NONLINEAR PROCESSING OF DISCRETE RADIOMETRIC INFORMATION WITH A SINGLE SAMPLE

V. G. Selivanov, K. A. Selivanova

(Presented by the scientific seminar of the Research Institute of Electronic Introscopy)

One of the characteristic features of radiometric monitoring systems is the presence of noise in the channel for processing radiometric information (noise of the radiation source, intrinsic noise of the photodetector device, etc.). In monitoring practice one often encounters problems of detecting a signal from a defect that lies at the noise level, so that the methods of processing radiometric information that exist in defectoscopy do not provide the required reliability of defect detection. Hence there arises the need to construct more effective detectors of the signal from a defect.

In works [1, 2] it was shown that, in order to construct more effective detectors of a signal from a defect, it is necessary to use a model of the signal from a defect with unknown parameters. From the point of view of mathematical statistics, this problem can be reduced to the problem of detection, in Poisson noise of intensity \(\lambda\), of a signal from a defect

\[ S(t,\tau_0,\tau_1,A)=A_s(t,\tau_0,\tau_1), \tag{1} \]

where \(A\) is the unknown amplitude of the signal from a defect;
\(s(t)\) is a known function of time;
\(\tau_0\) is the unknown parameter of the arrival time of the signal from a defect;
\(\tau_1\) is the unknown parameter of the duration of the signal from a defect.

Taking a trapezoidal approximation of the signal from a defect [3], we shall have

\[ s(t,\tau_0,\tau_1)= \begin{cases} \dfrac{t-\tau_0}{\tau_1}, & \tau_0 \leq t < \tau_0+\tau_1,\\[6pt] 1, & \tau_0+\tau_1 \leq t < \tau_0+T-\tau_1,\\[6pt] \dfrac{T-t+\tau_0}{\tau_1}, & \tau_0+T-\tau_1 \leq t \leq T+\tau_0. \end{cases} \tag{2} \]

Here \(T\) is the duration of the signal from a defect.

For a given monitoring velocity \(v\) and a given detector collimator-window length \(a\), we have

\[ T=t_0+\tau_1,\qquad t_0=\frac{a}{v}. \tag{3} \]

Assuming that the signal from a defect is observed over an interval of duration \(T\), for the probability that exactly \(n\) electrical pulses appear at the detector output during time \(T\) we shall have

\[ P_{\mathrm{ш}}(n)=\frac{(\lambda T)^n}{n!}e^{-\lambda T} \tag{4} \]

in the case when the signal from a defect is absent.

Fixing the values of the parameters \(A,\tau_0,\tau_1\) in the case where the defect signal is present, we obtain

\[ P_{\text{mix}}(n\mid A,\tau_0,\tau_1)= \frac{[(\lambda+A s(t,\tau_0,\tau_1))T]^n}{n!} e^{-(\lambda+A s(t,\tau_0,\tau_1))T}. \tag{5} \]

From (4) and (5), taking (3) into account, we find the logarithm of the likelihood ratio

\[ z=n\ln\left[1+\frac{A}{\lambda}s(t,\tau_0,\tau_1)\right] -A s(t,\tau_0,\tau_1)(\tau_1+t_0), \tag{6} \]

Expression (6) was obtained for fixed values of the parameters \(\tau_0,\tau_1,A\). Since the values of these parameters are unknown, it is required, on the basis of a single sample value \(n\), to construct estimates \(\hat{\tau}_0,\hat{\tau}_1,\hat{A}\).

The most effective method for estimating unknown parameters is the maximum-likelihood method. Therefore we shall seek estimates of the unknown parameters \(\hat{A},\hat{\tau}_0,\hat{\tau}_1\) from the conditions for the maximum of expression (5) or (6). Finding the maximum of (6) with respect to the parameter \(A\), we obtain

\[ \hat{A}=\frac{n-\lambda(t_0+\tau_1)} {(t_0+\tau_1)s(t,\tau_0,\tau_1)}. \tag{7} \]

From (7) we see that the maximum-likelihood estimate of the amplitude of the defect signal belongs to the class of linear estimates. Substituting (7) into (6), after simple transformations we obtain the optimal decision rule for the presence or absence of a defect signal for the model of a defect signal with unknown amplitude at any fixed value of the parameter \(\tau_0\). It is as follows: the decision \(\gamma_1\) is accepted (the defect signal is present) if the inequality

\[ n\left[\ln\frac{n}{\lambda(t_0+\tau_1)}-1\right]\geq K_{\hat{A}}, \tag{8} \]

is satisfied; conversely, the decision \(\gamma_0\) on the absence of a defect signal is accepted if (8) is not satisfied.

The value of the detection threshold \(K_{\hat{A}}\) and the detection-error probabilities \(\alpha\) (the probability of false-alarm error) and \(\beta\) (the probability of missed-signal error) in this case are easily found by the method set forth in [4].

However, the use of the optimal rule (8) in calculating the actual sensitivity of a flaw detector is associated with significant losses of useful information, since the uncertainty of the location of the defect in the inspected object leads to a violation of the homogeneity of the hypotheses being tested (the unknown parameter of the arrival time of the defect signal \(\tau_0\)).

