Comparing Edge Detection Methods based on Stochastic Entropies and Distances for PolSAR Imagery

Comparing Edge Detection Methods based on Stochastic Entropies and Distances for PolSAR Imagery

Abraão D. C. Nascimento, Michelle M. Horta, Alejandro C. Frery, , and
Renato J. Cintra, 
This work was supported by CNPq, Fapeal, FAPESP, and FACEPE, Brazil.A. D. C. Nascimento is with the Departamento de Estatística, Universidade Federal de Pernambuco, Cidade Universitária, 50740-540, Recife, PE, Brazil, e-mail: abraao@de.ufpe.br M. M. Horta is with the Departamento de Computação, Universidade Federal de São Carlos, Rod. Washington Luís km 235, 13565-905, São Carlos, SP, Brazil, e-mail: michellemh@gmail.com A. C. Frery is with the Instituto de Computação, Universidade Federal de Alagoas, BR 104 Norte km 97, 57072-970, Maceió, AL, Brazil, e-mail: acfrery@gmail.comR. J. Cintra is with the Signal Processing Group, Departamento de Estatística, Universidade Federal de Pernambuco, Cidade Universitária, 50740-540, Recife, PE, Brazil, e-mail: rjdsc@ieee.org
Abstract

Polarimetric synthetic aperture radar (PolSAR) has achieved a prominent position as a remote imaging method. However, PolSAR images are contaminated by speckle noise due to the coherent illumination employed during the data acquisition. This noise provides a granular aspect to the image, making its processing and analysis (such as in edge detection) hard tasks. This paper discusses seven methods for edge detection in multilook PolSAR images. In all methods, the basic idea consists in detecting transition points in the finest possible strip of data which spans two regions. The edge is contoured using the transitions points and a B-spline curve. Four stochastic distances, two differences of entropies, and the maximum likelihood criterion were used under the scaled complex Wishart distribution; the first six stem from the - class of measures. The performance of the discussed detection methods was quantified and analyzed by the computational time and probability of correct edge detection, with respect to the number of looks, the backscatter matrix as a whole, the SPAN, the covariance an the spatial resolution. The detection procedures were applied to three real PolSAR images. Results provide evidence that the methods based on the Bhattacharyya distance and the difference of Shannon entropies outperform the other techniques.

{keywords}

Image analysis, information theory, polarimetric SAR, edge detection.

I Introduction

Polarimetric synthetic aperture radar (PolSAR) has achieved a prominent position as a remote imaging method [1]. SAR images are contaminated by speckle, a typical noise present in data acquired with coherent illumination subject to multipath interference. Speckle noise introduces a granular aspect to the image, and its multiplicative nature makes SAR image analysis a challenging task [2]. Most of the classical image processing methods assume additive noise, which is ineffective for processing SAR imagery [3]. Thus, PolSAR image analysis require specifically tailored signal processing techniques.

Among image processing techniques, edge detection occupies a central position. In simple terms, its purpose is to identify boundaries between regions of different structural characteristics [4]. In PolSAR systems, such features are usually represented by different scattering characteristics, which are effected by surface reflectance and speckle noise.

Several edge detection approaches have been proposed for SAR imagery, among them gradient-based methods [5, 6, 7, 8]. Such approach consists in using a sliding window to define a measure map that highlights the edges, resembling the edge strength map technique [5]. Then a thresholding step is applied to perform the sought edge detection. In a different approach, Oliver et al. [6] proposed a maximum likelihood method aiming at two goals: (i) detecting the presence of an edge within a window and (ii) determining accurately the position of the edge. Another line of research is the specific study of edge detection based on physical properties from urban areas. As an example, Baselice and Ferraioli [9] utilized Markov random fields to model jointly the amplitude and interferometric phase of two complex SAR images.

Statistical procedures, including active contour models, have also been applied to edge detection on SAR images [10, 11, 3, 12, 4, 13]. Giron et al. [13] compare seven edge detectors following the general idea proposed by Gambini et al. [3]: finding transition points in strips of data. This latter method was successfully employed in [12] and [13] for a comparative study using other strategies based on the same active contour approach. In [4], Frery et al. extended it to multilook PolSAR data using the polarimetric distribution as model.

