Optical images-based edge detection in Synthetic Aperture Radar images
Gilberto P. Silva Junior, Alejandro C. Frery, Sandra Sandri, Humberto Bustince, Edurne Barrenechea, Cédric Marco-Detchart
OOptical images-based edge detection inSynthetic Aperture Radar images
Gilberto P. Silva Junior a , Alejandro C. Frery b , Sandra Sandri a, ∗ , HumbertoBustince c , Edurne Barrenechea c , C´edric Marco-Detchart c a Laborat´orio Associado de Computa¸c˜ao e Matem´atica Aplicada, Instituto Nacional dePesquisas Espaciais (LAC/INPE), Av. dos Astronautas, 1758, 12227–010,S˜ao Jos´e dos Campos, SP – Brazil b Laborat´orio de Computa¸c˜ao Cient´ıfica e An´alise Num´erica (LaCCAN/UFAL),Universidade Federal de Alagoas, Av. Lourival Melo Mota, s/n, 57072-970,Macei´o, AL – Brazil c Departamento de Autom´atica y Computaci´on, Universidad P´ublica de Navarra, CampusArrosad´ıa, 31006, Pamplona, Spain
Abstract
We address the issue of adapting optical images-based edge detection techniquesfor use in Polarimetric Synthetic Aperture Radar (PolSAR) imagery. We mod-ify the gravitational edge detection technique (inspired by the Law of UniversalGravity) proposed by Lopez-Molina et al, using the non-standard neighbour-hood configuration proposed by Fu et al, to reduce the speckle noise in polari-metric SAR imagery. We compare the modified and unmodified versions of thegravitational edge detection technique with the well-established one proposedby Canny, as well as with a recent multiscale fuzzy-based technique proposed byLopez-Molina et al. We also address the issues of aggregation of gray level im-ages before and after edge detection and of filtering. All techniques addressedhere are applied to a mosaic built using class distributions obtained from areal scene, as well as to the true PolSAR image; the mosaic results are as-sessed using Baddeley’s Delta Metric. Our experiments show that modifyingthe gravitational edge detection technique with a non-standard neighbourhoodconfiguration produces better results than the original technique, as well as theother techniques used for comparison. The experiments show that adapting edge ∗ Corresponding author
Email address: [email protected] (Sandra Sandri)
Preprint submitted to Knowledge-Based Systems August 15, 2018 a r X i v : . [ c s . C V ] A ug etection methods from Computational Intelligence for use in PolSAR imageryis a new field worthy of exploration. Keywords:
Edge detection, SAR images, Computational Intelligence,Gravitational method
1. Introduction
Edge detection seeks to identify sharp differences automatically in the infor-mation associated with adjacent pixels in an image [1]. Edge detection for opti-cal images is nowadays quite an established field. It is traditionally carried outusing gradient-based techniques, such as the well-known Canny algorithm [2].Techniques based on Computational Intelligence have also been proposed inthe recent literature. Sun et al [3] proposed the gravitational edge detectionmethod, inspired by Newton’s Universal Law of Gravity. Lopez-Molina et al [4]proposed a fuzzy extension for this technique, allowing the use of T-norms, alarge class of fuzzy operators; they also proposed small modifications in the basicformalism (see Section 3). D˘ankov´a et al [5] proposed the use of a fuzzy-basedfunction, the F-transform; the original universe of functions is transformed intoa universe of their skeleton models (vectors of F-transform components), mak-ing further computations easier to perform. Barrenechea et al [6] proposed theuse of interval-valued fuzzy relations for edge detection, using a T-norm anda T-conorm to produce a fuzzy edge image, that is then binarized. This ap-proach was extended by Chang and Chang [7]. First of all, two new imagesare created—one rather dark and the other rather bright—by applying two dif-ferent parameters on the linear combinations of the images obtained using minand max operators, respectively. Then, the fuzzy edge image is created bythe difference