IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO. 3, MAY 2011
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A Parzen-Window-Kernel-Based CFAR Algorithm for Ship Detection in SAR Images Gui Gao, Member, IEEE
Abstract—This letter proposes a Parzen-window-kernel-based algorithm for ship detection in synthetic aperture radar (SAR) images. First, the data-driving kernel functions of Parzen window are utilized to approximate the histogram of real SAR image, in order to complete the accurate modeling of SAR images. Then, a threshold of global constant false alarm rate is given theoretically, and the numerical solution of the threshold is also derived. The experimental results of the real data of typical targets demonstrate that the algorithm presented is effective. Index Terms—Parzen window, ship detection, synthetic aperture radar (SAR).
I. I NTRODUCTION
S
HIP detection in synthetic aperture radar (SAR) images is an important application of Earth observation for monitoring of fishing vessels, oil pollution, and warship reconnaissance [1], [2]. Many investigations have been carried out in the literature, such as constant false alarm rate (CFAR) [3], the wavelet transform-based [1], coherence images-based [2], etc. Among these algorithms, CFAR detection, which is famous for its constant false alarm probability and adaptive threshold, has been used widely. The precision of statistical modeling for clutter is crucial for the detection performance. Presently, the conventional CFAR detections mainly adopt parametric models, such as Gamma, Weibull, K, etc. [3], [4]. However, the modeling ability of a parametric model is doubtable for a complex unknown SAR scene. In fact, the nonparametric models are more flexible and can more accurately fit the real data [5]–[7]. The Parzen window kernel method [5]–[7] is a commonly used nonparametric probability density estimation strategy in the field of pattern recognition. It is a kind of nonparametric method and a datadriven model to estimate the probability density function (pdf) of SAR image data with excellent estimation accuracy. The Parzen window kernel method is suitable for estimating the complex unknown pdf. However, the acquirement of detection threshold is a hard task for the CFAR algorithm based on the Parzen window kernel. In this letter, a numerical solution of the threshold will be presented and tested over real data.
Manuscript received March 18, 2010; revised June 7, 2010 and August 28, 2010; accepted September 23, 2010. Date of publication December 16, 2010; date of current version April 22, 2011. This work was supported by the National Natural Science Foundation of China under Project 40801179. The author is with the School of Electronic Science and Engineering, National University of Defense Technology, Changsha, 410073, China (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LGRS.2010.2090492
II. S TATISTICAL M ODELING OF SAR I MAGE BASED ON PARZEN W INDOW K ERNEL The basic idea of the Parzen window kernel method is to utilize the weighted sum of different kernel functions for obtaining the estimation of the statistical distribution. Commonly used kernel functions include the uniform, triangle, cosine, and Gaussian. In this letter, we used the standard normal distribution as the kernel function 2 u 1 . (1) ϕ(u) = √ exp − 2 2π The corresponding cumulative distribution function (cdf) is given by 1 Φ(u) = √ 2π
u −∞
2 t exp − dt. 2
(2)
Therefore, the estimation of pdf for SAR image follows the approximation of the kernel functions as N x − xj 1 1 pˆN (x) = ϕ , N j=1 hN hN
x≥0
(3)
where x1 , x2 , . . . , xN denote the samples and are corresponded to the value of pixels in SAR image. N represents the number of sample points. hN (hN > 0) is the bandwidth that indicates the width of the kernel function. From (3), the Parzen window kernel method is actually a mixed distribution by accumulating different kernel functions. The estimation expression of image pdf is obtained by the weighted sum of the kernel functions in samples. Therefore, the characteristic of this method is suitable for the estimation of various complex and unknown pdf, in spite of single peak, multipeaks, regulation, or nonregulation. In (3), small hN will make the pdf estimate appear noisy and show spurious features, while big one will lead to smooth estimates where important structural features may be missed [5], [6]. The selection of bandwidth hN can adopt several methods, such as the plug-in estimators and data-driven manners [5], [6]. On the other hand, hN should decrease with the increase of N so as to make pˆN (x) be convergence. In this letter, we used the method in the literature [7] to set a fixed value of hN , which is defined as
1545-598X/$26.00 © 2010 IEEE
h hN = √ N
(4)
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IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 8, NO. 3, MAY 2011
where h is an adjustable constant. Commonly, h is an integer and determined empirically.
