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[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

Codec, Sept. 1998, Signal Process. Multimedia Lab., Univ. British Columbia, Vancouver, B.C., Canada. F. Kossentini and Y. Lee, “Computation-constrained fast MPEG-2 encoding,” Signal Processing Lett., vol. 4, pp. 224–226, Aug. 1997. W. Chung, F. Kossentini, and M. Smith, “Rate-distortion constrained statistical motion estimation for video coding,” in ICIP’95, Washington, DC, vol. 3, pp. 184–187. R. Arminato, R. Schafer, F. Kitson, and V. Bhaskaran, “Linear predictive coding of motion vectors,” in Proc. IS&T/SPIE EI’96. Y. Lee, F. Kossentini, M. Smith, and R. Ward, “Predictive RDconstrained motion estimation for very low bit rate video coding,” IEEE J. Select. Areas Commun., vol. 15, pp. 1752–1763, Dec. 1997. “TMN (H.263) encoder/decoder, version 2.0,” Telenor Res., TMN (H.263) Codec, June 1996. MPEG Software Simulation Group, “MPEG-2 encoder/decoder, version 1.0,” ISO/IEC DIS 13818-2 Codec, May 1995. T. Sikora, “The MPEG-4 video standard verification model,” IEEE Trans. Circuits Syst. Video Technol., vol. 7, pp. 19–31, Feb. 1997. M. Gallant, G. Cote, and F. Kossentini, “Computation constrained fastsearch motion estimation algorithm for TMN 7,” in Q15-A-45, ITU-T Q15/SG16, June 1997. B. Furht, J. Greenberg, and R. Westwater, Motion Estimation Algorithms for Video Compression. Boston: Kluwer, 1997. A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston, MA: Kluwer, 1992. J. Ribas-Corbera and S. Lei, “Optimal quantizer control in DCT video coding for low-delay video communications,” in Proc. Picture Coding Symp., Berlin, Germany, Sept. 1997.

Adaptive Target Detection in Foliage-Penetrating SAR Images Using Alpha-Stable Models A. Banerjee, P. Burlina, and R. Chellappa

Abstract—Detecting targets occluded by foliage in foliage-penetrating (FOPEN) ultra-wideband synthetic aperture radar (UWB SAR) images is an important and challenging problem. Given the different nature of target returns in foliage and nonfoliage regions and very low signal-toclutter ratio in UWB imagery, conventional detection algorithms fail to yield robust target detection results. A new target detection algorithm is proposed that 1) incorporates symmetric alpha-stable (SS) distributions for accurate clutter modeling, 2) constructs a two-dimensional (2-D) site model for deriving local context, and 3) exploits the site model for region-adaptive target detection. Theoretical and empirical evidence is given to support the use of the SS model for image segmentation and constant false alarm rate (CFAR) detection. Results of our algorithm on real FOPEN images collected by the Army Research Laboratory are provided. Index Terms—Alpha-stable models, SAR ATR, SAR segmentation.

I. INTRODUCTION Interest in the analysis of foliage-penetrating (FOPEN), ultrawideband synthetic aperture radar (UWB SAR) images stems from Manuscript received August 1, 1997; revised March 17, 1999. This paper was published in part in the Proceedings of the 1998 International Conference on Image Processing, Chicago, IL. This work was supported by the Army Federated Laboratories/Lockheed Sanders under Contract DAAL01-96-20001. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Dmitry B. Goldgof. A. Banerjee and R. Chellappa are with the Center for Automation Research, Department of Electrical Engineering, University of Maryland, College Park, MD 20742-3275 USA (e-mail: [email protected]; [email protected]). P. Burlina is with ImageCorp., Inc., College Park, MD 20740 USA (e-mail: [email protected]). Publisher Item Identifier S 1057-7149(99)09361-6.

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the radar’s ability to allow for detection of objects hidden beneath foliage. However, due to the very low signal-to-clutter ratio, conventional constant false alarm rate (CFAR) processing of UWB SAR data results in an unacceptably high number of false alarms. Since return signals from trees in foliage regions can be as strong as the responses from targets, typical cell-averaging or order-statistics CFAR’s perform poorly in foliage regions. Furthermore, the response of a target occluded by foliage is markedly different than if it were located in a clear region. Given the different nature of target returns in different clutter types, we wish to incorporate adaptability into our detection scheme. Clearly, image segmentation, which provides local clutter description, is of paramount importance in an adaptive system. The main contribution of this work lies in the design of an adaptive detection scheme which relies on modeling of the clutter statistics by symmetric alpha-stable (SS) distributions for image segmentation and CFAR detection. We use the SS pdf’s to accurately describe the highly impulsive, bursty behavior of FOPEN clutter. Theoretical and empirical evidence is provided to advocate the use of the SS model. Although the lack of explicit forms for their density functions inhibits their use, simple techniques for using SS distributions are developed for image segmentation and target detection. In particular, we derive a closed-form expression for the SS CFAR test that adjusts the detection threshold according to the local clutter type. Furthermore, the mixture SS CFAR test is introduced to reduce the number of false alarms along clutter boundaries. For segmentation purposes, we model a scene as being composed of distinct, contiguous regions of foliage and nonfoliage (or clear), each of which is distinguished by different statistical characteristics, such as the mean and variance. A natural approach to delineating these regions in UWB imagery is to estimate the statistical properties of the regions and use the descriptions to discriminate between the regions. Many such approaches to segmentation have been studied [7], [10], [17]. These methods generally proceed by formulating statistical model assumptions for region generation and image formation processes. Maximum likelihood (ML) or maximum a posteriori (MAP) estimation is then used for segmentation. Numerous examples of such approaches exist in the literature. Techniques that model images as Markov random fields (MRF’s) have been extensively investigated [7]. MRF’s attempt to represent spatial dependencies, and the MRF–Gibbs distribution equivalence allows for the computation of the MAP estimate of the original image [7]. For FOPEN SAR, image segmentation is achieved by modeling the conditional distribution of the clutter with a SS density and adopting an MRF model for the prior pixel label probability. The MAP estimate of the pixel label is the one that maximizes the product of the conditional and prior probabilities. Once the image is segmented, the labeled map enables the use of region-adaptive detectors. The region-adaptive algorithm is necessary for robust and efficient target detection. A common signal processing technique for automatic target detection (ATD) is CFAR processing. The goal of a CFAR detector is to provide detection thresholds that are impervious to the background clutter and noise while maintaining a constant false alarm rate. CFAR detectors attempt to characterize the local background statistics within a sliding window cell and determine a threshold based on signal detection theory. Since we model FOPEN clutter with SS distributions, we have designed the SS CFAR detector. Despite the lack of an explicit form of the density function, a simple, closed-form expression for the SS

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CFAR is derived. The alpha-stable CFAR improves upon the detection performance by employing a more accurate statistical model for the clutter. In addition, the SS CFAR exploits the segmented image to help estimate the local clutter statistics, and thereby determine a detection threshold that is a function of the clutter statistics and clutter type. Typically, CFAR processing is based on the assumption of homogeneous1 background clutter. Hence, when the CFAR window is intersected by regions of multiple clutter types, conventional CFAR’s do not provide an accurate threshold. Order statistics CFAR’s and its variations, such as the Greatest-of CFAR (GOCFAR) and the least-of CFAR (LOCFAR) have been designed to deal with multiple clutter types [6]. However, by exploiting the segmented image, we have derived an intelligent CFAR test, motivated by Finn’s heterogeneous clutter CFAR design [5], to account for nonhomogeneous background clutter. The SS mixture CFAR detector employs a probabilistic mixture model to describe the clutter statistics along boundaries of different clutter types. This model helps to reduce the false alarms resulting from incorrect statistical modeling due to the assumption of homogeneous background of conventional CFAR’s. An ROC curve comparing the performance of the SS, mixture SS, Weibull, and Gaussian CFAR detectors is provided. This work is organized as follows: a brief overview of UWB SAR phenomenology and the problems they present for an ATR algorithm is given in Section II. The usefulness of the SS model is explained in Section III. An algorithm for segmenting FOPEN SAR images using the SS model is described in Section IV (see Fig. 1 for a diagram of the algorithm). The region-adaptive SS CFAR detector is explained in Section V. Finally, experimental results are presented in Section VI. Section VII concludes this paper. II. UWB SAR PHENOMENOLOGY Ultra-wideband SAR’s differ from conventional SAR’s in the frequency range at which they operate. While high-frequency SAR’s typically operate with a narrowband signal at a carrier frequency between 10 and 33 GHz, UWB SAR’s emit a wideband signal covering frequencies from 25 MHz to 1.5 GHz [16], corresponding to a 1-ns transmit pulse. Indeed, due to the wide bandwidth, the UWB SAR is considered as a high-resolution, impulse radar. In addition, the UWB SAR is a real-valued radar.2 The pixel values of the FOPEN images are bipolar, since they are the back-projected values of the radar’s received voltage signal. The advantage of using the UWB SAR is that it offers the ability to detect objects partially or completely obscured by foliage. The long wavelength of the signal enables partial penetration of the leaves. Although the penetrating energy is able to excite objects hidden beneath the foliage, the return signal from the target is further attenuated by the leaves and branches of the trees. In addition, the high-frequency part of the incident signal produces significant reflections from the leaves, thereby increasing noise and speckle. This scattering also corrupts the response of occluded targets. Furthermore, the return from tree trunks, exhibiting multipath effects, is often as strong as the response of a target. The resulting increase in speckle and decrease in the signal-to-clutter ratio in UWB SAR imagery impairs the performance of conventional detection algorithms. ATD performance from the early-time or optical response is a challenging 1 Following Finn’s definition, homogeneous background clutter implies that the pixels within the CFAR window are independent and identically distributed (i.i.d.) [5]. This is due to the assumption that the CFAR window lies completely in one type of terrain. 2 The radar’s return signal is not modulated down to baseband, and hence the signal is not separated into the in-phase and quadrature components. Therefore, the ARL’s FOPEN SAR produces real-valued images.

