Classification of radar echoes using fractal geometry

Chaos, Solitons and Fractals 98 (2017) 130–144

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Review

Classification of radar echoes using fractal geometry Nafissa Azzaz∗, Boualem Haddad University of Science and Technology Houari Boumediene, Faculty of Electronic and Computer Science, Department of Telecommunication, BP N 32 El Alia, Bab Ezzouar, Algiers, Algeria

a r t i c l e

i n f o

Article history: Received 16 September 2015 Revised 26 January 2017 Accepted 7 March 2017

Keywords: Ground echoes Precipitations Radar Fractal dimension Fractal lacunarity Contourlets

a b s t r a c t This paper deals with the discrimination between the precipitation echoes and the ground echoes in meteorological radar images using fractal geometry. This study aims to improve the measurement of precipitations by weather radars. For this, we considered three radar sites: Bordeaux (France), Dakar (Senegal) and Me lbourne (USA). We showed that the fractal dimension based on contourlet and the fractal lacunarity are pertinent to discriminate between ground and precipitation echoes. We also demonstrated that the ground echoes have a multifractal structure but the precipitations are more homogeneous than ground echoes whatever the prevailing climate. Thereby, we developed an automatic classification system of radar using a graphic interface. This interface, based on the fractal geometry makes possible the identification of radar echoes type in real time. This system can be inserted in weather radar for the improvement of precipitation estimations. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Radar imaging is currently the most used technique for rainfall estimation by weather forecasters. However, by using electromagnetic waves, radars often receive echoes backscattered by the Earth’s surface. These echoes called ground echoes or fixed echoes are a source of noise and they reduce consequently the radar ability to estimate efficiently the precipitations [1]. The other kind of undesirable echoes is due to abnormal meteorological conditions, caused by gradients of refraction indices less than −157 × 10−6 km−1 . In this case, the electromagnetic waves coming from the radar are affected; as a result, they are propagated through atmospheric ducts and backscattered by the earth surface. These unwanted echoes are called Anomalous Propagations (APs) or anaprops. Also, the parameterization of precipitation fields in meteorological radars requires a prior knowledge of their structure and density at different scales [2,3]. Fractal geometry is of great interest for the analysis and processing of digital images for a very large number of applications. It is used in meteorology to study cells structure in radar and satellite images [4–6], in medicine [7–9], in geology [10], as well as in various other fields. Fractal geometry is used to describe selfsimilar sets called fractals and characterizes natural objects that cannot be described by classical geometry [11].

∗

Corresponding author. E-mail address: nafi[email protected] (N. Azzaz).

http://dx.doi.org/10.1016/j.chaos.2017.03.017 0960-0779/© 2017 Elsevier Ltd. All rights reserved.

Fractals may be described by their dimension [12,13], and their lacunarity [7,14]. These two fractal properties are used to discriminate between different types of structures with a fractal appearance, and also for classification and segmentation, due to their scale invariance, their rotation and their translation [9,15,16]. The major problem of weather radar measurement is related to the clutter coming from the Earth’s surface. These significantly reduce instrument performance by inducing significant errors in the estimation of precipitation, making the hydrological measurement very difficult [17,18]. The aim of this study is to identify the ground echoes in meteorological radar images. Several techniques are proposed in the literature for identifying and removing fixed echoes, for example Doppler filtering [19], or dual polarization filtering [20,21]. In order to eliminate clutters, the authors in [22] compare the statistical properties of the ground echoes to those of precipitation echoes, such as textural features clutter can also be removed by analyzing in real time the coefficient of the autocorrelation function of the radar signal [23]. Others applied the fuzzy logic technique, to classify the Doppler radar echoes types [24] or for identifying nonprecipitating echoes in radar scans [25,26]. We can also undertake the classification of radar echoes with a textural–fuzzy approach for the removal of ground clutter [17], and the technique of neurofuzzy to eliminate noise in Doppler radar signals [27]. Our work will carry on the implementation of two concepts of fractal geometry to classify two types of echoes, and this by using the Box-counting method to calculate the fractal dimension [11], and the method of Allain and Cloitre to determine the fractal lacunarity [29].

N. Azzaz, B. Haddad / Chaos, Solitons and Fractals 98 (2017) 130–144

In the recent past, it has been shown that the fractal dimension in single scale can’t be used as a discriminating parameter between precipitations and the fixed echoes [1,30]. The questions are: Does the fractal dimension provide a solution to discriminate between the two types of echoes? Can the lacunarity fractal be used as a parameter for identifying the ground echoes? To answer these questions, we propose a new approach, called a fractal dimension based on contourlets. This method consists in decomposing all images by the contourlets transform, to calculate the fractal dimension of the resultant contourlets coefficients through various scales and directions. The Contourlet Transform is a thorough representation for 2D and 3-D images. It is applied in various fields [31,32]. Using a rich set of images with oriented base at different directions and scales, the contourlets can actually capture the intrinsic contours in images and offer better anisotropy, multi-resolution, directivity and localization properties of the images than obtained with mono scales representation tools already existing [33]. The application of the fractal dimension on the resulting coefficients of the contourlets decomposition using the Box Counting method allows the extraction of the fractal characteristics with greater accuracy on different scales and orientations [33]. The purpose of our paper is to test that the fractal geometry can be successfully applied in discrimination between the fixed echoes and rain echoes. In addition, we propose an automatic system for real time identification of the two types of echoes. We considered in our analysis a database obtained from three radar sites with different climates and topographies, namely, Bordeaux (France), Dakar (Senegal) and Melbourne (United States). The paper is structured as follows: after an introduction, we present a theoretical study on the fractal dimension and the fractal lacunarity. The database is presented in the third section. In section four, we present the results and interpretations, and finally we end with a conclusion. 2. Theoretical concept In the real world, the Euclidean geometry does not characterize all forms of existing objects. Thus was born a new discipline called fractal geometry, which is actually complementary to the Euclidean geometry. The word “fractal” is a term proposed by Mandelbrot in 1974 from the Latin root “fractus” which is synonymous of irregular and fragmented [11]. The fractal objects are the result of an iterative process that presents a character of self-similarity and can be defined recursively [11,34]. In the field of image analysis and processing, fractal study provides rich information in content and represents a necessary element for the knowledge of their structure [34]. In the literature, there are several concepts used to characterize fractals. We mention as an example, the concept of fractal dimension [11,28,35–37], and the technique of fractal lacunarity [28]. In our case, we applied the box-counting method for calculating the fractal dimension [38], and the approach of Allain and Cloitre to estimate lacunarity [29]. The Lacunarity and Fractal dimension are two physical quantities using fractal geometry, and they must be complementary [39,40]. Indeed, Przemyslaw Borys [39] has mentioned that the Lacunarity is a measure designated to accompany the fractal analysis in case where the images have similar fractal dimensions. Lacunarity corresponds to the measurement of the distribution of the holes, whereas the fractal dimension is a measure of the masses distribution. These two measures should be inversely proportional, i.e., when the one decreases, the other increases [40]. In the literature, some articles suggest an equation for the relationship between the fractal dimension and lacunarity [41]:

