Clutter Patch Identification Based on Markov Random Field Models T. Kasetkasem and P.K. Varshney Department of Electrical Engineering and Computer Science Syracuse University Syracuse, NY 13244 Email:
[email protected] and
[email protected]
Abstract–This paper addresses the problem of clutter patch identification based on Markov random field (MRF) models. MRF has long been recognized by the image processing community to be an accurate model to describe a variety of image characteristics such as texture. Here, we use the MRF to model clutter patch characteristics, captured by a radar receiver or radar imagery equipment, due to the fact that clutter patches usually occur in connected regions. Furthermore, we assume that observations inside each clutter patch are homogenous, i.e., observations follow a single probability distribution. We use the Metropolis-Hasting algorithm and the reversible jump Markov chain algorithm to search for solutions based on the Maximum a Posteriori (MAP) criterion. Several examples are provided to illustrate the performance of our algorithm.
I. INTRODUCTION Accurate statistical characterization of clutter background is critical to the design of efficient target detection and identification algorithms for radar systems. The conventional model assumes that the return signal consists of a known signal in Gaussian noise. The performance of a radar signal detector based on this conventional model degrades significantly in the presence of a complex clutter background. To improve performance, it is imperative that the clutter background be modeled accurately. One needs to determine the homogenous patches of clutter that occur due to reflections from heterogeneous background. In addition, the underlying probability density functions (PDF) in each clutter patch need to be identified. Using this information, intelligent detection schemes can be designed that are expected to perform better. The goal of this paper is to address this important problem and develop an algorithm for clutter patch identification. Slamani [1] initiated the investigation of the clutter patch identification problem in which the surveillance volume is divided into homogenous regions based on PDFs of clutter. His methodology is composed of two steps. In the first step, a surveillance volume is partitioned into clutter patches and background noise regions using an appropriate threshold because noise power in background noise regions is lower than that in clutter patches. An appropriate technique for determining this threshold was also presented. Since clutter patches usually do not occur at isolated points, some misclassifications can be removed through a windowing method where the similarity among neighboring pixels is measured and compared. A change in the class of the pixel of
interest is made if the number of neighboring pixels conflicting with the pixel of interest exceeds a certain threshold. Clutter patches are divided using the above procedure according to their power levels until they cannot be divided any further. In the second step, each clutter patch is partitioned into more regions according to their underlying PDFs. Here, the Ozturk algorithm [2] is used to approximate the PDFs by generating a coordinate from an order statistics of the samples. Then, for a given neighborhood, the coordinate is plotted and distances to the list of possible PDFs are measured. Since the Ozturk plot of the border pixels between two adjacent clutter patches contains data from both patches, these pixels can be easily identified due to the large distance from possible distributions. Some promising results have been presented in [1]. However, this methodology is intuitive and lacks theoretical justification. Here, we extend the work presented in [1] and develop an algorithm under a statistical framework to identify clutter patches and estimate the underlying PDFs. Just like the algorithm in [1], our algorithm is automatic and does not require human intervention. Image segmentation techniques are expected to provide efficient solutions for this problem due to the similarities between segmentation and clutter patch identification problems. The main objective of both problems is to partition a nonhomogenous region into several homogenous regions according to some features. For image segmentation, these features may be textures or gray levels, whereas, for clutter patch identification, the PDFs are the main concern. In the past few decades, a number of image segmentation algorithms have been developed for a variety of problems. These fall into two general categories namely statistical and deterministic algorithms. Under the statistical framework, the Markov random field (MRF) model has received a great deal of attention because a MRF model can characterize the information contained among neighboring pixels quite accurately [3-8]. As a result, we will develop our algorithm based on a MRF model. Geman and Geman [3] employed a MRF model for image restoration. They modeled noiseless images as a MRF consisting of a certain number of possible gray levels. Observed images are distorted by the nonlinearity of equipment (such as cameras and films), and are also disturbed by the additive noise. The maximum a posteriori (MAP) criterion together with the simulated annealing algorithm
