Image Processing for Remote Sensing - Chapter 6

6 Change-Detection Methods for Location of Mines in SAR Imagery Maria Tates, Nasser Nasrabadi, Heesung Kwon, and Carl White CONTENTS 6.1 Introduction....................................................................................................................... 135 6.2 Difference, Euclidean Distance, and Image Ratioing Change-Detection Methods.............................................................................................................................. 137 6.3 Mahalanobis Distance–Based Change Detection......................................................... 137 6.4 Subspace Projection–Based Change Detection ............................................................ 138 6.5 Wiener Prediction–Based Change Detection................................................................ 139 6.6 Implementation Considerations..................................................................................... 140 6.6.1 Local Implementation Problem Formulation................................................... 140 6.6.2 Computing Matrix Inversion for Cÿ1 and Rÿ1................................................ 141 6.7 Experimental Results........................................................................................................ 142 6.8 Conclusions........................................................................................................................ 145 References ................................................................................................................................... 146 6.1 Introduction Using multi-temporal SAR images of the same scene, analysts employ several methods to determine changes among the set [1]. Changes may be abrupt in which case only two images are required or it may be gradual in which case several images of a given scene are compared to identify changes. The former case is considered here, where SAR images are analyzed to detect land mines. Much of the change-detection activity has been focused on optical data for specific applications, for example, several change-detection methods are implemented and compared to detect land-cover changes in multi-spectral imagery [2]. However, due to the natural limitations of optical sensors, such as sensitivity to weather and illumination conditions, SAR sensors may constitute a superior sensor for change detection as images for this purpose are obtained at various times of the day under varying conditions. SAR systems are capable of collecting data in all weather conditions and are unaffected by cloud cover or changing sunlight conditions. Exploiting the long-range propagation qualities of radar signals and utilizing the complex processing capabilities of current digital technology, SAR produces high-resolution imagery providing a wealth of infor-mation in many civilian and military applications. One major characteristic of SAR data is 135 © 2008 by Taylor & Francis Group, LLC 136 Image Processing for Remote Sensing the presence of speckle noise, which poses a challenge in classification. SAR data taken from different platforms also shows high variability. The accuracy of preprocessing tasks, such as image registration, also affects the accur-acy of the postprocessing task of change detection. Image registration, the process of aligning images into the same coordinate frame, must be done prior to change detection. Image registration can be an especially nontrivial task in instances where the images are acquired from nonstationary sources in the presence of sensor motion. In this paper we do not address the issue of image registration, but note that there is a plethora of information available on this topic and a survey is provided in Ref. [3]. In Ref. [4], Mayer and Schaum implemented several transforms on multi-temporal hyperspectral target signatures to improve target detection in a matched filter. The transforms implemented in their paper were either model-dependent, based solely on atmospheric corrections, or image-based techniques. The image-based transforms, includ-ing one that used Wiener filtering, yielded improvements in target detection using the matched filter and proved robust to various atmospheric conditions. In Ref. [5], Rignot and van Zyl compared various change-detection techniques for SAR imagery. In one method they implemented image ratioing, which proves more appropri-ate for SAR than simple differencing. However, the speckle decorrelation techniques they implemented outperform the ratio method under certain conditions. The results lead to the conclusion that the ratio method and speckle decorrelation method have compli-mentary characteristics for detecting gradual changes in the structural and dielectric properties of remotely sensed surfaces. In Ref. [6], Ranney and Soumekh evaluated change detection using the so-called signal subspace processing method, which is similar to the subspace projection tech-nique [7], to detect mines in averaged multi-look SAR imagery. They posed the