In the present work, we aim at extending the proposal of Gambini et al. [3]. In its original idea, this method consists in forming strips of data around regions that span from the centroid of the candidate region to points located outside the region under consideration. Resulting strips are then submitted to a screening phase, where each one is scanned looking for the point which defines two distinct regions under it based on maximization criterion, such as likelihood function. The point that satisfies a decision rule is called transition point. A pre-processing step may be necessary in order to define the initial regions. A post-processing step defines the edges of the regions by combining the transition points with B-spline curves.

For any type of approach, the edge detection problems can be related to three main aspects: (i) the procedure for detection, (ii) the determination of the most accurate edge position, and (iii) the specification of the window size (the window represents a square window or a strip of data). The latter may influence all others aspects in such manner that smaller windows may not convey enough information to identify the presence of edges, while bigger windows may contain more than two edges. Then, the ideal window size is the one that provides only a single edge within the window [6]. Following the approach discussed in [3], we assume that there is one edge within the window provided by the initial selection.

Our proposal improves or extends those procedures in three senses: (i) the finest possible strips of data are used, namely the ones of one pixel width, and (ii) stochastic distances and difference of entropies are employed as objective functions to be maximized; not only the likelihood function as in [3]. (iii) the influence of the spatial image resolution is performed.

The Kullback-Leibler and Bhattacharyya distances have been used by Morio et al. [14] as a scalar contrast measure between different channels of polarimetric and interferometric synthetic aperture radar (PolInSAR) images. These distances were calculated assuming that PolInSAR images follow the complex multidimensional Gaussian circular distribution. The distances were then compared by their discrimination ability. In the present paper, this work extended in two ways: (i) the complex Wishart model is utilized instead of the Gaussian circular distribution, and (ii) four distances, particular cases of the - class of divergences, were considered as well as two difference of entropies. Other information theory measures have been employed for change detection. Erten et al. [15] proposed a method based on mutual information for PolInSAR images assuming the scaled complex Wishart law. As a result, they could verify that the new detector is more efficient than the one based on the maximum likelihood ratio statistic [16].

Therefore, the contribution of this paper is two fold. Firstly, the paper presents a detailed discussion about information theory measures (using - divergences and entropies) as criteria for the edge detection problem [3]. Secondly, we compare their performance by edge accuracy and computational time, using contrast and spatial resolution as factors. Three real multilook PolSAR images are employed to show the application of the proposal. The best results were obtained using Rényi and Bhattacharya distances and difference of entropies.

This paper is structured as follows. Section II presents the seven assessed measures. An analysis of the proposed techniques using real and simulated data is presented in Section III. To that end, Monte Carlo experiments are performed in Section III-A to assess required computational time and probability of correct detection. Section III-B presents applications to real PolSAR images. Finally, the results are summarized in Section IV.

Ii Edge detection approaches under scaled complex Wishart distribution

In this section, we present seven approaches for detecting edges in PolSAR images based on the methodology employed by [3, 12, 4, 13]: finding a transition point in a strip of data which is an estimator of the edge position. In all cases, the estimator is the maximum of a given function. The first function is a likelihood, which is a multivariate extension of the approach shown in [12]. Remaining functions are - divergences and (difference of) entropies [17, 18, 19].

The edge detection methodologies considered in this paper operate in multiple stages: (i) identifying the initial centroid of the area of interest in a automatic, semiautomatic, or manual manner; (ii) casting rays from the centroid to the outside of the area; (iii) collecting data around the rays; (iv) detecting points in the strips of data which provide evidence of a change of properties, a transition; (v) defining the contour using a imputation method among the transition points, such as B-Splines [3]. We bring contributions to stages (iii) and (iv).

Initially, let us admit a region with centroid . Rays are traced from to control points , , located outside ; the resulting rays can be represented by segments in the form and the angle between consecutive rays is . Rays are converted into pixels using Bresenham’s midpoint line algorithm [20]. This representation provides the thinnest possible digital representation for a ray. This contrast with the 20-pixel wide strips employed in [3, 12, 4, 13]. This setup is illustrated in Fig. 1.

Fig. 1: Edge detection on the polarimetric strip from the centroid C of a region to the control point .