(SA) was used to estimate the most likely noiseless image from the observed image. The SA algorithm searches over the noiseless image space based on the Markov chain Monte Carlo (MCMC) procedure from which a new noiseless image is generated every iteration from a current noiseless image using the Gibbs sampling procedure [3,6]. The solutions of the MAP detector are obtained when the Markov chain converges. Similar algorithms have also been developed in [6-10]. In [6], authors replaced the SA algorithm by the Metropolis algorithm which are equivalent. But instead of obtaining a new noiseless image from the Gibbs sampling procedure, a new noiseless image is generated randomly and it is accepted with a certain probability derived from the Gibbs energy function (to be defined in the next section.) Begas [7] pointed out the high computational complexity of the SA algorithm, and introduced a suboptimum version where the algorithm converges to the closest local optimum instead of the global one. Thus, the accuracy of this algorithm depends on the initial guess. Hence, if an accurate initial noiseless image cannot be obtained, performance of this algorithm degrades significantly. Hasting [8] generalized the Metropolis algorithm to the case that we are more interested in namely in the a posteriori probability rather than the Gibbs energy function. Furthermore, the Metropolis-Hasting algorithm allows the space of noiseless images to be expanded or compacted [9-10]. Green [9] developed the reversible jump Markov chain (RJMC) algorithm based on the MetropolisHasting algorithm to permit jumps in the number of possible stages and/or the size of the corresponding parameters under the strict condition that the reversibility of an induced Markov chain must exist. Details of the RJMC algorithm are presented in the next sections. Applications of the RJMC algorithm have been presented in [9-10]. Barker and Rayner [10] used the RJMC algorithm for unsupervised image segmentation where the number of possible classes is unknown. In this paper, we develop the clutter patch identification algorithm based on statistical modeling, specifically on the MRF model. This approach builds on the work presented in [1,10] and is appropriate due to the fact that clutter patches are more likely to occur in connected regions. Moreover, the nature of the clutter patch identification problem is quite similar to image segmentation and classification problems. The clutter patch identification algorithm proposed in this paper is composed of two steps. In the first step, an initial clutter patch image (CI) is obtained by a procedure similar to the one introduced in [1]. Here, the initial number of possible clutter patches is set to the maximum number of clutter types assumed. In the second step, the MRF model is employed and the MAP criterion is selected for optimization. The MCMC procedure is employed to search for the solutions. In our proposed algorithm, a combination of the Metropolis-Hasting (MH) algorithm and the RJMC algorithm are employed. For a given iteration, a new CI is generated using the Gibbs sampling procedure, and it is accepted with probability one.
Then, the algorithm selects one clutter patch randomly and attempts to change the underlying distribution of this clutter patch to another by randomly choosing a new distribution from a list of possible distributions. The acceptance probability is calculated based on the MH procedure. Next, the algorithm randomly selects the next action that is to either “merge” or “split”. When “merge” is selected, the algorithm attempts to merge two clutter patches into a single patch by preserving some of their characteristics. Conversely, when “split” is chosen, the algorithm attempts to partition a clutter patch into two clutter patches. After a specific number of iterations, the algorithm updates all the parameters associated with PDFs of all the clutter patches. This paper is organized as follows. Section 2 provides the problem statement, and the corresponding optimum detector is developed in Section 3. The main algorithm based on the optimum detector is described in Section 4. Section 5 contains several examples and presents some numerical results to illustrate the performance of our algorithm. Some concluding remarks are provided in Section 6. II. PROBLEM STATEMENT Let S be a set of sites (pixels) s, and Λ = {0,1, K , L − 1} be
the phase space (intensity levels). Furthermore, let X(S ) ∈ ΛS denote a clutter patch (configuration) vector or a CI. Note that L ≥ 2 is the number of clutter patches in the scene in which the exact value of L is unknown. Here, we model L as a random variable characterized by the probability mass function PL (l ) . We assume that X(S) satisfies MRF properties with a Gibbs potential VC(x) [3-6]. Hence, we can write the marginal PDF of X(S ) [6] as
π X (x i li ) = ZX =
where
é ù 1 exp ê− VC (x i )ú , ZX êë C ⊂S úû
å
é
(1)
ù
å expêê− å VC (x)úú
is the normalizing ë C ⊂S û constant and li is the number of clutter patches in a CI xi. Note that X(S) is a realization of a clutter scene and
åVC (x)
x∈ΛS
is called the Gibbs energy function. These terms
C ⊂S
will be used extensively in our discussion. Let Y(S) ∈ ℜ S be the associated clutter or noise vector whose observations in pixels si and sj are statistically independent given the CI, i.e., p y ( s i ), y ( s j ) X( S ), L = (2) p( y ( s i ) X( S ), L ). p y ( s j ) X( S ), L
(
(
)
)
where p is the probability density function of clutter or noise. Furthermore, the probability density function (PDF) of clutter or noise at a site depends only on the type of clutter patch and observation at that site, i.e.,
p( y ( s i ) X( S ), L) = f ( y ( si ), x( s i ) ) . (3) where f(a,b) denotes a PDF. In radar and sonar systems, we usually assume that the clutter can be modeled as one of several known distribution types. For example, Slamani [1] assumed that noise can belong to one of the following distributions: Rayleigh, Weibull, lognormal or K distribution. Here, we also assume that f(a,b) can only come from a known family of distributions that is f ( y ( si ), x( si ) ) ∈ { f m ( y ( si ), θ m )}m={1,K,M } , (4) where M indicates the size of the family of PDFs, and θ m is the parameter vector corresponding to the underlying PDF. We formulate the clutter patch identification problem as an M-ary hypothesis testing problem where each hypothesis corresponds to a different CI. For a given CI (hypothesis), the observation volume is divided into several homogenous clutter regions. We note again that the term “homogenous” implies that the PDFs of observations at every pixel inside a clutter patch are identical and independent. Furthermore, since we formulate our problem as an M-ary hypothesis testing problem, techniques developed to solve signal detection problems can be employed and we provide our methodology in the next section.