D f = 2, 47 − 1, 4

(1)

131

Where: Df : is the fractal dimension. : is the fractal lacunarity In [42], they have also demonstrated that the lacunarity varies in the same sense with the fractal dimension, but they said that the first relation is the most used in the literature. 2.1. Fractal dimension The Box Counting method is based on the recovery of all analyzed image space, by a grid of squares (or "boxes") of side ε . The number N(ε ) is the number of boxes that are used to pave the cells presented in the analyzed image. The fractal dimension Df is then defined by [38]:

D f = lim

ln [N (ε )]

ε→0 ln(1/ε )

(2)

The number of boxes N(ε ) containing the pixels of rainfall support are counted. This operation is repeated for different values of ε (ε = 2, 4, 8, .., 2P ). The (2P ) value represents the maximum size of our images. By tracing for different values of ε , ln(N(ε )) versus ln(1/ε ), we get a points cloud {ln (N(ɛ)),ln (1/ɛ)}. The slope of the line that fits the points cloud gives us an estimation of the fractal dimension Df . 2.2. Fractal dimension based on contourlets 2.2.1. Contourlets transform In the literature, it is stated that the conventional multiresolution decompositions make a restricted and limited category of possibilities for multiscale image representations [43]. Recently, studies have shown that is possible to define new theoretical methods more appropriate for multi-scale representations, creating a new transform more suited to the extraction of cells geometric structures present in the images such as the objects contours [43,44]. We mention the new family of wavelet derivatives, namely: the Ridgelet transform, the Wedgelet transform, the Curvelet transform and the Contourlet transform [45,46]. The Contourlet transform which will be applied in our case, is a discrete version adapted to the digital images, and based on the use of directional pyramidal filters banks. It is a multi-scale decomposition, which operate in a multitude of directions and frequencies that offer a good compromise between the representation of the decomposed image characteristics and the perceptual quality of this latter reconstructed [43]. Contourlet transform was introduced by Minh Do and Martin Vetterli [45,47]. This is an image decomposition method, designed directly into the discrete domain, which provides a sparse representation of the data contained in the image, both in spatial and frequency resolutions. Contourlet transform is constructed by combining two distinct successive decomposition steps: a multi-scale decomposition followed by an oriented directional decomposition [43]. The first step uses a Laplacian pyramid (LP) [48] to transform the image into a series of bandpass LP levels and a single lowpass level (low frequency approximation of the image) [49]. The second step applies appropriate bidimensional filters and a sampling, in order to decompose each bandpass LP level into a number of directional strips, thus capturing the frequency information of the image [49,50]. The filters used in our contourlets decomposition are the 9/7 filters and the PKVA filters [51]. Finally, the image is represented by a set of multi-scales and oriented sub-bands [52]. These sets present the multi-scale and time-frequency characteristics of a decomposed image. Each scale can be divided into an arbitrary number of directions that is a power of 2 [52].

132

N. Azzaz, B. Haddad / Chaos, Solitons and Fractals 98 (2017) 130–144

Fig. 1. (a) The initial image of Dakar Radar, (b) Countourlet coefficients for a radar image of Dakar.

2.2.2. Fractal dimension based on contourlets When we have calculated the classical fractal dimension using box counting, we couldn’t discriminate between ground echoes and precipitation echoes. To achieve this, and knowing that the contourlets have important properties of spatial and frequency localization, we used a new method that combines two concepts: fractal dimension and contourlets transform. This method is called the fractal dimension based on contourlets [53], it is based on two steps: – The first step uses the decomposition of the image by the contourlets transform to obtain multi-level sub-bands. Each subband is characterized by a coefficient of contourlet. Those coefficients will be listed within 2-D matrix and each one represents a level of scale resulting from the decomposition of an image using the Laplacian pyramid. We illustrate as an example in Fig. 1(a) representation of countourlet coefficients for a radar image collected in Dakar. The figure is formed of several images which each one represents a level of contourlets decomposition of the initial image, it shows the embedded tree data structure for contourlet coefficients, for this reason there is no scale in the figure. The initial image is decomposed into pyramidal levels, which are then decomposed into directional subbands. Small coefficients are shown in black while large coefficients are shown in white.

– The second step is to compute the fractal dimension of the resultant contourlets coefficients, by applying the Box-counting method on the corresponding matrices of each scale level and calculating only the pixels that contain a non-null coefficient (i.e., the number of boxes containing relevant pixels at each scale is counted). For calculating the fractal dimension, it does not need the binarization of the image, but just considering only the non-zero pixels to count the number of boxes. In this approach, we don’t need a threshold to be set on the radar images. 2.3. Fractal lacunarity Fractal lacunarity is a second order fractal parameter that describes the texture of an image by measuring the distribution of holes in the image. A low lacunarity corresponds to a high homogeneity, and vice versa [28]. There are several methods in the literature for calculating the fractal lacunarity. In our case, we use Allain and Cloitre method, based on sliding boxes algorithm [29]. In this algorithm, we consider a box of size L centered on the first pixel of the image of size M × M as L ≤ M. This box is translated horizontally on each line of the image from left to right, with a step of 1. This operation is repeating for the M lines of the image by going from top to bottom with a step of 1, thereby treating the whole image. Thus, at the end of this operation, the structuring element reaches M − L + 1 possi-

N. Azzaz, B. Haddad / Chaos, Solitons and Fractals 98 (2017) 130–144

ble positions and a mass distribution n(m,L) is determined, for each line of the image. This latter represents the number of boxes with side L and mass m, m is the number of pixels contained in a box [54]. Dividing the quantity n(m,L) by the total number of boxes of side L, we obtain the distribution probability P(m,L) corresponding to the occurrence frequency of a box with a mass m and a side L. To study the properties of this distribution we consider the statistical moments of order q, denoted q∈N ∗ noted Z (q, L) such as [6,54]:

Z (q, L ) =

m max 

mq P (m, L )

The Lacunarity (L) considered at the scale L is defined as the relative moment of 2nd order:

Z 2 ( 2, L )

2

(4)

Z 1 ( 2, L )

We note that Z 1 (2, L ) = s¯(L ) and Z 2 (2, L ) = s2 (L ) + s¯2 (L ), where ¯s(L )is the mean of masses per box, denoted μ, and s2 is the variance of the masses by box, noted σ 2 . So, the lacunarity (L) will be defined by the following equation [29]:

(L ) =

s2 ( L ) +1= s¯2 (L )