problem as a trinary hypothesis-testing problem to determine whether no change has occurred, a target has entered a scene, or whether a target has left the scene. In the signal subspace processing method, one image is modeled as the convolution of the other image by a transform that accounts for variances in the image-point response between the refer-ence and test images due to varying conditions present during the different times of data collection. For the detection of abrupt changes in soil characteristics following volcanic eruptions, in Ref. [8] a two-stage change-detection process was utilized using SAR data. In the first stage, the probability distribution functions of the classes present in the images were estimated and then changes in the images via paradoxical and evidential reasoning were detected. In Ref. [9], difference images are automatically analyzed using two different methods based on Bayes theory in an unsupervised change-detection paradigm, which estimates the statistical distributions of the changed and unchanged pixels using an iterative method based on the expectation maximization algorithm. In Ref. [10], the performance of the minimum error solution based on the matrix Wiener filter was compared to the covariance equalization method (CE) to detect changes in hyperspectral imagery. They found that CE, which has relaxed operational requirements, is more robust to imperfect image registration than the matrix Wiener filter method. In this chapter, we have implemented image differencing, ratio, Euclidean distance, Mahalanobis distance, subspace projection-based, and Wiener filter-based change detec-tion and compared their performances to one another. We demonstrate all algorithms on co-registered SAR images obtained from a high resolution, VV-polarized SAR system. In Section 6.2 we discuss difference-based change detection, Euclidean distance, and ratio change-detection methods, which comprise the simpler change-detection methods imple-mented in this paper. In Section 6.3 through Section 6.5, we discuss more complex methods such as Mahalanobis distance, subspace projection–based change detection, © 2008 by Taylor & Francis Group, LLC Change-Detection Methods for Location of Mines in SAR Imagery 137 and Wiener filter–based change detection, respectively. We consider specific implemen-tation issues in Section 6.6 and present results in Section 6.7. Finally, a conclusion is provided in Section 6.8. 6.2 Difference, Euclidean Distance, and Image Ratioing Change-Detection Methods In simple difference–based change detection, a pixel from one image (the reference image) is subtracted from the pixel in the corresponding location of another image (the test image), which has changed with respect to the reference image. If the difference is greater than a threshold, then a change is said to have occurred. One can also subtract a block of pixels from one image from the corresponding block from a test image. This is referred to as Euclidean difference if the blocks of pixels are arranged into vectors and the L-2 norm of the difference of these vectors is computed. We implemented the Euclidean distance as in Equation 6.1, where we computed the difference between a block of pixels (arranged into a one-dimensional vector) from reference image, fX, and the corresponding block in the test image, fY, to obtain an error, eE: eE ¼ ðy ÿxÞT (6:1) where x and y are vectors of pixels taken from fX and fY, respectively. We display this error, eE, as an image. When no change has occurred this error is expected to be low and the error will be high when a change has occurred. While simple differencing considers only two pixels in making a change decision, the Euclidean distance takes into account pixels within the neighborhood of the pixel in question. This regional decision may have a smoothing effect on the change image at the expense of additional computational com-plexity. Closely related to the Euclidean distance metric for change detection is image ratioing. In many change-detection applications, ratioing proved more robust to illumination effects than simple differencing. It is implemented as follows: eR ¼ y (6:2) where y and x are pixels from the same locations in the test and reference images, respectively. 6.3 Mahalanobis Distance–Based Change Detection Simple techniques like differencing and image ratioing suffer from sensitivity to noise and illumination intensity variations in the images. Therefore, these methods may prove adequate only for applications to images with low noise and which show little illumin-ation variation. These methods may be inadequate when applied to SAR images due to the presence of highly correlated speckle noise, misregistration errors, and other unimportant © 2008 by Taylor & Francis Group, LLC 138 Image Processing for Remote Sensing changes in the images. There may be differences in the reference and test images caused by unknown fluctuations in the amplitude-phase in the radiation pattern of the physical radar between the two data sets