ì ùü éæ ö ï p( y ( s ) x j ( s )) ÷. úï êçç ÷ ïï ú ïï êè s∈S ø X k = arg ímax ê úý . æ ö ï Xj ê 1 úï ç VC (x j ) ÷.PL l j ú ï ï ê Z expç − ÷ ïî è C ⊂S ø û ïþ ë X Using (4), the above equation can be written as ì éæ ö ùü ï f m( x j ( s )) ( y ( s ), θ m ( x j ( s )) ) ÷. ú ï êçç ÷ úï ïï êè s∈S ø ï X k = arg ímax ê úý , Xj ö æ ï úï ê 1 ç VC (x j ) ÷.PL (l j )ú ï ï ê Z expç − ÷ ïî ø è C⊂S û ïþ ë X
∏
å
( )
(8)
∏
(9)
å
where
m( xl ( s )) denotes the mth type of PDF.
In practice, the direct maximization of equation (9) is not feasible due to the enormous number of possible CIs. Moreover, parameter vectors associated with each clutter patch are generally unknown. Therefore, there is a need for a more efficient way to obtain the solution of (9) and estimate the unknown parameters. Here, we will employ the Metropolis-Hasting algorithm and the reversible jump Markov chain algorithm [8-9] together with ML estimation to search for the solution of (9) and estimate unknown parameters simultaneously.
III. OPTIMUM DETECTOR IV. PROPOSED ALGORITHM
The maximum a posteriori (MAP) criterion [11-12] is used for identifying clutter patches in our work. This criterion is expressed as ü ì (5) X k = arg ímax P( X j , L j Y = y ) ý . þ î Xj
[
]
From Bayes’ rule, (5) can be rewritten as
ì é P(Y = y X j , l j ) P ( X j l j ) PL (l j ) ù ü ï ú ïý X k = arg ímax ê P(Y = y ) úï ïî X j êë ûþ
(6)
Since P(Y = y ) is independent of Xk, the above equation reduces to
[
]
ü ì X k = arg ímax p(Y = y X j , l j ) P( X j l j ) PL (l j ) ý = X þ î j
ìï éæ ù üï ö arg ímax êç p( y ( s ) x j ( s )) ÷ P( X j l j ) PL (l j )ú ý ç ÷ úû ïþ ïî X j êëè s∈S ø
∏
Substituting (1)-(3) into (7), we have
(7)
The main objective of this algorithm is to find the CI Xk and the estimated PDFs for each clutter patch that maximize (9). However, direct search techniques cannot be implemented with a reasonable computational cost due to the complexity of (9). Here, we propose an alternative technique that utilizes the MCMC procedure. This technique generates a sequence of random CIs that ultimately converges to the solutions (9). Note here that solutions of (9) may not be unique. In general, substantial amount of time is required for convergence to occur, if this procedure begins from an arbitrary CI. To expedite convergence, a fairly good estimation of CI must be obtained to be used as the initial guess. Here, we use the Slamani procedure and the Ozturk algorithm to estimate an intial CI when the number of clutter patches is set at the maximum number of allowable patches. Both techniques are computationally efficient, but lack systematic statistical framework for clutter patch identification. The Slamani procedure [1] partitions the observed region based on the average signal power whereas the Ozturk algorithm uses some statistical characteristics to measure the similarity among different samples. In our algorithm, we propose to first use the Slamani procedure in which the surveillance volume is partitioned based on the average signal power. The resulting CI is submitted to the next step where the Ozturk algorithm is employed to divide the clutter patches further. For a given pixel, a cell of size N × N pixels centered at the given pixel is created, and the Ozturk coordinate corresponding to this cell
is obtained. From [13], if the observations in different cells are generated from the same PDF, the distance between them in the Ozturk space is small. If the observations are generated from different PDFs, they result in a larger distance. Based on the above property, an existing clutter patch can be partitioned further by using a clustering algorithm [14]. Completion of this procedure yields the initial guess to be used in the main algorithm. The initialization procedure is summarized in Figure 1.
determined in a similar manner as in the split and change processes. Figure 2 displays our algorithm. Details of the algorithm are not presented here due to space limitations. There are available in [15].