σ2 +1 μ2

Table 1 Technical characteristics of the weather radar of Dakar region. Technical characteristic

Value

Transmission frequency Pulse repetition frequency Transmission power peak Spatial resolution Pulse duration Beam width to 3 dB

5.7 GHz 250 Hz 250 kW 1 km x 1 km 3 μs 1,3°

(3)

m=1

(L ) = 

133

(5)

3. Database We considered in this study three different weather radar sites namely, Dakar (Senegal), Bordeaux (France), Melbourne (Florida, USA). The database collected from these radars will be prepared before using it in our work. The images provided from the Dakar and Bordeaux radars are temporal sequences on which appear different kinds of echoes. The texture of the precipitation echoes is more homogeneous than that of ground echoes [17], these represent the unwanted signals. Using visual inspection, the images where precipitation echoes are distinctly separated from the ground echoes are selected, and the pixels representing the ground echoes are set to zero. The visual inspection consists in running sequential radar images with software, and then we get a kind of animation that shows moving cells representing the precipitation, and fixed cells describing the ground echoes. Then, the data base is divided into two parts; the first containing only the precipitation echoes and the second one the ground echoes. The images of ground echoes were collected in clear sky conditions. Melbourne radar is a Doppler radar, thus it proceeds to the elimination of a large part of the ground echoes. Nevertheless, it remains some residues that will be separated from the precipitation cells in the same way as for Dakar and Bordeaux radars.

Table 2 Technical characteristics of the weather radar of Bordeaux region. Technical characteristic

Value

Transmission frequency Pulse repetition frequency Transmission power peak Spatial resolution Pulse duration Beam width to 3 dB

3 GHz 300 Hz 500 kW 1 km x 1 km 5 μs 1,8°

acquisition of PPI radar data is made in polar coordinates. Radar signals are then digitized by the chain SANAGA and converted into Cartesian coordinates. The digital analysis of the reflectivity allows processing and data storage. This radar works only during storm periods because the criterion to turn "on" or "off" of the radar is related to the rainy season. In fact, the radar of Dakar is activated only during rainy events, and the rainy season in Dakar is reduced to about 3 months, from early July to late September [55]. The mean annual cumulative rainfall varies from 300 mm to 1500 mm and from Saint Louis to Cape Skirring [55]. The technical characteristics of the weather radar of Dakar are given in Table 1.

3.2. Region of Bordeaux The city of Bordeaux is located in the south west of France. The Bordeaux radar is located in the Bordeaux-Merignac airport, whose geographical coordinates are 44°49 54 North latitude and 0°41 30 West longitude. It is one of 24 metropolitan France radars. The acquisition of CAPPI radar data is made in polar coordinates. Radar signals are then digitized by the chain SANAGA and converted into Cartesian coordinates [18]. The radar is part of the French network (ARAMIS) managed by Météo-France. The area is almost flat. Table 2 gives us the technical characteristics of the radar of Bordeaux.

3.3. Region of Melbourne 3.1. Region of Dakar Dakar is the capital of the Republic of Senegal. It is located at the western extremity of Africa, on the narrow peninsula of Cape Verde. Being in a tropical semi-desert, Dakar has a microclimate of coastal type, influenced by the monsoon trade winds and the sea. The hot wet season extends from June to October with temperatures of about 27 °C and a peak of precipitations in August (250 mm). The radar of Dakar is installed on a tower of 30 m above the sea level at Yoff Airport, in the east of Dakar. Its geographical coordinates are 14° 44 North latitude and 17° 29 West longitude. It is operated by both the National Meteorological Office of Senegal and ASECNA (Association for the Safety of Air Navigation). The

Melbourne is a city of the United States of America located on the east coast of Florida. It is situated to the east of Orlando, and south of Cape Canaveral. It has a subtropical climate, typical of the Gulf and South Atlantic states, with hot and humid summers and cool winters. The used radar is a Nexrad WSR-88D, it works on the coherent Doppler principle. Its geographical position coordinates are 28° 6 32.4 N and 80° 39 0 W. It allows observing the precipitation in the oceanic, littoral and terrestrial part [56]. The technical characteristics of the weather radar of Melbourne are given in Table 3. The comparison of technical characteristics of the weather radar of Dakar, Bordeaux and Melbourne are given in Table 4. The dataset consists of 50 0 0 radar images for Bordeaux, 10 0 0 radar images for Dakar and 600 radar images for Melbourne.

134

N. Azzaz, B. Haddad / Chaos, Solitons and Fractals 98 (2017) 130–144 Table 3 Technical characteristics of the weather radar of Melbourne region [57]. Technical characteristic

Value

Transmission frequency Pulse repetition frequency Transmission power Spatial resolution Pulse duration Beam width to 3 dB

2.7- 3 GHz according to VCP from 320 to 1300 Hz 750 kW peak 1 km x 1 km according to VCP from 1.57 to 4.57 μs 0,96° at 2,7 GHz 0,88° at 3,0 GHz

3.4. Images preprocessing For Bordeaux and Dakar images, Ground echoes and precipitation are identified using visual inspection. In order to filter the spurious echoes in the precipitation images, we have used the masking method which consists of two steps: Creation of Mask and filtering the fixed echoes. Note that the radar images containing anaprops have not been pre-treated by masking method, because in these images we have only anaprops echoes that are not mixed with precipitation echoes. 3.4.1. Creation of mask To extract the clutter mapping in clear weather, we analyzed 500 images for Bordeaux and 100 images for Dakar. Then, we calculated the frequency of occurrence of each pixel’s color. A pointer will then determine the most frequent gray level from the first to the 262,144th pixel (512 × 512). An image is then reconstructed representing the mapping of the ground echoes in the study area. These images will be taken as a reference image or real maps. We found a proportion 95% of the black color in the image. This color means that there are no echoes. To reduce the computation time, we have added an instruction to the program by asking it to suspend the calculation for each pixel that would see the black color repeating over 50% of the database. This criterion can be justified by the fact that the Earth’s surface component obstacles have always the same coordinates. Fig. 2(a) and (b) represent respectively the mask in the Dakar and Bordeaux radars. 3.4.2. Filtering the fixed echoes The technique compares the colors of the pixels of the mask with those of the considered image pixel by pixel. If the image color is identical to the mask, we assign the black color to the pixel. Otherwise, we keep the initial color of the image. In addition, we used a feature of the ground echoes, namely, the small variation around the mean [58]. Then, we compare the pixels of the image treated with those of the interval [Nc-1; Nc+ 1], where Nc is the value of gray levels coding. The images where precipitation echoes are distinctly separated from the ground echoes are selected, and the pixels representing the ground echoes are set to zero. Then the data base is divided