Fig. 1. Block diagram of region-adaptive target detection algorithm. The FOPEN SAR image is first segmented. The CFAR detector exploits the segmented image to produce regions-of-interest (ROI’s) with fewer false alarms.

problem. Clearly, ATD algorithms need to be tailored to the region (i.e. foliage or nonfoliage) in which targets may be present. III. SYMMETRIC ALPHA-STABLE MODEL A. Motivation In signal detection and estimation problems, accurate statistical models are required for the signal and noise processes. In many practical applications, the behavior of random processes is assumed to be governed by Gaussian distributions, for the Gaussian assumption leads to linear and tractable solutions [1]. Since radar backscatter is the sum of many small, independent reflections from an irregular surface, it is tempting to invoke the central limit theorem to justify a Gaussian clutter model. However, this theory does not explain the impulsive, non-Gaussian nature of clutter observed in SAR data. Instead, a more powerful theorem, the Generalized central limit theorem, justifies the use of stable distributions for noisy data. The generalized CLT states that the sum of i.i.d. random variables with finite or infinite variance converges to a stable distribution. Thus, if an observed signal, such as radar clutter, can be viewed as the sum of many independent effects, the generalized CLT suggests that a stable model is appropriate. Moreover, it has been shown [18] that among Middleton’s statistical-physical models for impulsive interference (classes A, B, or C), the first order distribution of impulsive interference (class B) does follow an SS law. Hence, the SS model may be viewed as a limiting case of the class B model. A key feature of SS densities is that their tails decay algebraically, as opposed to Gaussian or generalized Gaussian whose tails decay exponentially. Thus, SS densities are heavy-tailed and can model highly impulsive phenomena. Two other popular heavy-tailed models include the K and Weibull distributions. However, the K-distribution is only valid for modeling the amplitude statistics. Therefore, it cannot model signals which contain positive and negative values such as the bipolar data of FOPEN SAR. Furthermore, since it is not symmetric, the K-distribution is not an attractive model when it is necessary to preserve the phase information [15]. A commonly used measure for impulsive noise is the amplitude probability distribution (APD), which is defined as the probability that the noise amplitude is above some threshold. Fig. 2(a) shows the plot of the theoretical APD for the SS, K, and Gaussian densities and the empirical APD of FOPEN clutter data. The plot demonstrates that for small amplitudes, all three models provide a good fit to the data. At larger amplitudes (i.e., the tails), the SS model proves to be a slightly better fit than the K-distribution while the Gaussian is a poor model at high amplitudes. Hence, there exists theoretical and empirical support for the use of alpha-stable distributions for modeling FOPEN SAR clutter. B. Background Let us now take a closer look at the alpha-stable model. The general form of the alpha-stable distribution is given by its characteristic

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for the SS density function. Hence, after estimating the parameters of the characteristic function, numerical methods for computing the inverse Fourier transform of 8(t) are required to approximate the density function. For detailed discussions of these methods, we refer the interested reader to [15]. As explained above, unless = 2 (the Gaussian pdf), SS random processes have infinite energy. For this reason, parameter estimation of the SS distributions is based on the theory of fractional order moments. Therefore, typical techniques such as least-squares estimation, Wiener filtering, and Kalman filtering are inappropriate for alpha-stable processes. Moreover, the general forms of spectral analysis are invalid, since alpha-stable processes are not squareintegrable. Although the covariance for such signals do not exist, an analogous measure known as covariation is used for alpha-stable estimation problems [15]. (a)

IV. ALPHA-STABLE IMAGE SEGMENTATION

(b) Fig. 2.

(a) APD plot. (b) Symmetric alpha-stable pdf’s.

function

8(t) = exp(jat 0 jtj [1 + j sign(t)!(t; )]):

(1)

= 0 implies a symmetric density and thus the SS characteristic function simplifies to

8(t) = exp(jat 0 jtj ):

(2)

The characteristic function is completely described by the following three parameters. • Characteristic exponent . It is a measure of the heaviness of the tails. The value of is restricted to the range of zero to two, and is inversely proportional to the thickness of the tails. • Dispersion parameter . It is analogous to the variance of the Gaussian distribution. It is a measure of the “spread” of the distribution. • Location parameter a. For 0 < 1, it is the median of the SS density, and for 1 < 2, it represents both the median and the mean, since SS random variables have finite moments of order less than or equal to . The SS density is symmetric about a. It is easily shown that = 1 corresponds to a Cauchy distribution and = 2 corresponds to a Gaussian pdf. In general, however, there is no closed-form expression

The pixel classification algorithm is derived by viewing image segmentation as a Bayesian estimation problem. Using the image as our observation, we estimate the label of each pixel. Two natural estimation techniques are the maximum likelihood (ML) and maximum a posteriori (MAP) methods. A stochastic model is used to characterize the conditional distribution of the image, i.e. the form of p(Xs =Cs = l) is assumed. Here, Xs is the given image data and Cs is the label for pixel s [7]. For our segmentation purposes, we estimate the pixel labels by assuming that the conditional probability of each pixel p(Xs =Cs = l) is governed by an SS distribution. Ground-truthed UWB SAR images serve as training data for foliage and clear regions. The training patches, of size M 2 N pixels, are extracted from which the location parameter a, characteristic exponent , and dispersion are estimated for each class. The lack of an explicit form for a SS pdf requires numerical methods for parameter estimation. Among the suggested algorithms are DuMouchel’s ML technique [3], Fama and Roll’s method of sample fractiles [4], and Koutrouvelis’s regression method based on sample characteristic functions [11]. Ma and Nikias proposed an alternative method using negative order moments [13]. For fast model parameter estimation, we use an algorithm devised by Tsihrintzis and Nikias [19]. After obtaining the parameters of the different classes, numerical methods are required to compute the inverse Fourier transform of the characteristic function to obtain the probability density function. Since, in general, closed-form expressions for the SS density do not exist, a look-up table is utilized to store the pdf for each class. The ML decision rule is

^l = arg max p(Xs =Cs = l) l

(3)

where ^ l takes values 0; 1 (foliage, clear) for a two-class problem. The label that maximizes the conditional probability of the pixel is the ML estimate. The conditional probability of each pixel value for both classes, determined from the look-up table, is used for the ML test. As the ML method does not incorporate the prior probabilities of the class labels, it produces a noisy segmentation. By assuming that each pixel label is equally likely throughout the image, it fails to capture the homogeneity of regions. In real images, this assumption is incorrect, for in a local region dominated by one class, the dominant class has a higher prior probability than the other classes. Such contextual information is taken into account in the maximum a posteriori (MAP) estimate of the pixel labels. The MAP estimate of the class label ^ l for a pixel given the observed image Xs is

^l = arg max p(X=C = L)p(C = L): l

(4)

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Here, C denotes the entire set of pixel labels and X represents the array of pixel values. By adopting an MRF model for the prior density function and exploiting the MRF-Gibbs distribution equivalence, the MAP estimate becomes

Fortunately, we are able to resolve these issues by exploiting the segmented image and the standard SS distribution. Suppose x is an SS random variable characterized by ; , and a nonzero a. By definition,

^l = arg max p(X=C = L) exp(0U (L))

f (x) $ 8(!) = exp(j!a 0 j!j )

(7)

f x 10=a = f (y) $ exp(0j!0 j ):