Initial CI
Set iteration number h = 0 SA algorithm
Observed CI
Slamani Procedure (Thresholding)
Atempt to change the underlying PDFs of clutter patches. (change process) x = random(1,2)
Ozturk algorithm (statistics)
If x = 1
If x = 2
Attempt to merge any two clutter patches. (merge process)
Attempt to split a clutter patch into two patches. (split process)
Initial CI Fig. 1. The initialization procedure
a new CI, h = h +1
The main algorithm employs an MCMC procedure, particularly the RJMC algorithm [9]. In this stage, the algorithm improves the accuracy of CI (especially in the border regions), allows the number of clutter patches to increase or to decrease, and also estimates the underlying PDFs of each clutter patch in an iterative manner. The SA algorithm is employed to increase the accuracy of the estimated CI by changing the clutter patch label assigned to a given pixel to another such that the new clutter patch label for a given pixel is more likely to be the one that gives a higher value in the argument of (9). In the SA step, the number of clutter patches remains fixed as well as the underlying PDFs of all clutter patches. Changes in the underlying PDFs are carried out by the “change” process. Here, the algorithm picks a clutter patch at random, and proposes a new PDF with a new set of associated parameters. If the proposed PDF yields a higher value of the argument in (9), it will be accepted with high probability whereas if the proposed PDF yields a smaller value of the argument in (9), it will be accepted with a low probability. In case of rejection, the algorithm will keep the old underlying PDF. Increase or decrease in the number of clutter patches is performed by “split” and “merge” processes, respectively. When the algorithm tries to increase the number of clutter patches, the “split” process is invoked. Here, a clutter patch is selected at random, and two disjoint clutter patches, whose union is the selected clutter patch, are proposed. Again, the algorithm accepts the split with high probability if the split yields a higher value of the argument in (9). When the algorithm tries to merge two clutter patches to a single clutter patch, two clutter patches are selected at random. Then, this merge is accepted with a probability
No
Is h > h_max Yes END Fig. 2. The main algorithm
V. EXPERIMENTAL RESULTS In this section, we choose T-2 Gumbel, Gamma, Pareto, Weibull, Lognormal, K, Beta, and Johnson-SU distributions as the set of allowable distributions. Their PDFs are given in Table 1. Example 1 In this example, the accuracy of our algorithm is illustrated by using the simulated CI displayed in Figure 3 whose intensity levels are one, two and three. This CI represents the ground truth. Here, intensity level three indicates background noise whereas one and two indicate clutter patches with different distributions. As mentioned in [1], the background noise region is usually characterized by the Rayleigh distribution which is equivalent to the Weibull distribution with shape parameter 2. We choose the location and scale parameters to be 0.88 and 0.46, respectively. For clutter patches, we choose the Lognormal distribution with location, scale and shape parameters 3.6, 4.0, and 0.89, respectively, for patch 1, and the Weibull distribution with location, scale and shape parameters, 4.5, 3.0 and 1.5, respectively, for patch 2. The resulting observed image is shown in Figure 4. Moreover, we assume that the number of
clutter patches is a uniform random variable between two and eight. Hence, the maximum number of clutter patches is picked to be eight. FORM OF THE PDFS PDF
(
1. T-2 Gumbel 2. Gamma
γ y −λ −1 exp − y −γ
)
(Γ(γ ))
γ −1
−1
exp(− y ) y
3. Pareto
y
4. Weibull
(yγ
5. Lognormal 6. Kdistribution 7. Beta 8. JohnsonSU
2π
− (γ +1)
−1
exp − 0.5(log( y ) / γ )
[
2(Γ(γ ) )
−1
(B(γ , δ ))
(2π (1 + y ))
2 −0.5
y
y>0 2
(0.5 y )
−1 γ −1
γ
)
γy
)
γ
3.04 1.51] ,
and
distributed
Weibull
with
distributed
with
0< y1
(
Weibull
T
−∞ < y < ∞
γ +1
exp − y
3.98 0.88]T ,
0.461 2.08]T , respectively, where [α β γ ]T is the parameter vector of location, scale and shape parameters respectively. As shown, the accuracy of parameter estimation is very promising.
TABLE1 Distribution
[3.64 [4.52 [0.88
K γ −1 ( y )
(1 − y )
δ −1
]
y>0
y>0
50
50
100
100
150
150
200
200 50
(
150
200
50
(a)
0
