Small target detection in clutter using recursive ... - IEEE Xplore

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Billingsley, J. B., Farina, A., Gini, F., Greco, M. V., and Verrazzani, L. (1999) Statistical analysis of measured radar ground clutter data. IEEE Transactions on Aerospace and Electronic Systems, 35 (Apr. 1999), 579—593.

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Small Target Detection in Clutter Using Recursive Nonlinear Prediction

I. INTRODUCTION Detecting small objects in a clutter scene is an important and challenging problem in many areas such as image processing [1], remote sensing [2], radar surveillance [3], manufacturing [4], infrared sensors [5], and robotics [6]. One popular approach to this problem uses a predictor to suppress the background clutter. More precisely, one uses some clutter data to train a predictor and decide the existence of a target in the received signal by checking the magnitude of the prediction error produced by the clutter-trained predictor. Since a target signal is usually quite different from a clutter process, a clutter-trained predictor will have small prediction error for a clutter process and will produce relatively large prediction error when it is applied to predict a target signal. Thus, the target detectability is increased. Let fn(1), n(2), : : : , n(N)g be a clutter sequence. A predictor f is used to model the clutter process by minimizing the prediction error: E=

Detecting small objects in clutter is usually carried out by using a predictor to suppress the background clutter. The idea is that a predictor which is trained using clutter data usually has small prediction error for the clutter process, but the prediction error will be relatively large if the signal fed into the predictor contains a target. While conventional approaches use a one-step-ahead predictor, we propose using a recursive predictor, which uses the predicted value to continue predicting the future points, to improve this predictive detection scheme. It is shown here that while the recursive prediction error of the clutter process is about the same as that of a one-step ahead predictor, the recursive predictor amplifies the prediction error of the target process. Therefore, the distance between the clutter and target processes is increased and the target detectability is enhanced. In addition, this recursive prediction approach has the same computational complexity as the one-step-ahead predictor since no extra training or modeling procedure is required. Real radar oceanic surveillance data are used to illustrate the effectiveness of the proposed detection method. Results show that the recursive prediction approach outperforms the one-step-ahead predictor in detecting small targets in the presence of strong clutter.

Manuscript received June 28, 1999; revised January 25, 2000; released for publication December 24, 1999. IEEE Log No. T-AES/36/2/05242. Refereeing of this contribution was handled by L. M. Kaplan.

c 2000 IEEE 0018-9251/00/$10.00 ° CORRESPONDENCE

N X

t=M+1

[n(t) ¡ nˆ (t)]2

(1)

with respect to the parameters of f where nˆ (t) = f(n(t ¡ 1)) = f(n(t ¡ 1), : : : , n(t ¡ M)), and M is the order of the predictor. When the trained predictor receives an incoming signal fx(t)g, the prediction error is used to decide whether x(t) contains a target or not. More precisely, let "(t) = (x(t) ¡ xˆ (t))2 = (x(t) ¡ f(x(t ¡ 1)))2 where x(t ¡ 1) = (x(t ¡ 1), : : : , x(t ¡ M)) and H1 : x(t) = s(t) + n(t)

H0 : x(t) = n(t): The decision is made by comparing "(t) with a prefixed threshold ´. That is, ½ H1 if "(t) > ´ : x(t) 2 H0 if "(t) · ´

(2)

(3)

The performance of this predictive detection scheme highly depends on the accuracy of the clutter prediction. If the clutter prediction error is not small enough, this detection scheme may not work well. Conventional approaches employ a linear predictor for clutter modeling [7]. However, since clutter processes are usually non-Gaussian [8], nonlinear and even chaotic [9], nonlinear predictors such as the neural networks [10], rational functions [11], and local approximations [3] have been proposed to improve the accuracy of clutter prediction. It is found that when f is nonlinear, it produces much smaller clutter prediction error and results in an enhanced target detection performance [10, 11]. One logical approach to further improve this nonlinear prediction method is to increase the accuracy of clutter prediction using a more 713