between these two new images. Another recent approach fromComputational Intelligence is the multiscale edge detection method proposedby Lopez-Molina et al [8], using Sobel operators for edge extraction and theconcept of Gaussian scale-space.SAR sensors are not as adversely affected by atmospheric conditions and the2resence of clouds as optical sensors. Moreover, unlike the optical counterparts,SAR sensors can be used at any time of day or night. For these reasons, remotesensing applications using SAR imagery have been growing over the years [9].SAR images, however, contain a great amount of noise, known as speckle , thatdegrades the visual quality of the images. Caused by inherent characteristics ofradar technology, this multiplicative non-Gaussian noise is proportional to theintensity of the received signal.Contrary to what happens with optical images, there are still few algorithmsspecifically dedicated to SAR images [10]. One interesting means to create edgedetection algorithms for SAR images is to modify those created for optical im-ages. However, the use of these methods on SAR images is not straightforward,due to speckle. One can either adapt optical image techniques to meet SARdata properties, or first preprocess the images using filters and then apply theoriginal optical techniques.The main purpose of our study is to investigate the application of the grav-itational edge detection, Here we modify the original 3 × × ×
2. Basic concepts on SAR images
Optical and SAR sensors measure the amount of energy reflected by a tar-get in various bands of the electromagnetic spectrum. The bands employedin most imaging radars use frequencies in the 2 MHz to 12 . . Polarimetric Synthetic Aperture Radar ),information about intensity and phase of the cross signals are also obtained,generating images relating to HV and VH polarizations. Usually, applicationsonly consider the HH, VV, and HV polarizations.
Figure 1: Horizontal and vertical signal polarizations transmitted by an antenna. Source: [17]
The imaging can be obtained by gathering all the intensity and phase in-formation data from the electromagnetic signal after it has been backscatteredby the target in a given polarization [18]. Each polarization in a given a scenegenerates a complex image, which can be thought of as two images, containingthe real and imaginary values for the pixels, respectively.We denote the complex images from HH, VV, and HV polarizations as S HH , S HV , and S V V . Multiplying the vector [ S HH S HV S V V ] by its transposedconjugated vector [ S ∗ HH S ∗ HV S ∗ V V ] t , we obtain a 3 × I HH , I HV , and I V V andtheir corresponding amplitude counterparts by A HH , A HV , and A V V . In thispaper, we only considered the amplitude images, such as those depicted inFigure 2. 5 HH A HV A V V
Figure 2: Amplitude images for polarizations HH, VV, and HV from the same scene
Speckle noise is multiplicative, non-Gaussian, and is proportional to theintensity of the received signal. Speckle degrades the visual quality of the dis-played image by sudden variations in image intensity with a salt and pepperpattern, as can be seen in Figure 2. It can be reduced with multiple looks inthe generation of the complex images, causing degradation in spatial resolution.Another way to reduce noise is to employ filters, as will be discussed in the nextsection.In SAR image classification, one often uses samples from the classes in orderto estimate the parameters of the distribution believed to underlie each class.Synthetic images can then be created using Monte Carlo simulation by takingthe realization of the random variable associated to the class of each classifiedpixel. This artifice is useful to choose the most apt classifier for a given appli-cation: instead of relying solely on the original image, one takes the classifierthat obtains the best average accuracy on the set of synthetic images. Thismethodology can also be used in other tasks, such as edge detection.
