Theorem Dimas Irion Alves , Bruna Gregory Palm , Hans Hellsten, Senior Member, IEEE, Viet Thuy Vu , Senior Member, IEEE, Mats I

5560 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 13, 2020 Wavelength-Resolution SAR Change Detection Using Bayes’ Theorem Dimas Irion Alves , Bruna Gregory Palm , Hans Hellsten, Senior Member, IEEE, Viet Thuy Vu , Senior Member, IEEE, Mats I. Pettersson , Renato Machado , Member, IEEE, Bartolomeu F. Uchôa-Filho , Senior Member, IEEE, and Patrik Dammert, Senior Member, IEEE

Abstract—This article presents Bayes’ theorem for wavelength- acquisitions separated in time can be found in SAR images with resolution synthetic aperture radar (SAR) change detection method a change detection method. development. Different change detection methods can be derived Among these applications, it is possible to highlight the using Bayes’ theorem in combination with the target model, clutter-plus-noise model, iterative implementation, and nonitera- detection of concealed targets in foliage-penetrating (FOPEN) tive implementation. As an example of the Bayes’ theorem use for applications, which has been of great interest for a long time [5]. wavelength-resolution SAR change detection method development, However, when change detection is employed in conventional we propose a simple change detection method with a clutter-plus- microwave SAR images for FOPEN applications, a large number noise model and noniterative implementation. In spite of simplicity, of false alarms are observed. Ultrahigh frequency and very high the proposed method provides a very competitive performance in terms of probability of detection and false alarm rate. The best frequency SAR systems are excellent options to overcome this result was a probability of detection of 98.7% versus a false alarm limitation [5]. rate of one per square kilometer. The usage of low frequencies for FOPEN applications is Index Terms—Bayes’ theorem, CARABAS, change detection, associated with large fractional bandwidth and a wide antenna synthetic aperture radar (SAR), wavelength resolution. bandwidth. Systems with such characteristics have resolutions in the order of the radar signal wavelengths. That is the reason why those are frequently named wavelength-resolution SAR I. INTRODUCTION systems [6]. The images generated by wavelength-resolution YNTHETIC aperture radar (SAR) plays an important role SAR systems do not suffer from the speckle noise. The scattering S in surveillance, geoscience, and remote sensing applica- process is related to scatterers with dimensions in the order of the tions. The capabilities to provide broad coverage and effectively signal wavelengths, which, for low-frequency SAR, are related operate in all weather conditions are major advantages of SAR to big objects (tree trunk, house, trucks) that are stable in time, in comparison to other sensor systems. SAR change detection and tend to be less affected by weather conditions [7]. These has been researched for many decades and it is an important characteristics indicate that wavelength-resolution SAR images research area used for different applications, such as detection are adequate for change detection in FOPEN applications. of concealed targets [1], ground scene monitoring [2], polarime- Wavelength-resolution SAR has been a research area since try [3], and even GMTI [4]. Generally, the changes between two 2002, and this time instance is marked by the measurement campaign in Vidsel, Sweden [8], [9]. The goal of the measurement campaign was to provide data to evaluate the performance Manuscript received February 13, 2020; revised June 23, 2020 and August 14, 2020; accepted September 9, 2020. Date of publication September 18, 2020; of change detection for targets obscured by foliage. Different date of current version September 28, 2020. This work was supported in part by passes with distinct flight headings have been tested. In the the Brazilian National Council for Scientific and Technological Development ground scene, the targets (vehicle) have been deployed at differ- (CNPq), in part by the Swedish-Brazilian Research and Innovation Centre (CISB), in part by Coordination for the Improvement of Higher Education ent locations and with distinct orientations. One of the measure- Personnel (CAPES), and in part by Saab AB. (Corresponding author: Dimas ment campaigns’ outcomes is 24 SAR images (calibrated and Irion Alves.) coregistered) that are available for the wavelength-resolution Dimas Irion Alves is with the Federal University of Pampa, Alegrete 97546- 550, Brazil, and also with the Federal University of Santa Catarina, Florianópolis SAR change detection research as a challenging problem [10]. 88040-900, Brazil (e-mail: [email protected]). A significant number of wavelength-resolution SAR change de- Bruna Gregory Palm and Renato Machado are with the Aeronautics In- tection methods have been developed. A common aspect among stitute of Technology, São José dos Campos 12228-900, Brazil (e-mail: [email protected]; [email protected]). them is that they are designed with the processing sequence: Hans Hellsten and Patrik Dammert are with the Saab Surveillance, Saab information extraction, thresholding, and false alarm minimiza- AB, 412 89 Gothenburg, Sweden (e-mail: [email protected]; tion. The information extraction can simply be achieved by a [email protected]). Viet Thuy Vu and Mats I. Pettersson are with the Blekinge Institute subtraction followed by a filter, e.g., adaptive noise canceler [11] of Technology, 37179 Karlskrona, Sweden (e-mail: [email protected]; and smoothing filter [12]. This is possible thanks to the dominant [email protected]). thermal noise present in wavelength-resolution SAR images. Bartolomeu F. Uchôa-Filho is with the Federal University of Santa Catarina, Florianópolis 88040-900, Brazil (e-mail: [email protected]). The information extraction can also be achieved but a likeli- Digital Object Identifier 10.1109/JSTARS.2020.3025089 hood ratio test with the help of clutter and noise models. The

