26
2 Voxel-Based Inversion Via Set-Theoretic Estimation
Of course, this results in non-unique solutions; each cell in the window pane has
two solutions. In fact, because the σ ’s are known to be real, we can separate the x’s
and y’s and E’s into their real and imaginary parts, and get four solutions for each
cell of the window pane. Each solution, then, is an outcome of an experiment, in the
language of set-theoretic estimation theory, and is a candidate for processing by the
robust statistical estimator that is described in the next chapter.
The integral relation that allows us to determine the anomalous current is derived
from the measurement process. Typically, we measure the perturbation of the probe
impedance, ΔZ, due to the flaw. Adopting the probe current J p as the phase
reference, this impedance is given by:
I
2 ΔZ = −
coil
E
(s) (r) · J p (r)dr
= −
f law
E
(i) (r) · J(r)dr,
(2.38)
where I is the total driving-current in the probe coil. In arriving at the final
expression, we used a reciprocity theorem (see [111, Chapter 5]) that relates the
scattered field, E (s) , at the primary source (the eddy-current probe coil) to the
incident field at the secondary source (the anomalous current source due to the flaw).
The integral expression, (2.38), is discretized by substituting the expansions,
(2.3), for the currents, and making use of the definitions of the field moments, (2.5),
(2.6). The result is quickly obtained:
ΔZ = −
KLM
J
(x)
KLM E
(i)(x)
KLM + J
(y)
KLM E
(i)(y)
KLM
,
(2.39)
where we assume that I = 1, and that the exciting coils produce an incident field
that is oriented in the (x, y)-plane; i.e., E
(i)(z)
KLM = 0.
2.2 Set-Theoretic Estimation
We quote Combettes [27, p. 202]
The basic philosophical motivation for the set theoretic approach is that more reliable
solutions can be obtained by exploiting known information rather than imposing an often
subjective notion of optimality. Thus, in the set theoretic framework, the emphasis is placed
on the feasibility of a solution rather than its optimality, as is done in the conventional
approach. The goal is not to produce a “best” solution but one that is consistent with all
available information. In set theoretic estimation, all the members of the feasibility set
are acceptable solutions. They can be regarded as the objects that, in light of all available
information, may have given rise to the observed data. The only way to restrict objectively
the feasibility set is to incorporate more information in the formulation. If some of the
feasible solutions are not acceptable, then it must be the case that the formulation fails to
2 Voxel-Based Inversion Via Set-Theoretic Estimation
Of course, this results in non-unique solutions; each cell in the window pane has
two solutions. In fact, because the σ ’s are known to be real, we can separate the x’s
and y’s and E’s into their real and imaginary parts, and get four solutions for each
cell of the window pane. Each solution, then, is an outcome of an experiment, in the
language of set-theoretic estimation theory, and is a candidate for processing by the
robust statistical estimator that is described in the next chapter.
The integral relation that allows us to determine the anomalous current is derived
from the measurement process. Typically, we measure the perturbation of the probe
impedance, ΔZ, due to the flaw. Adopting the probe current J p as the phase
reference, this impedance is given by:
I
2 ΔZ = −
coil
E
(s) (r) · J p (r)dr
= −
f law
E
(i) (r) · J(r)dr,
(2.38)
where I is the total driving-current in the probe coil. In arriving at the final
expression, we used a reciprocity theorem (see [111, Chapter 5]) that relates the
scattered field, E (s) , at the primary source (the eddy-current probe coil) to the
incident field at the secondary source (the anomalous current source due to the flaw).
The integral expression, (2.38), is discretized by substituting the expansions,
(2.3), for the currents, and making use of the definitions of the field moments, (2.5),
(2.6). The result is quickly obtained:
ΔZ = −
KLM
J
(x)
KLM E
(i)(x)
KLM + J
(y)
KLM E
(i)(y)
KLM
,
(2.39)
where we assume that I = 1, and that the exciting coils produce an incident field
that is oriented in the (x, y)-plane; i.e., E
(i)(z)
KLM = 0.
2.2 Set-Theoretic Estimation
We quote Combettes [27, p. 202]
The basic philosophical motivation for the set theoretic approach is that more reliable
solutions can be obtained by exploiting known information rather than imposing an often
subjective notion of optimality. Thus, in the set theoretic framework, the emphasis is placed
on the feasibility of a solution rather than its optimality, as is done in the conventional
approach. The goal is not to produce a “best” solution but one that is consistent with all
available information. In set theoretic estimation, all the members of the feasibility set
are acceptable solutions. They can be regarded as the objects that, in light of all available
information, may have given rise to the observed data. The only way to restrict objectively
the feasibility set is to incorporate more information in the formulation. If some of the
feasible solutions are not acceptable, then it must be the case that the formulation fails to