Analysis of the Process of Extracting Information on Local Changes in the Spatial Distribution of the Radiation Field Behind a Moving Object
Abstract Generated abstract
This paper analyzes how local inhomogeneities in the radiation field behind a moving object are converted into electrical signals by a scintillation detector in radiometric defectoscopy. Treating the detector and collimator as a linear filter, the authors derive response and spectral expressions showing that collimator dimensions and object velocity determine the effective transfer function, signal spectrum, and information losses. They argue that filter passbands should match the principal energy range of the detector signal, while recognizing the tradeoff between noise suppression and information loss. A cross-correlation recording scheme using spatially separated differential detector pairs is proposed as an optimal filtering approach, with the potential to suppress statistical, source, and environmental noise and to improve extraction of useful signals from local radiation-field changes.
Full Text
Analysis of the Process of Extracting Information on Local Changes in the Spatial Distribution of the Radiation Field Behind a Moving Object
Vorob’ev B. A., Diakonenko A. A., Pokrovskii A. B.
Presented by the joint seminar of the sectors of the DSM and GED, Scientific Research Institute of EI
A widely used method for studying the spatial distribution of the radiation field behind a moving object is based on receiving the radiation that has passed through the sample with a scintillation detector, converting it into an electrical signal, and subsequently analyzing this signal [1, 2].
The signal is determined mainly by local changes in the radiation field, but it also depends on the geometrical dimensions of the detector collimator. Detectors are usually characterized by their resolving power, since a detector with high resolution makes it possible to find differences between two closely spaced inhomogeneities of the field, whereas a detector with low resolution will register them as one. A detector with a smaller collimator area gives better resolution. Let us suppose that the inhomogeneity has a spherical form and is very small in comparison with the dimensions of the expected field inhomogeneities. We carry out the analysis in a rectangular coordinate system. The \(X\)-axis is located in the direction of motion of the object.
The field inhomogeneity falls entirely within the detector’s field of view. This determines the one-dimensional response function of the detector
\[ f(x)=\int_a^b f(x,y)\,dy, \tag{1} \]
where the limits of integration are taken according to the size of the collimator along the \(y\)-axis.
The function giving the total response to the elementary inhomogeneities of which an inhomogeneity of a definite type is composed will be written as \(h(x' - x)\).
where \(x'\) is the distance from \(x=0\) to the leading edge of the inhomogeneity along the direction of motion of the specimen onto the detector.
Then the detector response to an inhomogeneity at a point (Fig. 1) is described by a superposition integral of the form
\[ U(x')=\int_{-\infty}^{x'} f(x)\,h(x'-x)\,dx, \tag{2} \]
where
\[ h(x'-x) \begin{cases} A,\\ 0 \end{cases} \quad \begin{array}{ll} \text{for} & C \ge x'-x \ge 0,\\ & C < x'-x < 0, \end{array} \]
\[ f(x) \begin{cases} a,\\ 0 \end{cases} \quad \begin{array}{ll} \text{for} & B \ge x \ge 0,\\ & B < x < 0. \end{array} \]
Expression (2) is analogous to the integral describing the response of a bandpass filter to an input action [3], which has the form
\[ U_2(t)=\int_{-\infty}^{t_1} U_1(t)\,h(t_1-t)\,dt. \tag{3} \]
In equation (3), \(U_1(t)\) is the response to a unit impulse action, and \(h(t_1-t)\) is the pulse characteristic in time. Transferring this analogy to the case under consideration, the excitation at the detector output, at a constant velocity of displacement of the object, is written in the form
\[ U(t_1)=\int_{-\infty}^{t_1} f(vt)\,h(t_1-vt)\,v\,dt, \tag{4} \]
where \(x=vt\), \(t_1\) is the current time.
The spectral characteristic for an inhomogeneity of a specified length is
\[ S(\omega)=\int_{-\infty}^{+\infty} U(t_1)\,e^{-j\omega t_1}\,dt. \tag{5} \]
The analogy drawn makes it possible to conclude that, in the present case, the detector, from the standpoint of forming the envelope of the output signal, acts as a filter with a transfer function capable of changing the frequency components of the output signal either when the velocity of motion changes or when the collimator aperture changes. It is obvious that the detector-filter substantially changes both the shape and the frequency spectrum of the signal from the inhomogeneity \(h(x'-x)\).
Using the transfer function of the detector, let us consider further the operations of extracting the useful signal against the background of noise. In accordance with this, the filters of radiometric radiation-defectoscopy systems must be designed so that the passband of the filter corresponds to the frequency range in which the principal share of the detector-signal energy is concentrated. It is assumed here that the signal from inhomogeneities has a limited frequency range, whereas the noise has a continuous spectrum. If periodic noise is present, then in this case a considerable improvement in the signal-to-noise ratio is achieved by using a selective filter. The frequency spectrum of such noise has the form of a series of discrete lines of different amplitudes, including the fundamental noise frequency. It is important to emphasize that an inhomogeneity, like noise, is perceived by the detector, which itself is a kind of filter. A detector with large collimator dimensions is analogous to a filter with a narrower passband, and vice versa.