This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ ALVES et al.: WAVELENGTH-RESOLUTION SAR CHANGE DETECTION USING BAYES’ THEOREM 5561

simplest models for clutter and noise are the bivariate complex II. BAYES’THEOREM FOR CHANGE DETECTION normal distribution for coherent change detection [8], and, con- METHOD DEVELOPMENT sequently, bivariate Rayleigh distribution for incoherent change In this section, we present Bayes’ theorem in the SAR change detection [13]. detection scenario. The development of change detection meth- Oppositely as the current trend of using machine learning and ods based on Bayes’ theorem is also provided. neural networks in change detection [14]–[16], several change detection methods available in the literature are related to probability theory in general and Bayes’ theorem in particular [17]– A. Bayes’ Theorem in the SAR Change Detection Scenario [20]. These methods do not require the training stage, resulting Bayes’ theorem is used to describe the probability of an event, in less computational complexity and fewer application require- given some prior knowledge of conditions that could be related ments. They also tend to provide good detection performance for to this event. It can be stated as wavelength-resolution SAR image applications [11]–[13]. An P (B|A)P (A) example of Bayes-based technique can be found in [18], where P (A|B)= (1) P (B) an unsupervised change detection method with thresholding is presented. The method consists of a Bayes-based estimator, where A and B are two events, P (A) and P (B) probabilities which combines prior and current data knowledge for improving of events A and B, respectively, and P (A|B) is the conditional the change detection performance. Another example is presented probability of event A given that event B has occurred. in [20], where the authors proposed a change detection method A common SAR change detection scenario is considered for for polarimetric images using a Bayes classiﬁer. In this article, an interpretation of Bayes’ theorem. This scenario is character- we use Bayes’ theorem for wavelength-resolution SAR change ized by the use of two complex images; one is the surveillance detection method development. Based on Bayes’ theorem, the image, i.e., the image where it is desired to ﬁnd the targets, probability of change in a SAR scene can be estimated from and the other is the reference image, i.e., the image used to aid the data histogram of reference and surveillance images and to characterize the clutter. By considering Bayes’ theorem, the either a target model or a clutter-plus-noise model. This results in probability of detecting a change in one pixel under test, given two directions for change detection method development: one is the complex value of the tested pixel in the surveillance image zU based on the target model and the other on the clutter-plus-noise and given the complex value of the tested pixel in the reference model. From the implementation point of view, the probability image zR, can be expressed by of change in a SAR scene can be estimated either noniteratively or iteratively. For the iterative implementation, the method will P (zU |s ≡ sT ,zR) P (s ≡ sT |zR) P (s ≡ sT |zU ,zR)= (2) detect one by one target and update the data histogram iteratively P (zU |zR) until no new target is found. On the contrary, the noniterative where s ≡ sT is the statement that the given image pixel contains implementation helps to detect all possible targets at the same a change and can write shortly sT . In the opposite case, s ≡ sT time without updating the data histogram. is the statement that the given image pixel contains no change It is worth mentioning that part of this article has been and can write shortly sC . The conditional probability P (sT |zR) published in [21] and [22], focusing on the target model and is the target probability and since zR is independent of the the iterative implementation. Initial evaluation of the change statement on change, it can be written by detection performance considered very few images and targets, e.g., only two targets and two images in [22]. For this reason, the MK P (sT |zR)=P (sT )= (3) clutter and noise model and the noniterative implementation are N focused on this article. The evaluation of the change detection where K is the number of detected changes, M is the number performance is discussed in detail. The dataset for the assess- of pixels that a change occupies, and N is the number of image ment consists of 24 SAR images and 600 targets [10]. Since the pixels. dataset is available, it is easier to compare the change detection The conditional probability P (zU |zR) can be described by performance of different methods that use the same dataset. two mutually exclusive events, which are based on a presence The remainder of this article is organized as follows. Section II of a change sT or an absence of a change sC presents Bayes’ theorem for change detection method development. Two possible expressions that can be used for change P (zU |zR)=P (zU |sT ,zR) P (sT |zR) detection probability calculation are derived using Bayes’ theo- P z |s ,z − P s |z rem. One is based on the target model, and the other is based on + ( U C R)[1 ( T R)] the clutter-plus-noise model. Two possible ways to implement = P (zU |sT ,zR) P (sT ) the change detection methods: iterative and noniterative are also P z |s ,z − P s . discussed in this section. A change detection method is devel- + ( U C R)[1 ( T )] (4) oped in Section III using a clutter-plus-noise model, bivariate We can also rearrange (4) in a new form as Rayleigh, and the noniterative implementation. Section IV gives the data description, experimental results with CARABAS data, P (zU |zR) − P (zU |sC ,zR)[1− P (sT )] P (zU |sT ,zR)= . and experimental evaluation. Concluding remarks are provided P (sT ) in Section V. (5) 5562 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 13, 2020

