The FR images used as ground truth are given in

Figure 1.

Figure 1a is retrieved from an unrelated PolSAR scene using the B&B method to test the TV algorithm on real FR data, and

Figure 1b is constructed manually. There are nine slices of FR in

Figure 1b with values increasing from

${1}^{\circ}$ to

${9}^{\circ}$ and widths decreasing from 200 pixels to 1 pixels. The background of

Figure 1b is set to be zero. Using

Figure 1b, the performance of TV in keeping spatial resolutions can be directly viewed.

To test the performance of the proposed method against different levels of noise, we add to the scene Gaussian noise under SNR = 0 dB, SNR = 10 dB, and SNR = 20 dB, respectively. The SNR is defined as [

14]

In the experiment, the noise level of each channel is set to be equal [

14]

Therefore, the standard deviation of noise in each channel is calculated as:

As the true FR is known, if the method could remove all the noise, the bias between estimated FR and the true value is zero. We compared TV with the traditional method, i.e., average filtering (AF). In the AF algorithm, the size of the filtering window plays an important role. A larger window suppresses the noise more, but it also decreases the FR resolution. Instead of using a fixed window, we test AF under a different number of looks [

11] (the number of looks is defined as the number of pixels of the filtering window). In the experiment, all filtering windows are set to be square, that is to say,

$225$ looks correspond to a

$15\times 15$ window. Actually, during our test it is the number of looks that influences the result the most rather than the window shape. Therefore, if a

$5\times 45$ window is used, the same result is obtained as using a

$15\times 15$ window. Although the TV algorithm does not involve a filtering window, for a fair comparison we also added a filtering window after the TV algorithm was executed, that is to say, in the experiment below, the results of TV are obtained by first applying the TV algorithm and, then, adopting the AF algorithm. However, as described in the following, our main concern is the result of TV with a

$1\times 1$ window, i.e., the result from a pure TV algorithm. Results with all other windows are used for a fair comparison with the AF method.

#### 3.1.1. Result on the First FR Image

Figure 2 gives the result of two selected PolSAR scenes on the first FR image (shown in

Figure 1a) under three levels of noise. The left column of

Figure 2 shows the average of the FR bias

${\delta}_{f}$ of each scene changing with respect to different looks

and the right column shows the standard deviation of the FR bias

${\sigma}_{f}$ in each scene

In Equation (21) and Equation (22),

${\mathsf{\Omega}}_{E}$ and

${\mathsf{\Omega}}_{T}$ represent estimated the FR and the true FR, respectively. The closer the

${\delta}_{f}$ and

${\sigma}_{f}$ are to zero, the better the algorithm performs. For a better illustration of differences between the two methods, the logarithmic scale is used for the

y-axis and the linear scale is used for the

x-axis. Specifically, the coordinate of the

x-axis denotes the width of the filtering window, i.e., the square root of the number of looks. As we can see from

Figure 2, the bias of FR decreases as the SNR gets higher, but regardless of the noise level, the TV method performs much better than AF. For Scene 1, we find that the TV method with

$1\times 1$ looks performs similar or even better than AF with

$15\times 15$ looks in terms of both

${\delta}_{f}$ and

${\sigma}_{f}$, which means, in Scene 1, TV with

$1\times 1$ looks could achieve the same noise removing performance as AF with

$15\times 15$ looks.

For Scene 2, the superiority of TV in terms of

${\delta}_{f}$ is much more obvious. Specifically, the

${\delta}_{f}$ of TV with

$1\times 1$ looks is smaller than that of AF with

$30\times 30$ looks under the same noise level. However, the performance of TV in terms of

${\sigma}_{f}$ seems not satisfying. From

Figure 2d we can find for SNR = 20 dB, the

${\sigma}_{f}$ of TV with

$1\times 1$ looks is unexpectedly larger than that of AF with

$15\times 15$ looks. Recalling that

${\sigma}_{f}$ means the standard deviation of FR bias, a larger

${\sigma}_{f}$ means the bias will spread in a wider range. Nevertheless, when we examine the distribution of FR bias of TV and AF, we find that the bias of TV with