(8)

l

(5)

where p(X=C = L) is a SS density function and the energy function U (L) is defined in [17]. The function in (5) can be optimized by either simulated annealing or iterative conditional mode techniques [2]. V. ALPHA-STABLE CFAR DETECTOR Constant false alarm rate processors attempt to find clusters of bright pixels in SAR images which possibly indicate the bright back-scatter from targets. Due to the nonstationarity of the image, they adaptively set a threshold for detection based on the local characteristics of the background clutter. For example, a cell-average CFAR computes a weighted average of the samples within a window of N pixels to estimate the statistical properties of the background noise. An order-statistics (OS) CFAR arranges these N values and selects one or more of them to estimate the noise power. The cellaverage CFAR is handicapped by the assumption that the pixel under test is in a locally homogeneous clutter region [6]. Consequently, the performance of conventional CFAR detectors suffers when the guard-band of the CFAR is intersected by regions of clutter transitions, since they operate under the restrictive assumption that the pixels in the CFAR window are governed by a single statistical model and are i.i.d. [5]. Clearly, knowledge of the multiple clutter environment of the CFAR window is critical for improved performance. The CFAR detectors described below exploit the segmented image to determine a threshold that is a function of the clutter statistics and clutter type (i.e., foliage or clear). A. SS CFAR Hypothesis Test Another important property of a CFAR detector is the statistical model it employs to characterize the background clutter. As described above, simplified Gaussian assumptions do not constitute a robust model for impulsive signals. Recently, heavy-tailed distributions, such as K and Weibull, have been used to model such spiky clutter [12]. Since we have employed alpha-stable distributions to describe the UWB backscatter, we now develop the corresponding SS CFAR detector. It is important to note that CFAR processing with SS models is made difficult for two reasons. First, there is no closedform expression for an SS probability density function. Without an explicit form, it is difficult to form the CFAR test to relate the SS parameters to the given false alarm probability. The CFAR criterion is given as

1

T

f (x) dx = Pfa =2

(6)

where f (1) is the symmetric density function of a zero-mean random variable, T is the threshold, and Pfa is the desired false alarm probability. While it is possible to define the CFAR criterion using the characteristic function in the Fourier domain by Parseval’s relationship, it is not possible to integrate the resulting function except for the cases when = 1 and = 2. Second, it is difficult to capture the SS statistics of a local region with a reasonablysized window, since relatively large number of samples are necessary to accurately estimate the SS parameters. Tens of thousands of samples are required to appropriately characterize the heavy tails of the distribution [9].

where f (x) is the pdf of x and $ denotes a Fourier transform pair. Using the shifting and scaling properties of the Fourier transform,

The random variable Y has a characteristic function with a = 0 and is a zero-mean, unit-dispersion SS random variable with characteristic exponent . Hence, an SS random variable is standardized by

= 1, i.e. Y

Y = X 0 ai :

i

(9)

Examining the CFAR criterion,

1

1T 1

f (x) dx = Pfa =2

(10)

f (y) dy = P1fa= 2

(11)

f (y) dy = Pfa =2:

(12)

Since Y , or equivalently X 0a , is a test statistic for the hypothesis test, we can formulate the CFAR hypothesis test as

X 0 ai

i

i ; i = 0; 1:

(13)

The threshold i is determined offline by numerically integrating the standard density functions with parameter i . Note that this hypothesis test is a CFAR test, for the threshold is independent of the parameters estimated from the samples in the reference window. Furthermore, it can be seen that the SS CFAR test (13) is a generalization of the Gaussian CFAR test; when = 2, (13) simplifies to the Gaussian CFAR. In its sliding window, the SS CFAR counts the number of pixels of each class label and determines lmax as the label which has the majority (or plurality) of the class labels in the window. Instead of estimating from the reference window, the CFAR uses the value of the corresponding to lmax , thus avoiding the computational burden of estimating . The samples in the sliding window corresponding to lmax , used to estimate the location parameter a and dispersion

, provide the local description of the alpha-stable clutter statistics. The CFAR is region-adaptive since it exploits the segmented image by changing its value of (and therefore the distribution) according to the region in which pixel X lies, and thereby providing different thresholds for the foliage and clear regions. B. SS Mixture CFAR As mentioned above, a major drawback of conventional CFAR’s is the assumption of homogeneous background clutter. The CFAR implementation described above effectively masks out the pixels of the minority class in the determination of the clutter statistics. It is similar in flavor to order statistic CFAR’s and its derivatives, such as the GO CFAR and LO CFAR [6]. With the information from the

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Fig. 3.

Original UWB SAR images. Note that the trees are as bright as the targets.

segmented image, a more intelligent CFAR test is designed to incorporate all of the pixels in the CFAR window. From the segmented image, we know that the clutter type varies across the image and within the CFAR window. Using the law of total probability, we can rewrite the CFAR criterion in (6) as 1

T

i

f (x j Class i) Pr(Class i) dx = Pfa =2

(14)

where f (x j Class i) is the pdf for clutter class i and Pr(Class i) is the probability of clutter class i. We call (14) the mixture CFAR test. When the CFAR windows overlaps a region of clutter transition, the local statistics within the CFAR window can be considered as the output of multiple independent sources (i.e., the trees and the grass). Hence, a natural approach to modeling such sources is to use a mixture model. In our two-class problem, we consider the two distributions f (x j foliage) and f (x j clear). The probability of each class, Pr(foliage) and Pr(clear), is determined by computing the ratio of pixels of each type in the CFAR reference window. Let us define random variables Y and Z such that f (Y ) = f (X j foliage) and f (Z ) = f (X j clear). As before, we can standardize the SS variables Y and Z by

f (Ys ) = f Y 10=af

f f (Zs ) = f Z 10=ac

c

$ exp(0j! j 0

)

$ exp(0j! j ) 0

(15)

(16)

where i ; i , and ai are the estimated SS parameters estimated for each clutter type i, using only the pixels which have been labeled as class i. The SS mixture CFAR criterion then transforms to 1

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f (Ys ) + (1 0 )f (Zs ) dx = Pfa =2

(17)

where denotes Pr(foliage) and is determined off-line by numerically integrating a linear combination of the two standard SS density functions. As before, the Ys + (1 0 )Zs is a test statistic for the hypothesis test. Thus the mixture SS CFAR hypothesis test reduces to

X 0 af

f

+ (1

0 ) X 0 ac

c

:

(18)

The idea of the mixture CFAR is similar to Finn’s heterogeneous clutter CFAR [5], except here it is not necessary to perform hypothesis testing to determine the region of clutter transition; this information is provided by the segmented image. The mixture CFAR can be easily extended to improve detection performance in complex environments. For example, if a radar image contains m types of clutter (i.e., trees, grass, roads, bodies of water), each of the clutter types can modeled by various distributions. The mixture CFAR can accommodate the different clutter models and determine an optimal detection threshold along the boundaries of the clutter regions, as opposed to conventional CFAR’s which assume one statistical clutter model throughout the image.

VI. EXPERIMENTAL RESULTS The images used in this study were obtained by the UWB SAR developed at the Army Research Laboratory (ARL), Adelphi, MD. The images show the Aberdeen Proving Grounds, Aberdeen, MD, during the summer of 1996. The images are single-channel (HH) with downrange resolution of .0375 m and azimuthal resolution of 0.1125 m [14]. Fig. 2(b) shows the plots of the SS probability density functions, conditioned on the clutter type, using the parameters estimated from the foliage and clear training data. The estimated value of in the clear region is 1.98 and is 4.95 2 109 . Since 2, the SS density for the clear region has a smooth, Gaussian-like shape. For

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Fig. 4.

Comparison of MAP segmentation. The black pixels represent the foliage region, and the white pixels indicate the clear region.

the foliage region, is 1.55 2 109 and is estimated to be 1.57, resulting in a pdf with heavy tails. These estimates are justified, for we expect impulsive noise from the prominent scattering off of the leaves and branches in the foliage regions. In the clear regions the clutter response is due to the back-scatter from the grass and the ground. Since the blades of the grass are smaller than the resolution cell of the image, many such blades of grass in varying orientations contribute to the total response in each cell. By the central limit theorem, we expect the response to exhibit Gaussian characteristics [8]. Fig. 3 shows the UWB SAR image of the two scenes used in this study. They represent the log-magnitude of the bipolar data for viewing purposes. For both images, the radar signal had to penetrate through a line of trees to remotely sense the scenes. In other words,

all targets in the images are occluded by foliage. Note the numerous bright flashes in the foliage regions. The response of the trees are as strong as the target response, thus requiring an adaptive detection scheme with accurate clutter modeling. Illustrated in Fig. 4 is the segmentation obtained by the SS distribution. The Gaussian model cannot robustly estimate the pixel labels in the presence of impulsive noise. Clear pixels are misclassified as foliage pixels due to the high degree of speckle in the images. The SS assumption is able to handle to the impulsiveness of the clutter to obtain a more accurate segmentation. Performance of the Gaussian, Weibull, and the SS CFAR’s shown in Figs. 5 and 6. The CFAR’s are compared at the same false alarm probability of 1003 . For the target sizes and pixel resolution of these images, we use a rectangular CFAR window of height 300 pixels

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(a)

(b)

(c)

(d)

