**2. Basic properties of speckle and its modeling**

Speckle is a typical example for which pure multiplicative model is usually exploited (Touzi, 2002; Oliver&Quegan, 2004). This means that a dependence of signal dependent noise variance on true value *σsd* <sup>2</sup> = *f* (*I tr* ) is monotonically increasing proportionally to squared (true value). Speckle is not Gaussian and its probability density function (PDF) depends upon a way of image forming (amplitude or intensity) and number of looks (Oliver&Quegan, 2004). PDF of the speckle considerably differs from Gaussian if a single-look imaging mode is used and it is either Rayleigh (for amplitude images) or negative exponential (for intensity images) for the case of fully developed speckle. If multi-look imaging mode is applied, the speckle PDF becomes closer to Gaussian and depends upon the number of looks.

The histogram in Fig. 1(b) also shows one more aspect important for simulations. Speckle image values can be by 3...4 times larger than mean (which is close to *I tr* in homogeneous

represented in Fig. 1(b). Since both components of complex valued data are Gaussian with approximately the same variance, the amplitude values obey Rayleigh distribution. This one more time shows that for single-look amplitude images speckle can be simulated as pure multiplicative noise having Rayleigh PDF. Using modern simulation tools as, e.g., Matlab, this can be done easily, at least, for the case of independent identically distributed (i.i.d.), i.e.

Methods for Blind Estimation of Speckle Variance in SAR Images: Simulation Results and Verification for Real-Life Data

outside the limits 0…255 and, therefore, 16-bit representation of a simulated noisy image is to be used to preserve statistics of the speckle. In Section 3, we will show what might happen to the estimates provided by BENCs if clipping effects take place for simulated noisy SAR images,

The histogram in Fig. 1(b) also shows one more aspect important for simulations. Speckled image values can be by 3...4 times larger than mean (which is close to *tr I* in homogeneous image regions). Then, if the values of *tr I* are modelled as 8-bit data, the noisy values can be outside the limits 0…255 and, therefore, 16-bit representation of a simulated noisy image is to be used to preserve statistics of the speckle. In Section 3, we will show what might happen to the estimates provided by BENCs if clipping effects take place for simulated

*II I I I* ( *ij I* is an *ij*-th image pixel, *G*hom

<sup>2</sup> =0.273 for Rayleigh distributed speckle in amplitude single-look SAR

Fig. 1. Histograms of distributions for in-phase component (a) and amplitude (b) of

To additionally analyze statistics of the speckle, we have also tested several manually cropped homogeneous regions in different single-look images that correspond to either rather large (about 70x70 pixels) agricultural fields and water surface. The estimated speckle

**Figure 1.** Histograms of distributions for in-phase component (a) and amplitude (b) of complex-valued data in image

is a selected homogeneous region) has varied from 0.265 to 0.285. This is in a good

To additionally analyze statistics of the speckle, we have also tested several manually cropped homogeneous regions in different single-look images that correspond to either rather large (about 70x70 pixels) agricultural fields and water surface. The estimated speckle variance

> amplitude single-look SAR images. Higher order moments (skewness and kurtosis) for the studied homogeneous image regions are also in appropriate agreement with the theory (Oliver&Quegan, 2004). This means that for both simulated and real-life single-look amplitude SAR images any BENC should provide estimates of speckle variance close

homogeneous region) has varied from 0.265 to 0.285. This is in a good agreement with a

To consider and simulate speckle more adequately, we have also analyzed spatial correlation of speckle using TerraSAR-X data. This has been done in three different ways. First, standard 2D autocorrelation function (ACF) estimates have been obtained for 32x32

images. Higher order moments (skewness and kurtosis) for the studied homogeneous image regions are also in appropriate agreement with the theory (Oliver&Quegan, 2004). This means that for both simulated and real-life single-look amplitude SAR images any BENC should