Thus, since the defect signal \(S(t,\tau_0)\) is a function of the unknown parameter \(\tau_0\), some a priori knowledge of the parameter is necessary.

1. It is known a priori that the parameter \(\tau_0\) is the instant at which the defect begins to enter the zone of the detector window, i.e., the instant at which the mean intensity \(\lambda(T)\) begins to change in the direction of increase (the defect has a lower absorbing capacity for radiation than the defect-free absorber) or in the direction of decrease (the defect has a greater absorbing capacity). In the present work, without loss of generality, we shall assume that

\[ S(t,\tau_0)\geq 0, \tag{9} \]

if

\[ \tau_0\leq t\leq T+\tau_0. \]

1. The form of the function \(S(t,\tau_0)\) is known and is determined by expressions (1) and (2).

2. We assume that the duration \(T_s\) of the signal from the defect is known a priori.

3. On the basis of analysis of a large volume of empirical data on the nature of the arrangement of defects in objects of the same type under inspection, or from other practical considerations, it is always possible to establish the smallest (at least in probability) interval between two neighboring defects, such that on it the hypothesis \(H_0\) is satisfied with probability \(P(H_0)=1\). Let us assume that the probability of the occurrence of two defects at a distance closer than \(3T_s\) along the time axis is zero. This, in the most general case, is what limits the a priori knowledge of the parameter.

To construct an algorithm for detecting the signal from a defect, it is necessary to extract from the sample data certain a posteriori information about the parameter \(\tau_0\); that is, in other words, it is required to find the best estimate \(\hat{\tau}_0\) of the true parameter \(\tau_0\). If the estimate \(\hat{\tau}_0\) is unbiased, then, taking \(\hat{\tau}_0\) as the true value of the parameter \(\tau_0\), we arrive on average (with accuracy up to the variance of the estimate \(\hat{\tau}_0\)) at fulfillment of the homogeneity condition for the hypotheses \(H_1\) or \(H_0\), which consists in the fact that under the hypothesis \(H_0\) the sample values may be taken only from noise (of intensity \(\lambda(T)\)), while under the hypothesis \(H_1\) the sample values are taken from a signal-plus-noise mixture (of intensity \(\lambda(T)+S(T)>\lambda(T)\)).

It is easy to see that all useful information about the signal from a defect in this case can be obtained when two basic conditions are fulfilled:

a) the probability of fulfillment of the hypothesis \(H_1\)

\[ P(H_1)=1; \tag{10} \]

b) the duration \(T_s\) of the signal from the defect and the duration of the observation time interval \(T_0\) satisfy the equality

\[ T_0=T_s=T. \tag{11} \]

However, since the estimate \(\hat{\tau}_0\) can ensure fulfillment of the homogeneity condition for the hypotheses being tested only on average (that is, only with accuracy up to the variance of \(\hat{\tau}_0\)), and the probability of fulfillment of the hypothesis \(H_1\) when \(\hat{\tau}_0\) is used as the true value of \(\tau_0\), \(P(H_1)<1\), the decision rule obtained in this way for the presence or absence of a signal (even if \(\hat{\tau}_0\) is effective) from a defect will not be optimal.

Indeed, since \(\hat{\tau}_0\) has some finite, nonzero variance \(D(\hat{\tau}_0)>0\), in order to fulfill the conditions for reading off all information about the signal from a defect, it is necessary to require that \(T_0>T_s\), and this contradicts the homogeneity condition for the hypotheses being tested. On the other hand, since \(D(\hat{\tau}_0)>0\), a necessary condition for fulfillment of the homogeneity condition for the hypotheses being tested is the requirement \(T_0<T_s\), which contradicts the conditions necessary for reading off all useful information about the signal from a defect. And, to make matters worse, the equality \(T_0=T_s\) when \(D(\hat{\tau}_0)>0\) cannot be fulfilled, since this leads with probability \(P(\cdot)=1\) to violation of the homogeneity of the hypotheses being tested.

Finding the optimal structure of a detector of a signal from a de-

defect (1) is associated with the problem, unresolved in communication theory, of joint detection and estimation of signal parameters [5].

However, the use of the additional a priori information about the parameter $\tau_0$ indicated in items 1–4 makes it possible to construct a signal detector, close to the optimal one, for a defect with unknown parameters $\tau_0$ and $A$.

Decision rule (8) is optimal for a signal with unknown amplitude for any fixed values of the parameters $\tau_1$, $\tau_0$. However, a necessary condition for the decision rule (8) to be optimal is the equality $\tau_0=0$, that is, the instant at which information about the signal begins to be read must coincide with the instant of arrival of the signal. Otherwise the condition of homogeneity of the statistical hypotheses being tested is violated, and the problem cannot be reduced to the question of testing simple hypotheses [4].

On the other hand, if one takes into account the additional a priori information about the parameter $\tau_0$ indicated in items 1–4, one may conclude that an estimate of the parameter $\tau_0$, consisting in comparison with a threshold of the maximum of $m$ dependent statistics (8), obtained over the time interval $T$ and shifted relative to one another in the starting instant of the count by the time

\[ \Delta t=\frac{3T}{m}, \]

is close to optimal. We note that the estimate of $\tau_0$ in this case is made before detection, and the results of the estimate can be used only after the detector has made the decision $\gamma_1$ that a signal from a defect is present (Fig. 1), which is consistent with the latest results of the theory of joint estimation and detection [5].

