On the Issue of Choosing the Optimal Number of Introscope Sensors
Abstract Generated abstract
This paper addresses the selection of an optimal number of sensors in introscope systems for nondestructive testing, where defect detection must be accompanied by sufficient descriptive detail for human or automatic classification. It formulates the sensor block as a discretizing element in an information model and applies Shannon’s fidelity criterion to relate the probability distribution of defect images, reproduction quality, and the entropy of the sensor output. The proposed optimization problem is to minimize output entropy, interpreted as reducing the number and informational burden of sensors, while satisfying a specified quality requirement. The author notes that a general analytical solution is not available and that practical application is limited by the difficulty of determining both defect statistics and the subjective quality function.
Full Text
ON THE PROBLEM OF CHOOSING THE OPTIMAL NUMBER OF INTROSCOPE SENSORS
A. A. KOZLOV
(Presented by the scientific seminar of the automation sector of the Research Institute of Electrical Physics)
The increasing requirements for improving the methods and means of nondestructive testing of extended products have led to the emergence of a new area of inspection—introscopy.
The development of introscopy methods required a qualitatively new approach to the construction of inspection systems. In these systems it became necessary to satisfy not only the condition of defect detection, but also the condition of a detailed description of the defect, so that the operator could determine its class.
The problem arose of determining the requirements that the entire signal path of an introscope must satisfy in order to solve this problem.
This problem requires even greater attention when, instead of an operator, a machine capable of automatically recognizing defects is used.
The advantages of machines are that they make it possible to increase the reliability of inspection, eliminate the subjective nature of the operator’s assessment, and substantially increase the speed of product inspection.
However, obviously, the introduction of such a machine is associated with technical problems such as complexity and cost. Therefore the task arises of reducing the complexity and cost of machines. In an introscope, the signal from the radiation source, through the object under inspection, reaches a block of detectors or sensors; then, after the corresponding transformations (amplification, normalization, etc.), it is supplied either to a screen for the operator or to the decision-making block of the machine. The amount of information supplied for processing (about one defect), as we see, depends on the number of sensors. From the standpoint of recognition, it would be good if there were as many sensors as possible, but in that case the volume of equipment would increase sharply, enormous memory would be required, and, consequently, the cost of the introscope would increase. Therefore it is desirable to minimize the number of sensors (photomultipliers, scintillation crystals) while preserving the possibility of classifying defects.
For simplicity, let us assume that the discretization of the signal from the source is determined only by the sensors. We shall consider all other discretization operations that occur in the signal path of the introscope to be assigned to the sensor block. Such an approach, in our view, will make it possible to simplify the solution of the problem.
In order to assess the influence of the sensor block on the description of a defect, it is necessary to know the accuracy or fidelity of reproduction. The difficulty lies in describing the function performed by the sensor block: to what extent it (the description) minimizes the amount of entropy of the object image and introduces distortions into the description of the defect; therefore, a fidelity criterion (quality estimate) is necessary. In this case one may propose the following model of an introscope with the inclusion of a fidelity criterion in the decision block (Fig. 1).
Let us suppose that a certain signal in the form of a sequence continuous in amplitude and in time, with a limited spectrum of width \(2F\), arrives at the sensor field. Then the signal at the input of the sensor block
Fig. 1. Information model of an introscope with the inclusion of a fidelity criterion in the decision block
will be represented by a variable \(X\{x_1, x_2, \ldots, x_3\}\), having \(n\) independent or correlated samples (coordinates). This variable \(X\) can be represented by a point in an \(n\)-dimensional vector image space. As a result of this representation (discretization), an irreversible process takes place, i.e., the \(n\)-dimensional image space is divided into a finite number of regions. All images falling into a certain region are replaced by one characteristic value \(X_i^*\) (where \(i\) is the number of the region).
It then follows that, when transmitting from region \(i\) some message \(X\) (except for \(X = X^*\)), an error will arise, defined by the expression
\[ \delta = X_i^* - X. \tag{1} \]
For this discretization process, the expression determining the fidelity of reproduction, proposed by Shannon [1], can be written in the form:
\[ Q = \sum_{(i=1,2,\ldots n)} \int \cdots \int P(X)\, v(X, X^*)\, dX, \tag{2} \]
where \(P(X)\) is the probability density,
\(v\) is the quality criterion,
\(n\) is the number of regions.
Formula (2) makes it possible to evaluate the quality of any sensor system, proceeding from the known statistical characteristics of the defect and the function of the quality criterion (estimate).
The entropy of the signal at the output of the sensor system is determined by the formula
\[ H(X^*) = - \sum P(X_i^*) \log P(X_i^*), \tag{3} \]
where
\[ P(X_i^*) = \int_{i\text{-region}} \cdots \int P(X)\, dX. \]
Thus, the magnitude of the entropy is determined by the position of the boundaries of the regions, i.e., by the number of sensors, and is independent of the magnitude \(X_i^*\). By varying the number of sensors, one can minimize the magnitude \(H(X_i^*)\) and at the same time satisfy the adopted requirements with respect to quality. The quality in this case will depend on the position of \(X_i^*\).
The problem of minimizing \(H(X_i^*)\) under the adopted requirements on the quality of the output image will be called the problem of the optimal choice of the number of sensors. That is, we have the probability density function \(P(X)\) and some criterion (estimation) function \(v(X, X^*)\). Given that the inequality \(Q \geq Q_1\) is satisfied, where \(Q\) is determined from (2), and \(Q_1\) specifies the quality of the sensor system, it is required to find such a number of sensors as would minimize expression (3).
There is as yet no analytical solution to this problem, except for the degenerate case in which the probability density function is assumed constant and the mean-square error criterion is used. It is rather difficult to determine the function \(P(X)\) unambiguously; nevertheless, analysis of \(P(X)\) is in principle simpler than estimating the quality function \(v\). Estimation of the quality function is subjective in character and depends on the nature of the problem posed.
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