Data are assumed to follow a scaled complex Wishart distribution [21]. The scaled complex Wishart distribution has density given by

(1)

where is possible outcomes of , represents the covariance matrix, is the number of looks, is the number of polarization channels, is the multivariate gamma function, and and are the determinant and the trace, respectively. We refer to this distribution as

The strip of data collected around the th ray , , contains pixels. Each pixel of a given strip is assumed to be described by the return matrix , which is distributed according to the scaled complex Wishart law [4]. Denoting the correct boundary position as , we have the following configuration:

(2)

In other words, each strip is formed by two patches of samples, and each patch obeys a complex Wishart law with different parameter values. In this work, we assume that the number of looks is estimated beforehand and remains constant for the whole image. Thus, in every strip . This supposition is realistic, since the number of looks is obtained from radar upon reception of backscattered pulses [21].

The main idea is to estimate the edge position on the data strip around ray according to a specified decision rule. Gray lines in Fig. 1 illustrate different configurations when the edge position is shifted.

The model stated in (2) assumes that at most one transition occurs in any given ray. Issues may arise if multiple transitions take place; this is discussed further in Section III-B, Fig. 14.

In the following we present three different decision rules. For brevity, we omit the superscript index , since we focus our analysis on a single strip.

Ii-a Log-likelihood

The likelihood function of the sample described by (2) is given by

where is possible outcomes of the random matrices described in (2). The log-likelihood is more convenient for maximization purposes:

(3)

The maximum likelihood estimator of the index at the location where the transition occurs is given by

(4)

Using the model given by (1) in (4), with minor algebraic manipulation one obtains

(5)

where and are the maximum likelihood estimators for and , respectively, with respect to edge position . This estimator satisfies the following expression:

Ii-B Information Theory

Information theory measures have received considerable attention as tools for contrast quantification [22, 23]. In this section, we describe novel methodologies for detecting PolSAR boundaries based on such measures. We adopt the following notation.

Let and be two random matrices obeying the scaled complex Wishart distribution with same known numbers of looks. The associated probability densities are and , respectively, where and are their parameter vectors and is the vectorization operator. Both densities are defined over the set of Hermitian positive definite matrices .

Ii-B1 Stochastic distances

  • Kullback-Leibler:

  • Rényi of order :

  • Bhattacharyya:

  • Hellinger:

In the above expressions, the differential element is given by

where is the th entry of matrix , and and return the real and imaginary parts of their arguments, respectively. Provided that the densities characterize the same distribution and possibly differ only in the parameter, we use instead of as arguments of the considered distances. Frery et al. [24] derived analytical expressions for the above distances considering Wishart distributions for the general case and . Since we assume that the equivalent number of looks is the same, the distances can be expressed according to:

  • Kullback-Leibler:

  • Rényi:

  • Bhattacharyya

  • Hellinger

For , the Kullback-Leibler and Bhattacharyya distances become similar to the expressions derived by Morio et al. [14].

Above expressions are modified according to [18] in order to obtain test statistics with known asymptotic properties. Thus test statistics based on these distances can be readily derived for the null hypothesis :

(6)

where , , respectively, and are estimators for and using random samples of sizes and as well as , respectively.

Let be the number of elements of the parameter vectors, i.e. , as discussed in [25]. As derived by Frery et al. [24], the test statistics given in (6) are asymptotically -distributed with degrees of freedom. This result holds true as long as the maximum likelihood estimators are used with large samples [18].

Thus, four novel detectors can be proposed for finding edges on PolSAR imagery by seeking for the point that maximizes the test statistics between the two models, i.e.,

where .

Ii-B2 Stochastic entropies

The concepts of “information” and “entropy” received its fundamental mathematical treatment in the context of data communications by Shannon in 1948 [26]. Thenceforth, the proposition and the application of these measures have become active research fields in several areas. In particular, Frery et al. [27] derived entropies and associated test statistics for the scaled complex Wishart model.

As suggested by Pardo et al. [28], we consider the Rényi of order and Shannon entropies, denoted by and , respectively. A discrimination statistics based on stochastic entropies can be defined as

(7)

where , , , is the th element of vector , , , the superscript is the transposition operator, represents the conjugate transposition, is the Fisher information matrix, such that , is the size of vector , and

Frery et al. [27] showed that the expression given in (7) follows asymptotically a distribution, and it can be used to test whether two samples from the scaled Wishart distribution possess the same entropy, i.e., . Additionally, the Shannon and Rényi of order entropies are given by, respectively:

and

for . Applying these expressions to the density given in (1) the following entropies and variances are obtained:

  • Shannon:

    and

    where is the Kronecker product, “” is the operator which vectorizes its argument, and is the th-order multivariate polygamma function given by [25]

    and the ordinary polygamma function is [25]

    for .

