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Optimal Processing of the Information Available from NDE Measured Data V.Vengrinovich Institute of Applied Physics, Akademicheskaya str.16, 220072, Minsk, Belarus. G.-R.Tillack Federal Institute for Materials Research and Testing (BAM), Unter den Eichen 87, 12205 Berlin, Germany. Contact

Key words: NDE data processing, inverse problems, Bayesian restoration, data fusion

1. Introduction

From the information point of view the Materials Nondestructive evaluation (NDE) could be defined as the estimation of material's properties given the data measured by diverse sensors or a set of similar sensors (like matrix screens), which respond to the action of field, interacting with the material under test. Following this line one can predicate that NDE measurements are commonly indirect (the materials properties are never measured themselves), measured data are incomplete (regarding the restoration of material properties or defects structure) and noisy, while a model of NDE process is uncertain. These features turn the material's properties prediction from simple correlation estimate into a smart procedure for the fusion of the data received from the used sensors. Having so many uncertainties in the data as well as in the model one has to turn from the primitive desire for getting absolute result in the output to the reality of getting only a result with a final probability. The motivation of the consequent procedure should consist in the maximization of this probability. The present article is aimed to use Bayesian methodology and Bayesian restoration method like a procedure targeted to reach the maximum probability and to find the solution which satisfies to this maximum.
NDE data processing procedure can be taken as a "black box" reacting on an initial stimulus. The restoration of the unknown material quantities can be implemented by the Likelihood method yielding the materials properties by global minimization of the functional which involves both the data and the theoretical NDE process model. The disadvantage of the simple Likelihood estimation of the Material's properties is concluded in interpreting the noise and model uncertainty as a signal distortion. Finally one can not distinguish between them and the unknown properties under restoration. The authors deal in this article with the optimal information processing rule, which was deduced from more basic principles, particularly:
• Information Conservation Principle (ICP) as a basis for statistical concept of sufficiency;
• Optimal Information Processing Rule (IPR) which is the minimization of the ICP criterion functional.
Zellner [1] has shown that in order to deduce the optimal IPR, which is equivalent to the search for how to process the information in the optimal way, it is necessary and sufficient to apply the Bayesian restoration rule, which comes from the equalization of the information in the input and output of the data processing "box". Hence the exploitation of this rule ensures that all available information can be extracted from the given particular measured data and reliable prior information.

2. NDE information processing rule

Generally the transformation of the input data within NDE system, consisting of k-types of sensors, supposing the noise is additive, can be described by the following set of operator equations:


pk(yk)=Okm(x) + hk,

(1)

with subscript k giving the type of the sensor providing measured data, p_k the measured data in the time or spatial domain y_k, O_k the operator transforming the input into the output data or a kind of instrumental function, prescribed in the linear case by a matrix of order m“n with real elements a_il, m(x) the vector of unknown material properties (input data) depending on the argument x, and h_k the noise accompanying the measurements.
In practical NDE applications it is a common place, that the direct transformation matrix O_k is unknown or known but underestimated, such that it cannot be inverted and the inverse operator O_k^-1 is unknown. This and many other situations exist in which eq. (1) has no solutions m(x) in the classical sense what means that the solution is ambiguous. However while the classical solution of the eq. (1) mostly doesn't exist the general solution, which can be called pseudo-solution, is attainable after filtering by available filter. As previously shown the Bayesian filter is an optimal filter for this purpose. The pseudo-solution may not be unique, but it can becomes unique by introducing additional constraints like the minimal norm. This minimal norm can be deduced also from the maximization of the inference or the posterior probability defined by the Bayes formula:


(2)

with the inference or post data probability distributions P(m|p), the likelihood function P(p|m) which formally describes the response of the measuring system together with the noise, the evidence P(p) which is the global normalization constant which in its turn quantifies the quality of the question about the inference given data and prior, and the prior probability distribution P(m).
The maximization of the posterior probability P(m|p) from eq. (2) and applying eq. (1) for the NDE process simulation it is possible to find the pseudo-solution of the operator equation (1):