into two parts; the first containing only the precipitation echoes and the second one the ground echoes. The radar images of Figs. 3(a) and 4(a) from the region of Dakar and Bordeaux respectively, display the addition of a great number of ground echoes and a field of precipitating clouds. After filtering, the images obtained are those of Figs. 3(b) and 4(b), from which for the most part of the ground echoes have been removed and where the precipitation echoes have not been significantly modified. To get an estimation of the performance of this method, a recognition ratio has been calculated for each type of echo from the images, where rainfall echoes are distinctly separated from ground echoes. For a given type of echo, this parameter is defined as the ratio of the number of pixels obtained after filtering to that computed in the original image. We get that, 93% of ground echoes are removed and 95% of precipitation echoes are preserved in the radar images of Bordeaux. In the radar images of Dakar, only 90% of the ground echoes are eliminated and 93% of the precipitation echoes are faithfully reproduced. During this processing, the reflectivity of each pixel of the radar images still looks the same. In Figs. 5 and 6, we consider respectively in Dakar and Bordeaux, the case where the precipitations is moving over the ground echoes from west to east. In This case, it is more difficult to interpret the ground echoes by the masking method because the echoes of rainfall cover the clutter. To test the efficiency of our approach, we used the image animation program. We observe that the majority of clutter has been removed. It should be noted that for Melbourne images, we haven’t applied the masking method used for Bordeaux and Dakar, because the parasitic echoes have been filtered previously by the Doppler technique. Whatever the technique used in the treatment of parasitic echoes, residual echoes always remain [17,20]. Also, we can’t appreciate the quality of the Doppler technique because we don’t have the not-treated images before the application of the method. To identify the type of parasite echoes (fixed echoes, anaprops, insects …), we have proceeded to the animation of several series of five to ten successive images taken in clear weather. For all the data, we noticed that the parasitic echoes always appeared at the same coordinates, which allows us to conclude that we are in the presence of fixed terrestrial echoes.

4. Results and interpretations As an illustration, we present the results for the radar images in two sites, in Dakar (Senegal) and Bordeaux (France). Fig. 7(a) and (b) represent respectively the precipitation cells and the fixed echoes taken in the Dakar region. Fig. 8(a) and (b) describe respectively the precipitation cells and the fixed echoes cells collected in Bordeaux.

Table 4 Comparison of technical characteristics of the weather radar of Dakar [55], Bordeaux [55] and Melbourne. CAPPI is constant-altitude plan position indicator, PPI is plan position indicator, α is angle elevation, r is radar-target distance, and Z is radar reflectivity factor. VCP is Volume Coverage Patterns [57]. Sites

Dakar

Bordeaux

Melbourne

Scanning mode

PPI, α = 0.8°

VCP 11 and VCP21 α = 0.48° to α = 19,51°

Sampling interval Pixel size No. of steps for Z coding

Between 10 and 20 min 1 × 1 km2 256

CAPPI with α = 1.5° for r < 50km α = 0.4° for r > 50 km 5 min 1 × 1 km2 52

Azimuth Range resolution

0 to 360° 256 × 256 km2

0 to 360° 256 × 256 km2

5 min 1 × 1 km2 In clear air mode : from −28 to + 28 In precipitation mode : from 0 to 75 0 to 360° In Reflectivity : 460 km In Velocity : 230 km

N. Azzaz, B. Haddad / Chaos, Solitons and Fractals 98 (2017) 130–144

135

Fig. 2. (a) The mask of Dakar Radar, (b) The mask of Bordeaux radar. The scale is in km.

Fig. 3. Radar image of Dakar, (a) The image containing ground echoes and precipitation echoes. (b) The image containing only the precipitation echoes. The scale is in km.

Fig. 4. Radar image of Bordeaux, (a) The image containing ground echoes and precipitation echoes. (b) The image containing only the precipitation echoes. The scale is in km.

136

N. Azzaz, B. Haddad / Chaos, Solitons and Fractals 98 (2017) 130–144

Fig. 5. Radar image of Dakar, (a) The image containing ground echoes which are partially covered by the precipitation echoes. (b) The image containing only the precipitation echoes. The scale is in km.

Fig. 6. Radar image of Bordeaux, (a) The image containing ground echoes which are partially covered by the precipitation echoes. (b) The image containing only the precipitation echoes. The scale is in km.

4.1. Fractal dimension Fig. 9(a) and (b) give variations of the fractal dimensions calculated by the classical method of box-counting and by the proposed method based on contourlets, for both precipitation and ground echoes in Dakar Radar images. We note that there are several points for each resolution on the graph, because this last represents the result of a group of radar images. Similarly, Fig. 10(a) and (b) and Fig. 11(a) and (b) present the variations of classic fractal dimension and fractal dimension based on contourlets for precipitation cells and fixed echoes cells present in the radar images of Bordeaux and Melbourne respectively. The Curves of Figs. 9–11 can be described by the function y = ax + b where “a” is the slope or the fractal dimension and “b” the intercept. The values of the coefficients “a” and “b” are given in

the Table 5. The correlation coefficient and the root mean square error (RMSE) are used in approaches evaluation. Knowing that the fractal dimension is described by the slope of the trend curve, we can notice that by using the classical method, we obtain for the three regions, two approximately parallel slopes, and therefore two nearly identical fractal dimensions. According to the Table 5, the fractal dimension is estimated at 1.20 ± 0.03 for precipitation cells and 1.18 ± 0.042 for ground echoes cells for the Dakar region. Similarly for Bordeaux, the fractal dimension of the precipitation cells is equal to 1.15 ± 0.045 while that of the ground echoes cell is 1.17 ± 0.03. In Melbourne images, the fractal dimension is equal to 1.20 ± 0.025 for precipitation cells and 1.19 ± 0.019 for ground echoes cells. The correlation coefficient describing the quality of the fit is equal to 0.94. The different values of the fractal dimensions of precipitations overlaps with that of the ground echoes cells for all dataset studied. Therefore, we can deduce that

N. Azzaz, B. Haddad / Chaos, Solitons and Fractals 98 (2017) 130–144

137

Fig. 7. Radar image taken by the Dakar radar (a) Image containing precipitation cells, recorded on the 15th January 1999, and (b) Image containing fixed echoes cells, recorded on the 20th January 1999, Z is radar reflectivity factor. The scale is in km.

Fig. 8. Radar image, taken by the radar of Bordeaux, (a) image containing precipitation cells, recorded on the 13th November 1996, and (b) image containing echoes fixed cells, recorded on the 24th November 1996. The scale is in km.

Fig. 9. Variation of fractal dimension of precipitation echoes and ground echoes in Dakar, N(ε ) is number of boxes and ε is size of boxes, (a) classical method and (b) fractal dimension based on contourlet.