sophisticated nonlinear predictor, such as incorporating fuzzy logic or a global search technique into those nonlinear predictors [12]. The problem with this approach is a very heavy computational load, and the nonlinear predictor may no longer be suitable for real-time implementation. Besides, there is no guarantee that these complicated predictors have superior performance in predicting a clutter process, because of the unavoidable presence of measurement noise [13]. Notice that the basic idea of transforming the binary decision problem (2) to (3) is to increase the “distance” between signals from H0 and H1 and hence reduce the decision errors. According to (3), not only can decreasing the prediction error "(t) for H0 achieve this purpose, but increasing the prediction error for H1 can also improve the decision making. We, therefore, propose here to replace the one-step-ahead predictor used in the predictive detection method by a multistep-ahead recursive predictor [14]. By recursive prediction, we mean the predicted value is used as an input to the predictor to continue the prediction for future values. The success of this approach depends on whether a clutter-trained predictor can produce larger prediction error for an H1 signal when it is used to predict the signal recursively. We show that this is indeed the case in the following section. Although this recursive prediction approach can use any nonlinear predictor, the local method (or the memory-based approach) is used here [3, 15]. Note that the term “local” refers to forming a local neighborhood in the phase space of the delay vectors fn(t ¡ 1) = (n(t ¡ 1), : : : , n(t ¡ M))g instead of a local neighborhood in time. The main reason for employing the local method is that it is less dependent on the functional representation, which is unknown for a real clutter process. This characteristic is favorable in making recursive prediction since an inexact functional form produces representational errors which will be accumulated in a recursive prediction process. A brief description of the local predictor used in this study is also presented in Section II. To understand the effectiveness of this idea, real-life oceanic radar surveillance data are used in this study. The radar used to collect data is a coherent, dual-polarized, X-band radar, designed to be of instrumentation quality for research use. It was employed at a site at Osborne Head Gunnery Range, Nova Scotia, Canada (44± 36:710 N, 63± 25:410 W) in November, 1993. The radar site was located on a cliff roughly 100 ft above sea level facing the Atlantic Ocean. The database contains a variety of targets, including boats, buoys, and small floating targets consisting of reflective beach balls roughly a half meter in diameter and containing a marine radar enhancer [16]. Analyses based on the real data are reported in Section III. It is shown that the recursive 714

prediction approach outperforms the one-step-ahead prediction method in detecting small targets in a strong clutter environment. II. DETECTION USING NONLINEAR PREDICTION The predictive detection method uses the prediction error "(t) = (x(t) ¡ xˆ (t))2 where xˆ (t) = f(x(t ¡ 1)) and f is a predictor trained by clutter data to decide whether x(t) belongs to H0 or H1 as given in (3). An optimal threshold ´ can be chosen by different criteria, depending on whether the objective is minimizing the probability of false alarm and/or the probability of missing. Here, we employ the maximum likelihood criterion. More precisely, ´ is chosen as the intersection point of the distribution of the prediction error of signal from H0 and the distribution of the prediction error of signal from H1 . The simplest way to compute the probability distribution Pi for the hypothesis Hi from the prediction error is the histogram. The total detection error is then the sum of the probability of false alarm and the probability of missing. That is, Z 1 Z ´ P0 (x) dx + P1 (x) dx: (4) ´

0

When this overlapping area of the two probabilities P0 and P1 is large, it implies that the performance of the predictive detector is poor. In other words, a good predictive detector has a large distance between P0 and P1 , i.e., d(P0 , P1 ), to minimize the detection error in (4). Ideally, the more accurate the clutter prediction is, the smaller the detection error is expected to be since a more accurate predictor will make the first term of (4) smaller for a fixed ´. But as we can see, the detection error (4) can also be reduced by making the second term in (4) small. In other words, we can increase the prediction error of an H1 signal generated by a clutter-trained predictor. Geometrically, d(P0 , P1 ) is increased in this case by moving P1 apart from P0 , while improving the accuracy of clutter prediction increases d(P0 , P1 ) by moving P0 away from P1 . Since clutter has been shown to be chaotic [9, 10], it can be predicted recursively within a reasonable accuracy for a short time period bounded by its largest Lyapunov exponent [17]. This observation has been validated using real sea clutter data [18]. It was found that sea clutter data can be accurately predicted in a recursive manner for no less than 10 iterations. An accurate recursive prediction of clutter makes sure that while we try to increase the prediction error for the H1 signals, the prediction error for the H0 signals remains about the same as that of a one-step-ahead prediction. If both prediction errors increase, it is hard to know whether the detection error in (4) will be reduced or not. A fundamental question of this recursive prediction approach is whether or not a clutter-trained predictor