Let X represent the actual SAR image. Then X = {x1 , x2 , . . . , xN }.
(10)
III. CFAR D ETECTION If xmax denotes the maximum value of the samples, i.e.,
A. Theoretical Derivation of Detection Threshold Assuming that the pdf estimation of SAR image is pˆN (x), we combine (3) under the condition that the theoretical false alarm probability of detection is pfa , and then, the global detection threshold T is given by ∞ pfa =
pˆN (x)dx T
2 N ∞ 1 1 x − xj 1 1 √ exp − dx. (5) = hN N j=1 2 hN 2π T
√ Let t = (x − xj )/ 2hN . Then ∞ N 1 1 2 √ pfa = e−t dt. N π j=1
(6)
T −xj √ 2hN
According to the error function and the complementary √ x error 2 function whose expression is erfc(x) = 1 − (2/ π) 0 e−t dt, (6) can be written by pfa =
N T − xj 1 erfc √ . 2N j=1 2hN
(7)
The detection threshold T can be finally determined by (7). Thus, for a pixel, if its value exceeds T , this pixel is considered to be a target point; otherwise, it is declared to be a clutter point. The target detection is completed by comparing all pixels in the tested SAR image with T .
xmax = max(X)
(11)
we divide the range [0, xmax ] into equal intervals according to the step length τ and define the following equation:
(12) X = x(1) , x(2) , . . . , x(m) where x(l) , 1 ≤ l ≤ m, indicate the nodes of different intervals and x(1) = x1 . The number of nodes is given by m=
xmax + 1. τ
(13)
Because T ∈ [0, xmax ], the selection of τ is determined by the expected precision of T . For example, if the expected precision of T does not exceed 0.01, τ = 0.01. We compute the cdfs FN (x(l) ), 1 ≤ l ≤ m, separately. Then, FN = {FN (x(1) ), FN (x(2) ), . . . , FN (x(m) )}, where the expression of the numerical solution for FN (x(l) ) is given by ⎧ l−1 ⎨ pˆN (x(k) )+pˆN (x(k+1) ) · τ , m ≥ l ≥ 2 2 FN x(l) = k=1 ⎩ 0, l = 1. (14) Since the computation of the cdf FN (•) is a process of accumulation, the iterative relationship can be given by pˆN x(l−1) + pˆN x(l) ·τ, m ≥ l ≥ 2. FN x(l) = FN x(l−1) + 2 (15) Assume that QN is the complementary of FN , i.e., QN = 1 − FN = {QN (1), QN (2), . . . , QN (m)} .
(16)
B. Numerical Solution of the Threshold Considering (7), the relation of detection threshold T and false alarm probability is included in the accumulation of N complementary error functions. T is very difficult to derive theoretically given pfa . Consequently, the numerical solution of T is requested to be devised. According to (3), the cdf of pˆN (x) can be obtained by x FN (x) =
pˆN (t)dt.
(8)
0
Moreover, the relationship of FN (T ) and pfa can be expressed as ∞ pˆN (x)dx = 1 − FN (T ).
pfa = T
(9)
Obviously, QN is monotonously degressive, and QN (l) = 1 − FN (x(l) ). Summing up, Fig. 1 shows the detailed flowchart of calculating the detection threshold, which can be divided into seven steps. 1) Determining the step length τ by the expected precision of T and giving false alarm probability pfa . 2) Determining xmax , X , and m separately by (11)–(13). 3) Calculating pˆN (x(1) ), pˆN (x(2) ), . . . , pˆN (x(m) ) by (3) and (12). 4) Computing FN by (14) and (15). 5) Letting l = 1 and calculating QN by (16). 6) Calculating QN (l + 1) according to QN (l). If QN (l) > pfa > QN (l + 1), T = x(l) , and return. Otherwise, continue to the next step. 7) Letting l = l + 1 and returning to Step.6).