To consider and simulate speckle more adequately, we have also analyzed spatial correlation of speckle using TerraSAR-X data. This has been done in three different ways. First, standard 2D autocorrelation function (ACF) estimates have been obtained for 32x32 pixels size homo‐ geneous fragments. They have been inspected visually and have demonstrated the absence of far correlation and the presence of essential correlation for neighboring pixels in single-look amplitude images (see example in Fig. 2(a)). It is interesting that even higher correlation for neighboring pixels has been observed for multi-look images (see example in Fig. 2(b)). There are also ACF side lobes for azimuth direction that, most probably, arise due to peculiarities of

 

*Iij* (*Iij*

are modelled as 8-bit data, the noisy values can be

http://dx.doi.org/10.5772/57040

307

<sup>0</sup> <sup>100</sup> <sup>200</sup> <sup>300</sup> <sup>400</sup> <sup>0</sup>

(b)

<sup>2</sup> 0.273 for Rayleigh distributed speckle in

is an *ij*-th image pixel, *G*hom is a selected

image regions). Then, if the values of *I tr*

spatially uncorrelated speckle.

i.e. if they are represented as 8-bit data.

variance 

(*Iij* − *I* ̑ *Gmean*)<sup>2</sup> / *I* ̑ *Gmean* <sup>2</sup> , *<sup>I</sup>* ̑

theoretically stated *σ<sup>μ</sup>*

homogeneous region

*σ*̑ *μ* <sup>2</sup> = ∑ *G*hom

enough to 0.273.

the SAR response to a point target.

noisy SAR images, i.e. if they are represented as 8-bit data.


(a)

agreement with a theoretically stated

complex-valued data in image homogeneous region

<sup>2</sup> 2 2

provide estimates of speckle variance close enough to 0.273.

*G G*

*Gmean* <sup>=</sup> ∑ *G*hom

hom hom ( )/ , *ij Gmean Gmean Gmean ij*

To get an imagination on fully developed speckle PDF, consider real-life data produced by TerraSAR-X imager. Its attractive feature is that data (images) are freely available at the aforementioned site. These data have full description of parameters of the imaging system operation mode used for obtaining each presented image. Large size images (thousands to thousands pixels) for many different areas of the Earth are offered. Furthermore, a brief description of a territory, observed effects and cover types is given. This allows selecting and processing data with different numbers of looks, properties of a sensed terrain, a desired polarization, etc. Fragments of certain size as 512x512 pixels can be easily cut from large size data arrays and studied more thoroughly. Another positive feature is that single-look images are presented in the complex valued form. This allows obtaining single-look images in aforementioned forms (representations). It also makes possible to analyze distributions of real and imaginary part values for image fragments, etc. While considering single-look SAR images in this paper, we use amplitude images since it is the most common form and it provides convenient representation for visual analysis.

One more advantage is that TerraSAR-X is a high quality system designed by specialists from German Aerospace Agency DLR who have large experience in creation and management of spaceborne SAR systems (Herrmann et al., 2005). The TerraSAR-X imager provides a stability of noise characteristics and practical absence of an additive noise in formed images. Later, it will be explained why this is so important for further analysis.

To partly corroborate conformity of theory and practice for single-look SAR images, we have manually cropped several sub-arrays of complex valued data that correspond to homogeneous terrain regions. The histogram of real (in-phase) part values for one such a fragment is presented in Fig. 1(a) where sample data mean is close to zero. The histogram for imaginary (quadrature) part is very similar. Gaussianity tests hold for both data sub-arrays. The histo‐ gram of the amplitude single-look image for the same fragment is represented in Fig. 1(b). Since both components of complex valued data are Gaussian with approximately the same variance, the amplitude values obey Rayleigh distribution. This one more time shows that for single-look amplitude images speckle can be simulated as pure multiplicative noise having Rayleigh PDF. Using modern simulation tools as, e.g., Matlab, this can be done easily, at least, for the case of independent identically distributed (i.i.d.), i.e. spatially uncorrelated speckle.

approximately the same variance, the amplitude values obey Rayleigh distribution. This one