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will produce larger prediction errors for a target process when it is used for recursive prediction. The answer to this question is given in the following theorem. THEOREM 1 Recursive prediction of a target process x(t) (i.e., x(t) 2 H1 ), based on a predictor which is trained by clutter data, produces a larger prediction error than the one-step ahead prediction. PROOF Assuming that the signal power is much weaker than that of the clutter process, which is the case for small target detection, we can express the difference between the predictions of the received signal and the clutter process as jf K (x(t)) ¡ f K (n(t))j = js(t)jeK¸

h

xˆ (t + 1) = f(x(t)) = (5)

where x(t) = s(t) + n(t) is the received signal of H1 , s(t) is the target signal, ¸ is the Lyapunov exponent of the clutter process, and f K is the Kth iteration of f. Let xˆ r (t + K) and nˆ r (t + K) denote the K-stepahead recursive prediction of x(t) and n(t), respectively. Equation (5) can be expressed as (xˆ r (t + K) ¡ nˆ r (t + K))2 = s2 (t)e2K¸ :

Assume that the clutter-trained predictor f is accurate in predicting the clutter process. Then, nˆ r (t + K) = n(t + K) + "n (t + K) where "n (t) is the recursive clutter 2 = var("n ) is fairly prediction error at time t, and ¾"n small. Equation (6) becomes

(8)

Since the clutter process is chaotic, ¸ > 0. Equation (8) indicates that the recursive prediction error of x(t) is larger than the one-step-ahead predictor error of x(t), i.e., K = 1. Next, we briefly describe the local predictor f : RM ! R. The first step of the local prediction method is to construct a database from the training clutter sequence fn(1), n(2), : : : , n(N)g: n(M) = (n(M), n(M ¡ 1), : : : , n(1)) n(M + 1) = (n(M + 1), n(M), : : : , n(2)) .. .. . . n(N ¡ 1) = (n(N ¡ 1), n(N ¡ 2), : : : , n(N ¡ M)) CORRESPONDENCE

n(M + 1) n(M + 2) .. . n(N)

is employed since the local LS method is found to be relatively unstable [19]. To perform recursive prediction at t + 2, the predicted value xˆ (t) in (9) is used as an input to the predictor. That is,

where f is the averaged local predictor in (9). Of course, the h neighbors used for the recursive prediction at t + 2 will be different. After K iterations of this procedure, we get the K-step ahead recursive prediction xˆ r (t + K) from x(t). Note that the most time consuming part in implementing the local method is the nearest neighbor search through the entire training database. We applied the box-oriented approach given in [20] to speed up the computation.

(7)

Suppose further that the signal process s(t) is uncorrelated with the prediction error of x(t) and "n (t) is an identically independent noise process. We then apply the expectation operator to both sides of (7) to get E[(xˆ r (t + K) ¡ x(t + K))2 ] = ¾s2 (e2K¸ ¡ 1) ¡ ¾"2n :

h

1X 1X f(n(tk )) = n(tk + 1) h h k=1 k=1 (9)

xˆ r (t + 2) = f(xˆ (t + 1)) = f(xˆ (t + 1), x(t), : : : , x(t ¡ M + 2)) (10) (6)

(xˆ r (t + K) ¡ n(t + K) ¡ "n (t + K))2 = s2 (t)e2K¸ :

