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

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 following factors:


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 images, and the analysis of these results is performed.

### **3.1. Considered BENCs**

As it has been mentioned in Introduction, there are two basic approaches to blind estimation of *σ<sup>μ</sup>* 2 . The first approach presumes forming local estimates of speckle variance and robust processing of the obtained local estimates. The second approach is based on obtaining a scatterplot and robust regression line fitting into it.

Let us start from considering the former approach. It consists of the following stages. At the first stage, an analyzed image is divided into non-overlapping or overlapping blocks and local estimates are obtained as

$$\hat{\sigma}\_{\mu\ \ lm}^{2} = \sum\_{i=l}^{l+N-1} \sum\_{j=m}^{m+N-1} (I\_{ij} - \hat{\tilde{I}}\_{lm})^{2} / \left( (\mathbf{N}\,^{2} - \mathbf{1})\hat{\tilde{I}}\_{lm}^{2} \right), \quad \hat{\tilde{I}}\_{lm} = \sum\_{i=l}^{l+N-1} \sum\_{j=m}^{m+N-1} I\_{ij} / \left( \mathbf{N}\,^{2} \right) \tag{3}$$

properly (adaptively) set parameters. Since an improved minimal inter-quantile distance estimator provides the best accuracy (Lukin et al., 2007), we use it in our further studies. The technique based on obtaining the set of local estimates according to (3) and estimation of its mode by the improved minimal inter-quantile distance estimator (Lukin et al., 2007) is further referred as **Method 1**. A variable parameter of this method is the block size.

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

(a) (b)

The second group of BENC methods, as it has been mentioned above, is based on scatterplots. A traditional way of scatter-plot representation for signal-dependent noise is the following. For each block, a point in Cartesian system is obtained where its Y coordinate

(a) (b) Fig. 6. Histograms of local estimates (3) for the elementary images in Fig. 5(a) and Fig. 5(b)

(X axis coordinate *X I loc* ). An example of such a scatter-plot for the image in Fig. 5(b) is

The second group of BENC methods, as it has been mentioned above, is based on scatter-plots. A traditional way of scatter-plot representation for signal-dependent noise is the following. For each block, a point in Cartesian system is obtained where its Y coordinate corresponds to

*loc*). An example of such a scatter-plot for the image in Fig. 5(b) is presented in Fig. 8(a).

the main clusters of this scatter-plot (the cluster centers are indicated by red squares) where the clusters are formed by normal local estimates (see details below). However, there are also quite many points that are located far away from this curve and cluster centers. These points correspond to abnormal local estimates obtained in heterogeneous blocks. This means that if

> ^ *loc* <sup>2</sup> =*DI* ¯

*loc* is depicted in this scatter-plot. It is seen that it goes through the centers of

2ˆ *<sup>Y</sup> loc* and a local mean estimate is its argument

<sup>2</sup> and a local mean estimate is its argument (X axis coordinate

*loc* and then to obtain *σ*̑

*μ*

<sup>2</sup> =*D*, where *D* is

<sup>0</sup> 0.2 0.4 0.6 0.8 <sup>1</sup> 1.2 1.4 1.6 1.8 <sup>0</sup>

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

313

corresponds to a local variance estimate

^ *loc*

one presumes to fit a polynomial type curve *σ*

**Figure 6.** Histograms of local estimates (3) for the elementary images in Fig. 5(a) andFig. 5(b)

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> <sup>0</sup>

**Figure 5.** Two elementary single look amplitude SAR images for Rosenheim region

a local variance estimate *Y* =*σ*

*X* = *I* ¯

A curve *σ* ^ *loc* <sup>2</sup> <sup>=</sup>*σ<sup>μ</sup>* 2 *I* ¯

Fig. 5. Two elementary single look amplitude SAR images for Rosenheim region

where *N* denotes the block size under assumption that it has a square shape. According to a previous experience (Abramov et al., 2008), *N* is recommended to be from 5 till 9; *N*=5 is usually enough for i.i.d. noise whilst it is better to set *N* equal to 7, 8 or 9 for spatially correlated noise.

To understand the operation principles of the first group of methods, it can be useful to look at distributions of the local estimates (3). As examples, the two distributions of local estimates *σ*̑ *<sup>μ</sup> lm* <sup>2</sup> for the two real-life (TerraSAR-X) single-look elementary images (presented in Fig. 5(a) and Fig. 5(b)) are shown in Figs. 6(a) and 6(b), respectively (*N*=7 and non-overlapping blocks are used). It is easy to see that both distributions characterized by histograms have modes close to 0.273. Meanwhile, the percentages of "normal" local estimates (3) that produce quasi-Gaussian parts of distributions are considerably different – look at maximal values in histo‐ grams. Suppose that normal local estimates are those ones smaller than 0.5. Then, the probability *p* of occurrence of "normal" local estimates is approximately equal to 0.6 for the histogram in Fig. 6(a) and to 0.9 for the histogram in Fig. 6(b). For other tested real-life singlelook images (in particular, those ones shown in Fig. 7(a) and 7(b)), the estimated values of *p* are from 0.55 till 0.9. The same holds for the simulated test images presented in the previous Section.

