An Unsupervised Segmentation Method Based on MPM ... - IEEE Xplore

IEEE GEOSCIENCE AND REMOTE SENSING LETTERS, VOL. 2, NO. 1, JANUARY 2005

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An Unsupervised Segmentation Method Based on MPM for SAR Images Yongfeng Cao, Hong Sun, Member, IEEE, and Xin Xu

Abstract—An unsupervised segmentation method for synthetic aperture radar (SAR) images is proposed. It alternately approximates the maximization of the posterior marginals estimate of the pixel class labels and estimates all model parameters except the number of classes during segmentation. In this method, a multilevel logistic (MLL) model for the pixel class labels and Gamma distribution for the marginal distribution of each class in the observed SAR image are employed. In our implementation, the expectation–maximization algorithm is used to estimate parameters of the Gamma distributions, and the iterative conditional estimation algorithm is used to estimate the MLL model parameters. The segmentation results for synthetic and real SAR images show that the proposed method has a good performance. Index Terms—Hidden Markov models, image segmentation, parameter estimation, synthetic aperture radar (SAR).

I. INTRODUCTION

T

HIS LETTER addresses the problem of segmenting synthetic aperture radar (SAR) images. In image segmentation, maximum a posteriori (MAP) and maximization of the posterior marginals (MPM) criterions are usually used to estimate the pixel class labels from observed image. It has been shown that the MPM estimate is more appropriate for image segmentation than the MAP estimate [1]–[3]. This is because the MPM estimate assigns a cost to an incorrect segmentation based on the number of misclassified pixels in that segmentation, whereas the MAP estimate assigns the same cost to all incorrect segmentations. In image segmentation, the Markov random field (MRF) has been extendedly used as an image model, especially as a model of the underlying label image. The MRF-based segmentation methods usually use experiential hyperparameters of MRF models [2], [4] due to the difficulties of estimating them [5]. In this case, many experiments are needed in advance to get the hyperparameters, but these parameters for one image are not surely suitable for other images. It seems that estimating MRF model parameters in process is more practical and likely to get better results [6]. In this letter, aiming at the drawbacks of using experimental MRF hyperparameters, we propose an unsupervised segmentation method based on MPM (USMPM) for SAR images. The USMPM is an iterative scheme that consists of two alternating

Manuscript received June 15, 2004; revised October 14, 2004. This work was supported by the National Natural Science Foundation of China under Projects 60372057 and 40376051. The authors are with the Signal Processing Laboratory, Department of Communication Engineering, School of Electronic Information, Wuhan University, Wuhan 430079, China (e-mail: hongsun@whu.edu.cn). Digital Object Identifier 10.1109/LGRS.2004.839649

Fig. 1. Eight-neighborhood and cliques of two pixels.

steps: to approximate the MPM estimation of the pixel class labels and to estimate all model parameters except the number of classes. Multilevel logistic (MLL) model is used for the underlying label image. In Section II, models for label image and observed image are briefly described. In Section III, the method USMPM is presented. In Section IV, some segmentation results for synthetic and real SAR images are given. In the case of synthetic SAR image, the result of the proposed method, in which the hyperparameters of the label image model are estimated, is compared with the result of the MPM-based method using the true hyperparameters of the label image model. Section V contains the conclusions. II. IMAGE MODELS In this section, we consider a rectangular pixel lattice . The and the observed label field will be denoted image field will be denoted . Throughout this and will represent letter, and . We suppose there are difsample realizations of . ferent classes in SAR image, so It is well known that the Gamma distribution provides a good model for SAR intensity data. We suppose, therefore, that each pixel of SAR image is conditionally independent of all other pixels, conditioned on the knowledge of segmentation label at that pixel, , and satisfies gamma distribution. Formally (1) (2) is the where is the number of looks of the SAR image, and parameter of gamma distribution associated with segmentation label . For the label image we use a MLL model and consider only eight-neighborhood and cliques of two pixels (see Fig. 1). The local conditional distribution at site can be represented by Gibbs distribution as follows:

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(3)

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where neighborhood of site , Gibbs distribution

are hyperparameters, is the is the normalizing constant of

(4) for , where The posterior distribution of

given

otherwise. is

algorithms. Now, we propose an iterative unsupervised segmentation method called USMPM, which begins with an initial and , and ends until certain number of iterations has been finished. At the th iteration, is first obtained as shown above, is obtained using expression (11) using (8) and (7). Then, . and (14). Sections III-B and III-C will show how to get B. Expectation–Maximization (EM) Algorithm for Estimation of

(5) where . Based on image models of (1), (2), (3), and (5), we will show and below how to get the MPM estimation of label image how to estimate all model parameters by proposed method USMPM.

