the
20 domain. They are potentially modified during the inversion in order to improve the balance between the different terms
21 in the cost function.
22
23 $\chi$ is a given symmetric, non-negative cross-gradient function\index{cross-gradient
24 }. We use
25 \begin{equation}\label{EQU:REG:4}
26 \chi(a,b) = ( a_{,i} a_{,i}) \cdot ( b_{,j} b_{,j}) - ( a_{,i} b_{,i})^2
27 \end{equation}
28 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
30 minimizing the cost function will align the surface normals of the contours.
31
32 The coefficients $\omega^{(0)}_k$, $\omega^{(1)}_{ki}$ and $\omega^{(c)}_{lk}$ define weighting factors which
33 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
35 \begin{equation}\label{ref:EQU:REG:5}
36 \int_{\Omega} ( \omega^{(0)}_k + \frac{\omega^{(1)}_{ki}}{L_i^2} ) \; dx = \alpha_k
37 \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.
39 Similarly we set for $l<k$ we set
40 \begin{equation}\label{ref:EQU:REG:6}
41 \int_{\Omega} \frac{\omega^{(c)}_{lk}}{L^4} \; dx = \alpha^{(c)}_{lk}
42 \end{equation}
43 where $\alpha^{(c)}_{lk}$ defines the scale which is typically set to one and
44 \begin{equation}\label{ref:EQU:REG:6b}
45 \frac{1}{L^2} = \sum_i \frac{1}{L_i^2} \;.
46 \end{equation}
47
48 In some cases values for the level set functions are known to be zero at certain regions in the domain. Typically this is the region
49 above the surface of the Earths. This expressed using a
50 a characteristic function $q$ which varies with its location within the domain. The function $q$ is set to zero except for those
51 locations $x$ within the domain where the values of the level set functions is known to be zero. For these locations $x$
52 $q$ takes a positive value. for a single level set function one has
53 \begin{equation}\label{ref:EQU:REG:7}
54 q(x) = \left\{
55 \begin{array}{rl}
56 1 & \mbox{ if } m \mbox{ is set to zero at location } x \\
57 0 & \mbox{ otherwise }
58 \end{array}
59 \right.
60 \end{equation}
61 For multi-valued level set function the characteristic function is set componentwise:
62 \begin{equation}\label{ref:EQU:REG:7b}
63 q_k(x) = \left\{
64 \begin{array}{rl}
65 1 & \mbox{ if component } m_k \mbox{ is set to zero at location } x \\
66 0 & \mbox{ otherwise }
67 \end{array}
68 \right.
69 \end{equation}
70
71
72 \section{Usage}
73
74 \LG{Add example}
75
76 \begin{classdesc}{Regularization}{domain
77 \optional{, w0=\None}
78 \optional{, w1=\None}
79 \optional{, wc=\None}
80 \optional{, location_of_set_m=Data()}
81 \optional{, numLevelSets=1}
82 \optional{, useDiagonalHessianApproximation=\False}
83 \optional{, tol=1e-8}
84 \optional{, scale=\None}
85 \optional{, scale_c=\None}
86 }
87
88
89 initializes a regularization component of the cost function for inversion.
90 \member{domain} defines the domain of the inversion. \member{numLevelSets}
91 sets the number of level set functions to be found during the inversion.
92 \member{w0}, \member{w1} and \member{wc} define the weighting factors
93 $\omega^{(0)}$,
94 $\omega^{(1)}$ and
95 $\omega^{(c)}$, respectively. A value for \member{w0} or \member{w1} or both must be given.
96 If more then one level set function is involved \member{wc} must be given.
97 \member{location_of_set_m} sets the characteristic function $q$
98 to define locations where the level set function is set to zero, see equation~(\ref{ref:EQU:REG:7}).
99 \member{scale} and
100 \member{scale_c} set the scales $\alpha_k$ in equation~(\ref{ref:EQU:REG:5}) and
101 $\alpha^{(c)}_{lk}$ in equation~(\ref{ref:EQU:REG:6}), respectively. By default, their values are set to one.
102 Notice that weighting factors are rescaled to meet the scaling conditions. \member{tol} sets the
103 tolerance for the calculation of the Hessian approximation. \member{useDiagonalHessianApproximation}
104 indicates to ignore coupling in the Hessian approximation produced by the
105 cross-gradient term. This can speed-up an individual iteration step in the inversion but typically leads to more
106 inversion steps.
107 \end{classdesc}
108
109 \section{Gradient Calculation}
110
111
112 The cost function kernel\index{cost function!kernel} is given as
113 \begin{equation}\label{ref:EQU:REG:100}
114 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 )
116 + \sum_{l<k} \mu^{(c)}_{lk} \cdot \omega^{(c)}_{lk} \cdot \chi(m_l,m_k)
117 \end{equation}
118 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
120 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}
122 \begin{array}{rcl}
123 Y_k & = & \displaystyle{\frac{\partial K^{reg}}{\partial m_k}} \\
124 X_{ki} & = & \displaystyle{\frac{\partial K^{reg}}{\partial m_{k,i}}}
125 \end{array}
126 \end{equation}
127 For the case of a single valued level set function $m$ we get
128 \begin{equation}\label{ref:EQU:REG:202}
129 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}
139 and for $X$
140 \begin{equation}\label{ref:EQU:REG:303}
141 \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}
149 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}.
150 For the case of a single valued level set function $m$ we get
151 \begin{equation}\label{ref:EQU:REG:601}
152 \begin{array}{rcl}
153 A_{ij} & =& \mu \cdot \omega^{(1)}_i \cdot \delta_{ij} \\
154 D & = & \mu \cdot \omega^{(0)}
155 \end{array}
156 \end{equation}
157 For a two-valued level set function $(m_0,m_1)$ we have
158 \begin{equation}\label{ref:EQU:REG:602}
159 D_{kl} = \mu_k \cdot \omega^{(0)}_k \cdot \delta_{kl}
160 \end{equation}
161 and
162 \begin{equation}\label{ref:EQU:REG:603}
163 \begin{array}{rcl}
164 A_{0i0j} & = & \mu_0 \cdot \omega^{(1)}_{0i} \cdot \delta_{ij} + \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot
165 \left( (m_{1,j'}m_{1,j'} )\cdot \delta_{ij} - m_{1,i} \cdot m_{1,j} \right) \\
166 A_{0i1j} & = & \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot \left( 2 \cdot m_{0,i} \cdot m_{1,j}
167 - m_{1,i} \cdot m_{0,j} - ( m_{1,j'} m_{0,j'} ) \cdot \delta_{ij}
168 \right) \\
169 A_{1i0j} & = & \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot \left( 2 \cdot m_{1,i} \cdot m_{0,j}
170 - m_{0,i} \cdot m_{1,j} - ( m_{1,j'} m_{0,j'} ) \cdot \delta_{ij} \right) \\
171 A_{1i1j} & = & \mu_1 \cdot \omega^{(1)}_{1i} \cdot \delta_{ij} + \mu^{(c)}_{01} \cdot \omega^{(c)}_{01} \cdot
172 \left( (m_{0,j'}m_{0,j'} ) \cdot \delta_{ij} - m_{0,i} \cdot m_{0,j} ) \right)
173 \end{array}
174 \end{equation}
175
176

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