The histogram in Fig. 1(b) also shows one more aspect important for simulations. Speckle image values can be by 3...4 times larger than mean (which is close to *I tr* in homogeneous image regions). Then, if the values of *I tr* are modelled as 8-bit data, the noisy values can be outside the limits 0…255 and, therefore, 16-bit representation of a simulated noisy image is to be used to preserve statistics of the speckle. In Section 3, we will show what might happen to the estimates provided by BENCs if clipping effects take place for simulated noisy SAR images, i.e. if they are represented as 8-bit data. more time shows that for single-look amplitude images speckle can be simulated as pure multiplicative noise having Rayleigh PDF. Using modern simulation tools as, e.g., Matlab, this can be done easily, at least, for the case of independent identically distributed (i.i.d.), i.e. spatially uncorrelated speckle. The histogram in Fig. 1(b) also shows one more aspect important for simulations. Speckled image values can be by 3...4 times larger than mean (which is close to *tr I* in homogeneous image regions). Then, if the values of *tr I* are modelled as 8-bit data, the noisy values can be outside the limits 0…255 and, therefore, 16-bit representation of a simulated noisy image is to be used to preserve statistics of the speckle. In Section 3, we will show what might

happen to the estimates provided by BENCs if clipping effects take place for simulated

noisy SAR images, i.e. if they are represented as 8-bit data.

**2. Basic properties of speckle and its modeling**

becomes closer to Gaussian and depends upon the number of looks.

on true value *σsd*

<sup>2</sup> = *f* (*I tr*

306 Computational and Numerical Simulations

convenient representation for visual analysis.

will be explained why this is so important for further analysis.

Speckle is a typical example for which pure multiplicative model is usually exploited (Touzi, 2002; Oliver&Quegan, 2004). This means that a dependence of signal dependent noise variance

Speckle is not Gaussian and its probability density function (PDF) depends upon a way of image forming (amplitude or intensity) and number of looks (Oliver&Quegan, 2004). PDF of the speckle considerably differs from Gaussian if a single-look imaging mode is used and it is either Rayleigh (for amplitude images) or negative exponential (for intensity images) for the case of fully developed speckle. If multi-look imaging mode is applied, the speckle PDF

To get an imagination on fully developed speckle PDF, consider real-life data produced by TerraSAR-X imager. Its attractive feature is that data (images) are freely available at the aforementioned site. These data have full description of parameters of the imaging system operation mode used for obtaining each presented image. Large size images (thousands to thousands pixels) for many different areas of the Earth are offered. Furthermore, a brief description of a territory, observed effects and cover types is given. This allows selecting and processing data with different numbers of looks, properties of a sensed terrain, a desired polarization, etc. Fragments of certain size as 512x512 pixels can be easily cut from large size data arrays and studied more thoroughly. Another positive feature is that single-look images are presented in the complex valued form. This allows obtaining single-look images in aforementioned forms (representations). It also makes possible to analyze distributions of real and imaginary part values for image fragments, etc. While considering single-look SAR images in this paper, we use amplitude images since it is the most common form and it provides

One more advantage is that TerraSAR-X is a high quality system designed by specialists from German Aerospace Agency DLR who have large experience in creation and management of spaceborne SAR systems (Herrmann et al., 2005). The TerraSAR-X imager provides a stability of noise characteristics and practical absence of an additive noise in formed images. Later, it

To partly corroborate conformity of theory and practice for single-look SAR images, we have manually cropped several sub-arrays of complex valued data that correspond to homogeneous terrain regions. The histogram of real (in-phase) part values for one such a fragment is presented in Fig. 1(a) where sample data mean is close to zero. The histogram for imaginary (quadrature) part is very similar. Gaussianity tests hold for both data sub-arrays. The histo‐ gram of the amplitude single-look image for the same fragment is represented in Fig. 1(b). Since both components of complex valued data are Gaussian with approximately the same variance, the amplitude values obey Rayleigh distribution. This one more time shows that for single-look amplitude images speckle can be simulated as pure multiplicative noise having Rayleigh PDF. Using modern simulation tools as, e.g., Matlab, this can be done easily, at least, for the case of independent identically distributed (i.i.d.), i.e. spatially uncorrelated speckle.