  • Rényi of order : Denoting , the Rényi entropy is expressed by

    and

These and other results are further discussed in [27].

Therefore, two additional detectors can be defined by the points that maximize the test statistics based on entropies between the two distributions:

where .

The expressions given in (6) and (7) have known asymptotic distributions so, besides being discriminatory measures, they also have significance levels. This can be useful when the strip of data conveys a single class and, therefore, no detection should be made. Nevertheless, in the subsequent section these properties are not considered, due to the fact that the procedure is bounded to use small samples.

Iii Applications

We now apply the methodologies presented in the previous section to simulated and real data. Initially, in Section III-A, complex Wishart distributed scenarios were generated in order to quantify and to assess the accuracy and computational load of the detailed edge detection schemes. For such estimates of the detection error as well as the execution time for each method were compared. Finally, three applications to real data are performed in Section III-B. These applications illustrate the difficulties of determining an ideal window size.

Iii-a Precision and execution time

In order to measure and compare the accuracy of the discussed seven edge detection techniques, we follow the same methodology proposed by Frery et al. [4]. We estimate the probability of detecting the edge with an error less than pixels, for . This probability was estimated using strips of data of 400 pixels divided in halves, filled with samples from the Wishart distribution with two different covariance matrices:

(8)
(9)

which were observed by Frery et al. [4] in urban and forest imagery, respectively. In all cases four looks () were considered. Fig. 2 shows a simulated PolSAR image with this configuration according to the Pauli decomposition [29].

Fig. 2: False color of the Pauli decomposition of a simulated image from the scaled Wishart distribution with four looks and two halves defined by and .

Figure 3 shows the functions to be maximized as functions of the position for a typical simulation: the likelihood (Fig. 3(a)), distances (Fig. 3(b)), and entropies (Fig. 3(c)). The edge is at 200, and this position identified with a vertical solid line; the symmetric interval of twenty pixels around it is shown in vertical dash-dot lines. All of these distances exhibit a maximum which is close to the true edge position.

(a) Likelihood detector
(b) Distance detectors
(c) Entropy detectors
Fig. 3: Illustration of the proposed edge detection measures on images with halves which follow and distributions. The solid vertical line is at the edge and the dashed lines are at pixels from it.

We generated 1000 independent scene simulations to obtain the estimated boundaries positions (, , , ). The distance between these points and the true boundary (the absolute empirical bias) was evaluated for each replication:

and the probability of observing an error smaller than a certain number of pixels is estimated by relative frequencies as

where the number of replications for which the error is smaller than pixels.

Fig. 4 presents, in semilogarithmic scale, the estimated probability of finding the edge with an error equal or smaller than pixels, . Results show evidence that the methods based on entropies have the best results for . On the other hand, admitting an error , the method involving stochastic distances have the higher estimated detection probability. This fact suggests that a combination of these methodologies could lead to even more precise edge detection. Finally, the method based on the likelihood is outperformed by those based on stochastic distances; however, the former method provides better estimates when compared to and .

Fig. 4: Performance of detectors on phantom images with two halves distributed according to scaled complex Wishart laws.

Iii-A1 Influence of the backscatter matrix and time execution

Now we devise a simulation study where data is more similarly distributed; therefore offering a more difficult scenario. Also we aim at assessing the execution time of each method. Strips of data were defined with halves which follow distributions and , respectively. This situation is referenced by “case-”. Fig. 5 presents images sampled in cases , , , , and , i.e., with the same number of looks and varying covariance matrices.

Fig. 5: Images generated from , for , mapping [HHVV] – [HV] – [HHVV] onto the RGB channels.

The precision and execution time of the procedure were estimated using one thousand replications. The latter was measured in seconds of CPU cycles, while for the former we adopted a “hit-and-miss” criterion: a run is considered successful if it finds the true boundary, otherwise it is assumed wrong. The mean number of successful runs are reported in Fig. 6.