Let x(t) = (x(t), x(t ¡ 1), : : : , x(t ¡ M + 1)) be the prediction point. The local predictor forms a neighborhood of size h for x(t) from the database in the above table, that is, it picks h vectors n(t1 ), : : : , n(th ) which are Pclosest to x(t) in Euclidean norm. In other words, hk=1 kx(t) ¡ n(tk )k is a minimum. The predicted value, xˆ (t + 1), is found by either taking the average of the desired predicted values or performing a linear least squares (LS) fit to these h training vectors in the neighborhood. In this study the averaging approach given as

III. APPLICATION TO OCEAN SURVEILLANCE Two real radar data sets (D1 and D2) are used in the analysis. The raw data are stored as a series of 1 byte integers from 0 to 255, corresponding to a quantized voltage in the range [¡0:7, 0:7]. Like and cross polarizations are always available, with in-phase and quadrature values for each. The signal is generated by transmitting a pulse at a fairly low frequency, around 1 to 2 kHz. The returns from a single pulse are referred to as a sweep. The data sets were obtained by aiming the radar dish at a single azimuth for all sweeps. This type of data set can be considered as a staring radar image. These returns are range gated at a relatively high frequency, typically 5 to 10 MHz. The distance per range bin for both data sets is 15 m. While D1 has a range of 2200—3200 m, the range for D2 is 2000—2800 m. The target in the two data sets is a beach ball, roughly a half meter in diameter and containing a marine radar enhancer, which is located approximately at 2700 m in range. Figs. 1 and 2 show an image of the like polarization (horizontal) component of each data. When visible, the targets appear in the staring images 715

Fig. 1. Radar image of data set 1 (D1).

Fig. 3. Probability distributions of predicted clutter and target processes using recursive predictions (1-step, 2-step, 3-step, 4-step) for D1. Solid line is clutter process; dashed line is target signal.

Fig. 2. Radar image of data set 2 (D2).

as a dark horizontal line. The target was not anchored and so it fluctuated with the waves, causing it to be more easily detected at some times than at others. This is most noticeable when comparing D1 and D2 to each other. The former has a target that is more difficult to distinguish from the clutter; it was taken with stronger sea conditions. Wave conditions were sampled hourly using a buoy located 6 nmi offshore. The significant wave height, which is the average of the highest third of the waves recorded, is 0.89 m and 0.60 m for D1 and D2, respectively. The average wave period for D1 and D2 is 4.8 s and 3.9 s, respectively. In designing a suitable local predictor, the order M was determined by the false nearest neighbor (FNN) method [21]. The smallest order above which the number of FNN saturates is the optimum order for prediction. Due to space limit, readers are referred to [19] for the detailed FNN analysis of the sea clutter data used in this study. A suitable dimension for these two data sets is shown to be 5 for both cases. This value minimizes the number of FNN while also 716

Fig. 4. Probability distributions of predicted clutter and target processes using recursive predictions (1-step, 2-step, 3-step, 4-step) for D2. Solid line is clutter process; dashed line is target signal.

keeping the order relatively low and thus reduces the amount of training data needed to maintain the same amount of accuracy in prediction. We plot the distributions P0 and P1 of the prediction errors of D1 and D2 in Figs. 3 and 4, respectively. Recursive prediction is carried out up to four steps. The target prediction error for D1 is apparently increased by the recursive process. Since P0 remains about the same for the recursive prediction, d(P0 , P1 ) for D1 increases with a larger recursive prediction step. However, the difference in using a recursive prediction is not that obvious in D2. The distributions in all four cases in Fig. 4 look quite similar. It seems that the recursive prediction approach has a more significant improvement when the background clutter is strong.

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of a multistep-ahead recursive predictor is basically the same as that of a one-step-ahead predictor, the one-step-ahead predictor can be replaced entirely by a recursive one in the predictive detection method. Another interesting observation from Figs. 5 and 6 is that the performance of the one-step-ahead predictive detector for D1 is worse than that for D2. When a recursive predictor is applied to D1, it improves the detection performance and brings the ROC up to the same level as that in D2. The recursive prediction seems to be able to overcome the degradation caused by a rougher sea state condition as in D1 and produces a detection performance as with a higher signal-to-noise ratio. IV. CONCLUSION Fig. 5. Comparison of ROC using 1-step, 2-step, 3-step, 4-step recursive predictive detection for D1.