) is monotonically increasing proportionally to squared (true value).

complex-valued data in image homogeneous region To additionally analyze statistics of the speckle, we have also tested several manually **Figure 1.** Histograms of distributions for in-phase component (a) and amplitude (b) of complex-valued data in image homogeneous region

cropped homogeneous regions in different single-look images that correspond to either

Fig. 1. Histograms of distributions for in-phase component (a) and amplitude (b) of

rather large (about 70x70 pixels) agricultural fields and water surface. The estimated speckle variance <sup>2</sup> 2 2 hom hom ( )/ , *ij Gmean Gmean Gmean ij G G II I I I* ( *ij I* is an *ij*-th image pixel, *G*hom is a selected homogeneous region) has varied from 0.265 to 0.285. This is in a good agreement with a theoretically stated <sup>2</sup> 0.273 for Rayleigh distributed speckle in amplitude single-look SAR images. Higher order moments (skewness and kurtosis) for the studied homogeneous image regions are also in appropriate agreement with the theory (Oliver&Quegan, 2004). This means that for both simulated and real-life single-look amplitude SAR images any BENC should provide estimates of speckle variance close enough to 0.273. To consider and simulate speckle more adequately, we have also analyzed spatial correlation of speckle using TerraSAR-X data. This has been done in three different ways. First, standard 2D autocorrelation function (ACF) estimates have been obtained for 32x32 To additionally analyze statistics of the speckle, we have also tested several manually cropped homogeneous regions in different single-look images that correspond to either rather large (about 70x70 pixels) agricultural fields and water surface. The estimated speckle variance *σ*̑ *μ* <sup>2</sup> = ∑ *G*hom (*Iij* − *I* ̑ *Gmean*)<sup>2</sup> / *I* ̑ *Gmean* <sup>2</sup> , *<sup>I</sup>* ̑ *Gmean* <sup>=</sup> ∑ *G*hom *Iij* (*Iij* is an *ij*-th image pixel, *G*hom is a selected homogeneous region) has varied from 0.265 to 0.285. This is in a good agreement with a theoretically stated *σ<sup>μ</sup>* <sup>2</sup> =0.273 for Rayleigh distributed speckle in amplitude single-look SAR images. Higher order moments (skewness and kurtosis) for the studied homogeneous image regions are also in appropriate agreement with the theory (Oliver&Quegan, 2004). This means that for both simulated and real-life single-look amplitude SAR images any BENC should provide estimates of speckle variance close enough to 0.273.

To consider and simulate speckle more adequately, we have also analyzed spatial correlation of speckle using TerraSAR-X data. This has been done in three different ways. First, standard 2D autocorrelation function (ACF) estimates have been obtained for 32x32 pixels size homo‐ geneous fragments. They have been inspected visually and have demonstrated the absence of far correlation and the presence of essential correlation for neighboring pixels in single-look amplitude images (see example in Fig. 2(a)). It is interesting that even higher correlation for neighboring pixels has been observed for multi-look images (see example in Fig. 2(b)). There are also ACF side lobes for azimuth direction that, most probably, arise due to peculiarities of the SAR response to a point target.

First, these images have practically no self-noise that could later influence blind estimation of

(a) (b)

(c) (d)

Second, these are the images of natural scenes and, thus, they contain large quasihomogeneous regions, edges with different contrasts, various textures and small-sized

Second, these are the images of natural scenes and, thus, they contain large quasi-homogene‐

Note that we have simulated speckle with the same statistics for all pixels ignoring the fact that for small-sized targets it might differ from speckle in homogeneous image regions. This simplification is explained by the following two reasons. First, more complicated models of speckle are required for small-sized targets. Second, the percentage of pixels occupied by small-sized targets is quite small in real-life images (Lee et al., 2009) and local estimates of noise statistics in the corresponding scanning windows are anyway abnormal. Thus, these local estimates are "ignored" by the BENCs considered below which are robust (see Section 3 for

Fig. 4 gives two examples of noisy test image with fully developed speckle (single look). For one of them (Fig. 4(a)) speckle is i.i.d. whilst for the second (Fig. 4(b)) speckle is spatially

ous regions, edges with different contrasts, various textures and small-sized targets.