Besides, the distributions in Fig. 6 differ by heaviness of the right-hand tail. Recall that this tail stems from the presence of the so-called "abnormal" local estimates (3) that are obtained in heterogeneous image blocks (Vozel et al., 2009). For the elementary images that have a simpler structure (Figures 5(b) and 7), the tail heaviness is considerably less (see Fig. 7(b)).

The property that the distributions have maxima with the mode close to the true value of *σ<sup>μ</sup>* 2 has been put into the basis of several BENC methods for estimation of noise variance (Vozel et al., 2009). The task is then to find the distribution mode automatically, robustly and with a high enough accuracy. For this purpose, it is possible to exploit robust mode finders such as a sample myriad, bootstrapping and minimal inter-quantile distance with properly (adap‐ tively) set parameters. Since an improved minimal inter-quantile distance estimator provides the best accuracy (Lukin et al., 2007), we use it in our further studies. The technique based on obtaining the set of local estimates according to (3) and estimation of its mode by the improved minimal inter-quantile distance estimator (Lukin et al., 2007) is further referred as **Method 1.** A variable parameter of this method is the block size.

estimator provides the best accuracy (Lukin et al., 2007), we use it in our further studies. The technique based on obtaining the set of local estimates according to (3) and estimation of its 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 313

properly (adaptively) set parameters. Since an improved minimal inter-quantile distance

mode by the improved minimal inter-quantile distance estimator (Lukin et al., 2007) is further referred as **Method 1**. A variable parameter of this method is the block size.

Fig. 5. Two elementary single look amplitude SAR images for Rosenheim region

10000 14000 **Figure 5.** Two elementary single look amplitude SAR images for Rosenheim region

8000 9000

processing of the obtained local estimates. The second approach is based on obtaining a scatter-

Let us start from considering the former approach. It consists of the following stages. At the first stage, an analyzed image is divided into non-overlapping or overlapping blocks and local

> ¯ ̑ *lm* <sup>2</sup> ), *I* ¯ ̑ *lm* <sup>=</sup> ∑ *i*=*l l*+*N* −1

where *N* denotes the block size under assumption that it has a square shape. According to a previous experience (Abramov et al., 2008), *N* is recommended to be from 5 till 9; *N*=5 is usually enough for i.i.d. noise whilst it is better to set *N* equal to 7, 8 or 9 for spatially correlated noise.

To understand the operation principles of the first group of methods, it can be useful to look at distributions of the local estimates (3). As examples, the two distributions of local estimates

*<sup>μ</sup> lm* <sup>2</sup> for the two real-life (TerraSAR-X) single-look elementary images (presented in Fig. 5(a) and Fig. 5(b)) are shown in Figs. 6(a) and 6(b), respectively (*N*=7 and non-overlapping blocks are used). It is easy to see that both distributions characterized by histograms have modes close to 0.273. Meanwhile, the percentages of "normal" local estimates (3) that produce quasi-Gaussian parts of distributions are considerably different – look at maximal values in histo‐ grams. Suppose that normal local estimates are those ones smaller than 0.5. Then, the probability *p* of occurrence of "normal" local estimates is approximately equal to 0.6 for the histogram in Fig. 6(a) and to 0.9 for the histogram in Fig. 6(b). For other tested real-life singlelook images (in particular, those ones shown in Fig. 7(a) and 7(b)), the estimated values of *p* are from 0.55 till 0.9. The same holds for the simulated test images presented in the previous

Besides, the distributions in Fig. 6 differ by heaviness of the right-hand tail. Recall that this tail stems from the presence of the so-called "abnormal" local estimates (3) that are obtained in heterogeneous image blocks (Vozel et al., 2009). For the elementary images that have a simpler

The property that the distributions have maxima with the mode close to the true value of *σ<sup>μ</sup>*

has been put into the basis of several BENC methods for estimation of noise variance (Vozel et al., 2009). The task is then to find the distribution mode automatically, robustly and with a high enough accuracy. For this purpose, it is possible to exploit robust mode finders such as a sample myriad, bootstrapping and minimal inter-quantile distance with properly (adap‐ tively) set parameters. Since an improved minimal inter-quantile distance estimator provides the best accuracy (Lukin et al., 2007), we use it in our further studies. The technique based on obtaining the set of local estimates according to (3) and estimation of its mode by the improved minimal inter-quantile distance estimator (Lukin et al., 2007) is further referred as **Method 1.**

structure (Figures 5(b) and 7), the tail heaviness is considerably less (see Fig. 7(b)).

A variable parameter of this method is the block size.

∑ *j*=*m m*+*N* −1

*Iij* / *<sup>N</sup>* <sup>2</sup>

, (3)

2

*lm*)<sup>2</sup> / ((*<sup>N</sup>* <sup>2</sup> <sup>−</sup>1)*<sup>I</sup>*

plot and robust regression line fitting into it.

∑ *j*=*m m*+*N* −1

(*Iij* − *I* ¯ ̑

estimates are obtained as

312 Computational and Numerical Simulations

*σ*̑ *<sup>μ</sup> lm* <sup>2</sup> <sup>=</sup> ∑ *i*=*l l*+*N* −1

*σ*̑

Section.