Fig. 6 and Table I present the average estimates of precision for all methods under several situations; the first column of Table I shows the label for each case which is used in Figs. 6 and 7. The best estimators are highlighted in boldface.

Situations to , which correspond to cases where the covariance matrix does not vary or changes by a small amount, the hardest ones to perform edge detection. Thus they often lead to low precision regardless the technique. It is noticeable that entropy methods consistently outperform other techniques in these particularly challenging situations. The performance of all procedures is comparable at remaining situations. We could identify that the maximum likelihood and entropy-based estimators excelled, in three and in two situations, respectively, but for a small difference of the value of the true boundary.

Label Case
TABLE I: Estimates for boundary points (estimates closest to the true edge are highlighted)
Fig. 6: Precison of the detection on polarimetric images with halves which follow and distributions.

Fig. 7 presents the mean execution time of each method in seconds. Fig. 7(a) presents obtained detection times, where it is noticeable that the maximum likelihood method associated to is the slowest procedure. Fig. 7(b) shows only the times required by procedures based on stochastic distances. We notice that the method based on the Kullback-Leibler divergence, which furnishes , was the fastest one. The results can be summarized in the following inequalities of the required computational times for each technique:

The influence of the case on the computing time is very small. All implementations are in the R version 2.13.2 programming language, running on a PC with an Intel© Core® i7-2630QM GHz with 4  RAM.

(a) All times
(b) Information theoretic tool times
Fig. 7: Detection times of the proposed methods in terms of the situations presented in Table I.

Iii-A2 Influence of the backscatter matrix SPAN and image resolution

In the following we assess the relative information provided by the polarimetric data with respect to single-channel data for the problem of finding edges. Assuming the scaled complex Wishart with parameters and for the polarimetric data, each intensity channel is described by the gamma distribution with density given by

for , where is the th entry of and is the th entry of the random matrix  [30]. This density can be used in the log-likelihood of strips (cf. (3)) to derive the estimator of the edge position (cf. (4)).

We considered two random matrices and :

  • Case A: and , where , for a square matrix with order , and , . This case represents the influence of the SPAN [31], i.e., the trace of backscatter matrix when .

  • Case B: and . This aims at quantifying the influence of the covariance between polarization channels varying the values of .

A Monte Carlo simulation was performed to quantify the performance of the edge detectors under these conditions.

For Case A, images with pixels in each half were generated with . Edge detection was performed, and the probability of detecting within an error equal or smaller than pixels was estimated. Fig. 8 presents the results. As expected, all polarimetric methods outperformed the methods based on single-channel data. Additionally, these results provide evidence that the performance of the methods tends to be similar when the SPAN is significantly different, i.e., when the halves are quite different. Moreover, boundary detection based on entropy measures outperformed all other methods in Case A.

(a)
(b)
(c)
Fig. 8: Influence of the SPAN on the probability of detecting and edge with an error equal or smaller than a certain number of pixels.

Fig. 9 shows the performance of the discussed methods in Case B, for . As in Case A, polarimetric data led to better results than single-channel images. In this case, detectors based on stochastic distances and maximum likelihood outperformed those based on entropies. Fig. 9(a) legend presents the detectors ordered from best to worst from top to bottom and from left to right. The best result was obtained by , but small differences were observed among methods involving distances.

(a)
(b)
(c)
Fig. 9: Influence of the covariance between polarization channels on the probability of detecting and edge with an error equal or smaller than a certain number of pixels.

We now analyze the effect of spatial resolution on the precision of the edge detection procedures. To that end, we simulated a strip of 200 observations divided in two halves, each with different distributions, namely and , such that , with defined in (9). This strip is assumed to have the best spatial resolution, and is denoted “”. The position of the edge, which is at the middle of the strip, i.e. at 100, is estimated with all the available techniques. Then the spatial resolution of the strip is degraded by taking the mean of pairs of contiguous observations and downsampling one each two pixels obtaining, thus, a resolution image. The estimation of the edge position, which is now at 50, is performed. A new pyramid is formed, denoted “”, and the estimation of the edge (now at ) is again performed. The equivalent number of looks is assumed know, and at each resolution takes the values , and . This procedure was repeated generating independent initial strips, and the bias (, the expected value of the estimator minus the true value), the standard deviation (), the coefficient of variation () and the mean squared error () of each estimator were estimated. The results are shown in Table II.