1) Case 1: The ﬁrst case is straight ahead. If we insert (4) into the denominator of (2) and then divide the numerator and denominator with P (zU |sT ,zR)P (sT |zR), we get the conditional probability in the following form: 1 P (sT |zU ,zR)= . (6) P (zU |sC ,zR)[1− P (sT |zR)] 1+ P (zU |sT ) P (sT ) An approximate form of (6) is given by 1 P (sT |zU ,zR) ≈ (7) P (zU |sC ,zR) 1+ P (zU |sT ) P (sT ) due to the fact that N MK and hence MK − P s |z − ≈ . 1 ( T R)=1 N 1 (8) Apart from the number of targets assumed, the quotient η(z)= P (zU |sT )/P (zU |sC ,zR) will decide the conditional probability P (sT |zU ,zR). For this probability to be high, η(z) must be high and vice versa. The probability P (zU |sC ,zR), so-called clutter change distribution, is estimated by the histogram analysis of the SAR images, whereas the probability P (zT |sC ,zR), so-called target change distribution, is shown to be Fig. 1. Block diagram for iterative and noniterative implementation.

P (zT |sC ,zR) B. Implementation 1 dϕ = π2 z − z 2 2 (6) and (11) and their approximations (7) and (12) suggest 4 ( max min) Δϕ(zU ,zR) zU + zR − 2zU zR cos ϕ (9) to us two different ways to calculate the conditional probability P (sT |zU ,zR) for change detection: iterative and noniterative. where zmax and zmin are the upper and lower magnitude limits, In this part, we focus on the iterative and noniterative implemen- and Δφ(zU ,zR) is the angular span from arg(zU ) to arg(zR). tation of (11) and (12) in Case 2. A similar description for the 2) Case 2: The second case is motivated by exploiting the in- iterative and noniterative implementation of (6) and (7) in Case formation of the clutter-plus-noise statistical model into change 1 can be included. Fig. 1 shows the block diagram for iterative detection. Substituting (5) into (2) results in and noniterative implementation. For the iterative implementation, the number of assumed P (zU |sC ,zR) P (sT ) P (sT |zU ,zR)=1− . (10) targets will increase in each iteration. In the first iteration, P (zU |zR) K =1and the removal of the data surrounding the detected Applying Bayes’ theorem again to (10), we get the final expres- change is ignored. The parameters required by the statistical sion for (2) as distribution for the clutter-plus-noise of the SAR images to calculate P (zU ,zR|sC ) in Case 2 are estimated from whole P (zU ,zR|sC ) P (sT ) P (sT |zU ,zR)=1− . (11) reference and surveillance images. To form a histogram to find P (zU ,zR) P (zU ,zR) in Case 2, we arrange the modulus and phase of the Considering (8), the approximation for (11) is derived by reference image pixels in bins. The same arrangement is applied to the surveillance image. The frequency of a bin indicates the P (zU ,zR|sC ) P (sT |zU ,zR) ≈ 1 − . (12) simultaneous occurrence of a certain modulus and a certain P (zU ,zR) phase of the reference image and of a certain modulus and a The conditional probability P (zU ,zR|sC ) can be calculated certain phase of the surveillance image. The value of P (zU ,zR) by using an appropriate statistical distribution for the clutter- is given by the ratio of the frequency of the bin associated with plus-noise of the SAR images, whereas the joint probability z =[zU ,zR] to the sum of the frequency of all bins. We can P (zU ,zR) can get exact values from the SAR image histogram. get the values of the probability density function of P (zU ,zR) Under the assumption of a correct statistical model choice for by scaling with the inverses of the bin sizes. With such, we the clutter and noise in (11), it is expected that the occurrence of can easily calculate the conditional probability P (sT |zU ,zR) a change in the scene will lead to P (zU ,zR) P (zU ,zR|sC ), of each image pixel with K =1and the calculation results in resulting in P (sT |zU ,zR) ≈ 1. Conversely, a situation with the a matrix of probabilities with the same dimensions of the input absence of changes will lead to P (zU ,zR) ≈ P (zU ,zR|sC ), images. Under the assumption that there is a single change in the resulting in P (sT |zU ,zR) ≈ 0. SAR scene, the maximum value of P (sT |zU ,zR) is compared ALVES et al.: WAVELENGTH-RESOLUTION SAR CHANGE DETECTION USING BAYES’ THEOREM 5563