12000

(X axis coordinate *X I loc* ). An example of such a scatter-plot for the image in Fig. 5(b) is

**Figure 6.** Histograms of local estimates (3) for the elementary images in Fig. 5(a) andFig. 5(b)

The second group of BENC methods, as it has been mentioned above, is based on scatter-plots. A traditional way of scatter-plot representation for signal-dependent noise is the following. For each block, a point in Cartesian system is obtained where its Y coordinate corresponds to a local variance estimate *Y* =*σ* ^ *loc* <sup>2</sup> and a local mean estimate is its argument (X axis coordinate *X* = *I* ¯ *loc*). An example of such a scatter-plot for the image in Fig. 5(b) is presented in Fig. 8(a). A curve *σ* ^ *loc* <sup>2</sup> <sup>=</sup>*σ<sup>μ</sup>* 2 *I* ¯ *loc* is depicted in this scatter-plot. It is seen that it goes through the centers of the main clusters of this scatter-plot (the cluster centers are indicated by red squares) where the clusters are formed by normal local estimates (see details below). However, there are also quite many points that are located far away from this curve and cluster centers. These points correspond to abnormal local estimates obtained in heterogeneous blocks. This means that if one presumes to fit a polynomial type curve *σ* ^ *loc* <sup>2</sup> =*DI* ¯ *loc* and then to obtain *σ*̑ *μ* <sup>2</sup> =*D*, where *D* is 6000 7000

the parameter of the fitted curve, the method of curve fitting should be robust with respect to outliers. this curve and cluster centers. These points correspond to abnormal local estimates obtained in heterogeneous blocks. This means that if one presumes to fit a polynomial type curve 2ˆ*loc loc DI* and then to obtain <sup>2</sup> *<sup>D</sup>* , where *D* is the parameter of the fitted curve, the

indicated by red squares) where the clusters are formed by normal local estimates (see details below). However, there are also quite many points that are located far away from

2 2 <sup>ˆ</sup>*loc loc <sup>I</sup>* is depicted in this scatter-plot. It is seen that it

local mean estimate. Then one has *σ*̑

serves as the estimate *σ*̑

coordinates relate to *Yq* =*σ*̑

¯ ̑

estimates, *I*

cluster, and *Qcl*

as **Method 2**.

*loc* = *F I* ¯

> ¯ ̑

analyzed below, we have used the approach based on (6).

*norm q* <sup>2</sup> , *Xq* <sup>=</sup> *<sup>I</sup>*

inter-quantile distance estimator (Lukin et al., 2007).

look SAR image. Visual analysis of the scatter-plots in Figs. 8(b) and 8(c) shows that for them there are also some clusters of normal local estimates whereas abnormal estimates are present as well. An advantage of the latter two approaches is that it is, in general, simpler to fit a straight line than a higher order polynomial. In particular, there are standard means for this purpose as, e.g., the Matlab version of robustfit method (DuMouchel&O'Brien, 1989). For methods

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

Finally, there are also methods that exploit scatter-plot data to find cluster centers and curve (line) fitting using these scatter-plot centers (Zabrodina et al., 2011; Abramov et al., 2011). Cluster centers are indicated by red color dots in scatter-plots in Fig. 8. The cluster center

distribution mode of local variance estimates for a *q*-th cluster basically based on normal local

scatter-plot horizontal axis to a fixed number of intervals (we recommend to use ten intervals). The estimates of distribution modes for each cluster are obtained by the improved minimal

There is the straight line fitted into the cluster centers in Figs. 8(b) and 8(c). In this case, robustness with respect to abnormal local estimates is provided indirectly due to robust methods used for finding cluster centers. However, there can be also abnormal cluster centers. To reduce their influence, special techniques as RANSAC or double weighting (DW) LMSE fit can be applied (Zabrodina et al., 2011). Taking into account the comparison results (Abramov et al., 2011; Zabrodina et al., 2011), below we consider only the DW curve fitting to scatter-plot since this method, on the average, provides the best results. It is possible to use different sizes of blocks for local variance and local mean estimation in blocks. Below we study 5x5 and 7x7 pixel blocks. The technique based on forming a scatter-plot, its division into fixed number of clusters, finding cluster centers using mode estimation and DW line fitting is referred below

There are also other techniques based on curve fitting into cluster centers with improved robustness with respect to outliers. First, cluster centers can be determined without image presegmentation (as for the **Method 2** described above) and with pre-segmentation and further processing of the obtained segmentation map (Lukin et al., 2010). The result of image presegmentation is used in two ways. First, the number of image segments gives the number of clusters in the scatter-plot in a straightforward manner. Second, this information used for further image block discrimination into (probably) homogeneous and heterogeneous (Abramov et al., 2008; Lukin et al., 2010) allows diminishing the influence of abnormal errors on coordinate estimation of cluster centers. The next stages of the processing procedure are almost the same as in **Method 2**. However, **Method 3** also takes into account that the position of the last cluster(s) (the rightmost one(s)) can be erroneous due to clipping effects. They act so that the corresponding local estimates occur smaller than they should be in the case of

*norm <sup>q</sup>*, *q* =1, ..., *Qcl*

*norm <sup>q</sup>* denotes the estimate of distribution mode of the local mean estimates for this

is the number of clusters. Clusters are obtained by a simple division of the

*loc* where *F* is the fitted straight line parameter that

*μ*. This kind of a scatter-plot is shown in Fig. 8(c) for the same single-

where *σ*̑

*norm q*

<sup>2</sup> is the estimate of

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

315

 

 

method of curve fitting should be robust with respect to outliers.