The detection bias was always very small, being the largest values in the order of of a pixel and the smallest ones in the order of of a pixel. The absolute bias tended to exhibit a small increase when the spatial resolution was reduced, while the standard deviation and the mean squared error were reduced. The width of the Gaussian symmetric confidence intervals at the confidence level ranges between, approximately, half a pixel and one pixel, rendering, thus, that all techniques exhibit comparable accuracy. Changing the resolution had little effect on the coefficient of variation.

Boundary detection methods
Single channel data Full PolSAR data
Resolution Measures
TABLE II: Effect of the spatial resolution on the edge detection

Iii-B Application to real data

We applied the methods to three real images. The first example describes a simple situation: well separated regions with most of the strips including two regions only. The second and third examples present more sophisticated problems due to the size of the strips (larger strips may span more than two regions).

The first application was performed on a San Francisco Bay image obtained by the AIRSAR sensor in L-band with four nominal looks, which is available in [32]. Fig. 10(a) shows a scene of the image with three well defined regions: ocean (dark area – top and left part of the image), forest (gray area – top and right), and urban area (light area – bottom). Figs. 10 and 11 present the result of detecting the edges which separate urban and ocean regions from other classes, respectively. The following analysis was performed by visual inspection on edges reconstructed from the estimated transition points and B-splines of fourth degree.

In Fig. 10(b), estimator presents two mildly biased boundaries (highlighted with green squares), whereas and present only one. Remaining methods present similar behavior.

Fig. 11(b) also exhibits three situations where the detected edge was slightly biased, namely , and , each with one noticeable deviation. Again, other discussed methods behave alike.

(a) Urban area vs. sea and forest
(b) Detected egdes
Fig. 10: Performance of detectors with centroid on urban region from AIRSAR image of San Francisco Bay. Errors highlighted in green boxes.
(a) Sea vs. urban area and forest
(b) Detected egdes
Fig. 11: Performance of detectors with centroid on ocean region from AIRSAR image of San Francisco Bay. Errors highlighted in green boxes.

The Bhattacharyya and likelihood techniques did not miss any detection in these examples. This, and the simulation results on accuracy and computational time, leads us to suggest as the fastest and most efficient detector.

Fig. 12 shows an E-SAR image over the surroundings of Munich, Germany, which was obtained with (equivalent number of) looks. The nonconvex region to the center of Fig. 12(a) is of interest, for which an arbitrarily centroid and thirteen rays were defined. Fig. 12(b) summarizes the results. It shows the edges, and the label of the ray for which the transition point was identified far from the visual true point. Both maximum likelihood and Kullback-Leibler methods failed at detecting only one situation, highlighted with a green square. The other results are similar.

(a) HH channel of the ESAR image with selected rays 
(b) Detection in actual images
Fig. 12: Edge detection in an E-SAR image over the surroundings of Munich, Germany. Error highlighted in the green box.

There are practical situations in which all methods perform alike. Consider the EMISAR image of the agricultural area of Foulum, Denmark, presented in Fig. 13. This image was obtained in L-band and quad-pol; its HH channel is shown in Fig. 13(a). Notice that the method based on the joint likelihood (Fig. 13(b)) works similarly to and (Fig. 13(c)), (Fig. 13(d)), and (Fig 13(e)). However, the best technique regarding execution times is the one based on the Bhattacharyya distance.

(a) Real Images with selected axes
(b)
(c)
(d)
(e)
(f)
Fig. 13: Edge detection of an agricultural parcel in an EMISAR image of Foulum.

In order to analyze further the result obtained by the Hellinger distance, presented in Fig. 13(f), consider the ray zoomed in Fig. 14(b), and notice that it crosses three regions: gray, white and black leading, therefore, to two possible edges. As all the techniques are proposed in terms of the maximization of a discrimination measure, it is expected that the estimated boundary is the position of the highest distinction between two regions. In this case, it takes place in the second edge, i.e., between the white and black areas. This is confirmed by Fig. 14(c), which shows the likelihood, but not in Fig. 14(d), which shows the Hellinger distance. All other criteria behave alike, as shown in Fig. 14(e).

(a) A ray crossing three different regions 
(b) Zoom
(c)
(d)
(e)
Fig. 14: Details of of the edge detection in a hard-to-deal-with configuration from the EMISAR image of Foulum.