We use the EM algorithm to estimate model parameters of the observed image. The EM is a well-known algorithm for the estimation of parameters in incomplete-data problems. Our case is just an incomplete-data problem with the observed image and hidden label image . The EM algorithm has been successfully used for estimating Gauss parameters during the EM/MPM segmentation [2]. Now, we use it for estimating Gamma parameters of the observed SAR image. The EM algorithm is an iterative algorithm that consists of two alternating steps: the expectation is the estimation of at step and the maximization step. If the th iteration, then the expectation function at th iteration is

(9) III. USMPM A. Description of USMPM As shown in [1] and [2], finding the MPM estimation of is equal to maximizing each pixel’s posterior marginal probability mass function

Since the first term of (9) does not depend on , we only use the second term of (9). Let denote the number of pixel sites in . Substituting (1) into the second term of (9), we get

(6) over all , for every Then the MPM estimation of is

.

(7) Because exact computation of these marginal probability mass functions as in (6) is computationally infeasible, we use the approximation method proposed by Marroquin et al. [1]: use Gibbs sampler [4] to generate a discrete-time Markov , which converges in chain distribution to a random field with probability mass function (5), and then approximate function (6) by

(10) Substituting (2) into (10), differentiating, setting to zero, and solving for gives

(8) where for , otherwise, and is the number of visits to pixel made by Gibbs sampler. In supervised case, the MPM estimation of label image can be got using (8) and (7) with known model parameter and data . In unsupervised case, like the label image, the model parameter is unknown too, and should be estimated during the segmentation process. So, in the unsupervised case, the MPM segmentation should be integrated with some parameter estimation

(11) can be approximated by (8). It where is not surprising that (11) is the same as [2, eq. (12)], where the estimate was based on a Gaussian model hypothesis, if we consider that the ML estimation of Gamma parameter and Gaussian parameter are both the mean of samples, when data are complete.

CAO et al.: UNSUPERVISED SEGMENTATION METHOD BASED ON MPM FOR SAR IMAGES

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C. ICE Algorithm for Estimation of Using the EM algorithm to estimate model parameters of label image is difficult because of the intractable normalizing constant of Gibbs distribution (12) So, (12) usually is replaced by a tractable approximation such as the pseudolikelihood function (13) Yet, this approximation ignores the interactions between pixels and can perform unsatisfactorily unless the interactions in are rather weak [5]. To avoid dealing with the normalizing constant and at the same time get accurate parameter estimation, we chose the iterative conditional estimation (ICE) [7] algorithm for the estimation of . ICE is similar to the EM algorithm but results in a better estimation of the parameters, and the convergence is faster [5]. Let denote the discrete-time Markov chain that is generated by Gibbs sampler and which converges in distribution to the random field with probability mass function (5) characterized . Let denote the estimate of by parameters MLL model parameters from the label image sample . Then, at the th iteration, the estimate of MLL model parameters of the ICE algorithm is

Fig. 2. Synthetic three-look SAR image with two classes (size: 200

2 200).

(14) It can be seen that ICE algorithm can well use the Markov , which is also used for estimating and pachain rameter . For estimating from , we use the algorithm proposed by Drin et al. [8], which need only solve linear functions, but can get very accurate results. IV. EXPERIMENTAL RESULTS The proposed method USMPM is applied to synthetic and real SAR images. Fig. 2 shows a synthetic three-look SAR intensity image. The underlying label image of Fig. 2 was created using Gibbs sampler with the specified MLL model parameters . The USMPM (estimating hyperparameters during segmentation) and the same method using the true were both applied to hyperparameters the synthetic SAR image in Fig. 2, and the results are shown in Fig. 3. We calculated the percentage of misclassified pixels during segmentation for both methods, and the result is shown in Fig. 4. From the curves in Fig. 4, it can be seen that USMPM needs some iterations to get convergence and that once get convergence, it is with a percentage of misclassified pixels almost the same as that through the process using the true hyperparameters. As shown in Table I, the estimated values of are close