6000 7000

Fig. 7. The 512x512 pixels elementary single-look amplitude SAR images of Indonesia for (a) HH and (b) VV polarizations 8000 8000 90 **Figure 7.** The 512x512 pixels elementary single-look amplitude SAR images of Indonesia for (a) HH and (b) VV polari‐ zations

60 70 80

**Figure 8.** Different types of scatter-plots for the image in Fig. 5(b)

There are also other ways to obtain a scatter-plot. One variant is that a point Y coordinate corresponds to a local variance estimate *Y* =*σ*̑ *loc* 2 and a squared local mean estimate is its argument (X axis coordinate *X* = *I* ¯ *loc* <sup>2</sup> ). An example of such a scatter-plot obtained for the same single-look image is represented in Fig. 8(b). Then one has to fit a curve

same single-look image is represented in Fig. 8(b). Then one has to fit a curve

$$
\hat{\sigma}\_{loc}^2 = \mathbb{E}\,\bar{I}\_{loc}^2 \tag{4}
$$

i.e. straight line where the estimate *σ*̑ *μ* <sup>2</sup> =*E*, *E* is the parameter of the fitted line. Another option is to obtain a scatter-plot in such a way that a point coordinate Y relates to a local standard deviation estimate *Y* =*σ*̑ *loc* where its argument (X axis coordinate *X* = *I* ¯ *loc*) is the corresponding local mean estimate. Then one has *σ*̑ *loc* = *F I* ¯ *loc* where *F* is the fitted straight line parameter that serves as the estimate *σ*̑ *μ*. This kind of a scatter-plot is shown in Fig. 8(c) for the same singlelook SAR image. Visual analysis of the scatter-plots in Figs. 8(b) and 8(c) shows that for them there are also some clusters of normal local estimates whereas abnormal estimates are present as well. An advantage of the latter two approaches is that it is, in general, simpler to fit a straight line than a higher order polynomial. In particular, there are standard means for this purpose as, e.g., the Matlab version of robustfit method (DuMouchel&O'Brien, 1989). For methods analyzed below, we have used the approach based on (6).

the parameter of the fitted curve, the method of curve fitting should be robust with respect to

(a) (b) Fig. 7. The 512x512 pixels elementary single-look amplitude SAR images of Indonesia for (a)

**Figure 7.** The 512x512 pixels elementary single-look amplitude SAR images of Indonesia for (a) HH and (b) VV polari‐

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

There are also other ways to obtain a scatter-plot. One variant is that a point Y coordinate

argument (X axis coordinate <sup>2</sup> *X Iloc* ). An example of such a scatter-plot obtained for the

There are also other ways to obtain a scatter-plot. One variant is that a point Y coordinate

is to obtain a scatter-plot in such a way that a point coordinate Y relates to a local standard

*loc* where its argument (X axis coordinate *X* = *I*

*loc*

same single-look image is represented in Fig. 8(b). Then one has to fit a curve

(a) (b) (c)

**Fig. 8** Different types of scatter-plots for the image in Fig. 5(b)

¯ *loc*

single-look image is represented in Fig. 8(b). Then one has to fit a curve

*σ*̑ *loc* <sup>2</sup> =*EI* ¯ *loc*

*μ*

x 104

<sup>2</sup> *<sup>Y</sup> loc* and a squared local mean estimate is its

<sup>2</sup> ). An example of such a scatter-plot obtained for the same

<sup>2</sup> =*E*, *E* is the parameter of the fitted line. Another option

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>0</sup>

2 and a squared local mean estimate is its

<sup>2</sup> (4)

¯

*loc*) is the corresponding

goes through the centers of the main clusters of this scatter-plot (the cluster centers are indicated by red squares) where the clusters are formed by normal local estimates (see details below). However, there are also quite many points that are located far away from this curve and cluster centers. These points correspond to abnormal local estimates obtained in heterogeneous blocks. This means that if one presumes to fit a polynomial type curve

2 2 <sup>ˆ</sup>*loc loc <sup>I</sup>* is depicted in this scatter-plot. It is seen that it

<sup>2</sup> *<sup>D</sup>* , where *D* is the parameter of the fitted curve, the

 

 

method of curve fitting should be robust with respect to outliers.

outliers.