3. Related Work
One of the most successful edge detection algorithms for optical images wasproposed by Canny [2], based on the following guidelines: i) the algorithmshould mark as many real edges in the image as possible; ii) the marked edgesshould be as close as possible to the edge in the real image; iii) a given edge in6he image should only be marked once; and iv) image noise should not createfalse edges. It makes use of numerical optimization to derive optimal operatorsfor ridge and roof edges. The usual implementation of this method uses a 3 × f , = G × m × m (cid:107) (cid:126)r (cid:107) × (cid:126)r (cid:107) (cid:126)r (cid:107) , (1)where m and m are the masses of two bodies; (cid:126)r is the vector connecting them; (cid:126)f , is the gravitational force between them; (cid:107) . (cid:107) denotes the magnitude of avector; and G is the gravitational constant. In the analogy proposed by Sun etal [3]; the bodies are the gray level values of pixels in a grid; G is a functionof the values of the pixels in a given window; the distance between any twoadjacent pixels is equal to 1; and, when computing the resulting force of thepixel in the center of a window; the pixels outside that window are considerednegligible. Lopez-Molina et al [4] extended this technique, proposing the useof a Triangular Norm [19] in place of the product between the two masses , byfirst normalizing the gray level values to [0 , G as a normalization constant, calculated so asto guarantee that the resulting forces lie in [0,1]. Also, in the normalization of Triangular norm operators are mappings from [0 , to [0 , δq is added beforehand to both thenumerator and denominator so as avoid pixels with value 0, which would havetoo strong an effect on neighbouring pixels. The authors used 3 × × .
5% of the probability is concentrated within two standard deviationsfrom the mean. The filter estimates the mean and the standard deviation ofsamples around each pixel, and only those values within this interval are usedto compute the local mean. Lopes et al [11] proposed an adaptive version forthis filter, here referred to as “Enhanced Lee”.Torres et al [12] recently proposed a nonlocal means approach for PolSAR im-age speckle reduction based on stochastic distances; the method can be tailoredto any distribution, both univariate of multi-variate. It consists of comparingthe distributions which describe the central observation for each pixel, and eachof the observations which comprise a search region. The comparison is madethrough a goodness-of-fit test, and the p -value of the test statistic is used todefine the convolution matrix which will define the filter: the higher the p -valuethe larger the confidence and, thus, the importance, each observation will havein the convolution. In Torres et al’s proposal, the tests are derived from h - φ divergences between multi-look scaled complex Wishart distributions for fullyPolSAR data [21]. Their results are competitive with classical and advancedpolarimetric filters, with respect to usual quantitative measures of quality.Fu et al [10] proposed a statistical edge detector suitable for SAR imageswhich uses the squared successive difference of averages to estimate the edgestrength from the sliding window. An interesting feature of this paper is theproposal of a specific type of 9 × × Figure 3: Standard 3 ×
4. Materials and Methods