(3)

where is the vector of the restored material properties, Rⁿ the n-dimensional normalized space, the elements of which are column-vectors (m₁,...,m_n), Om(x)the simulated or mock data, P(m) the probability distribution function (pdf) for the prior expectation of the unknown material properties. The weighting factor g included in the summation is called regularization coefficient varying between 0 and 1. (For simplification the subscripts k describing the type of sensor for the data acquisition is omitted in the formula).
Now it is possible to formulate a procedure for the application of formula (3) to the solution of NDE inverse problems:

1. Description of the problem stressing on the definition of
   1. the process model or the operator O
   2. prior pdf P(m) and a priori knowledge about the norm space Rⁿ
   3. data acquisition arrangement
2. Estimation of the coefficient g like a measure of your belief either in the measured data or in the prior information
3. Data acquisition
4. Estimation of the noise characteristics (optional)
5. Estimation of the evidence P(p) (optional)
6. Application of minimization procedure to calculate the most probable solution for the unknown material properties.
Some examples for the application of proposed optimal IPR to concrete problems will be discussed below.

3. Hardness estimate through magnetic measurements

Let unknown material property m be the hardness H of the material, in this case steel, evaluated through magnetic coercive force meter which records the coercive force h in A/m with a measurement accuracy of 25 A/m. The correspondence between h and H is assumed to be linear with the proportionality coefficient 50 A/mHR_c while the error of this measurement is uniformly distributed in the confidence interval which is 7.5% of the minimum value of h: h_min=1000A/m, h_max=2000A/m. Suppose that H (in HR_c) can vary in the range 40HR_c < H £ 60HR_c and the probability distribution function (pdf) is uniform while the requested resolution is 1HR_c.
Now let ask about the inference after the measuring of H through h in two adjacent points. Suppose the hardness in two points are not independent what means that the hardness in any point may not differ from the hardness in the neighbor more then 2HR_c given measured data. Additionally it is supposed that the additive noise is Gaussian distributed yielding


(4)

where is an overall normalization constant; and denote the corresponding vectors which are the values of h and H in the sites 1 and 2: {h₁,h₂} and {H₁,H₂} correspondingly. h_k^m gives the current admissible measured value of h for fixed H value, and its standard deviations s _k.

2. Obviously the prior in this case can be represented like a probability of two joint events {H₁,H₂} : P()=0.05.0.05=0.0025, in the range 40HR_c < H £ 60HR_c, zero otherwise.
3. The estimation of the standard deviation and the normalization constant gives s _k = 70.71A/m and Z=8.088. The calculated likelihood pdf is shown in the Table 1.
4. Let the measured values in two sites correspondingly equal to: h₁=1025A/m; h₂=1150A/m. Seven admissible configurations {H₁,H₂} which satisfy to all constraints are shown in Figure 1.

   (50Hk-1000)- hkm	±250	±225	±200	±175	±150	±125	±100	±75	±50	±25	0
   P(h|H)	0.00024	0.000783	0.00226	0.00577	0.013	0.0259	0.0455	0.0705	0.0963	0.116	0.1236
   Table 1: Calculated likelihood pdf

Fig 1: Pdf for the configurations {H1,H2} given measured data and prior

5. The evidence was calculated from
   and estimated at the level 1.407×10^-4.
6. The pdf of the inference is shown in the fig.1. This diagram is the optimal answer which is resulted from the model, prior and measurements. If no prior is available only the likelihood can be used. In this case the resulting pdf would be much wider while the maximum probability is decreased. Additionally there appear also several local maximums in calculus of the pdf which mislead while searching for the most plausible solution.
   In real image restoration when the number of measurement points becomes several thousands such simple calculations are impracticable, but the procedure works definitely in the same manner.
   The influence of the prior information given limiting difference |H₁-H₂| between hardness in two adjacent points upon the inference is quite strong. The less is the difference the narrower is the freedom for the admissible solutions, the smaller is the evidence (quality of the question is worse as we have taken out of consideration many possible solutions) and hence the larger is the probability of most plausible solution The solution {H₁,H₂} with maximum inference and the whole inference pdf itself are changed as well. This example shows the importance of the prior and the dependence of the answer upon the statement of a question.