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Fig. 2. Representative SAR image.
Fig. 1.
Detailed flow of calculating the detection threshold.
IV. E XPERIMENTAL R ESULTS The proposed method is tested in real SAR images. Fig. 2 shows a representative SAR image used in this study; the horizontal and vertical axes are the directions of range and azimuth, respectively. The data were collected by the RS-1 spaceborne SAR of China near Kaohsiung harbor in Taiwan Strait, with the size of 3325 × 2877 pixels. The image is single look. As shown in Fig. 3, there are 16 ships in total, which are numbered 1–16, recorded by observers during the time of the RS-1 SAR image acquisition under the condition that wind speed is less than 1.05 m/s and sea state is calm. Table I lists the information about these ships. Fig. 4 shows the fitting result of the histogram in Fig. 2 based on the Parzen window kernel and K distribution [3], [8]. In order to quantitatively assess the fitting result, we adopt the Kullback–Leibler (KL) [9] distance and Kolmogorov–Smirnov (KS) [10] test as similarity measurements. 1. KL Distance Measurement Given the theoretical pdf p(w) and the actual pdf q(w), the KL distance or relative entropy between two densities p and q is defined as [9] q(w) dw. (17) D(qp) = q(w) log2 p(w)
Fig. 3. Illustration of scene content.
The approximated numerical calculation is given by
q(w)Δw p(w)Δw Q(w) = Q(w) · log2 P (w)
D(qp) =
q(w)Δw · log2
(18)
where Q(w) and P (w) denote the values of probability. Note that D(qp) is not symmetrical (i.e., D(qp) = (pq)). In our case, the symmetrized KL distance DKL = D(qp) + D(pq)
(19)
is adopted. When the actual density equals the theoretical density, DKL is zero. Otherwise, DKL is a positive value. The KL distance measurement reflects the overall similarity of the
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TABLE I LOCATION AND LENGTH OF SHIPS
TABLE II COMPARISON OF FITTING HISTOGRAM
Fig. 4. Fitting result of histogram.
actual and theoretical densities. The smaller the value of the KL distance measurement obtained, the higher similarity they have, which shows that fitting accuracy is better. Fig. 5. Detection results of ships. (a) CFAR using K distribution. (b) Proposed method.
2. KS Test Given a set of observations R1 , R2 , . . . , RN with identical distributions, statistical goodness-of-fit testing procedures can be used to choose between hypothesis H0 , an assertion that the data were drawn from distribution q, and an alternative hypothesis H1 , the one which was not drawn from distribution q. The test is based upon the empirical cdf of the observed data. The empirical cdf is a piecewise constant function PˆR (w) that is equal to zero at w = −∞ and increases by a value of 1/N at each observation Rk . That is [10] 1 PˆR (w) = |{Rk : Rk ≤ w}| N
(20)
where | · | denotes the cardinality of the set. The KS statistic DKS is defined as the supremum of the magnitude difference
between the empirical cdf and the cumulative distribution under H0 , QR (w). That is (21) DKS = sup PˆR (w) − QR (w) . w
Obviously, the KS test is complementary for the KL distance measurement. Fig. 4 shows the fitting results of the histogram in Fig. 2 using K distribution and the Parzen window kernel. In Fig. 4, the horizontal axis represents the amplitude of pixel, and the vertical axis represents the pdf. Based on (19) and (21), the KS value and the KL value of the fitting result shown in Fig. 4 are compared in Table II, where h is set empirically as 40. Thus, it is easy to find that the modeling method based on Parzen window kernel agrees well with the given SAR image,
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clutter pixels can be denoted as Ntotal_clutter , i.e., N0 × M0 − NL − Ntotal_t arg et . The actual false alarm ratio is defined as pFAR =
Ncd Ntotal_clutter
(23)
where Ncd represents the number of false alarm pixels generated by sea clutter. As shown in Fig. 6, combining (22) and (23), we obtain the receiver operating characteristic curves of the CFAR algorithm using K distribution and the presented method. These curves indicate that the method in this study has better detection precision than the CFAR algorithm based on K distribution. V. C ONCLUSION Fig. 6.