with a defined threshold λ.Ifmax{P (sT |zU ,zR)}≥λ, the first change is declared to be detected and we continue with the second iteration. In the second iteration, M pixels surrounding the detected change in the first iteration are removed from both the reference and surveillance images based on the assumption on the size and the shape of the change. The parameters required by the statistical distribution are updated due to the data Fig. 2. Simplified processing scheme for the proposed change detection removal. The histogram is also updated for the same reason. method. The conditional probability P (sT |zU ,zR) is calculated with K =2, and new values of P (sT |zU ,zR) and P (zU ,zR|sC ). If we still have max{P (sT |zU ,zR)}≥λ, the second change is SAR magnitude images. Among the clutter-plus-noise distri- claimed to have been detected, and we continue with the next bution models for wavelength-resolution images, the bivariate iterations. The same procedure is repeated for the next iterations Rayleigh distribution is the simplest one, since it belongs to a until max{P (sT |zU ,zR)} < λ. one-parameter family of probability distributions. It is worth For noniterative implementation, there is no assumption on mentioning that this distribution has already been used for the number of targets. We estimate one time the parameters CARABAS data and presented a good performance for change required by the statistical distribution for the clutter-plus-noise detection applications [13]. Although the Rayleigh distribution of the SAR images to calculate P (zU ,zR|sC ) from whole is not a perfect candidate for modeling the clutter-plus-noise of a reference and surveillance images. We also form a histogram wavelength-resolution SAR image, especially in the tail region, one time to find P (zU ,zR). With the values of P (zU ,zR|sC ) the model is still utilized due to its simplicity and efficiency in and P (zU ,zR), we can achieve the conditional probability SAR applications [25]. For these reasons, we select the bivariate P (sT |zU ,zR) with (12). All the values of P (sT |zU ,zR) are Rayleigh distribution here for SAR change detection method compared with a defined threshold λ. Only image pixels that development. meet the demand P (sT |zU ,zR) ≥ λ will be considered to be The probability density function of a bivariate Rayleigh dis- the changes. tribution can be written as [26] z z f z , z , |ρ 4 R U III. CHANGE DETECTION METHOD ZU ,ZR ( U ΩU ; R ΩR )= ΩRΩU (1 − ρ) In this section, we illustrate Bayes’ theorem for change detec- 1 z2 z2 tion method development presented in the previous section. The ×exp − R + U 1 − ρ ΩR ΩU illustration is given by developing a change detection method √ based on (12) and noniterative implementation for simplicity. 2 ρ zRzU × I0 √ According to (12) and the implementation, a statistical distribu- 1 − ρ ΩRΩU tion for the clutter-plus-noise of the SAR images is needed to (13) calculate P (zU ,zR|sC ). A statistical distribution is, therefore, 2 2 a prerequisite for the change detection method development. where ΩR = zr , ΩU = zU , I0(·) is the modified Bessel func- Bayes’ theorem for change detection method development pre- tion of the first kind with order zero, and ρ is the correlation sented in Section II can be used for developing both coherent and coefficient, which can be estimated by incoherent change detection algorithms. For simplicity, we also z2 ,z2 ρ cov( U r ) limit this development with incoherent SAR change detection = 2 2 (14) var(z )var(zr ) where zU and zR only represent the magnitude, i.e., zU = |zU | U and zR = |zR|. where cov(., .) and var(.) represent the covariance and the variance of random variables, and (.) represents the mean value. A. Clutter-Plus-Noise Distribution Model B. Processing Scheme For low-frequency wavelength-resolution SAR images, the correlation between images (different illuminations) is very high A simple processing scheme for the proposed change detec- in target-related pixels, and low for areas with the absence of a tion method is presented in Fig. 2. According to the processing scatter (water, open fields, small trees) [7]. The reason is that scheme, the input includes one surveillance image and one scatterers at low frequencies are rather large (>3 m) and that reference image that have been coregistered. The first processing the resolution cell contains only one scatter. Thus, we only use block is given by the noniterative implementation given in Fig. 1. single look images without using any type of clutter removal It consists of the statistical test based on (12), which is applied filtering technique. The large correlation is an effect of the to each pixel position of the images. The conditional probability high resolution in comparison to the wavelength and the fact P (zU ,zR|sC ) is calculated from the bivariate Rayleigh proba- that the CARABAS SAR system operates at low frequencies, bility density function (14), and P (zU ,zR) is estimated from the namely, 20−90 MHz. The clutter-plus-noise distribution can data histogram. The output of the first processing block will be a be modeled by bivariate Rayleigh [13], bivariate Gamma [23], binary data matrix. The value 1 is assigned to P (sT |zU ,zR) ≥ λ or K-distribution [24], which are usual distributions used for and the value 0 to P (sT |zU ,zR) < λ. 5564 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 13, 2020

