2.4 A Layer-Stripping Algorithm
29
be called e lmj . Hence, for this value of the view-index, v, we associate the triple,
(
J x , e x ) lmj (v), (
J y , e y ) lmj (v), (
J z , e z ) lmj (v), with the lmj th cell, and when we take
the ratio of the electric current to the electric field at the middle of the lmj th cell,
we arrive at the conductivity, σ lmj of the lmj th cell, which is our final goal. We
then perform another experiment by choosing another value of v, thereby generating
another triplet. This ensemble of triplets constitutes the feasible set for each cell.
2.3 Statistical Analysis of the Feasible Set
One of the principal efforts of research in Set-Theoretic Estimation is to determine
the number and nature of the experiments, (Z(v), E
(i)
lmj (v)), and receiver scans,
E
(n)
LMJ , that produce a good feasible set for statistical analysis. As might be expected,
this depends upon the complexity of the flaw that is to be reconstructed, and upon
the resolution desired in the reconstruction.
After deducing the feasibility set, we are then faced with the task of assigning a
single number for the conductivity of each cell; this is a problem of data analysis,
which uses robust regression, as described in [111, Chapter 13]. The aim of the
statistical analysis of the feasible set is to fit a constant through the data, which
are produced by the algorithm described in the preceding section, for all views (or
experiments), v.
A beneficial feature of this algorithm is that the analysis of the data set for each
cell is done independently of every other cell; i.e., a decision is made on a cell-bycell basis. Since the decision to be made for each cell involves a nonlinear (robust)
estimator, the computational burden is greatly reduced when compared to using a
nonlinear estimator to solve for many cells jointly. (The bilinear conjugate-gradient
algorithm is an exception.) Furthermore, this leads us to a constrained iterative
(‘layer-stripping’) algorithm, that uses the known and accepted results for some
cells to determine the results for others at a later stage of the iteration.
2.4 A Layer-Stripping Algorithm
A classical layer-stripping algorithm consists of solving an inverse problem layerby-layer, when the physical system permits such a reconstruction. If the excitation
and detection methods permit only one layer to be detected, say due to timing
arrangements in a pulsed system, then only that layer will be reconstructed. This
reconstruction, then, constitutes known data for the reconstruction of the next layer.
In our situation, we have discovered that we can reconstruct certain cells
accurately, similar to the reconstruction of a layer, and we then want to use this
result in the reconstruction of the remaining cells. We do not assume that an entire
layer has been reconstructed, nor that we will reconstruct on a layer-by-layer basis.
29
be called e lmj . Hence, for this value of the view-index, v, we associate the triple,
(
J x , e x ) lmj (v), (
J y , e y ) lmj (v), (
J z , e z ) lmj (v), with the lmj th cell, and when we take
the ratio of the electric current to the electric field at the middle of the lmj th cell,
we arrive at the conductivity, σ lmj of the lmj th cell, which is our final goal. We
then perform another experiment by choosing another value of v, thereby generating
another triplet. This ensemble of triplets constitutes the feasible set for each cell.
2.3 Statistical Analysis of the Feasible Set
One of the principal efforts of research in Set-Theoretic Estimation is to determine
the number and nature of the experiments, (Z(v), E
(i)
lmj (v)), and receiver scans,
E
(n)
LMJ , that produce a good feasible set for statistical analysis. As might be expected,
this depends upon the complexity of the flaw that is to be reconstructed, and upon
the resolution desired in the reconstruction.
After deducing the feasibility set, we are then faced with the task of assigning a
single number for the conductivity of each cell; this is a problem of data analysis,
which uses robust regression, as described in [111, Chapter 13]. The aim of the
statistical analysis of the feasible set is to fit a constant through the data, which
are produced by the algorithm described in the preceding section, for all views (or
experiments), v.
A beneficial feature of this algorithm is that the analysis of the data set for each
cell is done independently of every other cell; i.e., a decision is made on a cell-bycell basis. Since the decision to be made for each cell involves a nonlinear (robust)
estimator, the computational burden is greatly reduced when compared to using a
nonlinear estimator to solve for many cells jointly. (The bilinear conjugate-gradient
algorithm is an exception.) Furthermore, this leads us to a constrained iterative
(‘layer-stripping’) algorithm, that uses the known and accepted results for some
cells to determine the results for others at a later stage of the iteration.
2.4 A Layer-Stripping Algorithm
A classical layer-stripping algorithm consists of solving an inverse problem layerby-layer, when the physical system permits such a reconstruction. If the excitation
and detection methods permit only one layer to be detected, say due to timing
arrangements in a pulsed system, then only that layer will be reconstructed. This
reconstruction, then, constitutes known data for the reconstruction of the next layer.
In our situation, we have discovered that we can reconstruct certain cells
accurately, similar to the reconstruction of a layer, and we then want to use this
result in the reconstruction of the remaining cells. We do not assume that an entire
layer has been reconstructed, nor that we will reconstruct on a layer-by-layer basis.