We compare the edge detection methods proposed by Canny [2] and byLopez-Molina et al [8] to the modified gravitational approach using the prod-uct T-norm, followed by thresholding. The effect of preprocessing the imagesthrough filtering is also studied, using the filter described by Torres et al [12] andthe Enhanced Lee filter [11]. We study the behaviour of Lopez-Molina’s methodwith the usual 3 × { , . . . , } and [0 , , δq = 1 and making q (cid:48) = ( q + 1)(255 + 1) − , where q and q (cid:48) are the old and new value of a givenpixel, respectively. We apply the methods on data derived from a fully polarimetric image, pre-sented by Barreto et al [13], from an agricultural area in the Amazon regionin Brazil (see Figure 4). The authors describe a classification experiment usingclasses of interest from that area, such as water and different types of crops and9atural vegetation, at different stages of growth. Samples from the classes fromband L are used to estimate the parameters of the complex Wishart distributionassociated to each class. The results are assessed using a mosaic with the classesthat was created using the derived Wishart distributions. Figure 5 illustratesthe approach. For our study we apply the edge detection methods on twentyindependently simulated mosaics amplitude images, using the parameters esti-mated in [13] to assess the quality of the methods.a) b)
Figure 4: Images derived from a scene in Bebedouro in Brazil (not registered): a) LandsatRGB composition and b) SAR L-band RGB composition (source: [13]) a) b)
Figure 5: Images derived from a scene in Bebedouro in Brazil: a) training samples used togenerate Wishart distributions and b) synthetic mosaic images generated using the Wishartdistributions estimated in [13] from image samples (source: [13]) .2. Quality assessment The quality of the results is assessed by the Baddeley’s Delta Metric (BDM)[14], by comparison with what would be the perfect result, discarding thosepixels close to the outer frame.Let x and y be two binary images, seen as mappings from Λ to [0 , ρ be a metric on Λ, suchas the Euclidean distance, and d ( i, A ) be the distance between a site i and a set A ⊆ Λ, defined as d ( i, A ) = min j ∈ A ρ ( i, j ) . Let b ( x ) = { i ∈ Λ | x i = } denote the set of foreground sites in x . BDMbetween x and y , denoted as ∆ p,w. ( ., . ), is then defined as∆ p,w ( x , y ) = (cid:32) | Λ | (cid:88) i ∈ Λ | w ( d ( i, b ( x )) − w ( d ( i, b ( y )) | p (cid:33) p , ≤ p ≤ ∞ (2)where w is a strictly increasing concave function satisfying w (0) = 0. Here weuse w ( t ) = t and p = 2, as in [4].Throughout the text, we display BDM results in [0 , ,
5. Proposed methodologies
Edge detectors use a window around a center pixel to verify whether thatpixel belongs to an edge or not. When adapting optical image edge detectorsto radar imagery, we have to find the means to deal with speckle. The maincontribution of this study is to modify the original 3 × × • DAB (edge Detection on non-binary images, Aggregation of the resultingnon-binary images, Binarization) and • ADB (Aggregation of non-binary images, edge Detection on the resultingnon-binary image, Binarization). • DBA (edge Detection on non-binary images, Binarization, Aggregation ofthe resulting binary images).Options ABD, BAD and BDA are not considered, since that would mean ap-plying edge detectors on the binary images.In this work, we focus on the DAB and ADB strategies. For both of them,we use the arithmetic mean to aggregate gray level images. Strategy DBA,involving the aggregation of binary images, is left for future study.12hen no aggregation is considered, the strategies are reduced to only edgedetection and binarization. For example, for the HH, HV and VV polarizations,we obtain strategies DB-HH, DB-HV and DB-HH. Note that some methodsalready incorporate the binarization step in the edge detector. That is forinstance the case of all methods discussed previously. However, to be consistentwith the notation, we will denote by ADB the strategy in a method that includesbinarization, such as Canny’s and the multi-scale method, when it is applied tothe gray level image resulting from the aggregation of the images from the HH,HV and VV polarizations.
6. Experimental Results
The output of the Lopez-Molina gravitational method is an image with val-ues in [0 , . , .