In information theory one usually speaks of the entropy of a signal, which in the present case characterizes the expected magnitude of the signal. When a signal passes through a linear filter, a reduction of its entropy is inevitable. This reduction is equal to [4]
\[ \frac{1}{z}\int_{0}^{z} \ln h(f)\,df \tag{6} \]
where
\[ \begin{aligned} h(f) &\text{ is the transfer function of the filter,}\\ z &\text{ is the frequency range of the signal.} \end{aligned} \]
In other words, increasing the dimensions of the collimator leads to a reduction in the number of different frequency components of the detector signal. This contributes to energy losses, especially for small inhomogeneities. For transmission without losses, a transducer with a flat frequency characteristic is required. To further improve the signal-to-noise ratio, it is advisable, along with the corresponding selection of detector (collimator) parameters, to use in the recording circuit, immediately after the detector, a selective filter with the same passband. The purpose of this filter is to suppress any noise with frequencies outside the filter passband.
As follows from expression (6), any filtering at the output of the system entails an additional loss of information in exchange for an increase in the signal-to-noise ratio. In this case, to increase the informational
capabilities of the measuring channel, it is advisable to use a correlation filter, whose operation is based on the corresponding arrangement of the detectors (Fig. 2).
The recording device (Fig. 3) consists of a radiation source 1, two pairs of scintillation counters 3, 4 and 5, 6, connected differentially to one another. The pairs of scintillation counters are located at a distance \(l\) from one another along the direction of motion of the article 2. The output signals of the detectors pass through cathode followers 7, 8 and 9, 10 and are then subtracted, for example, on pulse transformers. The difference signals \(\Delta U(t)\) are amplified by amplifiers 13, 14. The voltage \(\Delta U(t)\) picked up from the secondary windings of transformers 11, 12 consists of the useful signal characterizing the quality of the material \(U_c(t)\) and the interference \(U_n(t)\)
\[ \Delta U(t)=U_c(t)+U_n(t) \tag{7} \]
Since detectors 3, 4 are displaced relative to detectors 5, 6 along the motion of the article, the signal characterizing the quality of the article in the region under counters 3, 4, \(U_c(t)\), will be taken from counters 5, 6 with a certain transport delay \(\tau = l/\upsilon\), where
\(l\) is the distance between the detector pairs,
\(\upsilon\) is the velocity of motion of the article.
A delay unit 15 is included in the circuit of amplifier 13. The signals from the transformers \(\Delta U(t)\) and \(\Delta U(t-\tau)\) are fed to a correlator consisting of a multiplying device 16 and an integrating device 17. The output signal of the correlator includes the following components [5]:
\[ R_{12}(\tau)=[U_c(t)+U_n(t)][U_c(t-\tau)+U_n(t+\tau)]= \]
\[ =U_c(t)U_c(t+\tau)+U_c(t)U_n(t+\tau)+U_n(t)U_c(t+\tau)+ \tag{8} \]
\[ +U_n(t)U_n(t+\tau) \]
The first and last terms are the autocorrelation functions of the useful signal and of the noise, respectively, while the remaining two terms are the functions of mutual correlation between the signal and the noise. The function \(R_{n_1 n_2}(\tau)\) decreases to zero when the shift \(\tau\) becomes so large that the correlation links of the fluctuational process are practically lost (Fig. 4). Mutual correlation between the signal and the noise is absent because of the absence of a statistical connection
Fig. 1.
Determination of the voltage at the output of a scintillation detector.
Fig. 3.
Block diagram of the recording device.
Fig. 2.
Layout of the radiation detectors.
Fig. 4.
Graph of the cross-correlation function of the signals \(\Delta U(t)\) and \(\Delta U(t-\tau)\).
between the signal and the noise. Thus, at the output of the correlator there remains only a component proportional to the cross-correlation function of the useful signal, of the form
\[ R_{12}(\tau)=\frac{1}{T}\int_{t-T}^{t} U_c(t)\,U_c(t-\tau)\,dt . \tag{9} \]
The recording device 18 records the cross-correlation function (9). This method of extracting information on local changes in the spatial radiation field gives a noise-suppression efficiency \(P_{\text{out}}/P_{\text{in}}\) of up to 40.
Thus, increasing the sensitivity of the radiometric recording method is possible by appropriately choosing the detector transfer function with optimal frequency characteristics. The threshold function is determined by the geometric dimensions of the collimator and by the monitoring speed. The optimal filter is a cross-correlation device, which reduces the influence not only of the statistical noise of the radiometric path, but also of the noise of the acquisition source and of the external medium, provided the radiation detectors are arranged appropriately.
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