and subtle inconsistencies in the data acquisition circuitry [5]; consequently, more robust methods are sought. One such method uses the Mahalanobis distance to detect changes in SAR imagery. In this change-detection application, we obtain an error (change) image by computing the Mahalanobis distance between x and y to obtain eMD as follows: eMD ¼ (x ÿy)TCÿ1(x ÿy) (6:3) The Cÿ1 term in the Mahalanobis distance is the inverse covariance matrix computed from vectors of pixels in fX. By considering the effects of other pixels in making a change decision with the inclusion of second-order statistics, the Mahalanobis distance method is expected to reduce false alarms. The Cÿ1 term should improve the estimate by reducing the effects of background clutter variance, which, for the purpose of detecting mines in SAR imagery, does not constitute a significant change. 6.4 Subspace Projection–Based Change Detection To apply subspace projection [11] to change detection a subspace must be defined for either the reference or the test image. One may implement Gram–Schmidt orthogonaliza-tion as in [6] or one can make use of eigen-decomposition to define a suitable subspace for the data. We computed the covariance of a sample from fX and its eigenvectors and eigenvalues as follows: CX ¼ (X ÿ mX)(X ÿ mX)T (6:4) where X is a matrix of pixels whose columns represent a block of pixels from fX, arranged as described in Section 6.6.1, having mean mX ¼ X1NN. 1NN is a square matrix of size NN whose elements are 1/N, where N is the number of columns of X. We define the subspace of the reference data, which we can express using eigen-decomposition in terms of its eigenvectors, Z, and eigenvalues, V: CX ¼ ZVZT (6:5) We truncate the number of eigenvectors in Z, denoted by Z, to develop a subspace projection operator, PX: PX ¼ ZZT (6:6) The projection of the test image onto the subspace of the reference image will provide a measure of how much of the test sample is represented by the reference image. Therefore, by computing the squared difference between the test image and its projection onto the subspace of the reference image we obtain an estimate of the difference between the two images: eSP(y) ¼ [yT(I ÿPX)y] (6:7) © 2008 by Taylor & Francis Group, LLC Change-Detection Methods for Location of Mines in SAR Imagery 139 We evaluated the effects of various levels of truncation and display the best results achieved. In our implementations we include the Cÿ1 term in the subspace projection error term as follows: eSP(y) ¼ yT(I ÿPX)TCÿ1(I ÿ PX)y (6:8) We expect it will play a similar role as it does in the Mahalanobis prediction and further diminish false alarms by suppressing the background clutter. Expanding the terms in (6.8) we get eSP(y) ¼ yTCÿ1y ÿyTPTCÿ1y ÿ yTCÿ1PXy þ yTPTCÿ1PXy (6:9) It can be shown that yTCÿ1PXy ¼ yTPXCÿ1y ¼ yTPXCÿ1PXy, so we can rewrite (6.9) as eSP(y) ¼ yTCÿ1y ÿ yCÿ1PXyT (6:10) 6.5 Wiener Prediction–Based Change Detection We propose a Wiener filter–based change-detection algorithm to overcome some of the limitations of simple differencing, namely to exploit the highly correlated nature of speckle noise, thereby reducing false alarms. Given two stochastic, jointly wide-sense stationary signals, the Wiener equation can be implemented for prediction of one data set from the other using the auto-and cross-correlations of the data. By minimizing the variance of the error committed in this prediction, the Wiener predictor provides the optimal solution in the linear minimum mean-squared error sense. If the data are well represented by the second-order statistics, the Wiener prediction will be very close to the actual data. In applying Wiener prediction theory to change detection the number of pixels where abrupt change has occurred is very small relative to the total number of pixels in the image; therefore, these changes will not be adequately represented by the correlation matrix. Consequently, there will be a low error in the prediction where the reference and test images are the same and no change exists between the two, and there will be a large error in the prediction of pixels where a change has occurred. The Wiener filter [11] is the linear minimum mean-squared error filter for second-order stationary data. Consider the signal yW ¼ Wx. The goal of Wiener filtering is to find W, which minimizes the error, eW ¼ k y ÿ yW k, where y represents a desired signal. In the case of change detection y is taken from the test image, fY, which contains changes as compared to the reference image, fX. The Wiener filter seeks the value of W, which minimizes the mean-squared error, eW. If the linear minimum mean-squared error esti-mator of Y satisfies the orthogonality condition, the following expressions hold: E{(Y ÿ WX)XT} ¼ 0 E{YXT} ÿ E{WXXT} ¼ 0 RYX ÿ WRXX ¼ 0 (6:11) where E{} represents the expectation operator, X is a matrix of pixels whose columns represent a block of pixels from fX, Y is a matrix of pixels from fY whose columns © 2008 by Taylor & Francis Group, LLC ... - tailieumienphi.vn