Model-based superpixel segmentation of SAR images

We propose a superpixel segmentation method for synthetic aperture radar (SAR) images. The method uses the SAR image amplitudes and pixels coordinates as features. The feature vectors are modeled statistically by taking into account the SAR image statistics. Nakagami and bivariate Gaussian distributions are used for amplitudes and position vectors, respectively. A finite mixture model (FMM) is proposed for pixel clustering. Learning and clustering steps are performed using posterior distributions. Based on the classification results obtained on real TerraSAR-X image, it is shown that the proposed method is capable of obtaining more accurate superpixels compared to state-of-the-art superpixel segmentation methods.


INTRODUCTION
Superpixels refer to irregular shaped large image regions that are obtained after the over-segmentation of an image. Superpixels are useful in reducing the complexity and the processing time of the images. In image understanding applications, superpixel segmentation is used as a pre-processing step to obtain a mid-level representation of the image. Superpixel segmentation methods are mostly proposed for natural optical images. In this study, we propose a superpixel segmentation method especially for SAR images. For this purpose, we use the theoretical statistics of the SAR images.
There are several superpixel segmentation methods proposed in the literature. One of the first proposed superpixel method [1] is a graph segmentation method that uses normalized cuts. In [2], another graph-based method is proposed. Beside the graph-based methods, clustering-based methods are recently proposed for superpixel segmentation [3], [4]. Quick-shift clustering algorithm is proposed in [3]. Quickshift assigns each data point to its closest neighbor regarding to a kernel function. In [4], a k-means clustering algorithm called SLIC (simple linear iterative clustering) is proposed for All the aforementioned methods can be used for superpixel segmentation of the color images. Most of the methods use the Euclidean distance or Gaussian kernels to measure the similarity between the pixel intensities. The Gaussian distribution may be a good approximation to model the color image statistics but this is not the same case in SAR images. SAR images have some special statistical properties unlike the optical color images. For instance, theoretical intensity and amplitude statistics of a multi-look SAR image follow gamma and Nakagami distributions, respectively [5]. Since the theoretical statistical models for SAR images are obtained under the multiplicative noise assumption, they are consequently more convenient models to deal with speckle in SAR images.
In [6], a superpixel segmentation method is proposed for polarimetric SAR images by using a Wishart distance instead of Euclidean. In this study, we propose a SAR image specific superpixel clustering method. We interpret the clustering method in the light of gestalt-based perceptual grouping rules listed in [7]. We only use two rules called similarity and proximity. For similarity, we use Nakagami distribution as a statistical measure, i.e. that two pixels in the same cluster are assumed to be generated from the same Nakagami distribution. For proximity, we use the bivariate Gaussian distribution to model the spatial distances between the pixels.
Both similarity and proximity statistics are combined into a finite mixture model (FMM). FMMs have been already used in SAR image intensity and amplitude classification [8], [9], [10], [11]. Using these statistical measures within an FMM, the pixels are clustered around the superpixels' centroids.
Organization of the paper is as follows. Section 2 and 3 respectively present the proposed FMM-based superpixel model and related segmentation algorithm. The simulation results are reported in Section 4. Section 5 summarizes the conclusion and future work.

FINITE MIXTURE MODEL FOR SUPERPIXELS
We denote the pixel amplitudes by ∈ R + and the coordi- We use Nakagami distribution to model the pixel amplitudes, that is a basic theoretical multi-look amplitude model for SAR images [5]. Nakagami distribution function is given by . (1) where and are the parameters of the th superpixel. We assume that the pixels are spatially distributed around the centroid of a superpixel according to the normal law as follows: (2) where m and Σ are the centroid and the covariance matrix of the th superpixel, respectively. For the th superpixel the parameter set is defined to be = { , , m , Σ }.
We aim to divide the image into mutually exclusive superpixels. Assuming that there are number of superpixels in the image, we define a -dimensional label vector z ∈ {0, 1} for each superpixel. The binary label vector z has the property that ∑ =1 , = 1. We define that z ∈ {[1, 0, . . . , 0], [0, 1, . . . , 0], . . . , [0, 0, . . . , 1]}. We also assume that f 's are conditionally independent given the labels, z 's. Since the natural prior for z is a multinomial distribution, we may define the following prior for z where 1: are the parameters of the multinomial distribution.
After these definitions, we may show that the marginalization of the joint density yields a finite mixture density as follows: where corresponds to mixture proportion. Graphical representation of the proposed FMM is given in Fig. 1.