Fig. 1. $\chi$ — channel for processing radiometric information; $n_j(T)$, $(j=1,2,\ldots,m)$ — summator of the number of electric pulses over the time $T$; $U_j(T)$, $(j=1,2,\ldots,m)$ — block for computing the statistic $U$; $\left(\dfrac{\max}{m}\right)$ — maximum-search block; $K$ — detection threshold; $\gamma_1$ — decision that a signal from a defect is present; $\gamma_0$ — decision that a signal from a defect is absent.

Therefore the algorithm for processing discrete radiometric information is constructed as follows: the radiometric-information processing channel $\chi$ contains $m$ identical summators $n_1(T)$, $n_2(T)$, ..., $n_m(T)$, at whose outputs we have a sequence of dependent sample values $n_l(T)$, $(l=1,2,\ldots)$, shifted relative to one another in the starting and ending instants of the count by the full time interval

\[ \Delta t=\frac{3T}{m}; \]

$m$ identical blocks $U_1(T)$, $U_2(T)$, ..., $U_m(T)$ for computing the statistic

\[ U=n(T)\left[\ln\frac{n(T)}{\lambda T}-1\right]; \tag{12} \]

a maximum-search block $\left(\dfrac{\max}{m}\right)$, selecting the largest of the $m$ statistics $U_j$ $(j=1,2,\ldots,m)$, and a detection threshold $K$.

To complete the solution of the problem of detecting a signal from a defect (1), it is necessary to find the estimate $\hat{\tau}_1$. It is not difficult to see that in

in this case it is not possible to find the maximum-likelihood estimate \(\hat{\tau}_{1\max}\) consistent with the estimate \(\hat{A}\).

On the other hand, from practical considerations one can indicate the limits of variation of the possible values of the duration parameter of the signal from the defect \(\tau_1\), that is,

\[ \tau_{1\min} \leq \tau_1 \leq \tau_{\max}. \tag{13} \]

Then, dividing the interval \([\tau_{1\min}, \tau_{1\max}]\) into \(M\) equal subintervals of duration

\[ \Delta \tau_1=\Delta t=\frac{\tau_{1\max}-\tau_{1\min}}{M}, \tag{14} \]

one may approximately assume that any value of the parameter is expressed as follows:

\[ \begin{gathered} \tau_{1k}=\tau_{1\min}+k\Delta\tau_1;\\ k=0,1,\ldots,M. \end{gathered} \tag{15} \]

Then the optimal decision rule for deciding on the presence or absence of a signal from a defect with unknown parameters \(A\) and \(\tau_1\) will be as follows: the decision \(\gamma_1\) on the presence of a signal from a defect is accepted if at least one of the inequalities

\[ n(T_k)\left[\ln\frac{n(T_k)}{\lambda(t_0+\tau_k)}-1\right]\geq C_k, \tag{16} \]

is satisfied, where \(n(T_k)\) is the sum of the number of electrical pulses from the photodetector output over the time \(T_k\).

\[ T_k=t_0+\tau_k. \tag{17} \]

The block diagram of a detector of a signal from a defect with unknown parameters \(A\), \(\tau_1\) is presented in Fig. 2. The channel for processing radiometric information \(\chi\) contains \(M\) adders \(n(T_1),\ldots,n(T_M)\), which compute the sum of the number of electrical pulses over time intervals \(T_1,\ldots,T_M\); \(M\) blocks \(U_1,U_2,\ldots,U_M\) for computing the statistic (16), and \(M\) detection thresholds \(C_1,\ldots,C_M\).

Fig. 2. \(\chi\) — channel for processing radiometric information; \(n_j(T)\), \((j=1,2,\ldots,M)\) — adder of the number of electrical pulses over time \(T_j\), \((j=1,2,\ldots,M)\); \(U_j\) — block for computing the statistic \(U_j\); \(C_j\), \((j=1,2,\ldots,M)\) — detection threshold; \(\gamma_1\) — decision on the presence of a signal from a defect; \(\gamma_0\) — decision on the absence of a signal from a defect.

Conclusions

1. Taking into account additional a priori information on the constraints imposed on the parameters \(A\) and \(\tau_0\) makes it possible to construct detectors of a signal from a defect that are close to optimal.

2. The optimal transfer characteristic of a nonlinear filter for detecting a signal from a defect with unknown amplitude has been obtained.

REFERENCES

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3. L. K. Tatochenko. Radioactive Isotopes in Instrument Engineering. Moscow, Atomizdat, 1960.

4. B. R. Levin. Theoretical Foundations of Statistical Radio Engineering. Vol. 11. Moscow, “Sovetskoe radio,” 1968.

5. D. Middleton, R. Esposito. Simultaneous Optimum Detection and Estimation of Signals in Noise. IEEE Trans. Inform. Theory, 1968, 14, 3, 434—444.