zations

314 Computational and Numerical Simulations

presented in Fig. 8(a). A curve

2ˆ*loc loc DI* and then to obtain

HH and (b) VV polarizations

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>0</sup>

corresponds to a local variance estimate

**Figure 8.** Different types of scatter-plots for the image in Fig. 5(b)

corresponds to a local variance estimate *Y* =*σ*̑

argument (X axis coordinate *X* = *I*

i.e. straight line where the estimate *σ*̑

deviation estimate *Y* =*σ*̑

Finally, there are also methods that exploit scatter-plot data to find cluster centers and curve (line) fitting using these scatter-plot centers (Zabrodina et al., 2011; Abramov et al., 2011). Cluster centers are indicated by red color dots in scatter-plots in Fig. 8. The cluster center coordinates relate to *Yq* =*σ*̑ *norm q* <sup>2</sup> , *Xq* <sup>=</sup> *<sup>I</sup>* ¯ ̑ *norm <sup>q</sup>*, *q* =1, ..., *Qcl* where *σ*̑ *norm q* <sup>2</sup> is the estimate of distribution mode of local variance estimates for a *q*-th cluster basically based on normal local estimates, *I* ¯ ̑ *norm <sup>q</sup>* denotes the estimate of distribution mode of the local mean estimates for this cluster, and *Qcl* is the number of clusters. Clusters are obtained by a simple division of the scatter-plot horizontal axis to a fixed number of intervals (we recommend to use ten intervals). The estimates of distribution modes for each cluster are obtained by the improved minimal inter-quantile distance estimator (Lukin et al., 2007).

There is the straight line fitted into the cluster centers in Figs. 8(b) and 8(c). In this case, robustness with respect to abnormal local estimates is provided indirectly due to robust methods used for finding cluster centers. However, there can be also abnormal cluster centers. To reduce their influence, special techniques as RANSAC or double weighting (DW) LMSE fit can be applied (Zabrodina et al., 2011). Taking into account the comparison results (Abramov et al., 2011; Zabrodina et al., 2011), below we consider only the DW curve fitting to scatter-plot since this method, on the average, provides the best results. It is possible to use different sizes of blocks for local variance and local mean estimation in blocks. Below we study 5x5 and 7x7 pixel blocks. The technique based on forming a scatter-plot, its division into fixed number of clusters, finding cluster centers using mode estimation and DW line fitting is referred below as **Method 2**.

There are also other techniques based on curve fitting into cluster centers with improved robustness with respect to outliers. First, cluster centers can be determined without image presegmentation (as for the **Method 2** described above) and with pre-segmentation and further processing of the obtained segmentation map (Lukin et al., 2010). The result of image presegmentation is used in two ways. First, the number of image segments gives the number of clusters in the scatter-plot in a straightforward manner. Second, this information used for further image block discrimination into (probably) homogeneous and heterogeneous (Abramov et al., 2008; Lukin et al., 2010) allows diminishing the influence of abnormal errors on coordinate estimation of cluster centers. The next stages of the processing procedure are almost the same as in **Method 2**. However, **Method 3** also takes into account that the position of the last cluster(s) (the rightmost one(s)) can be erroneous due to clipping effects. They act so that the corresponding local estimates occur smaller than they should be in the case of clipping absence. Then, an approach to improve estimation accuracy is to reject the rightmost cluster center(s) from further consideration. A practical rule for cluster rejection can be the following: if *I* ¯ ^ *norm q* >max(*Iij* )/ 4, *i* =1, ..., *I*Im, *j* =1, ..., *J*Im, then this cluster has to be rejected. This rule takes into account the fact that for Rayleigh distribution a random variable can be, with a small probability, by 3…4 times larger than the distribution mean.

## **3.2. Analysis of simulation results**

Let us analyze the obtained simulation results. The main properties and accuracy character‐ istics of the aforementioned methods based on finding a distribution mode have been inten‐ sively studied for the case of additive noise (Lukin et al., 2007). Although the multiplicative noise case is considered here, the conclusions drawn for the additive case might be still valid for **Method 1**. Recall that one of the main conclusions drawn in (Lukin et al., 2007) is that the final blind estimate of noise variance *σ*̑ *fin* <sup>2</sup> can be biased where the bias is mostly positive (i.e., the estimates are larger than the true value). The absolute value of bias is larger for images with more complex structure for which the parameter *p* introduced above is smaller.

Another conclusion is that the estimation bias (denoted as *Δμ* for the multiplicative noise case) usually contributes more to aggregate error *<sup>ε</sup>* <sup>2</sup> <sup>=</sup>*Δ<sup>μ</sup>* <sup>2</sup> <sup>+</sup> *<sup>θ</sup><sup>μ</sup>* 2 , where *θ<sup>μ</sup>* <sup>2</sup> denotes the variance of blind estimation of *σ<sup>μ</sup>* <sup>2</sup> . Here *Δ<sup>μ</sup>* <sup>=</sup> <sup>|</sup> *<sup>σ</sup>* ^ *μ* <sup>2</sup> <sup>−</sup>*σ<sup>μ</sup>* <sup>2</sup> | and *θ<sup>μ</sup>* <sup>2</sup> = (*σ* ^ *μ* <sup>2</sup> − *σ* ^ *μ* <sup>2</sup> )2 where notation • means averaging by realizations.

Let us check are these conclusions valid for the multiplicative noise case. Usually variance *θ<sup>μ</sup>* 2 is determined for a large number of realizations of the artificially added noise that corrupts a given test noise-free image. Thus, we have simulated 200 realizations of i.i.d. speckle with Rayleigh distribution. The obtained simulation results are presented in Table 1. Analysis shows that estimation bias is also positive for all four test images and for both studied sizes of blocks. The values of *θ<sup>μ</sup>* <sup>2</sup> are of the order 10-6. Thus, they are two magnitude order less than squared bias and have negligible contribution to *ε* <sup>2</sup> . This shows that, in fact, it is possible to analyze only the estimation bias or even the estimates obtained for only one realization of the speckle. At least, this is possible for the test images of the considered size of 512x512 pixels or larger (*θ<sup>μ</sup>* 2 decreases if a processed image size increases).

analysis of estimation accuracy. The results for 5x5 blocks for **Method 2** are slightly better than for 7x7 pixel blocks. Hence, the use of 5x5 pixel blocks is the better choice for the case of i.i.d.