138

N. Azzaz, B. Haddad / Chaos, Solitons and Fractals 98 (2017) 130–144

Fig. 10. Variation of fractal dimension of precipitation echoes and ground echoes in Bordeaux, N(ε ) is number of boxes and ε is size of boxes, (a) classical method of fractal dimension and (b) fractal dimension based on contourlet.

Fig. 11. Variation of fractal dimension of precipitation echoes and ground echoes in Melbourne, N(ε ) is number of boxes and ε is size of boxes, (a) classical method and (b) fractal dimension based on contourlet. Table 5 Fractals dimensions Df obtained for the three regions studied. Sites

Type of cells

Df

[Dfmin, Dfmax] (Classical method)

Df Uncertainty

The coefficient of intercept “b”

Df based on Contourlets

[Dfmin, Dfmax] (Df Based on contourlets)

Df Uncertainty (Df Based on contourlets)

The coefficient of intercept “b”

Dakar

Precipitation Fixed echoes

1,20 1,18

[1.17, 1.23] [1.138, 1.222]

±0.03 ±0.042

0 0

1.90 1.76

[1.898, 1.912] [1.740, 1.780]

±0.012 ±0.02

1.2 0

Bordeaux

Precipitation Fixed echoes

1,15 1,17

[1.105, 1.195] [1.14, 1.20]

±0.045 ±0.03

0 0

1.91 1.71

[1.909, 1.911] [1.685, 1.735]

±0.001 ±0.025

1.2 0

Melbourne

Precipitation Fixed echoes

1,20 1,19

[1.175, 1.225] [1.171, 1.209]

±0.025 ±0.019

1.25 1.15

1.92 1.76

[1.912, 1.928] [1.748, 1.772]

±0.008 ±0.012

1.2 0

classical fractal dimension can’t be considered as a discriminating parameter between precipitations and fixed echoes. However, the fractal dimension based on contourlets is equal to 1.90 ± 0.012 for precipitation cells and 1.76 ± 0.02 for ground echoes cells in the images collected in Dakar. For the region of Bordeaux, two distinct fractal dimension values equal respectively to 1.91 ± 0.001 for precipitation cells and 1.71 ± 0.025 for ground echoes cells are obtained. In Melbourne site, we find two different values of fractal dimension based on contourlets, 1.92 ± 0.008 for precipitation cells and 1.76 ± 0.012 for ground echoes cells. The

correlation coefficient describing the quality of the fit is equal to 0.96. We can conclude that the two cell types, precipitations and ground echoes are efficiently identified by the fractal dimension based on contourlets. Therefore, the classification using fractals becomes possible with this new method. The different values of the fractal dimensions of precipitations cells don’t overlap with that of the ground echoes cells for all regions studied. This shows the reliability of our method to identify the two types of cells. Note that the same situation is observed for all the sites studied.

N. Azzaz, B. Haddad / Chaos, Solitons and Fractals 98 (2017) 130–144

139

Table 6 z-test results obtained for the three regions studied.

Fig. 12. Radar image, taken by the radar of Bordeaux, containing Anomalous propagation cells, recorded on the 23th September 1996. The scale is in km.

Table 5 shows the fractal dimensions, calculated by the classical method of Box Counting and the new method of fractal dimension based on contourlets, and also the range of variation [Dfmin, Dfmax] and the uncertainty, calculated for each approach. Note that the uncertainty is the standard deviation “σ ”, computed for the points cloud by the relation given below:



σ=

n 1  (yi − y¯ )2 n

(6)

i=1

Where: yi is the true values of Df; y¯ is the fitted value of Df; n is the number of sequences . Df min is the minimum fractal dimension; Df min = Df - σ Df max is the maximum fractal dimension; Df max = Df + σ We illustrate in Fig. 12 anaprop echoes, present in radar image of Bordeaux. The Fig. 13 represents the variations of fractal dimension based on contourlets for precipitation cells and anomalous propagation “anaprop” cells obtained in the radar images of Bordeaux. The fractal dimension based on contourlets is equal to 1.91 ± 0.001 for precipitation cells and 1.76 ± 0.03 for Anaprop cells, in the images collected in Bordeaux. We conclude that the two cell types, precipitation and Anaprop echoes are distinctly identified by this parameter. In order to confirm the reliability of our results, we have used a statistical test called z-test. It is a parametric hypothesis test used to determine the statistical significance of the obtained results. The z-test puts two hypotheses: Hypothesis H0: The difference between the means is 0, it means that is no significant difference between the two samples A and B. Hypothesis H1: The difference between the means is different from 0, it means that is a significant difference between the two samples A and B. Since the calculated p-value is less than the level of significance alpha = 0.05, the null hypothesis H0 must be rejected and the alternative hypothesis H1 must be retained. It means that our data are independent and don’t overlap with each other. Note that p-value of the z-test is the probability of observing a statistic test as extreme as, or more extreme than, the observed value under the null hypothesis. Small values of p-value cast doubt

Sites

z (Observed value)

|z| (Critical value)

p-value

alpha

Dakar Bordeaux Melbourne

2.337 4.224 2.793

1.960 1.960 1.960

0.009 0.0025 0.003

0.05 0.05 0.05

on the validity of the null hypothesis. The results obtained by ztest are given in Table 6. For the three studied sites, the p-value is always less than the level of significance alpha = 0.05, and the z(Observed value) is higher than |z|(Critical Value) = 1.960, thus rejecting the hypothesis H0, and retaining the alternative hypothesis H1. The z-test shows that the fractal dimensions based on the contourlets of precipitations are not correlated with those of the ground echoes. This shows the reliability of this parameter to differentiate between these two types of echoes, and proves the statistical significance of the fractal dimensions based on contourlets results because the p-value < 0.05. 4.2. Fractal lacunarity Fig. 14(a)–(c) show the variations of lacunarity fractal of precipitation and ground echoes, for the sites of Dakar, Bordeaux and Melbourne. The trend curves representing the lacunarity (x) for each cell type has a shape of a hyperbole. Thus, these curves can be described by the function [59]:

(x ) = axb + c

(7)

where: a: is the homogeneity factor. b: represents the order of convergence of (x). c: represents the term of translation. x: represents the size of the boxes. From Fig. 14(a)–(c), we can observe that the lacunarity calculated for the ground echoes is greater than that calculated for the precipitations. Due to the climate prevailing in the study areas, one at midlatitude (Bordeaux), the second (Dakar) at tropical latitude where squall lines are frequently observed and the third (Melbourne) at the subtropical latitude, the structural characteristics of rain fields are different between Bordeaux, Dakar and Melbourne, which explains the differences in the vertical scale in Fig. 14. Also, the images are coded on different levels on the two sites. Similarly, the Dakar region is more flat than Bordeaux area, giving a difference in the distribution of spurious echoes [60]. For the identification of parasitic echoes by considering lacunarity, we used the homogeneity factor "a", and this by fitting the points cloud represented by different values of lacunarity. As well, we found that precipitation cells have a higher homogeneity factor than the fixed echoes for the three studied regions (Table 7): According to the Table 7, the values of homogeneity factor "a" don’t overlap between the different types of echoes for the considered sites. This shows the reliability of our method to identify the two types of cells by considering homogeneity factor as a discriminating parameter between the precipitation echoes and ground echoes. The Fig. 15 shows the ability to differentiate also between the echoes of anomalous propagation “anaprop” echoes and the precipitation echoes, using lacunarity method. We observe that the lacunarity values calculated for the precipitation echoes are greater than that calculated for the Anaprop

140

N. Azzaz, B. Haddad / Chaos, Solitons and Fractals 98 (2017) 130–144

Fig. 13. Variation of fractal dimension based on contourlet of precipitation echoes and anaprop echoes in Bordeaux, N(ε ) is number of boxes and ε is size of boxes.