Fig. 3. Noise-free (true) test images used for simulating SAR images

**Figure 3.** Noise-free (true) test images used for simulating SAR images

To simulate single- and multi-look SAR images, we have used four aerial optical images as *tr I* (all test images are of size 512x512pixels). These four images are presented in Fig. 3. Positive features of these images allowing to use them in simulation of SAR data are the following. First, these images have practically no self-noise that could later influence blind

Thus, we can state that speckle is spatially correlated in the considered TerraSAR-X images, both single- and multi-look ones. Then, this effect should be taken into account in

Methods for Blind Estimation of Speckle Variance in SAR Images: Simulation Results and Verification for Real-Life Data

http://dx.doi.org/10.5772/57040

309

speckle statistics.

targets.

more details).

simulations.

estimation of speckle statistics.

**Figure 2.** ACF estimates for 32x32 pixel homogeneous fragments in single-look (a) and multi-look (b) TerraSAR-X im‐ ages

Second, we have analyzed a normalized 8x8 DCT spectrum estimates obtained in a blind manner (Ponomarenko et al., 2010) for several considered images. These estimates also clearly indicate that speckle is spatially correlated, i.e., not i.i.d. (Lukin et al., 2011).

Third, we have also calculated a parameter *r* (Uss et al., 2012) able to indicate spatial correlation of noise for any type of signal dependent noise with spatially stationary spectral characteristics. For determination of *r*, two local estimates of noise variance are derived for each 8x8 pixel block with its left upper corner defined by indices *l* and *m*. The first estimate is calculated in the spatial domain

$$\hat{\sigma}\_{lm}^{2} = \sum\_{i=l}^{l+7} \sum\_{j=m}^{m+7} (I\_{ij} - \hat{\bar{I}}\_{lm})^2 / \left\langle \mathbf{63}\_{\prime} \right| \hat{\bar{I}}\_{lm} = \sum\_{i=l}^{l+7} \sum\_{j=m}^{m+7} I\_{ij} / \left\langle \mathbf{64}\_{\prime} \right\rangle \tag{1}$$

and the second estimate is calculated in the DCT domain as

$$(\hat{\sigma}\_{lm}^{sp})^2 = (1.483med(\left\lfloor D\_{qs}^{lm} \right\rfloor))^2 \tag{2}$$

where *Dqs lm*, *q* =0, ..., 7, *s* =0, ..., 7 except *q* =*s* =0 are DCT coefficients of *lm*-th block of a given image. Then, the ratio *Rlm* =*σ*̑ *lm* / *σ*̑ *lm sp* is calculated for each block. After this, the histogram of these ratios for all blocks is formed and its mode *r* is determined by the method (Lukin et al., 2007). For all considered real-life SAR images, the value of *r* was larger than 1.05 (Lukin et al., 2011b) for single-look SAR images and considerably larger for multi-look ones. This addition‐ ally gives an evidence in favor of the hypothesis that speckle is spatially correlated. Thus, we can state that speckle is spatially correlated in the considered TerraSAR-X images, both singleand multi-look ones. Then, this effect should be taken into account in simulations.