**Table 1.** Accuracy data for the considered test images corrupted by i.i.d. speckle (single-look case)

Finally, let us analyse data for **Method 3** (see Table 1). This method produces estimates that have very small absolute values of bias which is mostly negative for both 5x5 and 7x7 pixel

be considered as the most accurate. The results for 5x5 and 7x7 block sizes are comparable and

We have also obtained simulation results for 4-look test images corrupted by i.i.d. speckle

estimation bias is 0.0101, 0.0100 and 0.0005 for **Method 1**, **Method 2**, and **Method 3**, respec‐

other three test images are similar. Thus, we can state that **Method 3** again produces the best

2 is equal to 0.273/4=0.068). They are the following. For the first test image,

are smaller than for **Method 2** but larger than for **Method 1**. However,

**5x5 overlapping blocks 7x7 overlapping blocks**

**∙10-4 Δμ θμ**

**2**

**∙10-6 ε <sup>2</sup>**

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

**∙10-4**

317

<sup>2</sup> are equal to 0.21x10-6, 2.71x10-6, and 2.73x10-6 for these three methods.

are 1.028 x10-4, 1.027x10-4, and 0.030x10-4, respectively. The results for

among the studied BENCs and, thus, can

speckle.

**Method**

**Δμ θμ**

**2**

**∙10-6 ε <sup>2</sup>**

**Image Fr01** Method 1 0.017 1.39 2.95 0.031 2.05 9.47 Method 2 0.034 19.46 11.71 0.042 20.64 17.53 Method 3 -0.008 48.17 1.06 -0.003 53.07 0.60 **Image Fr02** Method 1 0.015 1.48 2.23 0.028 1.96 7.41 Method 2 0.030 10.60 8.99 0.042 9.99 17.58 Method 3 -0.008 57.63 1.27 -0.003 42.87 0.50 **Image Fr03** Method 1 0.014 1.05 1.90 0.027 1.35 7.31 Method 2 0.032 16.01 10.49 0.045 8.51 19.93 Method 3 -0.010 31.94 1.31 -0.003 34.08 0.41 **Image Fr04** Method 1 0.012 1.04 1.47 0.025 1.22 6.25 Method 2 0.016 50.39 3.11 0.017 33.40 3.30 Method 3 0.001 23.49 0.24 0.011 28.63 1.39

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

blocks. The values of *θ<sup>μ</sup>*

tively. The values of *θ<sup>μ</sup>*

Finally, the values of *ε* <sup>2</sup>

(theoretical *σ<sup>μ</sup>*

2

due to small bias, **Method 3** provides the smallest *ε* <sup>2</sup>

both block sizes can be recommended for practical use.

One more conclusion that follows from data analysis for **Method 1** in Table 1 is that the use of the block size 7x7 leads to more biased and, on the average, larger estimates than if 5x5 blocks are used. Nevertheless, the estimates for the fully developed speckle with *σ<sup>μ</sup>* <sup>2</sup> =0.273 are within the required limits (Vozel et al., 2009) from 0.8x0.273=0.218 to 1.2x0.273=0.328 with high probability (it is equal to *Δμ*≤ 0.055).

Consider now data for **Method 2**. They are, mostly, more biased than for **Method 1** for the same test image and block size (see data in Table 1). Moreover, the values of *θ<sup>μ</sup>* 2 and, thus, *ε* <sup>2</sup> are also sufficiently larger. However, estimation accuracy is still mainly determined by the estimation bias and, therefore, it is possible to consider only one realization of the speckle 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 317

clipping absence. Then, an approach to improve estimation accuracy is to reject the rightmost cluster center(s) from further consideration. A practical rule for cluster rejection can be the

This rule takes into account the fact that for Rayleigh distribution a random variable can be,

Let us analyze the obtained simulation results. The main properties and accuracy character‐ istics of the aforementioned methods based on finding a distribution mode have been inten‐ sively studied for the case of additive noise (Lukin et al., 2007). Although the multiplicative noise case is considered here, the conclusions drawn for the additive case might be still valid for **Method 1**. Recall that one of the main conclusions drawn in (Lukin et al., 2007) is that the

the estimates are larger than the true value). The absolute value of bias is larger for images

Another conclusion is that the estimation bias (denoted as *Δμ* for the multiplicative noise case)

<sup>2</sup> | and *θ<sup>μ</sup>*

Let us check are these conclusions valid for the multiplicative noise case. Usually variance *θ<sup>μ</sup>*

is determined for a large number of realizations of the artificially added noise that corrupts a given test noise-free image. Thus, we have simulated 200 realizations of i.i.d. speckle with Rayleigh distribution. The obtained simulation results are presented in Table 1. Analysis shows that estimation bias is also positive for all four test images and for both studied sizes of blocks.

only the estimation bias or even the estimates obtained for only one realization of the speckle. At least, this is possible for the test images of the considered size of 512x512 pixels or larger

