SAR Image Segmentation using Wavelets and Gaussian ... - IEEE Xplore

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SAR Image Segmentation using Wavelets and Gaussian Mixture Model Anirban Dutta and Kandarpa Kumar Sarma Department of Electronics and Communication Engineering, Gauhati University Guwahati, Assam - 781014, India Email: [email protected], [email protected]

Abstract—Synthetic Aperture Radar (SAR) segmentation is often acknowledged as a difficult task due to the presence of speckle noise because of which traditional segmentation algorithm fail to give satisfactory results. In this paper, Gaussian Mixture Model (GMM) along with the combination of wavelets is proposed for noisy image segmentation. First, texture feature are abstracted in the wavelet domain and according to the features of its distribution, it is filtered. Finally, the SAR image is segmented using GMM, the parameters of which are estimated by EM algorithm. The pixels are classified into different classes according to their probability belonging to each Gaussian distribution. Index Terms—GMM, DWT, Speckle, EM, Threshold

I. I NTRODUCTION Environmental surveillance, earth resource mapping and defense organizations require broad area imaging at high resolutions. Very often, the imagery must be acquired in severe weather conditions or during night as well as day. Such a competence is provided by Synthetic Aperture Radar (SAR). Because of the ability to use in inclement weather conditions and high spatial resolution, SAR imaging systems are perhaps the most popular remote sensing technique used in the past decades [1]. It has been widely applied in surface surveillance, disaster monitoring etc., which make it urgent to realize the automatic interpretation of the obtained SAR images. In computer vision, image segmentation is generally the first stage to analyze or interpret an image automatically. It aims to partition the image into multiple segments having certain similar properties such as homogeneity, uniformity, texture etc. The objective of SAR image segmentation is to divide an image into regions of different characteristics. The main scope of image segmentation and classification is to group the pixels into obvious image regions that are easier to scrutinize. SAR images cannot be segmented successfully by using traditional image segmentation algorithms that perform well on optical images because of the existence of speckle noise in SAR images [2]. Speckle phenomenon can be modeled as a multiplicative noise, with standard deviation equal to pixel reflectivity value. The probability density function (PDF) of the pixel intensities in SAR images is also impressed by speckle noise. This phenomenon can be expressed by the nonlinear intensity inhomogeneity in SAR images [3]. Recently, a number of SAR image segmentation methods have been studied and categorized into different groups [4]

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viz global thresholding technique [5], segmentation using Artificial Neural Network (ANN) [6], wavelet transform based technique [7], statistical based approach [8] to name a few. The method based on statistical learning model is a critical one and intensive research is going on in this direction. Its main aim is to develop a probabilistic model first, and then train the model parameters using the given samples. After that, the model assigns each pixel to the class [4]. GMM are among the most basic and popular used statistical models. GMM is an effective algorithm suitable for image segmentation based on the Bayes theorem. It is a powerful, flexible and efficient statistical modeling tool for multivariate data. They can model any density function that contain enough mixture components because of their unique universal approximation ability. Expectation Maximization (EM) algorithm is the standard method used to learn GMM to observed data, which converges to a Maximum Likelihood (ML) estimate of the mixture parameters [9]. SAR image segmentation should take speckle noise into account and hence, the existing segmentation method are on the basis of studying the statistical behavior of SAR image using GMM. The outline of this paper is as follows. In Section II, technical details of Wavelet transform is discussed along with its de-noising capability. In Section III, the conventional GMM algorithm for image segmentation is introduced. Section IV gives the proposed model for SAR image segmentation. In Section V, the experimental results are shown. Finally, the conclusions are drawn in Section VI. II. WAVELET F ILTER Recently, wavelet transform has become a very popular method in the event of analyzing and de-noising of signals and images. The discrete wavelet transform (DWT) is an implementation of the wavelet transform using a discrete set of wavelet scales and translations, thereby separating the data into different frequency components with each component resolution matched to its scale. In fact, image analysis can not only be processed in space domain, but can also be performed in the frequency domain. The main feature of DWT is multi scale representation of the function. The 2-D DWT decomposes an image 𝑓 (𝑥, 𝑦) into high and low frequency components along each image dimension according eq. (1) and (2).

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2014 International Conference on Signal Processing and Integrated Networks (SPIN)

−1 𝑀 −1 𝑁 ∑ ∑ 1 𝑊𝜑 (𝑗, 𝑚, 𝑛) = √ 𝑓 (𝑥, 𝑦)𝜑𝑗,𝑚,𝑛 (𝑥, 𝑦) 𝑀 𝑁 𝑥=0 𝑦=0

𝑊𝜓 𝑖 (𝑗, 𝑚, 𝑛) = √

(1)

−1 𝑀 −1 𝑁 ∑ ∑ 1 𝑓 (𝑥, 𝑦)𝜓 𝑖 𝑗,𝑚,𝑛 (𝑥, 𝑦) (2) 𝑀 𝑁 𝑥=0 𝑦=0

for 𝑖 = 𝐻, 𝑉, 𝐷 where 𝑊𝜑 (𝑗, 𝑚, 𝑛) is the approximate coefficients and 𝑊𝜓 (𝑗, 𝑚, 𝑛) are the detail coefficients along horizontal, vertical and diagonal direction respectively as shown in Figure 1.