Two standard morphological operations, erosion and dilation are used in the second processing block. The ﬁrst one aims to remove the small detections, e.g., less than the resolution of the SAR system, that might be false alarms. The latter merges the adjacent detections that are separated, e.g., less than the resolution of the SAR system or even less than four times the resolution of the SAR system. This avoids fragment of targets. The element structure for these morphological operations is, therefore, connected to the size of the system resolution cell. The output of the second processing block is the detected changes with false alarms (if any) presented by a binary data matrix. There are also optional processing steps. For example, to avoid P (sT |zU ,zR) relating to isolated instances, we can calculate the average probabilities of P (sT |zU ,zR) using an averaging ﬁlter considering a window given by the size of the system resolution cell. Then, these average probabilities are placed to thresholding. Fig. 3. CARABAS image of ground scene.

IV. EXPERIMENTAL RESULTS To evaluate the proposed change detection algorithm, we used data, and it is similar to the one used in [13] or [23]. Throughout the same CARABAS II dataset used in [8] and [9]. This dataset this article, we use the same set of values Δz ∈ [0.2, 0.3, 0.4] as was provided by the Swedish Defence Research Agency and is having been used in [23], in order to guarantee a fair comparison made available by AFRL in [10]. The dataset is composed of 24 between the tested methods. magnitude images that are already calibrated, preprocessed, and There might be mismatches between the selected distribution geo-coded, according to the Swedish reference system RR92 [9]. and the data histogram that lead (12) to negative values, i.e., Each magnitude image includes 3000×2000 pixels, contains P (sT |zU ,zR) < 0. In this situation, we simply set the condi- 2 25 testing targets, and covers the same ground area of 6 km tional probability to zero, i.e., P (sT |zU ,zR)=0. (3 × 2 km). The tested targets consist of ten TGB11 model A fixed threshold λ is applied to all the results of the statistical vehicles, eight TGB30 model vehicles, and seven TGB40 model tests. The threshold should be selected according to the charac- vehicles [9]. All target deployments were positioned in a forest teristics of each specific application and should lie in the range to test the capability to detect concealed targets, i.e., FOPEN. (0, 1). In this article, a wide range of thresholds is considered to The 24 images consider four different target deployments obtain the receiver operating characteristic (ROC) curves. (missions), divided into two geographically different forest sites. To perform a fair comparison between the evaluated meth- For each deployment, it was realized six image acquisition with ods, we apply similar morphological operations, such as those different flight geometries (passes). All measurements adopted considered in [9], after the thresholding step. The morphological HH polarization used the strip map SAR mode, and adopted an operations used herein are one erosion followed by two dilations. incidence angle of 58◦, for a range of 12 km to a common aim The erosion uses a square structuring element with the same size point [9]. Also, the flights were conducted with the radar looking as the system resolution cell (3 × 3 m). Hence, the operation left. More details on the data can be obtained in [8] and [9]. For removes the isolated detected changes. The dilations use square simplicity, we will follow the same image classification as given structuring elements whose sizes enable merging any detected in [9]. To exemplify, Fig. 3 shows one CARABAS II image of samples that are separated by up to 10 m. the ground scene of interest. An example of change detection results presented by a binary data matrix with the detected changes and the false alarms A. Implementation Aspects is given in Fig. 4. The detected changes are marked with a rectangle, and the false alarms are marked with a circle. The experimental evaluation was performed using all the 24 images and considering the processing scheme provided in B. Method Evaluation Fig. 2. Using an image pair, we can calculate the probabilities for all image pixels based on (12). However, to minimize The evaluation of the proposed method is realized in terms the processing time and to reduce the number of evaluations of the detection