To simulate single- and multi-look SAR images, we have used four aerial optical images as *I tr* (all test images are of size 512x512pixels). These four images are presented in Fig. 3. Positive features of these images allowing to use them in simulation of SAR data are the following. First, these images have practically no self-noise that could later influence blind estimation of speckle statistics. To simulate single- and multi-look SAR images, we have used four aerial optical images as *tr I* (all test images are of size 512x512pixels). These four images are presented in Fig. 3. Positive features of these images allowing to use them in simulation of SAR data are the following. First, these images have practically no self-noise that could later influence blind

Thus, we can state that speckle is spatially correlated in the considered TerraSAR-X images,

Second, these are the images of natural scenes and, thus, they contain large quasihomogeneous regions, edges with different contrasts, various textures and small-sized **Figure 3.** Noise-free (true) test images used for simulating SAR images

Fig. 3. Noise-free (true) test images used for simulating SAR images

estimation of speckle statistics.

**Figure 2.** ACF estimates for 32x32 pixel homogeneous fragments in single-look (a) and multi-look (b) TerraSAR-X im‐

Second, we have analyzed a normalized 8x8 DCT spectrum estimates obtained in a blind manner (Ponomarenko et al., 2010) for several considered images. These estimates also clearly

Third, we have also calculated a parameter *r* (Uss et al., 2012) able to indicate spatial correlation of noise for any type of signal dependent noise with spatially stationary spectral characteristics. For determination of *r*, two local estimates of noise variance are derived for each 8x8 pixel block with its left upper corner defined by indices *l* and *m*. The first estimate is calculated in

indicate that speckle is spatially correlated, i.e., not i.i.d. (Lukin et al., 2011).

(*Iij* − *I* ¯ ̑

and the second estimate is calculated in the DCT domain as

(*σ*̑ *lm sp* )

*lm* / *σ*̑ *lm* *lm*)<sup>2</sup> / 63, *I* ¯ ̑ *lm* <sup>=</sup>∑ *i*=*l l*+7 ∑ *j*=*m m*+7

<sup>2</sup> =(1.483*med*(| *Dqs*

and multi-look ones. Then, this effect should be taken into account in simulations.

*lm* |))

*sp* is calculated for each block. After this, the histogram of

*lm*, *q* =0, ..., 7, *s* =0, ..., 7 except *q* =*s* =0 are DCT coefficients of *lm*-th block of a given

these ratios for all blocks is formed and its mode *r* is determined by the method (Lukin et al., 2007). For all considered real-life SAR images, the value of *r* was larger than 1.05 (Lukin et al., 2011b) for single-look SAR images and considerably larger for multi-look ones. This addition‐ ally gives an evidence in favor of the hypothesis that speckle is spatially correlated. Thus, we can state that speckle is spatially correlated in the considered TerraSAR-X images, both single-

To simulate single- and multi-look SAR images, we have used four aerial optical images as *I tr* (all test images are of size 512x512pixels). These four images are presented in Fig. 3. Positive features of these images allowing to use them in simulation of SAR data are the following.

*Iij* / 64, (1)

<sup>2</sup> (2)

ages

the spatial domain

308 Computational and Numerical Simulations

where *Dqs*

image. Then, the ratio *Rlm* =*σ*̑

*σ*̑ *lm* <sup>2</sup> =∑ *i*=*l l*+7 ∑ *j*=*m m*+7

> targets. Second, these are the images of natural scenes and, thus, they contain large quasi-homogene‐ ous regions, edges with different contrasts, various textures and small-sized targets.

> Note that we have simulated speckle with the same statistics for all pixels ignoring the fact that for small-sized targets it might differ from speckle in homogeneous image regions. This simplification is explained by the following two reasons. First, more complicated models of speckle are required for small-sized targets. Second, the percentage of pixels occupied by small-sized targets is quite small in real-life images (Lee et al., 2009) and local estimates of noise statistics in the corresponding scanning windows are anyway abnormal. Thus, these local estimates are "ignored" by the BENCs considered below which are robust (see Section 3 for more details).