Fig. 5. Gaussian and Weibull CFAR detection results overlaid on the segmented image. The red pixels highlight the CFAR hits. (a) and (b) Gaussian CFAR. (c) and (d) Weibull CFAR.

and width 100 pixels with a guard-band of size 298 pixels by 98 pixels. Fig. 6(c) shows the improvement in detection performance when using the SS mixture CFAR. Note the reduction of false alarms along the boundary of the foliage and clear regions. Fig. 6(d) is a plot of the experimental (as opposed to theoretical) ROC curve comparing the detection capabilities of the four CFAR’s. The curve is based on tests of the CFAR detectors over several square kilometers (eight images of size 5376 2 2048) of the Aberdeen Proving Grounds. The pixels declared as CFAR “hits” are clustered. The amount by which the CFAR statistic exceeds the threshold at every pixel, i.e. (for the

SS CFAR),

in cluster

C

s(i) = Xi 0 a

0T

(19)

are summed to form the feature value for the cluster

F (C ) =

i2C

s(i):

(20)

The feature value F (C ) will emphasize the brightest and largest clusters in order to reject the false alarms, since we assume that the targets will form larger clusters than the false alarms due to

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(a)

(b)

(c)

(d)

Fig. 6. SS and mixture SS CFAR detection results. The green boxes are correct detections and the black boxes are false alarms. The ROC curve compares the performance of the four CFAR detectors. (a) and (b) SS CFAR. (c) Mixture CFAR. (d) ROC curve.

the trees. The results indicate that for 100% detection, the alphastable model, when compared to the Gaussian and Weibull models, improves detection performance by a factor of ten. The mixture SS CFAR offers further reduction in false alarms when compared to the SS CFAR detector. Many of the false alarms are due to man-made objects which are not considered targets, such as telephone poles as shown in Fig. 6.

information by which the CFAR detector is able to achieve improved performance over conventional CFAR detectors. The mixture CFAR is developed to reduce false-alarms along clutter boundaries. We have also shown how SS distributions can be profitably used to model the image statistics of UWB SAR imagery. The model provides better segmentation and detection results when compared to Gaussian models.

VII. CONCLUSION

ACKNOWLEDGMENT

In this work, we have shown that basic pattern recognition and signal processing techniques for image segmentation and target detection, namely MAP segmentation and CFAR detection, are useful when analyzing FOPEN SAR images. The segmentation provides additional

The authors thank the researchers of the Army Research Laboratory, Adelphi, MD, for providing UWB SAR imagery, in particular, R. Kapoor of ARL for the APD plot; and the reviewers, whose input has helped to improve this work.

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A Moment-Based Approach for Deskewing Rotationally Symmetric Shapes

REFERENCES [1] A. Banerjee, P. Burlina, and F. Alajaji, “Image segmentation and labeling using the Polya urn model,” IEEE Trans. Image Processing, vol. 8, pp. 1243–1253, Sept. 1999. [2] A. Banerjee, P. Burlina, and R. Chellappa, “Adaptive target detection in foliage-penetrating SAR images using alpha-stable models,” Tech. Rep. CAR-TR-870, Ctr. Automat. Res., Univ. Maryland, College Park, Sept. 1997. [3] W. DuMouchel, “Stable distributions in statistical inference,” Ph.D. dissertation, Dept. Statist., Yale Univ., New Haven, CT, 1971. [4] E. Fama and R. Roll, “Parameter estimates for symmetric stable distributions,” J. Amer. Stat. Assoc., vol. 66, pp. 331–338, 1971. [5] H. M. Finn, “A CFAR design for a window spanning two clutter fields,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-22, 1986. [6] P. Gandhi and S. Kassam, “Analysis of CFAR processors in nonhomogeneous background,” IEEE Trans. Aerosp. Electron. Syst., vol. 24, 1988. [7] S. Geman and D. Geman, “Stochastic relaxation, Gibbs distribution, and Bayesian restoration of images,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp. 721–741, 1984. [8] J. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Amer., vol. 66, pp. 1145–1150, Nov. 1976. [9] R. Kapoor, G. Tsihrintzis, and N. Nandhakumar, “Detection of obscured targets in heavy-tailed radar clutter using an ultra-wideband (UWB) radar and alpha-stable clutter models,” in Proc. 30th Ann. Asilomar Conf. Signal and Systems, 1996. [10] P. A. Kelly, H. Derin, and K. D. Hartt, “Adaptive segmentation of speckeled images using a hierarchical radom field model,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 1, pp. 1628–1641, 1988. [11] I. Koutrouvelis, “Regression-type estimation of the parameters of stable laws,” J. Amer. Stat. Assoc., vol. 75, pp. 918–928, 1980. [12] S. Kuttikad and R. Chellappa, “Non-Gaussian CFAR techniques for target detection in high-resolution SAR images,” in Proc. IEEE Int. Conf. Image Processing, 1995. [13] X. Ma and C. Nikias, “On blind channel identification for impulsive signal environments,” in Proc. ICASSP, May 1995. [14] J. McCorkle and L. Nguyen, “Focusing of dispersive targets using synthetic aperture radar,” Tech. Rep., Army Research Lab. Adelphi, MD, 1994. [15] C. Nikias and M. Shao, Signal Processing with Alpha-Stable Distributions and Applications. New York: Wiley, 1995. [16] M. Ressler, L. Happ, L. Nguyen, T. Ton, and M. Bennett, “The army research laboratory ultra-wideband testbed radars,” in Proc. IEEE Int, Radar Conf., 1995. [17] E. Rignot and R. Chellappa, “Segmentation of polarimetric synthetic aperture radar data,” IEEE Trans. Image Processing, vol. 1, pp. 281–300, 1992. [18] G. Tsihrintzis and C. Nikias, “Incoherent receivers in alpha-stable impulsive noise,” IEEE Trans. Signal Processing, vol. 43, pp. 2225–2229, 1995. , “Fast estimation of the parameters of alpha-stable impulsive [19] interference,” IEEE Trans. Signal Processing, vol. 44, pp. 1492–1503, 1996.

Soo-Chang Pei and Ji-Hwei Horng

Abstract—The covariance matrix of a pattern is composed by its second order central moments. For a rotationally symmetric shape, its covariance matrix is a scalar identity matrix. In this work, we apply this property to restore the skewed shape of rotational symmetry. The relations between the skew transformation matrix and the covariance matrices of original and skewed shapes are derived. By computing the covariance matrix of the skewed shape and letting the covariance matrix of the original shape be a scalar identity matrix, the skew transformation matrix can be solved. Then, the rotationally symmetric shape can be recovered by multiplying the inverse transformation matrix with the skewed shape. The method does not rely on continuous contours and is robust to noise, because only the second-order moments of the input shape are required. Experimental results are also presented. Index Terms—Covariance matrix, reflective symmetry, rotation matrix, rotational symmetry, shear matrix, skewed symmetry.

I. INTRODUCTION A rotationally symmetric shape is a shape that repeats itself after being rotated around its centroid through any multiple of a certain angle. To satisfy this property, the minimum repeating angle must be 2=n; where n is a positive integer. This shape is called an nfold rotationally symmetric shape. Many researchers on the area of image analysis have pay attention to the importance of rotational symmetry. Lots of methods have been proposed to normalize the rotationally symmetric planar shapes (RSS) [1]–[5]. However, none of them has dealt with the skewed shape of rotational symmetry. In real applications, the view direction is not always perpendicular to the plane containing RSS, it results in a skewed RSS on the image plane. Thus, a complete normalization system must includes the deskew procedure. Many methods have been proposed to recover the skewed shape of reflective symmetry [6]–[8]. But they are not directly applicable to the case of rotational symmetry. Although the set of reflectively symmetric shapes overlaps with the set of rotationally symmetric shapes, they are not identical. The relation of these two sets are shown in Fig. 1, where a typical example is given for each subset. An alternative procedure for generating a skewed shape is to perform a shear operation with parameter followed by a rotation in the image plane with parameter : Friedberg proposed a momentbased method to recover the skewed reflective symmetry [7]. He used the property that the covariance matrix for a reflectively symmetric shape is a diagonal matrix, that is, the moment m11 is necessarily equal to zero. For a set of assumed parameters and of the skew coordinate transformation matrix, the moment m11 of the recovered shape can be written as a function of the set of parameters and the second-order moments of the skewed shape. Let the moment m11 of the deskewed shape be equal to zero, can be described as a function of : The constraint reduces the search space from Manuscript received January 27, 1998; revised March 10, 1999. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Josiane B. Zerubia. S.-C. Pei is with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 106, R.O.C. (e-mail: [email protected]). J.-H. Horng is with the Department of Information Management, Jin-Wen Institute of Technology, Taipei, Taiwan, R.O.C. Publisher Item Identifier S 1057-7149(99)09356-2.