Fig. 14. Comparison between fractal lacunarity calculated for precipitation cells and that calculated for ground echoes cells, (a) for Dakar, (b) for Bordeaux and (c) for Melbourne.

N. Azzaz, B. Haddad / Chaos, Solitons and Fractals 98 (2017) 130–144

141

Table 7 Fractals lacunarity obtained for the three regions studied. Sites

Type of cells

a (factors of homogeneity)

[a min ; a max]

Uncertainty of “a”

Dakar

Precipitation Fixed echoes Precipitation Fixed echoes Precipitation Fixed echoes

238,9 −159 −6423 −416.4 312,4 269.9

[221.9 ; 255.9] [(−216.5; −101.5] [−6.627 ; −6.22] [(−431.2; −401.3] [297,1 ; 327.7] [264.9 ; 274.9]

±17.0 ±57.5 ±0.204 ±24.8 ±15.3 ±5

Bordeaux Melbourne

Fig. 15. Comparison between fractal lacunarity calculated for precipitation cells and that calculated for Anaprop cells for Bordeaux.

echoes. Then, we conclude that we can easily differentiate between these two kinds of cells.

4.3. Automatic classification system To facilitate the use of classification approaches described above, we proposed an automatic system that identifies the precipitations in radar images. For this, we designed a graphical user interface composed of several menus. Fig. 16 describes the functions provided by this interface. For processing, we first select the images from the toolbar "Data Selection" and then we choose the studied radar site. The GUI accepts all input images formats, like: jpg, tif, gif, png and bmp. Then we apply one of the two process contained in the menu bar "type of processing", by clicking on the button "Fractal Dimension based contourlet" or on the button "Fractal Lacunarity". The variations of the fractal dimension based on contourlets and those of the fractal lacunarity are then displayed on the interface (Fig. 17(a) and (b)). The interface allows us to identify the type of radar echo. The cells type like precipitations, fixed echoes or anaprops will be displayed in the menu bar "Results Fractal Dimension based contourlets" or in the menu bar "Result Fractal Lacunarity" by clicking on the button "Type Cells", then allowing the user to ignore false echoes after identifying them. Identification of precipitations processed through the "Cells Type" button was made for both treatments.

This graphical interface can be implanted on all radars, whatever the studied site. In order to adapt the use of the GUI to a new site, or to identify others types of echoes such as anomalous echoes or other non-precipitation echoes, a prior study is necessary for calculating the fractal dimension and fractal lacunarity of a group of radar images collected from these new sites. Then, we have only to add the comparison factor in the GUI algorithm, to obtain our new results. 5. Conclusion An automatic system for identification of spurious radar echoes was proposed using fractal geometry. The sought goal is to identify the parasite echoes backscattered by the Earth’s surface to improve the performance of non-Doppler and non-polarimetric weather radar that equip most of weather stations. In this work, we considered non-treated images collected in Bordeaux and Dakar. To eliminate the ground echoes mixed with precipitation, we proposed a technique based on masking parasite echoes. For the site of Melbourne, we haven’t applied the masking method used for Bordeaux and Dakar, because the parasitic echoes are filtered by the Doppler technique. We established that the fractal lacunarity and the fractal dimension based on contourlets which is applied for the first time for the analysis of radar echoes structure, are two discriminant parameters, and allow identifying unambiguously the two types of echoes in three regions where different climates prevail. Similarly, we have shown that the two parameters make possible to identify

142

N. Azzaz, B. Haddad / Chaos, Solitons and Fractals 98 (2017) 130–144

Fig. 16. Precipitation image collected in Bordeaux.

Fig. 17. (a) Variation of the fractal dimension based on contourlets of the processed image, and (b) Variation of the fractal lacunarity of the processed image.

the anaprops observed only in the Borderaux region. In the literature, the fractal dimension and fractal lacunarity are often used separately. We also checked that the classic fractal dimension based on the Box-Counting technique does not allow the discrimination between the types of echoes. In addition, we showed that the structure of precipitation is more homogeneous than that of parasitic echoes, and then we can characterize each type of echo by the factor of homogeneity determined by the fractal lacunarity. Furthermore, we proposed a graphical interface that directly identifies the precipitation for Doppler radars, such as the one of Melbourne, whereas parasites echoes like ground echoes or insects are identified in another class. From here we can easily estimate the precipitations. Also, the GUI unambiguously identify spurious echoes for nonDoppler and non-polarimetric radars, such as Bordeaux and Dakar radars, thus facilitating task to the radar operator.

This graphical interface, easy to handle can be implanted on all radars. To extend the use of this Graphical Interface to other sites, a prior study is necessary for calculating the fractal dimension and fractal lacunarity of a group of radar images collected from these new sites. The learning phase only requires a twenty to thirty radar images for each site. This pretreatment is justified by the multi fractal structure of precipitation and topographic differences of the various sites. In the masking method, we will need also a hundred of radar images taken in the studied region throughout all the year to establish a mask characterizing the clutter, to eliminate them. The GUI can also be used for other identifications, such as for discrimination between anomalous propagation echoes and precipitation echoes, and also for identification of convective and stratiform cells in radar images, by using the fractal geometric analysis, thus improving the short-term forecasts of natural disasters [61].