probability Pd, i.e., the ratio of the number of of situations with a mismatch between model and data, only detected targets to the known number of targets, and the number pixels with possible positive changes in the surveillance image of false alarms per square kilometer (FAR). For results repro- are tested. One magnitude constraint is performed in the first ducibility, every object detected by the algorithm was considered processing block. If the change in the magnitude of an image as a change, even knowing that some of them could be related to pixel is too small, they are not considered to contain a change. structures caused by the back-lobes, which are associated with Hence, the probability P (sT |zU ,zR) issetto0ifzU

Fig. 4. Change detection results with detected changes and false alarms. Fig. 6. ROC performance for proposed and Vu et al. [23] and Gomes et al. [27] methods.

for Δz =0.4. Note that increasing Δz improves the detection performance, but with diminishing returns. On the other hand, a large value of Δz may have the detrimental effect of removing targets, as mentioned before. As observed, Δz =0.4 is the best option tested for characterizing the target-like pattern in this dataset, although the intermediate value of Δz =0.3 has also proved to be a good choice. Optimal values of Δz could be achieved through an investigation about the statistics of the targets dataset. Thus, based on the numerical results, it is appropriate to state that prior knowledge about the targets in the dataset can be incorporated into the application constraints to improve the proposed method’s performance. Also, we compare the proposed method with [23] and [27], in which the ﬁrst presents one of the best, already published, Fig. 5. ROC performance for proposed and Ulander et al. [8] methods. detection performances for this dataset, and the second is one of the most recent results published using the CARABAS II dataset. The technique published in [23] consists of likelihood- this evaluation, we considered the same 24 image pairs as used based change detection method considering the clutter-plus- in [9] in comparison to the best ROC curve presented in [8]. noise statistics modeled as a bivariate Gamma distribution. Sim- Fig. 5 shows that the proposed method outperforms the reference ilarly, the techniques published in [27] consists of a likelihood- method for Δz =0.2, Δz =0.3, and Δz =0.4. based change detection methods considering the clutter-plus- It is important to highlight that the use of higher values for noise statistics modeled as a bivariate Rayleigh distribution and the magnitude constraint reduces the impact of the tail mismatch k-distribution. For the sake of simplicity, only the results of the between the data and the model distribution, reducing the oc- k-distribution are considered for the evaluation, given that it currence of the false alarm. However, the combination of high presented the best performances. Fig. 6 shows the performance magnitudes for the constraint and the erosion morphological comparison between the proposed method and the best ones operation could result in the erasure of some targets, making presented in [23] and [27]. it impossible to obtain 100% of detection with any threshold. From the results presented in Fig. 6, it is observable that Although this phenomenon has not been observed for the con- the proposed method excels in terms of false alarm rate and straints used in Fig. 5, it is expected to occur for higher Δz detection probability, the reference method [27] for all the values. Finally, the possibility of this phenomenon’s occurrence evaluated scenarios. In this same ﬁgure of merit, we compare can be reduced by selecting a better-matched distribution model the proposed method with the reference method [23]. We can or using different signal processing schemes. note that the proposed method with Δz =0.4 outperforms Analyzing the results presented in Fig. 5, for the points the reference method in all tested points with Δz =0.3, and 0 with FAR =10 , the following probabilities of detection are outperforms the reference one with Δz =0.4 for Pd > 95%. observable: Pd = 90% for the reference method; Pd = 96.8% Moreover, the proposed method with Δz =0.3 outperforms for Δz =0.2; Pd = 98.2% for Δz =0.3; and Pd = 98.7% the reference method with Δz =0.3 and with Δz =0.4 for 5566 IEEE JOURNAL OF SELECTED TOPICS IN APPLIED EARTH OBSERVATIONS AND REMOTE SENSING, VOL. 13, 2020