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[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

Codec, Sept. 1998, Signal Process. Multimedia Lab., Univ. British Columbia, Vancouver, B.C., Canada. F. Kossentini and Y. Lee, “Computation-constrained fast MPEG-2 encoding,” Signal Processing Lett., vol. 4, pp. 224–226, Aug. 1997. W. Chung, F. Kossentini, and M. Smith, “Rate-distortion constrained statistical motion estimation for video coding,” in ICIP’95, Washington, DC, vol. 3, pp. 184–187. R. Arminato, R. Schafer, F. Kitson, and V. Bhaskaran, “Linear predictive coding of motion vectors,” in Proc. IS&T/SPIE EI’96. Y. Lee, F. Kossentini, M. Smith, and R. Ward, “Predictive RDconstrained motion estimation for very low bit rate video coding,” IEEE J. Select. Areas Commun., vol. 15, pp. 1752–1763, Dec. 1997. “TMN (H.263) encoder/decoder, version 2.0,” Telenor Res., TMN (H.263) Codec, June 1996. MPEG Software Simulation Group, “MPEG-2 encoder/decoder, version 1.0,” ISO/IEC DIS 13818-2 Codec, May 1995. T. Sikora, “The MPEG-4 video standard verification model,” IEEE Trans. Circuits Syst. Video Technol., vol. 7, pp. 19–31, Feb. 1997. M. Gallant, G. Cote, and F. Kossentini, “Computation constrained fastsearch motion estimation algorithm for TMN 7,” in Q15-A-45, ITU-T Q15/SG16, June 1997. B. Furht, J. Greenberg, and R. Westwater, Motion Estimation Algorithms for Video Compression. Boston: Kluwer, 1997. A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Boston, MA: Kluwer, 1992. J. Ribas-Corbera and S. Lei, “Optimal quantizer control in DCT video coding for low-delay video communications,” in Proc. Picture Coding Symp., Berlin, Germany, Sept. 1997.

Adaptive Target Detection in Foliage-Penetrating SAR Images Using Alpha-Stable Models A. Banerjee, P. Burlina, and R. Chellappa

Abstract—Detecting targets occluded by foliage in foliage-penetrating (FOPEN) ultra-wideband synthetic aperture radar (UWB SAR) images is an important and challenging problem. Given the different nature of target returns in foliage and nonfoliage regions and very low signal-toclutter ratio in UWB imagery, conventional detection algorithms fail to yield robust target detection results. A new target detection algorithm is proposed that 1) incorporates symmetric alpha-stable (SS) distributions for accurate clutter modeling, 2) constructs a two-dimensional (2-D) site model for deriving local context, and 3) exploits the site model for region-adaptive target detection. Theoretical and empirical evidence is given to support the use of the SS model for image segmentation and constant false alarm rate (CFAR) detection. Results of our algorithm on real FOPEN images collected by the Army Research Laboratory are provided. Index Terms—Alpha-stable models, SAR ATR, SAR segmentation.

I. INTRODUCTION Interest in the analysis of foliage-penetrating (FOPEN), ultrawideband synthetic aperture radar (UWB SAR) images stems from Manuscript received August 1, 1997; revised March 17, 1999. This paper was published in part in the Proceedings of the 1998 International Conference on Image Processing, Chicago, IL. This work was supported by the Army Federated Laboratories/Lockheed Sanders under Contract DAAL01-96-20001. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Dmitry B. Goldgof. A. Banerjee and R. Chellappa are with the Center for Automation Research, Department of Electrical Engineering, University of Maryland, College Park, MD 20742-3275 USA (e-mail: [email protected]; [email protected]). P. Burlina is with ImageCorp., Inc., College Park, MD 20740 USA (e-mail: [email protected]). Publisher Item Identifier S 1057-7149(99)09361-6.

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the radar’s ability to allow for detection of objects hidden beneath foliage. However, due to the very low signal-to-clutter ratio, conventional constant false alarm rate (CFAR) processing of UWB SAR data results in an unacceptably high number of false alarms. Since return signals from trees in foliage regions can be as strong as the responses from targets, typical cell-averaging or order-statistics CFAR’s perform poorly in foliage regions. Furthermore, the response of a target occluded by foliage is markedly different than if it were located in a clear region. Given the different nature of target returns in different clutter types, we wish to incorporate adaptability into our detection scheme. Clearly, image segmentation, which provides local clutter description, is of paramount importance in an adaptive system. The main contribution of this work lies in the design of an adaptive detection scheme which relies on modeling of the clutter statistics by symmetric alpha-stable (SS) distributions for image segmentation and CFAR detection. We use the SS pdf’s to accurately describe the highly impulsive, bursty behavior of FOPEN clutter. Theoretical and empirical evidence is provided to advocate the use of the SS model. Although the lack of explicit forms for their density functions inhibits their use, simple techniques for using SS distributions are developed for image segmentation and target detection. In particular, we derive a closed-form expression for the SS CFAR test that adjusts the detection threshold according to the local clutter type. Furthermore, the mixture SS CFAR test is introduced to reduce the number of false alarms along clutter boundaries. For segmentation purposes, we model a scene as being composed of distinct, contiguous regions of foliage and nonfoliage (or clear), each of which is distinguished by different statistical characteristics, such as the mean and variance. A natural approach to delineating these regions in UWB imagery is to estimate the statistical properties of the regions and use the descriptions to discriminate between the regions. Many such approaches to segmentation have been studied [7], [10], [17]. These methods generally proceed by formulating statistical model assumptions for region generation and image formation processes. Maximum likelihood (ML) or maximum a posteriori (MAP) estimation is then used for segmentation. Numerous examples of such approaches exist in the literature. Techniques that model images as Markov random fields (MRF’s) have been extensively investigated [7]. MRF’s attempt to represent spatial dependencies, and the MRF–Gibbs distribution equivalence allows for the computation of the MAP estimate of the original image [7]. For FOPEN SAR, image segmentation is achieved by modeling the conditional distribution of the clutter with a SS density and adopting an MRF model for the prior pixel label probability. The MAP estimate of the pixel label is the one that maximizes the product of the conditional and prior probabilities. Once the image is segmented, the labeled map enables the use of region-adaptive detectors. The region-adaptive algorithm is necessary for robust and efficient target detection. A common signal processing technique for automatic target detection (ATD) is CFAR processing. The goal of a CFAR detector is to provide detection thresholds that are impervious to the background clutter and noise while maintaining a constant false alarm rate. CFAR detectors attempt to characterize the local background statistics within a sliding window cell and determine a threshold based on signal detection theory. Since we model FOPEN clutter with SS distributions, we have designed the SS CFAR detector. Despite the lack of an explicit form of the density function, a simple, closed-form expression for the SS

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CFAR is derived. The alpha-stable CFAR improves upon the detection performance by employing a more accurate statistical model for the clutter. In addition, the SS CFAR exploits the segmented image to help estimate the local clutter statistics, and thereby determine a detection threshold that is a function of the clutter statistics and clutter type. Typically, CFAR processing is based on the assumption of homogeneous1 background clutter. Hence, when the CFAR window is intersected by regions of multiple clutter types, conventional CFAR’s do not provide an accurate threshold. Order statistics CFAR’s and its variations, such as the Greatest-of CFAR (GOCFAR) and the least-of CFAR (LOCFAR) have been designed to deal with multiple clutter types [6]. However, by exploiting the segmented image, we have derived an intelligent CFAR test, motivated by Finn’s heterogeneous clutter CFAR design [5], to account for nonhomogeneous background clutter. The SS mixture CFAR detector employs a probabilistic mixture model to describe the clutter statistics along boundaries of different clutter types. This model helps to reduce the false alarms resulting from incorrect statistical modeling due to the assumption of homogeneous background of conventional CFAR’s. An ROC curve comparing the performance of the SS, mixture SS, Weibull, and Gaussian CFAR detectors is provided. This work is organized as follows: a brief overview of UWB SAR phenomenology and the problems they present for an ATR algorithm is given in Section II. The usefulness of the SS model is explained in Section III. An algorithm for segmenting FOPEN SAR images using the SS model is described in Section IV (see Fig. 1 for a diagram of the algorithm). The region-adaptive SS CFAR detector is explained in Section V. Finally, experimental results are presented in Section VI. Section VII concludes this paper. II. UWB SAR PHENOMENOLOGY Ultra-wideband SAR’s differ from conventional SAR’s in the frequency range at which they operate. While high-frequency SAR’s typically operate with a narrowband signal at a carrier frequency between 10 and 33 GHz, UWB SAR’s emit a wideband signal covering frequencies from 25 MHz to 1.5 GHz [16], corresponding to a 1-ns transmit pulse. Indeed, due to the wide bandwidth, the UWB SAR is considered as a high-resolution, impulse radar. In addition, the UWB SAR is a real-valued radar.2 The pixel values of the FOPEN images are bipolar, since they are the back-projected values of the radar’s received voltage signal. The advantage of using the UWB SAR is that it offers the ability to detect objects partially or completely obscured by foliage. The long wavelength of the signal enables partial penetration of the leaves. Although the penetrating energy is able to excite objects hidden beneath the foliage, the return signal from the target is further attenuated by the leaves and branches of the trees. In addition, the high-frequency part of the incident signal produces significant reflections from the leaves, thereby increasing noise and speckle. This scattering also corrupts the response of occluded targets. Furthermore, the return from tree trunks, exhibiting multipath effects, is often as strong as the response of a target. The resulting increase in speckle and decrease in the signal-to-clutter ratio in UWB SAR imagery impairs the performance of conventional detection algorithms. ATD performance from the early-time or optical response is a challenging 1 Following Finn’s definition, homogeneous background clutter implies that the pixels within the CFAR window are independent and identically distributed (i.i.d.) [5]. This is due to the assumption that the CFAR window lies completely in one type of terrain. 2 The radar’s return signal is not modulated down to baseband, and hence the signal is not separated into the in-phase and quadrature components. Therefore, the ARL’s FOPEN SAR produces real-valued images.