> Fig. 4 gives two examples of noisy test image with fully developed speckle (single look). For one of them (Fig. 4(a)) speckle is i.i.d. whilst for the second (Fig. 4(b)) speckle is spatially

3 for more details).

in Fig. 4(b) is much closer to practice.

correlated (see details of its simulation below). Even visual analysis of these two noisy images allows noticing the difference in speckle spatial correlation. As it will become clear from the visual analysis of real-life SAR images presented later in Section 4, the case shown in Fig. 4(b) is much closer to practice. Fig. 4 gives two examples of noisy test image with fully developed speckle (single look). For one of them (Fig. 4(a)) speckle is i.i.d. whilst for the second (Fig. 4(b)) speckle is spatially correlated (see details of its simulation below). Even visual analysis of these two noisy images allows noticing the difference in speckle spatial correlation. As it will become clear from the visual analysis of real-life SAR images presented later in Section 4, the case shown

The source code in Matlab that realizes the described algorithm is presented below:

Methods for Blind Estimation of Speckle Variance in SAR Images: Simulation Results and Verification for Real-Life Data

Here M, N correspond to IIm and JIm (that is to the simulated image size), and all other notations are the same as described above. The obtained 2D array has a Rayleigh distribution and has practically the same spatial correlation properties as GCN. Then the values of RES(i,j) are pixel-

The image presented in Fig. 4(b) has been obtained in the way described above. Moreover, this allows getting multi-look images if several realizations of the speckle with desired spectrum

**3. Considered blind estimation techniques and their accuracy analysis for**

Describing the considered BENC methods, one should keep in mind that blind estimates of speckle characteristics obtained for a given method can differ from each other due to the

**•** properties and parameters (if they can be varied or user defined) of a method applied;

Because of this, we first describe BENCs used in our studies and the main principles put into their basis. Then, simulation results are presented for simulated single- and multi-look SAR

As it has been mentioned in Introduction, there are two basic approaches to blind estimation

. The first approach presumes forming local estimates of speckle variance and robust

to obtain the corresponding speckle values

http://dx.doi.org/10.5772/57040

311

C=GCN(:);

[CC,CI]=sort(C); [BB,BI]=sort(B);

RES=reshape(C,M,N);

wise multiplied by *Iij*

*<sup>n</sup>*, *<sup>i</sup>* =1, ...*I*Im, *<sup>j</sup>* =1, ..., *<sup>J</sup>*Im.

**simulated data**

following factors:

are generated and then averaged.

**•** method robustness with respect to outliers;

**•** an observed speckle realization in the considered image;

images, and the analysis of these results is performed.

**•** content of an analyzed image;

**3.1. Considered BENCs**

of *σ<sup>μ</sup>* 2

**•** clipping effects (if they take place).

*true*

C(CI)=B(BI);

*Iij*

B=random('rayleigh',1,1,M\*N)/1.26;

local estimates are "ignored" by the BENCs considered below which are robust (see Section

Note that we have simulated speckle with the same statistics for all pixels ignoring the fact that for small-sized targets it might differ from speckle in homogeneous image regions. This simplification is explained by the following two reasons. First, more complicated models of speckle are required for small-sized targets. Second, the percentage of pixels occupied by

Thus, from a practical point of view, it is more reasonable to simulate spatially **Figure 4.** The first test image corrupted by i.i.d. (a) and spatially correlated (b) speckle

correlated speckle. This can be done in different ways. In our study, we have employed the following simulation algorithm: 1. Generate 2D array of a required size *I J* Im Im for Gaussian zero mean spatially correlated noise (GCN – Gaussian correlated noise) with a desired spatial spectrum (this is a Thus, from a practical point of view, it is more reasonable to simulate spatially correlated speckle. This can be done in different ways. In our study, we have employed the following simulation algorithm:

Fig. 4. The first test image corrupted by i.i.d. (a) and spatially correlated (b) speckle


The source code in Matlab that realizes the described algorithm is presented below:

C=GCN(:);

correlated (see details of its simulation below). Even visual analysis of these two noisy images allows noticing the difference in speckle spatial correlation. As it will become clear from the visual analysis of real-life SAR images presented later in Section 4, the case shown in Fig.