N. Azzaz, B. Haddad / Chaos, Solitons and Fractals 98 (2017) 130–144

References [1] Khider M, Haddad B, Taleb-Ahmed A. Analyse multi-fractale des échos radar par la méthode des maximums des modules de la transformée en ondelette (MMTO) 2D pour les sites de Bordeaux (France), Setif (Algérie) : application à l’élimination des échos parasites. Teledetection, Editions scientifiques GB 2010;8(4):271–83. [2] Mesnard F, Sauvageot H. Structural characteristics of rain fields. J Geophys Res 2003;108(13):4385. doi:10.1029/2002JD002808. [3] Prabhakara C, Iacovazzi R, Yoo JM, Kim KM, Bell TL. A method to estimate rain rate over tropical oceans with TRMM Microwave Imager radiometer. J Meteorol Soc Jpn 2008;86(1):203–12. doi:10.2151/jmsj.83.595. [4] Malamou A, Pandis C, Frangos P. Application of the modified fractal signature method for terrain classification from synthetic aperture radar images. Comput Sci 2014;20:118–21. [5] ZheXian L, Ying W, GeLan M. Possible causes of the variation in fractal dimension of the perimeter during the tropical cyclone Dan motion. Sci China- Earth Sci 2014;57:1383–92. doi:10.1007/s11430- 013- 4745- 8. [6] Comisso M. On the use of dimension and lacunarity for comparing the resonant behavior of convoluted wire antennas. In: Progress in electromagnetics research, PIER, South Carolina, USA, 96; 2009. p. 361–76. doi:10.2528/ PIER09082505. [7] Ivanovici M, Richard N, Paulus D. Color Image Segmentation. In: Advanced color image processing and analysis. Christine Fernandez-Maloigne; 2012. p. 199–202. [8] Katsaloulis P, Ghosh A, Philippe AC, Provata A, Deriche R. Fractality in the neuron axonal topography of the human brain based on 3-D diffusion MRI. Eur Phys J B 2012;85:1383–92. doi:10.1140/epjb/e2012- 30045- y. [9] Lee WL, Chen YC, Hsieh KS. Ultrasonic liver tissues classification by fractal feature vector based on m-band wavelet transform. IEEE Trans Med Imaging 2003;22:382–92. doi:10.1109/TMI.2003.809593. [10] GDi Martino, Iodice A, Riccio D. On the fractal nature of volcano morphology detected via SAR image analysis: the case of Somma-Vesuvius volcanic complex. Eur J Remote Sens 2012;45:177–87. doi:10.5721/eujrs20124517. [11] Mandelbrot BB. The fractal geometry of nature. San Francisco: CA: Freeman; 1982. p. 468. doi:10.10 02/esp.3290 080415. [12] Quodbach J, Moussavi A, Tammer R, Frahm J, Kleinebudde P. Assessment of disintegrant efficacy with fractal dimensions from real-time MRI. Int J Pharm 2014;475:605. doi:10.1016/j.ijpharm.2014.09.021. [13] Saraswat P, Raj V, Sen MK, Narayanan A. Multiattribute seismic analysis with fractal dimension and 2d and 3d continuous wavelet transform. SPE Reservoir Eval Eng 2014;17:436–43. doi:10.2118/164417-PA. [14] Tolle CR, Mc Junkin TR, Rohrbaugh DT, LaViolette RA. Lacunarity definition for ramified data sets based on optimal cover. Physical D 2003;179(3) 129-115. doi:10.1016/S0167-2789(03)0 0 029-0. [15] Ca˘ liman A, Ivanovici M. Psoriasis image analysis using color lacunarity. In: The 13th international conference on optimization of electrical and electronic equipment, OPTIM’12; 2012. p. 1401–6. doi:10.1109/OPTIM.2012.6231850. [16] Frazer GW, Wulder MA, Niemann KO. Simulation and quantification of the fine-scale spatial pattern and heterogeneity of forest canopy structure: a lacunarity -based method designed for analysis of continuous canopy heights. Forest Ecol Manage 2005;214:65–90. doi:10.1016/j.foreco.2005.03.056. [17] Sadouki L, Haddad B. Classification of radar echoes with a textural–fuzzy approach: an application for the removal of ground clutter observed in Sétif (Algeria) and Bordeaux (France) sites. Int J Remote Sens 2013;33(21):7447–63. doi:10.1080/01431161.2013.823522. [18] Sauvageot H, Despaux G. SANAGA: Un systeme d’acquisition numerique et de visualisation des données radar pour la validité des estimations satellitaires de precipitations. Veille climatique Satellitaire 1990;31:51–5. [19] Doviak RJ, Zrnic DS. Doppler radar and weather observations. Academic Press; 1993. p. 562. [20] Islam T, Rico-Ramirez MA, Han D, Srivastava PK. Artificial intelligence techniques for clutter identification with polamitric radar signatures. Atmos Res 2012;109-110:95–113. doi:10.1016/j.atmosres.2012.02.007. [21] Chandrasekar V, Keränen R, Lim S, Moisseev D. Recent advances in classification of observations from dual polarization weather radars. Atmos Res 2013;119:97–111. doi:10.1016/j.atmosres.2011.08.014. [22] Haddad B, Adane A, Sauvageot H, Sadouki L. Identification and filtering of rainfull and ground radar echoes using textural features. Int J Remote Sens 2004;25(21):4641–56. doi:10.1080/01431160310001654455. [23] Hamuzu K, Wakabayashi M. Ground clutter rejection. In: Collier Clukie an, editor. Hydrological applications of weather radar. Ellis Horwood Ltd; 1991. p. 13– 142. doi: 10.1175/1520-0426(1994)0111026:SDOAPI2.0.CO%3B2. [24] Hubert JC, Dixon M, Ellis SM. Weather radar clutter. Part II: real-time identification and filtering. J Atmos Oceanic Technol 2009;26:1181–97 http://dx.doi. org/10.1175/2009JTECHA1160.1. [25] Berenguer M, Sempere-Torres D, Corral C, Sanchez-Diezma R. A fuzzy logic technique for identifying nonprecipitating echoes in radar scans. J Atmos Oceanic Technol 2006;23:1157–80 http://dx.doi.org/10.1175/JTECH1914.1. [26] Cho YH, Lee G, Kim KE, Zawadzki I. Identification and removal of ground echoes and anomalous propagation using the characteristics of radar echoes. J Atmos Oceanic Technol 2006;23:1206–22 http://dx.doi.org/10.1175/ JTECH1913.1.