Fig. 3. (Left) Segmentation results and (right) misclassified pixels of (first row) the USMPM and (second row) the same method using the true hyperparameters.

for both methods. Segmentation result of USMPM for real X-SAR image is given in Fig. 5, in which three classes (lake, city zone, and others) are considered and the corresponding estimate of is [268, 16516, 5315]. Segmentation results of both methods were got after 100 iterative times. , the length at each iteration is 20. All computations of Markov chain were carried out using programs written in C and running on a Pentium-based personal computer (2.4 GHz). USMPM needs about 3 min for the synthetic SAR image of size 200 200, and about 0.7 min for the real X-SAR image of size 85 150. We make a comparison with the classic -mean method. size nonoverlapping small Fig. 2 was first divided into windows, and each window was denoted by its intensity mean. Then, these windows were grouped into two classes with

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TRUE VALUE

Fig. 4. Percentage of misclassified pixels versus the times of iteration.

AND

TABLE I FINAL ESTIMATION OF PARAMETER FOR TWO CLASSES IN FIG. 2

AND

used in MRF-based segmentation, because of the difficulty to estimate them. In this case, some drawbacks are obvious, such as many experiments are needed to get hyperparameters in advance, and the experiential hyperparameters for one image are not surely suitable for other images. Aiming at the drawbacks of using experiential hyperparameters, we proposed an unsupervised segmentation method using MPM criterion for SAR images, which estimates all model parameters except the number of classes during the process instead of in advance. MLL model is used for the label image, and the marginal distribution of each class for SAR image segmentation is supposed to be Gamma distribution. For iteratively estimating all model parameters, the EM and ICE algorithms are used. Segmentation results for synthetic SAR image show that in the sense of the percentage of misclassified pixels, USMPM performs almost as well as the same method using the true hyperparameters.

REFERENCES

Fig. 5. Real three-look X-SAR image of Switzerland (size: 85 segmentation result (three classes: lake, city zone, and others).

2 150) and

-mean method according to intensity mean. When the size of nonoverlapping window became 8 8, -mean method got the least percentage of misclassified pixels, 5.3, which is far larger than the percentage of 2.6 obtained by USPMP proposed in this letter. V. CONCLUSION In the sense of cost function, MPM is better than MAP for images segmentation. Experiential hyperparameters are usually

[1] J. Marroquin, S. Mitter, and T. Poggio, “Probabilistic solution of illposed problems in computational vision,” J. Amer. Statist. Assoc., vol. 82, pp. 76–89, Mar. 1987. [2] M. L. Comer and E. J. Delp, “The EM/MPM algorithm for segmentation of textured images: Analysis and further experimental results,” IEEE Trans. Image Process., vol. 9, pp. 1731–1744, Oct. 2000. [3] J. Zhang, J. W. Modestino, and D. A. Langan, “Maximum-likelihood parameter estimation for unsupervised stochastic model-based image segmentation,” IEEE Trans. Image Process., vol. 3, pp. 404–420, 4 1994. [4] S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell., vol. PAMI-6, no. 6, pp. 721–741, Nov. 1984. [5] M. V. Ibáñez and A. Simó, “Parameter estimation in Markov random field image modeling with imperfect observations: A comparative study,” Pattern Recognit. Lett., vol. 24, no. 14, pp. 2377–2389, 2003. [6] D. Melas and S. Wilson, “Double Markov random fields and Bayesian image segmentation,” IEEE Trans. Signal Process., vol. 50, no. 2, pp. 357–365, Feb. 2002. [7] W. Pieczynski, “Statistical image segmentation,” in Proc. Machine Graphics and Vision (GKPO), Naleczow, Poland, May 1992, pp. 261–268. [8] H. Derin and H. Elliot, “Modeling and segmentation of noisy and textured images using Gibbs random fields,” IEEE Trans. Pattern Anal. Mach. Intell., vol. PAMI-9, no. 1, pp. 39–55, Jan. 1987.