2.2 Set-Theoretic Estimation
27
include some constraint that has not been identified. Once the set based on this constraint
is incorporated, any point in the feasibility set should be acceptable; if not the cycle is
repeated. Usually, there is more than one solution, which may be counterintuitive from the
standpoint of conventional point estimation theory where, to extract a single solution, an
objective function with a unique extremum is employed. On the other hand, because of the
arbitrariness in the selection of such an objective function, the result is, at best, nothing but
a qualitative selection of a feasible solution.
In set theoretic estimation, we do not seek a unique solution, as in optimization,
but we seek the set of feasible solutions; all such solutions are acceptable [27]. A
feasible solution is one that is consistent with all available information, such as the
field equation, (2.41), and the data equation, (2.40), both shown below.
The algorithm that we are developing is based on statistical decision theory
applied to the outcomes of a number of “experiments.” Each experiment is labeled
by the view-index v that is associated with the frequency of excitation and position
of the exciting probe in the eddy-current NDE process. The outcomes of the
experiments are elements of the “feasibility set,” in the language of set theoretic
estimation; that is, they satisfy all known information about the problem.
We write the data equation and the field equation as
Z(v) = −
LMJ
E
(i)
LMJ (v) · J LMJ (v)
(2.40)
E
(i)
lmj (v) = E lmj (v) −
LMJ
G
(ee)
jJ (l − L, m − M; ω) · J LMJ (v) , (2.41)
respectively, where G (ee) is an ‘electric-electric’ Green’s dyadic, that transforms an
electric current into an electric field moment. 1 Note that (2.40) and (2.41) are linear
in all the variables.
We define the vth “experiment” to be the pair (Z(v), E
(i)
lmj (v)), and the “outcomes” of this experiment to be the pair (J lmj (v), E lmj (v)), which satisfy (2.40) and
(2.41). Thus, the outcomes are feasible because they satisfy all known information
about the problem. Clearly, we cannot talk about a unique solution, because the
feasible set contains many points.
In the problems that are described in this chapter, we model a single transmitting
coil to excite the system, and a single receiver coil that is scanned over the region
of interest. In place of a single, movable, receiver coil, it is possible to use a fixed
array of receivers. In either case, this is an example of a transmit-receive (T/R)
configuration, which is becoming more widely used in the NDE industry. The use
of a T/R configuration allows us to gain more information from each experiment
(i.e., from each viewing).
For example, if we have a single excitation source (the transmitter), and a single
receiver sensor that is scanned over N s points, then each view, v, produces N s
results, {Z 1 (v), . . . , Z N s (v)}. The actual current J LMJ (v), however, is associated
1 We consider only problems in which the host and anomalies are nonmagnetic.
27
include some constraint that has not been identified. Once the set based on this constraint
is incorporated, any point in the feasibility set should be acceptable; if not the cycle is
repeated. Usually, there is more than one solution, which may be counterintuitive from the
standpoint of conventional point estimation theory where, to extract a single solution, an
objective function with a unique extremum is employed. On the other hand, because of the
arbitrariness in the selection of such an objective function, the result is, at best, nothing but
a qualitative selection of a feasible solution.
In set theoretic estimation, we do not seek a unique solution, as in optimization,
but we seek the set of feasible solutions; all such solutions are acceptable [27]. A
feasible solution is one that is consistent with all available information, such as the
field equation, (2.41), and the data equation, (2.40), both shown below.
The algorithm that we are developing is based on statistical decision theory
applied to the outcomes of a number of “experiments.” Each experiment is labeled
by the view-index v that is associated with the frequency of excitation and position
of the exciting probe in the eddy-current NDE process. The outcomes of the
experiments are elements of the “feasibility set,” in the language of set theoretic
estimation; that is, they satisfy all known information about the problem.
We write the data equation and the field equation as
Z(v) = −
LMJ
E
(i)
LMJ (v) · J LMJ (v)
(2.40)
E
(i)
lmj (v) = E lmj (v) −
LMJ
G
(ee)
jJ (l − L, m − M; ω) · J LMJ (v) , (2.41)
respectively, where G (ee) is an ‘electric-electric’ Green’s dyadic, that transforms an
electric current into an electric field moment. 1 Note that (2.40) and (2.41) are linear
in all the variables.
We define the vth “experiment” to be the pair (Z(v), E
(i)
lmj (v)), and the “outcomes” of this experiment to be the pair (J lmj (v), E lmj (v)), which satisfy (2.40) and
(2.41). Thus, the outcomes are feasible because they satisfy all known information
about the problem. Clearly, we cannot talk about a unique solution, because the
feasible set contains many points.
In the problems that are described in this chapter, we model a single transmitting
coil to excite the system, and a single receiver coil that is scanned over the region
of interest. In place of a single, movable, receiver coil, it is possible to use a fixed
array of receivers. In either case, this is an example of a transmit-receive (T/R)
configuration, which is becoming more widely used in the NDE industry. The use
of a T/R configuration allows us to gain more information from each experiment
(i.e., from each viewing).
For example, if we have a single excitation source (the transmitter), and a single
receiver sensor that is scanned over N s points, then each view, v, produces N s
results, {Z 1 (v), . . . , Z N s (v)}. The actual current J LMJ (v), however, is associated
1 We consider only problems in which the host and anomalies are nonmagnetic.