4. Crack localization with Eddy-Current (EC) probe.

Commonly a crack localization problem appears in the context of the operator recording the increments of electromagnetic impedance, caused by the crack appearance in the material under test and attempting to find the 3D location of this crack. The experimental data are typically incomplete what means that they are not sufficient to deduce the source location directly. The uncertainties are introduced also from the signal measurement error, gap instability, mixture with the signals from false sources, etc.


Fig 2: Clarifications to the problem of crack localization with one ECS.

Another sort of uncertainty is caused by the fuzzy model of NDE process. In this case the operator O in eq. (1) is supposed to have uncertainties in the description of the relation between the signal increment and the distance of the crack from the sensor. This situation is typically caused by the fact that the detected signal additionally depends upon many other factors like shape, sizes and orientation of the crack while the mixing matrix is unknown. Herein the problem of optimal processing of the data acquired by Eddy-Current Sensors (ECS) is considered to determine crack location.
Let sensor A (compare fig. 2) be capable to scan the surface of the welding joint with a crack C. Its current localization relatively to the origin of coordinates r_A is measured synchronously with the signal A^*. Corresponding data sets are stored for subsequent processing. Finally the optimal radius-vector r_C in the 3D space has to be calculated.
The Bayes formula for the conditional probability of r_C estimate given a set of data A_i^*(r_A) has the following form:

(5)


where P(r_c) gives the prior probability for the crack at position r_c which depends upon the welding technique. The simplest prior hypothesis assumes that the crack has equal probability to appear in any site of the weld which is restricted by the parallelepiped with the bounds sizes X, Y and d:


(6)

If it is supposed that the probability P(A^*|r_A + r_AC )) is independent of r_A, then the Bayes formula (6) transforms to


(7)

Additionally assumptions have to be made about the NDE process model. As a first approximation it can be assumed that the signal is inverse proportional to the square distance between the sensor and defect:

A* = R /(|rAC|)2

(8)

As in the previous example it is assumed that the measurement error is Gaussian distributed with a standard deviation s, what corresponds to the condition of errors random origin and independence. Then the pdf for getting the signal value A^* given the distance of the sensor from the crack |r_AC| (forward problem) can be written in the form:


(9)

Substituting eqs. (6) and (9) into eq. (7), ignoring the constant evidence P(A^*), taking the logarithm of both sides, and removing the probabilities of constant values the optimal most probable solution for r_C is received by simple maximization of the probability P(r_c | A*) in eq. (8) given the measured data A_i^*:


(10)

It can be seen from eq. (10), that the optimal position of the crack can be found with the help of simple likelihood algorithm for the supposed constraints. The main problem with the algorithm is its convergence which - without any additional constraints - can be very slow or even inaccessible if the discrepancy will be large enough due to fuzzy model or strongly noisy data. That means that the real configuration space for searching the optimal solution r_C becomes too large and the expected global minimum is either unattainable or hidden by a set of local minimums.
This situation can be noticeably improved by introducing the prior knowledge about the possible crack location. Usually the arc welding technology provides the higher probability for crack appearance in the root of the welding joint and close to the bead, while the central weld part sounds more healthy. It will be shown below how this additional information can be fused with the experimental data to improve the 3D localization of the crack.