Fig. 1.

Decomposition using DWT

The 2D-DWT technique can be implemented by using digital filters and downsamplers [10]. We define four different filters by using two separate 1D transform given by eq. (3) to (6). 𝜑(𝑥, 𝑦) = 𝜑(𝑥)𝜑(𝑦)

(3)

𝜓 𝐻 (𝑥, 𝑦) = 𝜓(𝑥)𝜑(𝑦)

(4)

𝜓 𝑉 (𝑥, 𝑦) = 𝜑(𝑥)𝜓(𝑦)

(5)

𝜓 𝐷 (𝑥, 𝑦) = 𝜓(𝑥)𝜓(𝑦)

(6)

This operation spits the image into four bands viz. Sub band LL1 represents the horizontal and vertical low frequency components of the image. ∙ Sub band HH1 represents the horizontal and vertical high frequency components of the image. ∙ Sub band LH1 represents the horizontal low and vertical high frequency components and. ∙ Sub band HL1 represents the horizontal high and vertical low frequency components. To obtain the next level in the decomposition the two filters are again applied, but only to the LL1 sub band. Again, we obtain four sub bands labeled LL2, HH2, LH2, and HL2 with ∙

Fig. 2.

2 level DWT decomposition

representations similar to the first level sub bands shown in Figure 2. The low frequency content contain much of the information and are called the approximate coefficients (LL band) while high frequency content which contributes to noise and edges are called the detailed coefficients (LH,HL,HH band). Wavelet de-noising tries to restore the image corrupted with noise by preserving the signal characteristics, regardless of its frequency content. In wavelet domain, most of the wavelet coefficients have contribution to noise while a small part of wavelet coefficients have contribution to the signal energy. The wavelet coefficients can thus be divided into two categories; the first kind of wavelet coefficients are obtained by noise transformation, the amplitude of this type is small, but the number is more. The second type of wavelet coefficients is obtained by transformation of signal, and contains the result of signal transformation, the amplitude of which is great, but fewer in number. A general wavelet based procedure for denoising the image based on threshold algorithm [11] is as follows: 1) Calculate the DWT of the noisy image to produce the noisy wavelet coefficients. 2) Apply appropriate threshold limit to the detail wavelet coefficients (threshold may be universal or sub band adaptive). 3) Compute the Inverse DWT of the threshold wavelet coefficients to get the de-noised estimate. 4) There are two thresholding functions frequently used, i.e. hard threshold (setting to zero the elements whose absolute values are lower than the threshold) and soft thresholding (first setting the elements whose absolute values are lower than the threshold to zero and then scaling the non zero coefficients towards zero). With this way, noise can not only be reduced, but also the details of images can be preserved better. III. G AUSSIAN M IXTURE M ODEL Image is a matrix where each element is a pixel. In standard GMM, each pixel 𝑥𝑖 is considered to be a random variable whose probability density function Φ(𝑥𝑖 ∣Θ𝑗 ) is a Gaussian function [9]. For the case of a single real-valued variable 𝑥, the Gaussian distribution has its own mean 𝜇 and standard deviation 𝜎 and is defined by eq. (7).

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2 1 Φ(𝑥∣Θ) = √ exp(− (𝑥−𝜇) ) (7) 2 2𝜎 2𝜋𝜎 2 where Θ = {𝜇, 𝜎}. For the case of a D-dimensional vector 𝑥, each Gaussian distribution Φ(𝑥∣Θ) can be written in the form of eq. (8).

} 1 𝑇 − (𝑥 − 𝜇) Σ−1 (𝑥 − 𝜇) 2 (8) where Θ = (𝜇, Σ), 𝜇 =mean, Σ =covariance matrix. 1

1 exp Φ(𝑥∣Θ) = 𝐷/2 ∣Σ∣1/2 (2𝜋)

𝐾 ∑

] 2 (𝑡+1)

𝜎𝑗

(9)

where Π = {𝜋1 , 𝜋2 , ..., 𝜋𝑘 } and 𝜋𝑗 is the prior distribution of the pixel 𝑥𝑖 belonging to the label Ω𝑗 . Each Gaussian distribution Φ(𝑥𝑖 ∣Θ𝑗 ) is called a component of the mixture. The complete GMM is parameterized by the mean vectors, co-variance matrices and mixture weights from all component densities. In order to estimate the parameters, ML estimation is used. Eq. (10) gives the log likelihood function of the standard GMM. ⎧ ⎫ 𝑁 𝐾 ⎨∑ ⎬ ∑ log 𝜋𝑗 Φ(𝑥𝑖 ∣Θ𝑗 ) (10) 𝐿(Θ∣Π, 𝑋) = ⎩ ⎭ 𝑖=1

=

𝑖=1

)2 (𝑡) ( 𝑧𝑖,𝑗 𝑥𝑖 − 𝜇𝑗 (𝑡+1) 𝑁 ∑ 𝑖=1

(𝑡+1)