One more conclusion that follows from data analysis for **Method 1** in Table 1 is that the use of the block size 7x7 leads to more biased and, on the average, larger estimates than if 5x5 blocks

the required limits (Vozel et al., 2009) from 0.8x0.273=0.218 to 1.2x0.273=0.328 with high

Consider now data for **Method 2**. They are, mostly, more biased than for **Method 1** for the

are also sufficiently larger. However, estimation accuracy is still mainly determined by the estimation bias and, therefore, it is possible to consider only one realization of the speckle in

are used. Nevertheless, the estimates for the fully developed speckle with *σ<sup>μ</sup>*

same test image and block size (see data in Table 1). Moreover, the values of *θ<sup>μ</sup>*

<sup>2</sup> <sup>+</sup> *<sup>θ</sup><sup>μ</sup>* 2

<sup>2</sup> = (*σ* ^ *μ* <sup>2</sup> − *σ* ^ *μ*

<sup>2</sup> are of the order 10-6. Thus, they are two magnitude order less than squared

with more complex structure for which the parameter *p* introduced above is smaller.

with a small probability, by 3…4 times larger than the distribution mean.

)/ 4, *i* =1, ..., *I*Im, *j* =1, ..., *J*Im, then this cluster has to be rejected.

<sup>2</sup> can be biased where the bias is mostly positive (i.e.,

. This shows that, in fact, it is possible to analyze

<sup>2</sup> denotes the variance of

<sup>2</sup> =0.273 are within

and, thus, *ε* <sup>2</sup>

2

2

<sup>2</sup> )2 where notation • means

, where *θ<sup>μ</sup>*

following: if *I*

¯ ^ *norm q*

316 Computational and Numerical Simulations

**3.2. Analysis of simulation results**

final blind estimate of noise variance *σ*̑ *fin*

bias and have negligible contribution to *ε* <sup>2</sup>

probability (it is equal to *Δμ*≤ 0.055).

decreases if a processed image size increases).

blind estimation of *σ<sup>μ</sup>*

The values of *θ<sup>μ</sup>*

(*θ<sup>μ</sup>* 2

averaging by realizations.

usually contributes more to aggregate error *<sup>ε</sup>* <sup>2</sup> <sup>=</sup>*Δ<sup>μ</sup>*

<sup>2</sup> . Here *Δ<sup>μ</sup>* <sup>=</sup> <sup>|</sup> *<sup>σ</sup>*

^ *μ* <sup>2</sup> <sup>−</sup>*σ<sup>μ</sup>*

>max(*Iij*


**Table 1.** Accuracy data for the considered test images corrupted by i.i.d. speckle (single-look case)

analysis of estimation accuracy. The results for 5x5 blocks for **Method 2** are slightly better than for 7x7 pixel blocks. Hence, the use of 5x5 pixel blocks is the better choice for the case of i.i.d. speckle.

Finally, let us analyse data for **Method 3** (see Table 1). This method produces estimates that have very small absolute values of bias which is mostly negative for both 5x5 and 7x7 pixel blocks. The values of *θ<sup>μ</sup>* 2 are smaller than for **Method 2** but larger than for **Method 1**. However, due to small bias, **Method 3** provides the smallest *ε* <sup>2</sup> among the studied BENCs and, thus, can be considered as the most accurate. The results for 5x5 and 7x7 block sizes are comparable and both block sizes can be recommended for practical use.

We have also obtained simulation results for 4-look test images corrupted by i.i.d. speckle (theoretical *σ<sup>μ</sup>* 2 is equal to 0.273/4=0.068). They are the following. For the first test image, estimation bias is 0.0101, 0.0100 and 0.0005 for **Method 1**, **Method 2**, and **Method 3**, respec‐ tively. The values of *θ<sup>μ</sup>* <sup>2</sup> are equal to 0.21x10-6, 2.71x10-6, and 2.73x10-6 for these three methods. Finally, the values of *ε* <sup>2</sup> are 1.028 x10-4, 1.027x10-4, and 0.030x10-4, respectively. The results for other three test images are similar. Thus, we can state that **Method 3** again produces the best accuracy and the influence of estimation variance *θ<sup>μ</sup>* 2 can be ignored in further studies. One more observation is that the values of *ε* <sup>2</sup> for multi-look test images have become smaller than for single-look test images. This does not mean that accuracy has improved since, in fact, accuracy has to be characterized not by *<sup>ε</sup>* <sup>2</sup> but by *ε*/*σ<sup>μ</sup>* <sup>2</sup> . In fact, accuracy characterized by *ε*/*σ<sup>μ</sup>* 2 has the tendency to make worse if *σ<sup>μ</sup>* 2 diminishes. This means that it is more difficult to accurately estimate speckle variance *σ<sup>μ</sup>* 2 for multi-look SAR images than for single-look ones. Analysis shows that it is worth using the block size 7x7 pixels for **Method 3** which is the most

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

Finally, simulation results for four-look test images corrupted by spatially correlated speckle

=0.068. Overestimation is observed for **Method 1** for all four test images and overestimation is larger for 7x7 blocks. Even larger overestimation takes place for **Method 2**, especially if 7x7 blocks are used. **Method 3** usually produces small under-estimation, the errors are, on the average, the smallest among the considered BENCs and 7x7 block size seems to be a proper