Fig. 1. Block diagram of region-adaptive target detection algorithm. The FOPEN SAR image is first segmented. The CFAR detector exploits the segmented image to produce regions-of-interest (ROI’s) with fewer false alarms.

problem. Clearly, ATD algorithms need to be tailored to the region (i.e. foliage or nonfoliage) in which targets may be present. III. SYMMETRIC ALPHA-STABLE MODEL A. Motivation In signal detection and estimation problems, accurate statistical models are required for the signal and noise processes. In many practical applications, the behavior of random processes is assumed to be governed by Gaussian distributions, for the Gaussian assumption leads to linear and tractable solutions [1]. Since radar backscatter is the sum of many small, independent reflections from an irregular surface, it is tempting to invoke the central limit theorem to justify a Gaussian clutter model. However, this theory does not explain the impulsive, non-Gaussian nature of clutter observed in SAR data. Instead, a more powerful theorem, the Generalized central limit theorem, justifies the use of stable distributions for noisy data. The generalized CLT states that the sum of i.i.d. random variables with finite or infinite variance converges to a stable distribution. Thus, if an observed signal, such as radar clutter, can be viewed as the sum of many independent effects, the generalized CLT suggests that a stable model is appropriate. Moreover, it has been shown [18] that among Middleton’s statistical-physical models for impulsive interference (classes A, B, or C), the first order distribution of impulsive interference (class B) does follow an SS law. Hence, the SS model may be viewed as a limiting case of the class B model. A key feature of SS densities is that their tails decay algebraically, as opposed to Gaussian or generalized Gaussian whose tails decay exponentially. Thus, SS densities are heavy-tailed and can model highly impulsive phenomena. Two other popular heavy-tailed models include the K and Weibull distributions. However, the K-distribution is only valid for modeling the amplitude statistics. Therefore, it cannot model signals which contain positive and negative values such as the bipolar data of FOPEN SAR. Furthermore, since it is not symmetric, the K-distribution is not an attractive model when it is necessary to preserve the phase information [15]. A commonly used measure for impulsive noise is the amplitude probability distribution (APD), which is defined as the probability that the noise amplitude is above some threshold. Fig. 2(a) shows the plot of the theoretical APD for the SS, K, and Gaussian densities and the empirical APD of FOPEN clutter data. The plot demonstrates that for small amplitudes, all three models provide a good fit to the data. At larger amplitudes (i.e., the tails), the SS model proves to be a slightly better fit than the K-distribution while the Gaussian is a poor model at high amplitudes. Hence, there exists theoretical and empirical support for the use of alpha-stable distributions for modeling FOPEN SAR clutter. B. Background Let us now take a closer look at the alpha-stable model. The general form of the alpha-stable distribution is given by its characteristic

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for the SS density function. Hence, after estimating the parameters of the characteristic function, numerical methods for computing the inverse Fourier transform of 8(t) are required to approximate the density function. For detailed discussions of these methods, we refer the interested reader to [15]. As explained above, unless = 2 (the Gaussian pdf), SS random processes have infinite energy. For this reason, parameter estimation of the SS distributions is based on the theory of fractional order moments. Therefore, typical techniques such as least-squares estimation, Wiener filtering, and Kalman filtering are inappropriate for alpha-stable processes. Moreover, the general forms of spectral analysis are invalid, since alpha-stable processes are not squareintegrable. Although the covariance for such signals do not exist, an analogous measure known as covariation is used for alpha-stable estimation problems [15]. (a)

IV. ALPHA-STABLE IMAGE SEGMENTATION

(b) Fig. 2.

(a) APD plot. (b) Symmetric alpha-stable pdf’s.

function

8(t) = exp(jat 0 jtj [1 + j sign(t)!(t; )]):

(1)

= 0 implies a symmetric density and thus the SS characteristic function simplifies to

8(t) = exp(jat 0 jtj ):

(2)

The characteristic function is completely described by the following three parameters. • Characteristic exponent . It is a measure of the heaviness of the tails. The value of is restricted to the range of zero to two, and is inversely proportional to the thickness of the tails. • Dispersion parameter . It is analogous to the variance of the Gaussian distribution. It is a measure of the “spread” of the distribution. • Location parameter a. For 0 < 1, it is the median of the SS density, and for 1 < 2, it represents both the median and the mean, since SS random variables have finite moments of order less than or equal to . The SS density is symmetric about a. It is easily shown that = 1 corresponds to a Cauchy distribution and = 2 corresponds to a Gaussian pdf. In general, however, there is no closed-form expression

The pixel classification algorithm is derived by viewing image segmentation as a Bayesian estimation problem. Using the image as our observation, we estimate the label of each pixel. Two natural estimation techniques are the maximum likelihood (ML) and maximum a posteriori (MAP) methods. A stochastic model is used to characterize the conditional distribution of the image, i.e. the form of p(Xs =Cs = l) is assumed. Here, Xs is the given image data and Cs is the label for pixel s [7]. For our segmentation purposes, we estimate the pixel labels by assuming that the conditional probability of each pixel p(Xs =Cs = l) is governed by an SS distribution. Ground-truthed UWB SAR images serve as training data for foliage and clear regions. The training patches, of size M 2 N pixels, are extracted from which the location parameter a, characteristic exponent , and dispersion are estimated for each class. The lack of an explicit form for a SS pdf requires numerical methods for parameter estimation. Among the suggested algorithms are DuMouchel’s ML technique [3], Fama and Roll’s method of sample fractiles [4], and Koutrouvelis’s regression method based on sample characteristic functions [11]. Ma and Nikias proposed an alternative method using negative order moments [13]. For fast model parameter estimation, we use an algorithm devised by Tsihrintzis and Nikias [19]. After obtaining the parameters of the different classes, numerical methods are required to compute the inverse Fourier transform of the characteristic function to obtain the probability density function. Since, in general, closed-form expressions for the SS density do not exist, a look-up table is utilized to store the pdf for each class. The ML decision rule is

^l = arg max p(Xs =Cs = l) l

(3)

where ^ l takes values 0; 1 (foliage, clear) for a two-class problem. The label that maximizes the conditional probability of the pixel is the ML estimate. The conditional probability of each pixel value for both classes, determined from the look-up table, is used for the ML test. As the ML method does not incorporate the prior probabilities of the class labels, it produces a noisy segmentation. By assuming that each pixel label is equally likely throughout the image, it fails to capture the homogeneity of regions. In real images, this assumption is incorrect, for in a local region dominated by one class, the dominant class has a higher prior probability than the other classes. Such contextual information is taken into account in the maximum a posteriori (MAP) estimate of the pixel labels. The MAP estimate of the class label ^ l for a pixel given the observed image Xs is

^l = arg max p(X=C = L)p(C = L): l

(4)

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Here, C denotes the entire set of pixel labels and X represents the array of pixel values. By adopting an MRF model for the prior density function and exploiting the MRF-Gibbs distribution equivalence, the MAP estimate becomes

Fortunately, we are able to resolve these issues by exploiting the segmented image and the standard SS distribution. Suppose x is an SS random variable characterized by ; , and a nonzero a. By definition,

^l = arg max p(X=C = L) exp(0U (L))

f (x) $ 8(!) = exp(j!a 0 j!j )

(7)

f x 10=a = f (y) $ exp(0j!0 j ):