143

[27] Xiang L. Adaptative Network fuzzy inference system used in interference cancellation of radar seeker. In: IEEE international conference on intelligent computing and intelligent systems (ICIS), 2; 2010. p. 93–7. doi:10.1109/ICICISYS. 2010.5658850. [28] Falconer K. Fractal Geometry: mathematical foundations and applications. Chichester, England: John Wiley & Sons; 1990. p. 288. doi:10.10 02/0470 013850. ch1. [29] Allain C, Cloitre M. Characterizing the lacunarity of random and deterministic fractal sets. Phys Rev A 1991;44:3552–8. doi:10.1103/ PhysRevA.44.3552. [30] Haddad B, Sadouki L, Sauvageot H, Adane A. Analyse de la dimension fractale des échos radar en Algérie, France et Sénégal. Télédétection 2006;5(4):299–306. [31] Wenzhuo L, Kaimin S, Hongya Z. Algorithm for relative radiometric consistency process of remote sensing images based on object-oriented smoothing and contourlet transforms. J Appl Remote Sens 2014;8(1):083607. doi:10.1117/ 1.JRS.8.083607. [32] Kiani M, Ghofrani S. Two new methods based on contourlet transform for despeckling synthetic aperture radar images. J Appl Remote Sens 2014;8. doi:10. 1117/1.JRS.8.083604. [33] Zhang Q, Gao L. Medical image retrieval based on nonsubsampled contourlet transform and fractal dimension. 3rd International conference on bioinformatics and biomedical engineering, ICBBE; 2009. [34] Kpalma K. Caractérisation de textures par l’anisotropie de la dimension fractale. 2e Colloque africain sur la recherche en informatique (CARI). Skilling J, editor. Burkina Faso: Ouagadougou; 1994. [35] Feder J. Fractals. Plenum Press, New York, 1988, p. 283. [doi:10.1007/ 978- 1- 4899- 2124- 6]. [36] Barnsley M. Fractal everywhere. San Diego, CA, U.S.A: Academy Press Professional; 1988. p. 394. doi:10.1002/actp.1990.010410617. [37] Hasting H, Sugihara G. Fractals: a user’s guide for the natural sciences. Oxford, U.K.: Oxford University Press; 1993. p. 248. doi:10.1086/418972. [38] Lovejoy S, Schertzer D. Multifractals, universality classes and satellite and radar measurement of cloud and rain fields. J Geophys Res 1990;95(D3):2021–34. doi:10.1029/ JD095iD03p02021. [39] Borys P. On the relation between lacunarity and fractal dimension. Acta Physica Polonica B 2009;40(5):1485–90. [40] Krasowska M, Borys P, Grzywna ZJ. Lacunarity as a measure of texture. Acta Physica Polonica B 2004;35(4):1519–34. [41] Armatas GS, Kolonia KM, Pomonis PJ. Morphometry of porous solids: lacunarity, fractal dimensions. Connect Some Topological Similarities Neurons Langmuir 2002;18(16):10421–9. doi:10.1021/la026213e. [42] Smith TG, Lange GD, Marks WB. Fractal methods and results in cellular morphology- dimensions, lacunarity and multifractals". J Neurosci Methods 1996;69(2):123–36. doi:10.1016/S0165-0270(96)0 0 080-5. [43] Baaziz N, Abahmane O, Missaoui R, Ikuta C, Nishio Y. Texture feature extraction in the spatial frequency domain for content-based image retrieval. J CoRR 2010, abs/1012.5208. [44] Candès EJ, Donoho DL. Recovering edges in ill-posed inverse problems: optimality of Curvelet frames. Ann Stat 2002;30(3):784–842. [45] Do MN, Vetterli M. Contourlets: a directional multiresolution image representation. In: International conference on image processing, IEEE, 1; 2002. p. 357– 60. doi:10.1109/ICIP.2002.1038034. [46] Starck JL, Candès EJ, Donoho DL. The Curvelet transform for image denoising. Trans Image Process IEEE, Rochester, USA 2002;11(6):670–84. doi:10.1109/TIP. 2002.1014998. [47] Do MN, Vetterli M. Pyramidal directional filter banks and Curvelets. In: International conference on image processing, 3. Thessaloniki, Greece: IEEE; 2001. p. 158–61. doi:10.1109/ICIP.2001.958075. [48] Burt PJ, Adelson EH. The Laplacian pyramid as a compact image code. IEEE Trans. Commun, New York, USA 1983;31(4):532–40. doi:10.1109/TCOM.1983. 1095851. [49] Serief C. Extraction Automatique de Points d’intérêt à base de la transformée de contourlets non sous échantillonnée pour le réglage des images Doctoral thesis, Algeria; 2009. [50] Bamberger RH, Smith MJT. A filter bank for the directional decomposition of images: theory and design. IEEE Trans. Signal Process., Atlanta, USA 1992. doi:10.1109/78.127960. [51] Zhang S, Yang G, Cheng Z, Wetering H, Ikuta C, Nishio Y. A novel optimization design approach for contourlet directional filter banks. IEICE Electron Express 2014;11:1–11. doi:10.1587/elex.11.20140556. [52] Do MN, Vetterli M. The contourlet transform: an efficient directional multiresolution image representation. IEEE Trans. Image process Urbana-Champaign, USA 2005;14(12):2091–106. doi:10.1109/TIP.2005.859376. [53] Pan X, Ruan Q, Wang Y. Palmprint recognition using contourlets-based local fractal dimensions. In: 9th International conference on signal processing, ICSP, Beijing, China, 35; 2008. p. 2108–11. doi:10.1109/ICOSP.2008.4697561. [54] Lehamel M. segmentation d’images texturées à partir des attributs fractals, Magister memory. 1979, p. 64–66. [55] Nzeukou A, Sauvageot H. Distribution of rainfall parameters near the Coasts of France and Senegal. J Appl Meteorol 2002;41(1):69–82. doi:10.1175/ 1520-0450(2002)0410069:DORPNT2.0.CO;2. [56] Wolff DB, Marks DA, Amitai E, Silberstein DS, Fisher BL, Tokay A. Ground validation for the tropical rainfall measurement mission (TRMM). J. Atmos. Ocean. Tech. 2005;22(4):365–80. http://dx.doi.org/10.1175/JTECH1700.1.

144

N. Azzaz, B. Haddad / Chaos, Solitons and Fractals 98 (2017) 130–144

[57] Taylor G, Evans R, Manobianco J, Schumann R, Wheeler M and Yersavich A. Report on the comparison of the scan strategies employed by the Patrick Air Force Base WSR-74C/McGill radar and the NWS Melbourne WSR-88D radar. Ensco, Inc., Melbourne, FL, United States. 1994. [doi:19950 0 05179]. [58] Darricaud J. Physique et théorie de radar. Tom 1 et 2. Paris: Sodipe; 1981. [59] Zaia A, Eleonori R, Maponi P, Rossi R, Murri R. Medical Imaging and Osteoporosis: Fractal’s Lacunarity Analysis of Trabecular Bone in MR Images. In: Proc. of the 18th IEEE symposium on computer-based medical systems (CBMS’05), Dublin, Ireland, 35; 2005. p. 1063–7125. doi:10.1109/ICOSP.2008.4697561.

[60] Djafri A, Haddad B. Mapping of weather radar ground clutter using the digital elevation model (SRTM). Signal Image Process (SIPIJ) 2012;3(4):135–51. doi:10. 5121/sipij.2012.3411.135. [61] Azzaz N, Haddad B. Automatic system for classification of precipitation cells. Int J Appl Math Electron Comput 2015;3(3):189–93. http://dx.doi.org/10.18100/ ijamec.77967.