**Method Method 1 Method 2 Method 3 Block size 5x5 7x7 5x5 7x7 5x5 7x7** Image Fr01 0.0025 0.0081 0.0063 0.0104 -0.0067 -0.0070 Image Fr02 0.0009 0.0055 0.0027 0.0103 -0.0076 -0.0031 Image Fr03 0.0037 0.0099 0.0042 0.0127 -0.0075 -0.0015 Image Fr04 0.0038 0.0096 0.0004 0.0109 -0.0071 0.0013

2 for the test four-look images corrupted by spatially correlated noise (σμ

First, we will verify our BENCs for the single-look real-life TerraSAR-X images presented in Figures 5 and 7. The obtained data will be considered in subsection 4.1. Besides, in subsection 4.2, we will verify our BENCs for multi-look SAR images of urban area in Canada (Toronto) (these images are presented later). All of them are acquired for HH polarization. As it is stated in file description, approximate number of looks is about 6. Thus, the expected

0.036…0.054 for blind estimates that can be considered appropriate in practice. Let us keep

Let us start from data obtained for **Method 1**. The estimates for block sizes 5x5, 7x7 and 9x9 pixels are collected in Table 4. We decided to analyse 9x9 blocks (not exploited in simulations) to understand practical tendencies and to be sure in our recommendations. Analysis shows that the estimates for 9x9 blocks are larger than for 7x7 and 5x5 blocks. Moreover, for the image in Fig. 5(a) the blind estimate is outside the desired limits. This happens because this image has complex structure and a large percentage of local estimates are abnormal. Although **Method 1** is robust with respect to outliers, its robustness is not enough to keep the blind

^ *μ* <sup>2</sup> -*σ<sup>μ</sup>*

2 similarly to the previous case *σ<sup>μ</sup>*

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

2=0.068)

<sup>2</sup> =0.045 for multi-look data, we can get the limits

2

319

accurate according to simulation data.

choice.

**Table 3.** The values of σ

*σμ*

^ μ <sup>2</sup> -σμ

<sup>2</sup> <sup>≈</sup>0.273 / <sup>6</sup>≈0.045. Similarly, assuming *σ<sup>μ</sup>*

**4.1. Verification results for single-look SAR images**

these limits in mind in further analysis.

estimate within the required limits.

**4. Verification results for real-life SAR images**

are represented in Table 3. The data are presented as *σ*

Consider now the case of spatially correlated noise. We have carried out preliminary simula‐ tions and established that estimation bias contributes considerably more than estimation variance to the *ε* <sup>2</sup> . Thus, below we present only the errors determined as the difference between the obtained estimate *σ* ^ *μ* 2 and the true value *σ<sup>μ</sup>* 2 for single (only one) realization. The simulation results for single-look images are collected in Table 2.


**Table 2.** The values of σ ^ μ <sup>2</sup> -σμ 2 for the test single-look images corrupted by spatially correlated noise (σμ 2=0.273)

An interesting observation that follows from data analysis in Table 2 is that the differences are mostly negative, at least, for 5x5 block size, i.e. speckle variance is underestimated. This can be explained as follows. One factor that influences blind estimation is distribution mode position. Normal local estimates in blocks that form this mode are mostly smaller than *σ<sup>μ</sup>* 2 (Lukin et al., 2011b). Because of this, speckle variance estimates tend to smaller values for **Method 1**, cluster centers tend to smaller values for **Method 2** and **Method 3** as well. Another factor is the method robustness with respect to abnormal local estimates which are, recall, larger than normal estimates. These abnormal local estimates "draw" the final estimates to another side, i.e. "force" them to be larger. Thus, these two factors partly compensate each other. Since **Method 3** is more robust with respect to outliers (a large part of them is rejected due to pre-segmentation), this method provides smaller estimates *σ* ^ *μ* 2 .

As it can be also seen from analysis of data in Table 2, the estimates *σ* ^ *μ* 2 for 7x7 blocks are larger than the corresponding estimates for 5x5 blocks. This is because mode position for normal local estimates shifts to right (to larger values) if the block size increases. This effects have been illustrated for spatially correlated speckle (Lukin et al., 2011b) and for spatially correlated additive noise (Abramov et al., 2008). Then, the final estimates for all BENCs also increase. Analysis shows that it is worth using the block size 7x7 pixels for **Method 3** which is the most accurate according to simulation data.

Finally, simulation results for four-look test images corrupted by spatially correlated speckle are represented in Table 3. The data are presented as *σ* ^ *μ* <sup>2</sup> -*σ<sup>μ</sup>* 2 similarly to the previous case *σ<sup>μ</sup>* 2 =0.068. Overestimation is observed for **Method 1** for all four test images and overestimation is larger for 7x7 blocks. Even larger overestimation takes place for **Method 2**, especially if 7x7 blocks are used. **Method 3** usually produces small under-estimation, the errors are, on the average, the smallest among the considered BENCs and 7x7 block size seems to be a proper choice.


**Table 3.** The values of σ ^ μ <sup>2</sup> -σμ 2 for the test four-look images corrupted by spatially correlated noise (σμ 2=0.068)