15] interval which produces the best BDM. For Canny’smethod, we search for the best value for the noise standard deviation parameter σ in the interval [0 . , . δ σ = 0 . σ ∈ { . , . , . , . , . , . , . , . , . , . , . , . , . , . , . } .We applied two filters (Torres [12] and Enhanced Lee [11]) on intensity val-ues, which were then transformed in amplitude before further processing. Table 1: Average BDM results for Canny’s method, with standard deviation inside parentheses
Strategy No filter Torres filter Enh. Lee filterDB-HH 26.72 (1.22) 23.40 (1.92) 28.85 (1.86)DB-HV 30.70 (1.39) 26.74 (1.33) 66.40 (0.42)DB-VV 29.81 (2.17) 26.28 (1.09) 24.83 (1.98)ADB 27.87 (1.16) 48.16 (0.003) 30.24 (1.20)13 able 2: Average BDM results for the multi-scale method, with standard deviation insideparentheses
Strategy No filter Torres filter Enh. Lee filterDB-HH 28.23 (0.89) 28.00 (0.93) 25.36 (2.11)DB-HV 28.34 (1.13) 24.55 (2.97) 19.56 (2.36)DB-VV 25.62 (1.10) 28.42 (0.56) 28.89 (2.25)ADB 25.15 (3.23) 24.37 (1.52) 20.71 (3.97)
Table 3: Average BDM results for the gravitational method; with standard deviation insideparentheses
Strategy No filter Torres filter Enh. Lee filterDB-HH 33.89 (1.98) 26.61 (2.00) 38.97 (0.79)DB-HV 31.95 (0.46) 27.14 (1.22) 32.26 (2.78)DB-VV 32.35 (1.47) 28.95 (0.95) 43.65 (1.13)DAB 29.26 (1.62) 25.91 (1.55) 27.71 (2.62)ADB 31.50 (0.82) 26.63 (1.27) 18.24 (3.41)Tables 1, 2, 3, and 4 show the results for BDM mean and standard devia-tion after applying four methods to twenty simulated mosaic images: Canny’smethod, Lopez-Molina et al’s multi-scale method, Lopez-Molina et al’s originalgravitational method, and Lopez-Molina et al’s method modified using Fu’s 9 × able 4: Average BDM results for the gravitational method modified by Fu’s neighbourhood,with standard deviation inside parentheses Strategy No filter Torres filter Enh. Lee filterDB-HH 25.27 (0.76) 22.18 (0.48) 17.79 (3.05)DB-HV 26.48 (1.00) 24.21 (0.65) 18.40 (5.75)DB-VV 21.41 (1.97) 18.14 (0.77) 17.83 (2.54)DAB 22.67 (2.29) 18.97 (1.62) 5.43 (1.68)ADB 23.80 (2.23) 22.74 (0.50) 5.16 (0.36)Molina gravitational method modified with Fu’s neighbourhood. In particular,the best results are obtained for strategies DAB and ADB with preprocessingwith the Enhanced Lee filter.Figure 6 shows the negative images corresponding to the best results, accord-ing to BDM, obtained by the edge detection methods and the filtering strategieswith the best average values; note that the image boundaries are depicted onlyfor illustrative purposes. We see that according to BDM the best binary image(depicted in Figure 6b) presents little noise and most of the regions are sepa-rated, even though the lines are rather thick. We also see that BDM was ableto distinguish the best image from the others.Figure 6b shows the best results from the methods come from filtered images,which raises the question of how important preprocessing by filtering is. In whatfollows we discuss the details of the gravitational method using the original 3 × × × × Figure 6: Best BDM results obtained from the best methods (average): a) Canny (DB-HH,with Torres filtering, BDM = 18.51), b) Multi-scale (DB-HV, with Enh. Lee filtering, BDM= 14.94), c) Gravitational (ADB with Enh. Lee filtering, BDM = 10.96) and d) Gravitationaland Fu (ADB with Enh. Lee filtering, BDM = 3.05) of noise (some edges were detected using Torres filter with an increase of noisewhen compared to the unfiltered image).When we compare the results in Figures 7 and 8 we see that the gravitationalmethod modified with Fu’s 3 × × Figure 7: Results for the gravitational method with original 3 × a) b) c) Figure 8: Results for the gravitational method modified with Fu’s neighbourhood, on a singlesimulated image from ADB: a) no filtering (BDM = 23.80), b) Torres (BDM = 23.76) and c)Enh. Lee (BDM = 3.05)
DBA aggregation method produced results with less noise for the gravitationalmethod, with and without modification, than the results obtained with theindividual polarizations. The best results for the gravitational method, bothwith and without modification with Fu’s neighbourhood, were obtained withthresholds around the same interval that produced the best results using themosaics.