(8)

l

(5)

where p(X=C = L) is a SS density function and the energy function U (L) is defined in [17]. The function in (5) can be optimized by either simulated annealing or iterative conditional mode techniques [2]. V. ALPHA-STABLE CFAR DETECTOR Constant false alarm rate processors attempt to find clusters of bright pixels in SAR images which possibly indicate the bright back-scatter from targets. Due to the nonstationarity of the image, they adaptively set a threshold for detection based on the local characteristics of the background clutter. For example, a cell-average CFAR computes a weighted average of the samples within a window of N pixels to estimate the statistical properties of the background noise. An order-statistics (OS) CFAR arranges these N values and selects one or more of them to estimate the noise power. The cellaverage CFAR is handicapped by the assumption that the pixel under test is in a locally homogeneous clutter region [6]. Consequently, the performance of conventional CFAR detectors suffers when the guard-band of the CFAR is intersected by regions of clutter transitions, since they operate under the restrictive assumption that the pixels in the CFAR window are governed by a single statistical model and are i.i.d. [5]. Clearly, knowledge of the multiple clutter environment of the CFAR window is critical for improved performance. The CFAR detectors described below exploit the segmented image to determine a threshold that is a function of the clutter statistics and clutter type (i.e., foliage or clear). A. SS CFAR Hypothesis Test Another important property of a CFAR detector is the statistical model it employs to characterize the background clutter. As described above, simplified Gaussian assumptions do not constitute a robust model for impulsive signals. Recently, heavy-tailed distributions, such as K and Weibull, have been used to model such spiky clutter [12]. Since we have employed alpha-stable distributions to describe the UWB backscatter, we now develop the corresponding SS CFAR detector. It is important to note that CFAR processing with SS models is made difficult for two reasons. First, there is no closedform expression for an SS probability density function. Without an explicit form, it is difficult to form the CFAR test to relate the SS parameters to the given false alarm probability. The CFAR criterion is given as

1

T

f (x) dx = Pfa =2

(6)

where f (1) is the symmetric density function of a zero-mean random variable, T is the threshold, and Pfa is the desired false alarm probability. While it is possible to define the CFAR criterion using the characteristic function in the Fourier domain by Parseval’s relationship, it is not possible to integrate the resulting function except for the cases when = 1 and = 2. Second, it is difficult to capture the SS statistics of a local region with a reasonablysized window, since relatively large number of samples are necessary to accurately estimate the SS parameters. Tens of thousands of samples are required to appropriately characterize the heavy tails of the distribution [9].

where f (x) is the pdf of x and $ denotes a Fourier transform pair. Using the shifting and scaling properties of the Fourier transform,

The random variable Y has a characteristic function with a = 0 and is a zero-mean, unit-dispersion SS random variable with characteristic exponent . Hence, an SS random variable is standardized by

= 1, i.e. Y

Y = X 0 ai :

i

(9)

Examining the CFAR criterion,

1

1T 1

f (x) dx = Pfa =2

(10)

f (y) dy = P1fa= 2

(11)

f (y) dy = Pfa =2:

(12)

Since Y , or equivalently X 0a , is a test statistic for the hypothesis test, we can formulate the CFAR hypothesis test as

X 0 ai

i

i ; i = 0; 1:

(13)

The threshold i is determined offline by numerically integrating the standard density functions with parameter i . Note that this hypothesis test is a CFAR test, for the threshold is independent of the parameters estimated from the samples in the reference window. Furthermore, it can be seen that the SS CFAR test (13) is a generalization of the Gaussian CFAR test; when = 2, (13) simplifies to the Gaussian CFAR. In its sliding window, the SS CFAR counts the number of pixels of each class label and determines lmax as the label which has the majority (or plurality) of the class labels in the window. Instead of estimating from the reference window, the CFAR uses the value of the corresponding to lmax , thus avoiding the computational burden of estimating . The samples in the sliding window corresponding to lmax , used to estimate the location parameter a and dispersion

, provide the local description of the alpha-stable clutter statistics. The CFAR is region-adaptive since it exploits the segmented image by changing its value of (and therefore the distribution) according to the region in which pixel X lies, and thereby providing different thresholds for the foliage and clear regions. B. SS Mixture CFAR As mentioned above, a major drawback of conventional CFAR’s is the assumption of homogeneous background clutter. The CFAR implementation described above effectively masks out the pixels of the minority class in the determination of the clutter statistics. It is similar in flavor to order statistic CFAR’s and its derivatives, such as the GO CFAR and LO CFAR [6]. With the information from the

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Fig. 3.

Original UWB SAR images. Note that the trees are as bright as the targets.

segmented image, a more intelligent CFAR test is designed to incorporate all of the pixels in the CFAR window. From the segmented image, we know that the clutter type varies across the image and within the CFAR window. Using the law of total probability, we can rewrite the CFAR criterion in (6) as 1

T

i

f (x j Class i) Pr(Class i) dx = Pfa =2

(14)

where f (x j Class i) is the pdf for clutter class i and Pr(Class i) is the probability of clutter class i. We call (14) the mixture CFAR test. When the CFAR windows overlaps a region of clutter transition, the local statistics within the CFAR window can be considered as the output of multiple independent sources (i.e., the trees and the grass). Hence, a natural approach to modeling such sources is to use a mixture model. In our two-class problem, we consider the two distributions f (x j foliage) and f (x j clear). The probability of each class, Pr(foliage) and Pr(clear), is determined by computing the ratio of pixels of each type in the CFAR reference window. Let us define random variables Y and Z such that f (Y ) = f (X j foliage) and f (Z ) = f (X j clear). As before, we can standardize the SS variables Y and Z by

f (Ys ) = f Y 10=af

f f (Zs ) = f Z 10=ac

c

$ exp(0j! j 0

)

$ exp(0j! j ) 0

(15)

(16)

where i ; i , and ai are the estimated SS parameters estimated for each clutter type i, using only the pixels which have been labeled as class i. The SS mixture CFAR criterion then transforms to 1

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f (Ys ) + (1 0 )f (Zs ) dx = Pfa =2

(17)

where denotes Pr(foliage) and is determined off-line by numerically integrating a linear combination of the two standard SS density functions. As before, the Ys + (1 0 )Zs is a test statistic for the hypothesis test. Thus the mixture SS CFAR hypothesis test reduces to

X 0 af

f

+ (1

0 ) X 0 ac

c

:

(18)

The idea of the mixture CFAR is similar to Finn’s heterogeneous clutter CFAR [5], except here it is not necessary to perform hypothesis testing to determine the region of clutter transition; this information is provided by the segmented image. The mixture CFAR can be easily extended to improve detection performance in complex environments. For example, if a radar image contains m types of clutter (i.e., trees, grass, roads, bodies of water), each of the clutter types can modeled by various distributions. The mixture CFAR can accommodate the different clutter models and determine an optimal detection threshold along the boundaries of the clutter regions, as opposed to conventional CFAR’s which assume one statistical clutter model throughout the image.

VI. EXPERIMENTAL RESULTS The images used in this study were obtained by the UWB SAR developed at the Army Research Laboratory (ARL), Adelphi, MD. The images show the Aberdeen Proving Grounds, Aberdeen, MD, during the summer of 1996. The images are single-channel (HH) with downrange resolution of .0375 m and azimuthal resolution of 0.1125 m [14]. Fig. 2(b) shows the plots of the SS probability density functions, conditioned on the clutter type, using the parameters estimated from the foliage and clear training data. The estimated value of in the clear region is 1.98 and is 4.95 2 109 . Since 2, the SS density for the clear region has a smooth, Gaussian-like shape. For

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Fig. 4.

Comparison of MAP segmentation. The black pixels represent the foliage region, and the white pixels indicate the clear region.

the foliage region, is 1.55 2 109 and is estimated to be 1.57, resulting in a pdf with heavy tails. These estimates are justified, for we expect impulsive noise from the prominent scattering off of the leaves and branches in the foliage regions. In the clear regions the clutter response is due to the back-scatter from the grass and the ground. Since the blades of the grass are smaller than the resolution cell of the image, many such blades of grass in varying orientations contribute to the total response in each cell. By the central limit theorem, we expect the response to exhibit Gaussian characteristics [8]. Fig. 3 shows the UWB SAR image of the two scenes used in this study. They represent the log-magnitude of the bipolar data for viewing purposes. For both images, the radar signal had to penetrate through a line of trees to remotely sense the scenes. In other words,

all targets in the images are occluded by foliage. Note the numerous bright flashes in the foliage regions. The response of the trees are as strong as the target response, thus requiring an adaptive detection scheme with accurate clutter modeling. Illustrated in Fig. 4 is the segmentation obtained by the SS distribution. The Gaussian model cannot robustly estimate the pixel labels in the presence of impulsive noise. Clear pixels are misclassified as foliage pixels due to the high degree of speckle in the images. The SS assumption is able to handle to the impulsiveness of the clutter to obtain a more accurate segmentation. Performance of the Gaussian, Weibull, and the SS CFAR’s shown in Figs. 5 and 6. The CFAR’s are compared at the same false alarm probability of 1003 . For the target sizes and pixel resolution of these images, we use a rectangular CFAR window of height 300 pixels

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(a)

(b)

(c)

(d)

Fig. 5. Gaussian and Weibull CFAR detection results overlaid on the segmented image. The red pixels highlight the CFAR hits. (a) and (b) Gaussian CFAR. (c) and (d) Weibull CFAR.