TABLE I PERFORMANCE RESULTS OF THE PROPOSED METHOD FOR A THRESHOLD λ =0.5 AND Δz =0.3

Note: The rows highlighted with gray color are related to images that contain back-lobe structures.

Pd > 92.5% and Pd > 96.3%, respectively. However, for the say that more sophisticated techniques in the simple processing reference method, when Δz =0.4, the algorithm was unable chain considered in this article are required. However, the results to detect some targets, resulting in a maximum Pd < 100%. presented in Table I show that the proposed method works well It is important to emphasize that these performance gains with most of the experimental data without the aid of any other are directly related to the target/clutter statistics selected in processing techniques. the evaluated methods. Finally, considering the points with Finally, it is essential to mention that back-lobe structures FAR =10◦, the following probabilities of detection are obtained are related to system and image formation issues, and they for the method proposed in [23]: Pd = 97.8% for Δz =0.3 have a target-like pattern. Thus, detections related to back-lobe and Pd = 97.6% for Δz =0.4. For the proposed method, the structures do not represent a method error and should not be following probabilities of detection are obtained: Pd = 98.2% counted as false alarms in our evaluation. By analyzing the for Δz =0.3 and Pd = 98.7% for Δz =0.4. results of the four image pairs, it is veriﬁed that only two of the To further investigate the performance of the proposed de- ten previously observed false alarms are not associated with this tection method, we present the results for a speciﬁc test setup type of structure. Disregarding false alarms related to back-lobe in Table I, where the gray highlighted rows refer to images that structures, the number of false alarms presented in Table I would contain back-lobe structures. By comparison with the results pre- drop to a total of 68, resulting in FAR =0.47. sented in [9], Table I reveals that the proposed method presents better performance for both detection probability (585 target detections against 579) and false alarm rate (76 false alarms V. C ONCLUSION against 96). Furthermore, Table I shows that the majority of false Bayes’ theorem for wavelength-resolution SAR change de- alarms 75% and a large number of the missed detections 46.66% tection method presented in this article can be used for the are related to the pairs 18 and 20. In fact, these images were development of new change detection algorithms. As an ex- studied in [11] being characterized by the presence of some low ample, we have developed a simple change detection method amplitude targets and by high amplitude elongated structures, using a clutter-plus-noise model with a noniterative approach. which may cause false alarms. Based on the study presented The bivariate Rayleigh distribution was considered for mod- in [11], aiming to reduce false alarms in these image pairs, we can eling the clutter-plus-noise in the used wavelength-resolution ALVES et al.: WAVELENGTH-RESOLUTION SAR CHANGE DETECTION USING BAYES’ THEOREM 5567