7. Conclusions
Contrary to what happens with optical imagery, few algorithms are specif-ically dedicated to PolSAR image edge detection [10]. One interesting meansto create edge detection algorithms for SAR images is to modify those createdfor optical images, in such a way as to reduce the non-Gaussian noise. Here17 igure 9: HV Bebedouro binary images, with Enh. Lee filter: the first row depicts resultsobtained using the Canny and the Multi-scale methods; and the second row depicts results ob-tained using the original Gravitational method and its modification with Fu’s neighbourhood,(the latter two methods use binarization threshold=.2) we have investigated the modification of a method issued from ComputationalIntelligence for optical imagery, the gravitational edge detection method exten-sion proposed in [4] (see also [3]), to synthetic aperture radar imagery. In orderto deal with speckle, we modified the gravitational method with a non-standard9 × × igure 10: ADB Bebedouro binary images, with Enh. Lee filter: the first row depicts resultsobtained using the Canny and the Multi-scale methods; and the second row depicts results ob-tained using the original Gravitational method and its modification with Fu’s neighbourhood,(the latter two methods use binarization threshold=.2) joint use may compensate for the presence of speckle, we also proposed a ty-pology of experiments regarding aggregation of these images. In particular, weaddressed two procedures: DAB (edge Detection on non-binary images, Aggre-gation of the resulting non-binary images, Binarization) and ADB (Aggregationof non-binary images, edge Detection on the resulting non-binary image, Bina-rization).For means of comparison, we also addressed the use of two other edge de-tector methods stemming from the realm of optical images: the traditionalmethod proposed by Canny [2] and a recent multi-scale one coming from Com-19 igure 11: DAB Bebedouro binary images, with Enh. Lee filter: the first and second rowsrespectively depict results using the original Gravitational and Gravitational modified withFu’s neighbourhood, the first and second columns respectively depict results obtained withbinarization thresholds .1 and .2 putational Intelligence, based on Sobel operators for edge extraction and theconcept of Gaussian scale-space [8].We studied the effect of filtering the images prior to edge detection by twoprocedures: Enhanced Lee [11] and Torres [12] filters. The methods were appliedon twenty samples of a scene, which were simulated using Wishart distributionsderived from a fully polarimetric image [13]. Using both visual inspection andthe Baddeley Delta metric [14] we verified that the combination with the Lopez-Molina technique with the 9 × cknowledgements The authors are grateful for Wagner Barreto Silva, Leonardo Torres and CorinaFreitas for help in the preparation of this manuscript. They are also thankful for theeditor and reviewers for comments and suggestions. The Brazilian authors acknowl-edge support from CNPq (Projeto Universal 487032/2012-8). The Spanish authorshave been supported by project TIN2013-40765-P of the Spanish Government.
ReferencesReferences [1] R. C. Gonzalez, R. E. Woods, Digital Image Processing (3rd Ed.), Prentice-Hall,Inc., Upper Saddle River, NJ, USA, 2006.[2] J. Canny, A computational approach to edge detection, IEEE Transactions onPattern Analysis and Machine Intelligence 8 (6) (1986) 679–698. doi:10.1109/TPAMI.1986.4767851 .[3] G. Sun, Q. Liu, C. Ji, X. Li, A novel approach for edge detection based on thetheory of universal gravity theory of universal gravity, Pattern Recognition 40 (10)(2007) 2766–2775.[4] C. Lopez-Molina, H. Bustince, J. Fernandez, P. Couto, B. D. Baets, A gravi-tational approach to edge detection based on triangular norms, Pattern Recog-nition 43 (11) (2010) 3730–3741. doi:http://dx.doi.org/10.1016/j.patcog.2010.05.035 .[5] M. Dankov´a, P. Hod´akov´a, I. Perfilieva, M. Vajgl, Edge detection using F-transform, in: 2011 11th International Conference on Intelligent Systems Designand Applications (ISDA), IEEE, 2011, pp. 672–677.[6] E. Barrenechea, H. Bustince, B. D. Baets, C. Lopez-Molina, Construction ofinterval-valued fuzzy relations with application to the generation of fuzzy edgeimages, IEEE Transactions on Fuzzy Systems 19 (5) (2011) 819–830.[7] J.-Y. Chang, Y.-H. Chang, Applying weighted mean aggregation to edge detectionof images, in: 2013 International Conference on System Science and Engineering(ICSSE), IEEE, 2013, pp. 153–157.