Comparison of performance curves for detection.
which implies the higher precision of fitting using the Parzen window method than using K distribution. With pfa = 0.01 and τ = 0.1, we compare the proposed algorithm with the CFAR using K distribution which has been successfully applied in the Ocean Monitoring Workstation of Canada [3], [4]. Fig. 5 shows the detection results of the two algorithms. From Fig. 5, there are more clutter false alarms that occurred in the CFAR algorithm based on K distribution than in our method under the same false alarm probability. This is because the higher fitting precision can be acquired using the Parzen window method than K distribution and the mismatch of statistical modeling results in significant CFAR loss. Utilizing nonoptimized matlab7.1 codes under a hardware environment of 2-GHz CPU and 1-GB memory, the time consumptions are 2.0167 and 64.3682 s, respectively, subjected to the CFAR algorithm using K distribution and the proposed method. In order to evaluate the performance of the algorithms in a quantitative manner, the estimation of the target detecting probability is defined as pd =
Ntd Ntotal_t arg et
(22)
where Ntd is the number of target pixels detected and Ntotal_t arg et denotes the number of all target pixels. Ntotal_t arg et is determined by a manual segmentation result of targets. Meanwhile, in order to remove the land disturbance, we utilize Fig. 3 to mask Fig. 2. If the size of image is N0 × M0 and the number of land pixels is NL , then the number of sea
Aiming at ship detection in SAR images, this letter has proposed a Parzen-window-kernel-based CFAR algorithm. The idea is first using nonparametric methods based on Parzen window kernel to estimate the pdf of SAR image data with high estimation accuracy. Then, a numerical solution of the threshold is derived. The analysis of the detection performance over the typical real SAR images confirms the effectiveness of the proposed algorithm. Further investigations on whether some simpler techniques such as lookup tables can be employed need to be done, which may greatly reduce the time consumption of detection. R EFERENCES [1] M. Tello, C. López-Martínez, and J. J. Mallorqui, “A novel algorithm for ship detection in SAR imagery based on the wavelet transform,” IEEE Geosci. Remote Sens. Lett., vol. 2, no. 2, pp. 201–205, Apr. 2005. [2] K. Ouchi, S. Tamaki, H. Yaguchi, and M. Iehara, “Ship detection based on coherence images derived from cross correlation of multilook SAR images,” IEEE Geosci. Remote Sens. Lett., vol. 1, no. 3, pp. 184–187, Jul. 2004. [3] R. A. English, S. J. Rawlinson, and N. M. Sandirasegaram, “Development of an ATR workbench for SAR imagery,” Defense R&D Canada, Ottawa, ON, Canada, Tech. Rep., DRDC Ottawa, TR2002-155, 2002. [4] P. W. Vachon, “Validation of ship detection by the RADARSAT synthetic aperture radar and the ocean monitoring workstation,” Can. J. Remote Sens., vol. 26, no. 3, pp. 200–212, Mar. 2000. [5] M. P. Wand and M. C. Jones, Kernel Smoothing. London, U.K.: Chapman & Hall, 1995. [6] M. Silveira and S. Heleno, “Classification of water region in SAR images using level sets and non-parametric density estimation,” in Proc. IEEE ICIP, 2009, pp. 1685–1688. [7] J. Sun, Modern Pattern Recognition. Changsha, China: Publ. House Nat. Univ. Defense Technol., 2002. [8] S. Erfanian and V. T. Vakili, “Introducing excision switching-CFAR in K distributed sea clutter,” Signal Process., vol. 89, no. 6, pp. 1023–1031, Jun. 2009. [9] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley Interscience, 1991. [10] M. D. DeVore and J. A. O’Sullivan, “Quantitative statistical assessment of conditional models for synthetic aperture radar,” IEEE Trans. Image Process., vol. 13, no. 2, pp. 113–125, Feb. 2004.