> (a) (b) Fig. 4. The first test image corrupted by i.i.d. (a) and spatially correlated (b) speckle

Thus, from a practical point of view, it is more reasonable to simulate spatially correlated speckle. This can be done in different ways. In our study, we have employed the following

**1.** Generate 2D array of a required size *I*Im × *J*Im for Gaussian zero mean spatially correlated noise (GCN – Gaussian correlated noise) with a desired spatial spectrum (this is a standard

**2.** Transform the 2D GCN data array into 1D array C of size K=*I*Im × *J*Im in a pre-selected way,

**3.** Generate 1D array B of i.i.d. Rayleigh distributed unity mean random variables of size K=

**4.** For the array C, form an array of indices CI in such a manner that CI(1) is the index of the element in C which is the largest, СI(2) is the index of the element in C which is the second largest, and so on. Finally, CI(K) is the last element of the array CI which is the index of

**6.** For k=1..K make valid the condition C(CI(i))=B(BI(i)). Then, noise with Gaussian distri‐ bution is replaced by noise with the required distribution (Rayleigh in our case).

**7.** The obtained array C is transformed to 2D array RES of size K=*I*Im × *J*Im in the way inverse

Thus, from a practical point of view, it is more reasonable to simulate spatially correlated speckle. This can be done in different ways. In our study, we have employed the

1. Generate 2D array of a required size *I J* Im Im for Gaussian zero mean spatially correlated noise (GCN – Gaussian correlated noise) with a desired spatial spectrum (this is a

2. Transform the 2D GCN data array into 1D array C of size K= *I J* Im Im in a pre-

3. Generate 1D array B of i.i.d. Rayleigh distributed unity mean random variables of

Fig. 4 gives two examples of noisy test image with fully developed speckle (single look). For one of them (Fig. 4(a)) speckle is i.i.d. whilst for the second (Fig. 4(b)) speckle is spatially correlated (see details of its simulation below). Even visual analysis of these two noisy images allows noticing the difference in speckle spatial correlation. As it will become clear from the visual analysis of real-life SAR images presented later in Section 4, the case shown

Note that we have simulated speckle with the same statistics for all pixels ignoring the fact that for small-sized targets it might differ from speckle in homogeneous image regions. This simplification is explained by the following two reasons. First, more complicated models of speckle are required for small-sized targets. Second, the percentage of pixels occupied by small-sized targets is quite small in real-life images (Lee et al., 2009) and local estimates of noise statistics in the corresponding scanning windows are anyway abnormal. Thus, these local estimates are "ignored" by the BENCs considered below which are robust (see Section

4(b) is much closer to practice.

in Fig. 4(b) is much closer to practice.

following simulation algorithm:

task solution of which is omitted).

e.g., by row-by-row scanning.

size K= *I J* Im Im .

the smallest element of C.

to it has been done in step 2.

simulation algorithm:

*I*Im × *J*Im.

standard task solution of which is omitted).

**Figure 4.** The first test image corrupted by i.i.d. (a) and spatially correlated (b) speckle

selected way, e.g., by row-by-row scanning.

**5.** Similarly, form an index array BI for the array B.

3 for more details).

310 Computational and Numerical Simulations

B=random('rayleigh',1,1,M\*N)/1.26;

[CC,CI]=sort(C);

[BB,BI]=sort(B);

C(CI)=B(BI);

RES=reshape(C,M,N);

Here M, N correspond to IIm and JIm (that is to the simulated image size), and all other notations are the same as described above. The obtained 2D array has a Rayleigh distribution and has practically the same spatial correlation properties as GCN. Then the values of RES(i,j) are pixelwise multiplied by *Iij true* to obtain the corresponding speckle values *Iij <sup>n</sup>*, *<sup>i</sup>* =1, ...*I*Im, *<sup>j</sup>* =1, ..., *<sup>J</sup>*Im.

The image presented in Fig. 4(b) has been obtained in the way described above. Moreover, this allows getting multi-look images if several realizations of the speckle with desired spectrum are generated and then averaged.