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Hans Hellsten (Senior Member, IEEE) started his Renato Machado (Member, IEEE) received the B.S. radar career with the Swedish Defence Research degree in electrical engineering from the São Paulo Agency, subsequently becoming the Director of Re- State University, Ilha Solteira, Brazil, in 2001, and search. Around 1990, he formulated and patented the M.Sc. and Ph.D. degrees in electrical engineering the basic principles for meter wave SAR imaging. In from the Federal University of Santa Catarina, Flori- 1992, he and his colleagues’ work resulted in success- anópolis, Brazil, in 2004 and 2008, respectively. ful ﬂights with VHF SAR, leading to worldwide in- He was a Visiting Ph.D. Researcher with the De- terest. The continuing good results motivated signif- partment of Electrical Engineering, Arizona State icant funding, allowing a second enhanced system— University, Tempe, AZ, USA, from August 2006 to CARABAS II—to be built. Transferring to the radar June 2007. He was with the Research and Devel- industry in 2001, he was the main responsible for opment Department, Nokia Institute of Technology, the Saab CARABAS III demonstrator radar system becoming operative around Brasília, Brazil, in 2007 and 2008. Between 2009 and 2017, he has served 2010. He is currently a Senior Radar Expert, Product Manager, and Principal as a Professor with the Department of Electronics and Computing, Federal Engineer with Saab Electronic Defence Systems, working to further enhance University of Santa Maria, Santa Maria, Brazil. From December 2011 to October meter wave SAR. He also holds an Adjunct Professorship with Halmstad 2013, he was the Director of the Santa Maria Space Science Laboratory— Technical University, Halmstad, Sweden. He is the inventor of about 30 patents LACESM/CRS/INPE, Santa Maria, Brazil. Between November 2013 and March and more than 50 scientiﬁc and technical publications, including a recently 2015, he spent his sabbatical license, as a Visiting Researcher Fellow, with the published book on meter wave SAR technology. Blekinge Institute of Technology, Karlskrona, Sweden, working in partnership with Saab Electronic Defence Systems, SAAB AB. Since December 2017, he has been a Professor with the Department of Telecommunications, Aeronautics Institute of Technology (ITA), São José dos Campos, Brazil, and with the Graduate Program in Electronics and Computer Engineering, ITA. His research interests include SAR image processing, wireless communications, digital signal processing, and radar signal processing. Dr. Machado is a member of the IEEE Geoscience and Remote Sensing Viet Thuy Vu (Senior Member, IEEE) was born in Society, IEEE Aerospace and Electronic Systems Society, and the Brazilian Hanoi, Vietnam. He received the Diploma degree in Telecommunications and Signal Processing Society. electronics from the Hanoi University of Technol- ogy, Hanoi, Vietnam, in 1999, the M.Sc. degree in communication engineering from the University of Bartolomeu F. Uchôa-Filho (Senior Member, IEEE) Duisburg-Essen, Duisburg, Germany, in 2004, and the was born in Recife, Brazil, in 1965. He received Licentiate and the Doctoral degrees in applied signal the B.S. degree from the Federal University of Per- processing from the Blekinge Institute of Technol- nambuco (UFPE), Recife, Brazil, in 1989, the M.Sc. ogy (BTH), Karlskrona, Sweden, in 2009 and 2011, degree from the State University of Campinas (UNI- respectively. CAMP), Campinas, Brazil, in 1992, and the Ph.D. de- Currently he is an Associate Professor at BTH. His gree from the University of Notre Dame, Notre Dame, research interests include mono- and bistatic-SAR signal processing, applica- Indiana, U.S.A., in 1996, all in electrical engineering. tions of SAR in change detection, moving target detection and terahertz 3D He has held Postdoctoral/Visiting Scholar posi- imaging, and radio occultation. tions with UNICAMP from 1977 to 1999, The Uni- versity of Sydney, Camperdown, NSW, Australia, from 2009 to 2010, Centrale-Suplec, and CNAM, both in France, in 2016. His research interests include coding and information theory, with applications to digital communications systems.

Patrik Dammert (Senior Member, IEEE) received Mats I. Pettersson received the M.Sc. degree in the M.Sc. degree in electrical engineering and the engineering physics, the Licentiate degree in radio Ph.D. degree in the ﬁeld of applications of spaceborne and space science, and the Ph.D. degree in signal SAR interferometry from the Chalmers University of processing from the Chalmers University of Technol- Technology, Gothenburg, Sweden, in 1993 and 1999, ogy, Gothenburg, Sweden, in 1993, 1995, and 2000, respectively. respectively. He joined Saab, Gothenburg, Sweden, in 2000. At For some years, he was with Mobile Communica- Saab, he has been responsible for the development of tion Research, Ericsson, and for ten years, he was with high-performance SAR systems for airborne radars, Swedish Defence Research Agency (FOI). At FOI, spanning from VHF-band to X-band systems (with he focused on ultrawide band low-frequency SAR ﬂat plate antennas and with AESA antennas). He has systems. Since 2005, he has been with the Blekinge also been the Project Manager and Associate Doctoral Student Supervisor for Institute of Technology, Karlskrona, Sweden, where he is currently a Full research projects with Saab in collaborations with the Chalmers University of Professor. His research is related to remote sensing and his main interests include Technology and Blekinge Institute of Technology. His research interests include SAR processing, space-time adaptive processing, high-resolution SAR change high-resolution radar and SAR systems, radar modeling, algorithms and signal detection, automotive radar, radio occultation, and computer vision. processing, autofocus, and target detection in heavy-tailed radar clutter.