𝜋𝑗

=

(13) (𝑡) 𝑧𝑖,𝑗

𝑁 1 ∑ (𝑡) 𝑧 𝑁 𝑖=1 𝑖,𝑗

(14)

where t indicates the iteration step. Step 4: Evaluate the log-likelihood and check the convergence of either the log-likelihood function, or the parameter values. If the convergence criterion is not satisfied, then go to step 2. Once, the parameters are optimized, maximum aposteriori is used to assign labels to each pixel in the image. ∙

IV. P ROPOSED M ODEL 𝜋𝑗 Φ(𝑥𝑖 ∣Θ𝑗 )

𝑗=1

𝑗=1

The aim of ML estimation is to find the model parameters which maximizes the likelihood of the GMM given the data set 𝑋. ML parameters estimates can be obtained iteratively using a special case of EM algorithm. The EM algorithm for Gaussian mixture model can be summarized as follows: ∙ Step 1: Initialize the parameters {Θ, Π} = {𝜇𝑗 , 𝜎𝑗 , 𝜋𝑗 } ∙ Step 2: Evaluate the posterior probability 𝑧𝑖,𝑗 using the current parameter values according to eq. (11). (𝑡) 𝑧𝑖,𝑗

[

{

Let 𝑥𝑖 ; 𝑖=1,2,..,N; denote the observation at the 𝑖-th pixel of an image. Labels are denoted by Ω1 , Ω2 , ..., Ω𝑘 . If the distribution of 𝑥 can be modeled by a mixture of k gaussians, then the standard GMM assumes that the density function at an observation 𝑥𝑖 is given by eq. (9). 𝑓 (𝑥𝑖 ∣Π, Θ) =

𝑁 ∑

(𝑡)

The stochastic peculiarity of SAR image speckle distribution is a kind of multiplicative noise and is modeled by eq. (15). 𝑍 = 𝑋.𝑉

(15)

In it, Z is the intension of SAR image with noise, X is the stochastic peculiarity of radar backscattering for the target on the ground, V is the stochastic process of speckle [12]. When SAR image data is transformed with logarithmic transformation, speckle noise is approximately the independent additive Gauss white noise [13] as shown in eq. (16). 𝑙𝑜𝑔𝑍 = 𝑙𝑜𝑔𝑋 + 𝑙𝑜𝑔𝑉

(16)

By taking advantage of the logarithmic transformation, homomorphic de-speckling methods convert the multiplicative noise to an additive one. As a result, the problem of despeckling is reduced to the problem of rejecting an additive noise, where wavelet based de-noising is applied to perform this task.

(𝑡)

𝜋𝑗 Φ(𝑥𝑖 ∣Θ𝑗 ) = 𝐾 ∑ (𝑡) (𝑡) 𝜋𝑗 Φ(𝑥𝑖 ∣Θ𝑗 )

(11) Fig. 3.

Block diagram of proposed model

𝑙=1

∙

Step 3: Re-estimate the parameters {Θ, Π} = {𝜇𝑗 , 𝜎𝑗 , 𝜋𝑗 } and update the means, covariance values and prior distributions according to eq. (12) to (14). 𝑁 ∑ (𝑡+1)

𝜇𝑗

=

Figure 3 shows the conceptual flowchart of proposed method, which mainly consists two parts: 1) Preprocessing: Speckle Removal using DWT and 2) Postprocessing: Image Segmentation using GMM.

(𝑡) 𝑧𝑖,𝑗 𝑥𝑖

𝑖=1 𝑁 ∑

𝑖=1

(12) (𝑡) 𝑧𝑖,𝑗

In the preprocessing part, a 2 level DWT is applied to the transformed image and soft thresholding technique is adopted

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Fig. 4.

Fig. 5.

Original image(China Lake airport)

Fig. 6.

Original image(pipeline over the Rio Grande river)

Fig. 7. Segmented image(pipeline over the Rio Grande river) with 4 classes

Segmented image(China Lake airport) with 4 classes

to suppress the noise. In the post-processing segment, GMM is applied to the filtered image, the parameters of which are determined by EM algorithm. Finally, the pixels are classified according to their maximum probability belonging to each gaussian distribution class to get the segmented output. V. E XPERIMENTAL R ESULTS

VI. CONCLUSION Speckle noise inherent in SAR images is one of the major cause hampering good segmentation results. This paper proposes a method by combining the advantages of statistical based GMM and multi resolution analysis of DWT. Experimental results show that that the proposed method gives prompt results in SAR image segmentation mitigating the effect of speckle noise.

Two primary SAR images are taken from Sandia National Laboratories SAR imagery database. One is the pipeline over the Rio Grande river near Albuquerque, New Mexico and the other is the China Lake airport, California. Figures 4, 6 show the original image and Figures 5, 7 show the segmented image of the first and second SAR image respectively using the proposed method. From the segmentation results, it can be seen that different objects such as the runway of the airport or the bridge above the river is clearly segmented out without the speckle noise effect. However, due to the sensitivity of GMM with parameter initialization and loss of edge information on account of wavelet denoising; some false classification points may exist near the boundaries which need further refinement which is also our future work.

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