5. Crack localization given prior information.

Let introduce additional prior knowledge about the crack location in the form of Normal Generated Prior Distribution (NGPD) proposed by Chiu [2] for use in reliability analysis. It can be applied to the probable crack location by the following reasons: The crack can appear at every position in the weldment if it is assumed that its characteristic parameter - the length - is normally distributed in every layer. A crack, which length is larger than a certain specified value, is classified as inadmissible defect, otherwise it is admissible. Furthermore it is assumed that the mean crack length l in the layer has presumably normal prior distribution Ń(m,s²). As m is positive then for s ²>1 the NGPD, which is described by the following expression


(11)

is U-shaped as it is shown in fig. 3. In eq. (11) F^-1(z) is such a value of z that F(q)=z and F(q) denotes the standard normal cumulative distribution function given by


(12)

Fig 3: Introduced prior distribution of the crack location under the surface in the form of Normal Generated Prior Distribution (NGPD) with m=0.2, s=1.2.


which is tabulated [3]. As the z-coordinate is defined, the position of the crack is given by


(13)

while the x and y-coordinates are determined by the position of the sensor on the surface if the signal A^* reaches its maximum value. Now it is possible to localize the crack by minimizing the weighted functional, one part of which is the likelihood like in eq. (10) and the other consists of two cofactors:
1. the regularization coefficient g and
2. the prior functional after taking its logarithm and removing the constant summand.
This result is received after substituting eqs. (9) and (11) into eq. (7), taking its logarithm and maximizing the posterior probability, what has the same meaning as minimizing the sum of both functionals (likelihood and prior probability) with the opposite sign. Finally the following optimal solution for the crack position is given by:


(14)

Eq. (14) defines an optimization problem for determining the optimal crack location, which gives the best fitting to the measured data from one hand and to the prior hypothesis from the other (so called default model). The belief in the superiority of one of those trade-off functionals is controlled by means of the regularization coefficient g .

6. Discussion

It was found that the Bayesian information processing rule yields a procedure for the fusion of all information which are available about the object under testing. The remaining question is: What advantage has been achieved by incorporating prior information into the solution given by eq. (10)? Disposing only the measured data and thus using only the likelihood restoration without or with negligible constraints, the dimensionality of the problem and accordingly the configuration space become larger and the chance to find the solution among a large number of relative minima/maxima is decreased. By incorporating prior information with its real weight, the configuration space for search is decreased and as well as the dimensionality of the problem. This behavior was shown in terms of the example for hardness detection.
The second advantage arising from the introduction of prior knowledge is the regularization of the problem providing the convergence of the iteration procedure to ensure the a stable solution of the problem represented by a t - sequentially compact.
Returning back to the task for crack localization one can find that the solution with the prior is slightly preferable also with respect to the remote cracks, located close to the weld root, because a "searching space" in this region during iteration will be extended (prior probability is higher) due to the prior pdf.

7. Conclusions

It is shown that Bayesian methodology for unknown material properties and defects restoration is the optimal way for the processing of NDE data, which are commonly incomplete and noisy while the process model is fuzzy.
This approach is also available as an optimal technique for the fusion of data acquired by similar or dissimilar NDE sensors.
The formalized procedure for the implementation of the optimal Information Processing Rule to practical problems is proposed and illustrated using two examples from NDE.
In previous articles [4,5] Bayesian based techniques were developed for x-ray image restoration from incomplete noisy data which also demonstrates the capabilities of the discussed method.

References

1. A. Zellner: Optimal Information Processing and Bayes Theorem. The American Statistitian 42 4, pp. 278-284 (1998)
2. W. K. Chiu: A New prior Distribution for Attributes Sampling. Technometrics 16, pp. 93-102 (1974).
3. H. F. Martz and R. A. Waller: Bayesian Reliability Analysis. John Wiley&Sons, New York, 1982.
4. V. Vengrinovich, Yu. Denkevich, and G.-R. Tillack: Reconstruction of Three-Dimensional Binary Structures from an Extremely Limited Number of Cone-Beam x-ray Projections. Choise of Prior. J. Phys. D: Appl. Phys. 32 pp. 2505-2514 (1999)
5. V. Vengrinovich, Yu. Denkevich, and G.-R. Tillack. Bayesian 3D x-ray reconstruction from incomplete noisy data. In book: Maximum Entropy and Bayesian Methods, ed. by W. von der Linden et. al., Kluwer Academic Publishers, pp. 73-83 (1999)

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