In this article we propose using a recursive predictor in the predictive detection method for small target detection in clutter. While the recursive predictor produces prediction error of the same magnitude as the one-step-ahead predictor for clutter signals, it increases the prediction error for a signal containing a target and hence reduces the total detection error of the predictive detection method. This idea is justified theoretically and is validated using real radar oceanic surveillance data. Real data results show that the improvement from the recursively prediction approach is more significant when the target is weak and the background clutter is strong. Since the recursive prediction technique has the same computational complexity as the one-step-ahead prediction method, this new approach is suitable for real-time applications. HENRY LEUNG Dept. of Electrical and Computer Engineering University of Calgary Calgary, Alberta Canada T2N 1N4

Fig. 6. Comparison of ROC using 1-step, 2-step, 3-step, 4-step recursive predictive detection for D2.

A more rigorous evaluation is carried out by computing the receiver operating characteristic (ROC). The ROCs based on 1-step to 4-step prediction for D1 and D2 are plotted in Figs. 5 and 6, respectively. Fig. 5 shows that using a recursive prediction, the probability of detection can be increased by 0.15 for a fixed probability of false alarm. This large improvement is observed when a 2-step recursive predictor is used instead of the one-step-ahead predictor. The improvement is less significant when the prediction step is further increased to 3 or 4. The improvement appears to saturate after a certain level. For D2, the recursive prediction approach does not really have any significant improvement as shown in Fig. 6. However, this analysis confirms that the recursive prediction approach will not degrade the detection performance, as even the target-to-clutter ratio is relatively higher. Since the computational load CORRESPONDENCE

ALYSSA YOUNG Dept. of Mathematics University of British Columbia Vancouver, BC Canada V6T 1Z2 REFERENCES [1]

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French, P. A., Zeidler, J. R., and Ku, W. H. (1997) Enhanced detectability of small objectives in correlated clutter using an improved 2-D adaptive lattice algorithm. IEEE Transactions on Image Processing, 6 (1997), 383—397. Carlotto, M. J. (1997) Detection and analysis of change in remotely sensed imagery with application to wide area surveillance. IEEE Transactions on Image Processing, 6 (1997), 189—202. Leung, H. (1996) Nonlinear clutter cancellation and detection using a memory-based predictor. IEEE Transactions on Aerospace and Electronic Systems, 32 (1996), 1249—1256. 717

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Correcting the Pspice Large-Signal Model for PWM Converters Operating in DCM

It is shown that an incorrect parameter limit in the existing Pspice pulsewidth modulated (PWM) switch model has been detected. The main objective of this work is to alert users of such models to a potential source of simulation error under certain parameter specifications as shown here. Simulation of the boost converter operating in discontinuous conduction mode (DCM) based on this incorrect model results in either convergence problems or wrong results. Correction of the model is provided and its validity is verified using a specific converter example. The results illustrate how the old and corrected models produce different results when the boost converter is subjected to some transients under sudden load variations.

I. INTRODUCTION In the modeling of power converters, the state-space average technique developed by Middlebrook and Cuk [1] in the 1970s provided a generalized approach for a large class of pulsewidth modulated (PWM) converters. Since then, the concept of averaging has been explored extensively and various improvements have been made in studying the transient responses and stability problems of dc-dc converters including PWM and resonant converters. One of the most important results obtained was the three-terminal PWM switch model developed by Vorperian [2]. This approach is based on the observation that in a PWM converter, the switching action of the single-pole double-throw switch can be realized by an active switch (usually a controlled transistor) and a passive switch (usually a diode), and it is the only nonlinear component in the converter. Once its invariant property is determined, the average equivalent circuit model of the converter can be Manuscript received September 20, 1999; revised December 6, 1999; released for publication February 2, 2000. IEEE Log No. T-AES/36/2/05243. Refereeing of this contribution was handled by W. M. Polivka.

c 2000 IEEE 0018-9251/00/$10.00 °

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APRIL 2000