Sar Paper Jordi

Change Detection on SAR Images by using a Parametric Estimation of the Kullback-Leibler Divergence Jordi Inglada CNES, French Space Agency, DSO/OT/QTIS, bpi 811 CNES, 18, Avenue Edouard Belin, 31401 Toulouse Cedex 4, France Tel: +33 (0)5 61 27 33 97, Fax: +33 (0)5 61 27 31 67 e-mail: [email protected]

Abstract – This communication presents a method for performing change detection using a pair of SAR images acquired at different dates. The main difficulty with SAR images is the presence of speckle noise which may produce noisy change images if they are acquired with slightly different angles. The technique proposed in the present paper uses a parametric estimation of the probability distributions locally in each image as a characterization of the surfaces. The change is measured as a distance between these probability laws. The dissimilarity measure between the statistical distributions used here is a symmetric version of the Kullback-Leibler divergence. I. INTRODUCTION While the detection of changes between two optical images can easily be addressed by the use of indexes based on differences, this task is much more difficult in radar images because of the speckle noise. The speckle makes very difficult to use images with even slightly different acquisition angles. It is now accepted that the statistics of SAR images can be well modeled by the family of probability distributions known as the Pearson system [1]. This family of distributions is very easy to use, since the analytical expression of the probability density function for a given area in the image can be obtained by computing two parameters, β1 and β2 which only depend on the four first statistical moments of the data. Hence, the Pearson system allows for easily finding the statistical distribution of a given set of samples (a neighborhood).

expression for the statistical distributions, the computation of the Kullback-Leibler divergence is straightforward. Results on real data (volcanic eruption of the mount Nyiragongo, D.R. Congo, January 2001) are shown and compared to ground truth and other classical change detection algorithms currently used for SAR images. II. THE ALGORITHM In the literature, there exist plenty of measures used for change detection. Most of them are based on radiometry evolutions : pixel difference or ratios. Even if the intensity ratio between radar images could be thought as a good choice for change detection (Touzi et al. [2] have used it for edge detection), it doesn’t make sense when images are acquired with different incidence angles. However, in many operational applications of remote sensing, as for example natural hazards management, it is difficult to use images acquired with the same incidence angles, since the multiplicity of modes for current SAR sensors (Radarsat, Envisat) reduce the probability of finding an archive acquisition which fits the first available acquisition after the event of interest. We make the assumption that the statistical distribution of the pixels will remain more stable than the pixel values themselves. Thus, we use a dissimilarity measure based on the change of pixel statistics. A. Kullback-Leibler divergence

The change detection algorithm proposed in this paper is based on the modification of the statistics between the two acquisition dates of each pixel’s neighborhood. A pixel will be considered as having changed if its statistical distribution changes from one image to the other. In order to quantify this change, we need a measure which maps the two estimated statistical distributions (one for each date) for each pixel to a scalar index of change. Several approaches could be taken, like computing the mean square error between the two distributions, the norm of a vector of moments, etc.

Let P and Q be two probability laws. The Kullback-Leibler divergence from Q to P, in the case where these two laws have the densities p et q, is given by :

We have chosen to use a measure derived from the information theory called Kullback-Leibler divergence. Since, thanks to the use of the Pearson system, we have an analytical

One can easily show that K(Q|P) ≥ 0 with equality when the two laws are identical. One can thus use K(Q|P) as a measure of the divergence from Q to P. This measure is not

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K(Q|P) =




log

p(x) p(x)dx q(x)

(1)

In the case of discrete probability laws, defined on the finite set of points {x1 , x2 , · · · }, this divergence is defined as follows : P({xi }) P({xi }) (2) K(Q|P) = ∑ log Q({x i }) i

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symmetric : K(Q|P) = K(P|Q). One can define a symmetric version by writing: D(P, Q) = D(Q, P) = K(Q|P) + K(P|Q)