INFERENCE
We need to perform a posterior inference for the proposed probabilistic model. In the model, there are two key variables to be inferred, namely the superpixel labels z 1: , and the parameters of the Nakagami and Gaussian distributions 1: . We assume the mixture proportions to be fixed = 1/ . We resort to iterated conditional mode (ICM) algorithm because the inference from the joint posterior (z 1: , 1: , 1: |f 1: ) is not tractable. In ICM algorithm, conditional densities of variables are maximized iteratively. The joint posterior of the variables is factorized as follows (z 1: , 1: , 1: |f 1: ) ∝ (f 1: |z 1: , 1: ) (z 1: | 1: ) (4) where the likelihood term (the first term on the righthand side) can be factorized as follows: The joint prior distribution of the labels (z 1: | 1: ) in 23rd European Signal Processing Conference (EUSIPCO) (4) is given by In order to perform a posterior inference, we use the block ICM algorithm. Unlike the ICM algorithm, block ICM updates the same kind of variables together at a time. In this way, we aim to obtain a faster algorithm than conventional ICM [12]. We update the variables along the iterations in the following order: where = 1, .., , = 1, .., and is the pseudo time index.

EXPERIMENTAL RESULTS
We present a comparison of the proposed FMM-based SAR superpixel and two state-of-the-art superpixel methods. These algorithms are SLIC [4] and Quickshift [3].
We use VLFeat open source library [13] for implementation of SLIC and Quickshift algorithms.

Practical Issues
In this section, we give the details of the practical implementation of the proposed algorithm. We initialize the superpixels by dividing the image into × ( < ) sub-image blocks. We follow the same strategy used in [4], i.e. when maximizing the superpixel labels, we only take into account the neighboring pixels but not the entire superpixel. The neighborhood is defined to be a 2 × 2 region around the superpixel center.
Since the methods are based on clustering, they do not take into consideration the spatial dependency of the pixel. Therefore, a superpixel may include more than one connected component. At the end of the each algorithm, we remove the small connected components by merging them with the adjacent big segment. For this purpose, we use the code provided in [14].

SAR Image
We have tested the algorithms on the test image TSX (see Fig. 2(a)): 500 × 500 pixels, HH polarized, TerraSAR-X SpotLight (8.2 m ground resolution) which was acquired over Rosenheim in Germany on January 27, 2008 ( c ⃝Infoterra). The ground-truth map is generated manually as seen in Fig.  2(b) by assuming three kind of land cover classes. One of them is urban zone. The remaining two regions are two different land zones which are seen dark and bright in Fig. 2(a).

Comparisons
Superpixel segmentation results of three algorithms are given in Fig. 3. The numbers of superpixels obtained by FMM, SLIC and QS are 100, 102 and 102, respectively. As seen from the segmentation maps in Fig. 3, the superpixels found by proposed FMM-based method fits the real SAR image boundaries better than those found by SLIC and Quick-shift.
We use under-segmentation error (UE), boundary recall (BR) and boundary precision (BP) for the numerical evaluation of the performances of the algorithms. Although BR is the mostly used measure for superpixel evaluation, high BR values do not always correctly measure the goodness of fit of the edges. If the superpixel boundaries are unnecessarily overfitted to the images details, BR may indicate a good edge segmentation result despite a bad edge representation. Therefore, we use BP along with UE and BR. The UE used in this paper is defined by where is a superpixel and is the complement of the   ground-truth segment of th class. BR is the fraction of the ground-truth edges that hit the superpixel boundaries within at least two pixels. BP is the fraction of the superpixel boundaries hit by the ground-truth edges. Fig. 4 shows the UE values obtained by three algorithms for different number of superpixels. As seen from the UE and BP plots in Fig. 4 and 6, the proposed FMM performs significantly better than the other two methods. According to BR plots in Fig. 5, the performance of the proposed algorithm is better when the number of superpixels is less than 300.

CONCLUSION
In this study, we have proposed a FMM-based superpixel segmentation method that takes into account the SAR image statistics. The proposed method yields significantly better superpixel segmentation for SAR images compared to stateof-the-art superpixel methods. By using proposed method, without any preprocessing as noise removing, we are able to obtain superpixel segmentation of SAR images. The proposed method provides a mid-level representation of the images that may be used for higher-level understanding of SAR images. We use theoretical Nakagami distribution to model the SAR image statistics but other empirical distributions may be used as well.