The Hellinger distance is limited to the interval. Additionally, let , empirical results provide evidence that there is a subset of the parametric space of the scaled complex Wishart distribution for which for all . This fact renders the statistics inappropriate for quantifying distinctions between distributed Wishart random fields when .

This behavior opens new research lines based on a redefinition of the proposed methods in order to cope with situations in which rays cross three or more distinct regions.

Iv Conclusions

In this paper, boundary detection procedures using information theory measures (stochastic distances and entropies) and a likelihood function were considered. Due to its analytic tractability, these techniques are derived under the assumption that the Wishart distribution is an acceptable model for full PolSAR data. In order to quantify and compare their performances, phantoms and real PolSAR images were used.

The performance of each procedure was quantified by the probability of correctly detecting the edge position within at most pixels, as well as by the execution time. Problems with different levels of complexity were assessed in order to quantify the influence of the number of looks, the backscatter matrix as a whole, the SPAN, the covariance, and the spatial resolution. Also, the information provided by the full polarimetric format with respect to each intensity channel was assessed, leading to the conclusion that the complete data set is recommended for edge detection. The results provide evidence that all methods are able to detect edges correctly within four pixels, with the exception of entropy-based techniques which fail in more complex scenarios with similar covariance matrices.

The discriminatory function based on the Kullback-Leibler distance is the fastest to compute, while the likelihood function is the slowest one among the considered measures.

The simulation studies revealed that information theory measures are consistently closer to the true boundary than the joint likelihood detector. In particular, Shannon and Kullback-Leibler detectors provided the most accurate detection and the smallest execution time, respectively.

Finally, three applications to real data obtained by E-SAR, EMISAR, and AIRSAR sensors were performed. The seemingly poor performance of the edge detection based on the Hellinger distance was identified and explained by its properties.

As a conclusion, techniques which employ Rényi and Bhattacharyya distances and entropies outperform other methods and are recommended for edge detection in PolSAR imagery.

Further research aims at considering models which include heterogeneity [21, 33, 4], robust, improved and nonparametric inference [34, 35, 36, 37], and small samples techniques [38].

Acknowledgment

The authors are grateful to CNPq, Facepe, Fapeal, FAPESP, and Capes for their support.

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Abraão D. C. Nascimento holds B.Sc.  M.Sc.  and D.Sc. degrees in Statistics from Universidade Federal de Pernambuco (UFPE), Brazil, in 2005, 2007, and 2012, respectively. In 2013, he joined the Department of Statistics at Universidade Federal da Paraíba. His research interests are statistical information theory, inference on random matrices, and asymptotic theory.

Michelle M. Horta received the B.S. degree in computer sciences from Universidade Católica de Pernambuco, Recife, Brazil, in 2002, the M.Sc. degree in computer sciences from Universidade Federal de Pernambuco, Recife, Brazil, in 2004, and the Ph.D. degree in applied physics from Universidade de São Paulo, São Carlos, Brazil, in 2009. She is currently working toward the Postdoctoral Research in the Departamento de Computação, Universidade Federal de São Carlos, São Carlos, Brazil. Her research interests are remote sensing, pattern recognition and statistical models.

Alejandro C. Frery graduated in Electronic and Electrical Engineering from the Universidad de Mendoza, Argentina. His M.Sc. degree was in Applied Mathematics (Statistics) from the Instituto de Matemática Pura e Aplicada (Rio de Janeiro) and his Ph.D. degree was in Applied Computing from the Instituto Nacional de Pesquisas Espaciais (São José dos Campos, Brazil). He is currently with the Instituto de Computação, Universidade Federal de Alagoas, Maceió, Brazil. His research interests are statistical computing and stochastic modeling.

Renato J. Cintra earned his B.Sc., M.Sc., and D.Sc. degrees in Electrical Engineering from Universidade Federal de Pernambuco, Brazil, in 1999, 2001, and 2005, respectively. In 2005, he joined the Department of Statistics at UFPE. During 2008-2009, he worked at the University of Calgary, Canada, as a visiting research fellow. He is also a graduate faculty member of the Department of Electrical and Computer Engineering, University of Akron, OH. His long term topics of research include theory and methods for digital signal processing, communications systems, and applied mathematics.

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