and width 100 pixels with a guard-band of size 298 pixels by 98 pixels. Fig. 6(c) shows the improvement in detection performance when using the SS mixture CFAR. Note the reduction of false alarms along the boundary of the foliage and clear regions. Fig. 6(d) is a plot of the experimental (as opposed to theoretical) ROC curve comparing the detection capabilities of the four CFAR’s. The curve is based on tests of the CFAR detectors over several square kilometers (eight images of size 5376 2 2048) of the Aberdeen Proving Grounds. The pixels declared as CFAR “hits” are clustered. The amount by which the CFAR statistic exceeds the threshold at every pixel, i.e. (for the

SS CFAR),

in cluster

C

s(i) = Xi 0 a

0T

(19)

are summed to form the feature value for the cluster

F (C ) =

i2C

s(i):

(20)

The feature value F (C ) will emphasize the brightest and largest clusters in order to reject the false alarms, since we assume that the targets will form larger clusters than the false alarms due to

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(a)

(b)

(c)

(d)

Fig. 6. SS and mixture SS CFAR detection results. The green boxes are correct detections and the black boxes are false alarms. The ROC curve compares the performance of the four CFAR detectors. (a) and (b) SS CFAR. (c) Mixture CFAR. (d) ROC curve.

the trees. The results indicate that for 100% detection, the alphastable model, when compared to the Gaussian and Weibull models, improves detection performance by a factor of ten. The mixture SS CFAR offers further reduction in false alarms when compared to the SS CFAR detector. Many of the false alarms are due to man-made objects which are not considered targets, such as telephone poles as shown in Fig. 6.

information by which the CFAR detector is able to achieve improved performance over conventional CFAR detectors. The mixture CFAR is developed to reduce false-alarms along clutter boundaries. We have also shown how SS distributions can be profitably used to model the image statistics of UWB SAR imagery. The model provides better segmentation and detection results when compared to Gaussian models.

VII. CONCLUSION

ACKNOWLEDGMENT

In this work, we have shown that basic pattern recognition and signal processing techniques for image segmentation and target detection, namely MAP segmentation and CFAR detection, are useful when analyzing FOPEN SAR images. The segmentation provides additional

The authors thank the researchers of the Army Research Laboratory, Adelphi, MD, for providing UWB SAR imagery, in particular, R. Kapoor of ARL for the APD plot; and the reviewers, whose input has helped to improve this work.

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A Moment-Based Approach for Deskewing Rotationally Symmetric Shapes

REFERENCES [1] A. Banerjee, P. Burlina, and F. Alajaji, “Image segmentation and labeling using the Polya urn model,” IEEE Trans. Image Processing, vol. 8, pp. 1243–1253, Sept. 1999. [2] A. Banerjee, P. Burlina, and R. Chellappa, “Adaptive target detection in foliage-penetrating SAR images using alpha-stable models,” Tech. Rep. CAR-TR-870, Ctr. Automat. Res., Univ. Maryland, College Park, Sept. 1997. [3] W. DuMouchel, “Stable distributions in statistical inference,” Ph.D. dissertation, Dept. Statist., Yale Univ., New Haven, CT, 1971. [4] E. Fama and R. Roll, “Parameter estimates for symmetric stable distributions,” J. Amer. Stat. Assoc., vol. 66, pp. 331–338, 1971. [5] H. M. Finn, “A CFAR design for a window spanning two clutter fields,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-22, 1986. [6] P. Gandhi and S. Kassam, “Analysis of CFAR processors in nonhomogeneous background,” IEEE Trans. Aerosp. Electron. Syst., vol. 24, 1988. [7] S. Geman and D. Geman, “Stochastic relaxation, Gibbs distribution, and Bayesian restoration of images,” IEEE Trans. Pattern Anal. Machine Intell., vol. PAMI-6, pp. 721–741, 1984. [8] J. Goodman, “Some fundamental properties of speckle,” J. Opt. Soc. Amer., vol. 66, pp. 1145–1150, Nov. 1976. [9] R. Kapoor, G. Tsihrintzis, and N. Nandhakumar, “Detection of obscured targets in heavy-tailed radar clutter using an ultra-wideband (UWB) radar and alpha-stable clutter models,” in Proc. 30th Ann. Asilomar Conf. Signal and Systems, 1996. [10] P. A. Kelly, H. Derin, and K. D. Hartt, “Adaptive segmentation of speckeled images using a hierarchical radom field model,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 1, pp. 1628–1641, 1988. [11] I. Koutrouvelis, “Regression-type estimation of the parameters of stable laws,” J. Amer. Stat. Assoc., vol. 75, pp. 918–928, 1980. [12] S. Kuttikad and R. Chellappa, “Non-Gaussian CFAR techniques for target detection in high-resolution SAR images,” in Proc. IEEE Int. Conf. Image Processing, 1995. [13] X. Ma and C. Nikias, “On blind channel identification for impulsive signal environments,” in Proc. ICASSP, May 1995. [14] J. McCorkle and L. Nguyen, “Focusing of dispersive targets using synthetic aperture radar,” Tech. Rep., Army Research Lab. Adelphi, MD, 1994. [15] C. Nikias and M. Shao, Signal Processing with Alpha-Stable Distributions and Applications. New York: Wiley, 1995. [16] M. Ressler, L. Happ, L. Nguyen, T. Ton, and M. Bennett, “The army research laboratory ultra-wideband testbed radars,” in Proc. IEEE Int, Radar Conf., 1995. [17] E. Rignot and R. Chellappa, “Segmentation of polarimetric synthetic aperture radar data,” IEEE Trans. Image Processing, vol. 1, pp. 281–300, 1992. [18] G. Tsihrintzis and C. Nikias, “Incoherent receivers in alpha-stable impulsive noise,” IEEE Trans. Signal Processing, vol. 43, pp. 2225–2229, 1995. , “Fast estimation of the parameters of alpha-stable impulsive [19] interference,” IEEE Trans. Signal Processing, vol. 44, pp. 1492–1503, 1996.

Soo-Chang Pei and Ji-Hwei Horng

Abstract—The covariance matrix of a pattern is composed by its second order central moments. For a rotationally symmetric shape, its covariance matrix is a scalar identity matrix. In this work, we apply this property to restore the skewed shape of rotational symmetry. The relations between the skew transformation matrix and the covariance matrices of original and skewed shapes are derived. By computing the covariance matrix of the skewed shape and letting the covariance matrix of the original shape be a scalar identity matrix, the skew transformation matrix can be solved. Then, the rotationally symmetric shape can be recovered by multiplying the inverse transformation matrix with the skewed shape. The method does not rely on continuous contours and is robust to noise, because only the second-order moments of the input shape are required. Experimental results are also presented. Index Terms—Covariance matrix, reflective symmetry, rotation matrix, rotational symmetry, shear matrix, skewed symmetry.

I. INTRODUCTION A rotationally symmetric shape is a shape that repeats itself after being rotated around its centroid through any multiple of a certain angle. To satisfy this property, the minimum repeating angle must be 2=n; where n is a positive integer. This shape is called an nfold rotationally symmetric shape. Many researchers on the area of image analysis have pay attention to the importance of rotational symmetry. Lots of methods have been proposed to normalize the rotationally symmetric planar shapes (RSS) [1]–[5]. However, none of them has dealt with the skewed shape of rotational symmetry. In real applications, the view direction is not always perpendicular to the plane containing RSS, it results in a skewed RSS on the image plane. Thus, a complete normalization system must includes the deskew procedure. Many methods have been proposed to recover the skewed shape of reflective symmetry [6]–[8]. But they are not directly applicable to the case of rotational symmetry. Although the set of reflectively symmetric shapes overlaps with the set of rotationally symmetric shapes, they are not identical. The relation of these two sets are shown in Fig. 1, where a typical example is given for each subset. An alternative procedure for generating a skewed shape is to perform a shear operation with parameter followed by a rotation in the image plane with parameter : Friedberg proposed a momentbased method to recover the skewed reflective symmetry [7]. He used the property that the covariance matrix for a reflectively symmetric shape is a diagonal matrix, that is, the moment m11 is necessarily equal to zero. For a set of assumed parameters and of the skew coordinate transformation matrix, the moment m11 of the recovered shape can be written as a function of the set of parameters and the second-order moments of the skewed shape. Let the moment m11 of the deskewed shape be equal to zero, can be described as a function of : The constraint reduces the search space from Manuscript received January 27, 1998; revised March 10, 1999. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Josiane B. Zerubia. S.-C. Pei is with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan 106, R.O.C. (e-mail: [email protected]). J.-H. Horng is with the Department of Information Management, Jin-Wen Institute of Technology, Taipei, Taiwan, R.O.C. Publisher Item Identifier S 1057-7149(99)09356-2.

1057–7149/99$10.00 1999 IEEE