8] C. Lopez-Molina, B. D. Baets, H. Bustince, J. Sanz, E. Barrenechea, Multiscaleedge detection based on Gaussian smoothing and edge tracking, Knowledge-BasedSystems 44 (5) (2013) 010–111.[9] H. Mott, Remote Sensing with Polarimetric Radar, John Wiley & Sons, 2006.[10] X. Fu, H. You, K. Fu, A statistical approach to detect edges in SAR images basedon square successive difference of averages, IEEE Geoscience and Remote SensingLetters 9 (6) (2012) 1094–1098. doi:10.1109/LGRS.2012.2190378 .[11] A. Lopes, R. Touzi, E. Nezry, Adaptive speckle filters and scene heterogeneity,IEEE Transactions on Geoscience and Remote Sensing 28 (6) (1990) 992–1000.[12] L. Torres, S. J. S. Sant’Anna, C. C. Freitas, A. C. Frery, Speckle reduction inpolarimetric SAR imagery with stochastic distances and nonlocal means, PatternRecognition 47 (2014) 141–157. doi:10.1016/j.patcog.2013.04.001 .[13] W. B. da Silva, C. da Costa Freitas, S. J. S. Sant’Anna, A. C. Frery, Classifica-tion of segments in PolSAR imagery by minimum stochastic distances betweenWishart distributions, IEEE Journal of Selected Topics in Applied Earth Obser-vations and Remote Sensing 6 (3) (2013) 1263–1273.[14] A. J. Baddeley, An error metric for binary images, in: W. F¨orstner, H. Ruwiedel(Eds.), Robust Computer Vision: Quality of Vision Algorithms, Wichmann, Karl-sruhe, 1992, pp. 59–78.[15] G. P. Silva Junior, A. Frery, S. Sandri, Synthetic aperture radar edge detec-tion with canny’s procedure and a gravitational approach, in: 11th InternationalFLINS Conference on Decision Making and Soft Computing (FLINS), 2014, pp.149–154. doi:10.1142/9789814619998_0027 .[16] J. Richards, Remote Sensing with Imaging Radar, Signals and CommunicationTechnology, Springer, 2009.[17] P. R. Meneses, T. d. Almeida, Intodu¸c˜ao ao Processamento de Imagens de Sen-soriamento Remoto, Universidade de Braslia - CNPq, Brasilia, Br, 2012.[18] J. Lee, E. Pottier, Polarimetric Radar Imaging: From Basics to Applications,Optical Science and Engineering, Taylor & Francis, 2009.URL http://books.google.com.br/books?id=1nAvp2HW_gwC
19] D. Dubois, H. Prade, Possibility Theory: An Approach to Computerized Process-ing of Uncertainty, Plenum Press, New York, USA, 1988.[20] J.-S. Lee, A simple speckle smoothing algorithm for synthetic aperture radarimages, IEEE Transactions on Systems, Man and Cybernetics 13 (1) (1983) 85–89. doi:10.1109/TSMC.1983.6313036 .[21] A. C. Frery, A. D. C. Nascimento, R. J. Cintra, Analytic expressions for stochasticdistances between relaxed complex Wishart distributions, IEEE Transactions onGeoscience and Remote Sensing 52 (2) (2014) 1213–1226. doi:10.1109/TGRS.2013.2248737 .[22] M. Nagao, T. Matsuyama, Edge preserving smoothing, Computer Graphics andImage Processing 9 (4) (1979) 394–407.[23] C. Lopez-Molina, B. D. Baets, H. Bustince, Generating fuzzy edge images fromgradient magnitudes, Computer Vision and Image Understanding 115 (11) (2011)1571–1580. doi:http://dx.doi.org/10.1016/j.cviu.2011.07.003 .[24] M. E. Buemi, A. C. Frery, H. S. Ramos, Speckle reduction with adaptive stackfilters, Pattern Recognition Letters 36 (2014) 281–287. doi:10.1016/j.patrec.2013.06.005 .URL http://dx.doi.org/10.1016/j.patrec.2013.06.005http://dx.doi.org/10.1016/j.patrec.2013.06.005