(3)

that we will call Kullback-Leibler distance, KLD. B. Parametric estimation of the Kullback-Leibler distance In order to achieve a robust estimation of the KLD, we will use a parametric estimation of the two probability densities, and we will numerically compute the KLD. 1) Probability density estimation : the Pearson system: The Pearson system consists on a set of eight distributions (gaussian, gamma, beta, etc.). Details on the theory about the Pearson system can be found in [3]. All the parameters of these distributions can be obtained as a function of two parameters : β1 =

µ23 , µ32

(4a)

β2 =

µ4 . µ22

(4b)

That means that any distribution of the Pearson system can be identified from a given sample by computing the 4 first statistical moments. That also means that any distribution can be represented by a point on the (β1 , β2 ) plane, the Pearson plane. For instance, the gaussian distribution has (β1 , β2 ) = (0, 3). Figure 1 shows an example of distribution estimation. The histogram of the image patch is plotted with dots and the corresponding estimated Pearson distribution is shown in continuous line.

Fig. 1.

Example of Pearson density estimation.

The main advantage of working with the Pearson system is that a robust estimate of the first four moments can be obtained using only a few pixels. Thus, one can locally estimate the probability distribution using a small window on the image.

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Fig. 2.

Kullback-Leibler distance on the Pearson plane.

Fig. 3.

Kullback-Leibler distance on the Pearson plane.

2) Kullback-Leibler distance on the Pearson plane: Once the probability densities of the two image patches that we want to compare are estimated, we numerically can compute the KLD. Actually, one could argue that give the fact that we have a parametric representation of each probability distribution, (β1 , β2 ), an euclidean distance on the Pearson plane could be used in order to asses the change of probability law. In order to show that the KLD gives a different information than this euclidean distance, we propose the following experience : we set a given probability density, that is a point in the Pearson plane; then we compute the DKL between this distribution and any other distribution of the Pearson system (any other point on the Pearson plane). To examples of this experience are shown on figures 2 and 3, where each point of the Pearson plane has been given a gray level which is proportional to the DKL. If the information given by the DKL was analogous to an euclidean distance on the (β1 , β2 ) vector, we should see an isotropic gradient of the gray levels centered on the distribution to which we compute the KLD. We see that this 4105

is not the case. III. E XAMPLE OF APPLICATION We show here an example of application of these technique to a real case. A couple of Radarsat images after and before the eruption of the Nyiragongo volcano (D.R. of Congo) which occurred in January 2002 have been used. The first Radarsat available acquisition was a F2 beam for which no archive data was available. A F5 archive beam was used instead. In the figure 4 on can see the images and the comparison between the result obtained with the classical image intensity ratio approach and the method proposed here. One can see that the changes detected with the KLD method have sharp edges and the image is less noisy. However, the type of change detected is the same using both methods. IV. C ONCLUSIONS A method for performing change detection on SAR images has been proposed. The main advantage of this method, when compared to the classical image intensity ratio, is that images acquired with slightly different incidence angles can be used. Our method estimates the evolution of the probability law on the neighborhood of each pixel. Actually, our method can be considered as an extension of the classical one, which actually measures the evolution of the first statistical moment (the mean intensity on the pixel neighborhood). The KLD method measures the evolution of the probability distribution, which also includes the evolution of the mean. One such method, usually would need a lot of samples (large image window, which implies a loss of geometrical resolution) in order to achieve a robust estimate of the probability distributions. We can avoid these problem by using a parametric estimation of the laws thanks to the Pearson system.

(a) Before

(b) After

(c) Intensity ratio

(d) Kulback-Leibler distance

R EFERENCES [1] Y. Delignon, R. Garello, and A. Hillion, “Statistical modelling of ocean SAR images,” IEE Proc. on Radar, Sonar and Navig., vol. 44, no. 66, pp. 348–354, 1997. [2] R. Touzi, A. Lopes, and P. Bousquet, “A Statistical and Geometrical Edge Detector for SAR Images,” IEEE Transactions on Geoscience and Remote Sensing, vol. 26, no. 6, pp. 764–773, 1988. [3] N. Johnson and S. Kotz, Distributions in statistics: continuous univariate distributions. Wiley Interscience, 1969.

Fig. 4. Change detection : comparison between the image intensity ratio and the KLD using the same window size (35 × 35 pixels).

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