$1\times 1$ looks is still more centralized than the bias of AF with

$15\times 15$ looks, as shown by the bias histograms in

Figure 3. Obviously we can see from

Figure 3 the FR bias of TV with

$1\times 1$ looks (

Figure 3b) is distributed more concentrated around zero than bias of AF with

$15\times 15$ looks (

Figure 3a). The larger

${\sigma}_{f}$ of TV with

$1\times 1$ looks is mainly caused by a very few pixels whose values are abnormally large due to remaining noise. The number of abnormal pixels is so small (about 50 in a million) that they do not influence the image precision and resolution too much. Therefore, in Scene 2, TV with

$1\times 1$ looks still performs better than AF with

$15\times 15$ looks. In conclusion, TV could obtain FR in high precision as AF achieves with

$15\times 15$ looks.

Another phenomenon is that the increase of the window width also reduces the bias of FR, which accords with the conclusion given in [

11]. However, an interesting fact could be observed from both scenes under noise level SNR = 20 dB that the FR bias increases a little after it reaches to a minimum. This phenomenon is more obvious in TV than in AF. At first sight, it seems contradictory to the conclusions in [

11]. However, the conclusions in [

11] are based on the assumption that the FR angles adding to each pixel are equal. Recalling Equation (3) here, if the FR in each pixels is equal to each other, then a large window will suppress the noise only, but if the FR varies pixel by pixel, a large window will also distort the original FR in each pixel. For example, assume there is a FR image only consisting of two pixel, then according to Equation (3) we would get the two following equations to derive the FR of the two pixels:

and

To eliminate the noise, a 2 looks filtering window is assumed to be used, then, according to Equation (5) the derived FR would be

From Equation (25) we can easily find that if

${\mathsf{\Omega}}_{1}={\mathsf{\Omega}}_{2}$, the precision of

$\mathsf{\Omega}$ is only affected by the noise component. However when

${\mathsf{\Omega}}_{1}\ne {\mathsf{\Omega}}_{2}$, even if the noise is perfectly eliminated, resulting

$\mathsf{\Omega}$ would still be biased by the difference between

${\mathsf{\Omega}}_{1}$ and

${\mathsf{\Omega}}_{2}$. This result explains why under SNR = 20 dB the resulting FR bias with a large window is larger than that with a small window. A large window suppressed the noise component well but suffers more from the FR variation. Therefore, in real scenes a large filtering window is not always a good option.

From Equation (25), we know if the noise is well removed, we will have

where the

$\widehat{M}$ matrix is the noise free scattering matrix that could be calculated in the synthetic experiment as:

When FR is precisely estimated, the resulting

${\widehat{S}}_{vh}$ and

${\widehat{S}}_{hv}$ from Equation (26) will be reciprocal, that is to say the differences between

${\widehat{S}}_{vh}$ and

${\widehat{S}}_{hv}$
will be zero. Therefore, the reciprocal bias

${\epsilon}_{re}$ could also be used as an indicator for the algorithm’s performance. The smaller

${\epsilon}_{re}$ is, the better the corresponding algorithm is.

Using Equation (27) and

Figure 1a, we can obtain the noise free scattering matrix for Scene 1 and Scene 2. Subsisting the noise free matrix and the estimated FR into Equation (26), we can obtain the reciprocal bias for each scene.

Figure 4 gives the result of the reciprocal bias with respect to different noise levels and different window widths, where the left column shows the average of the reciprocal bias

${\delta}_{re}$ and the right column corresponds to the standard deviation

${\sigma}_{re}$ of reciprocal bias. The closer the

${\delta}_{re}$ and

${\sigma}_{re}$ is to zero, the better the corresponding method is. It can be observed that the trend of the reciprocal bias for both scenes in

Figure 4 is exactly the same as with that of FR bias in

Figure 2. The reason for the above result is obvious. The precision of the resulting

$\widehat{S}$ matrix in Equation (26) is only determined by the precision of the estimated FR when the noise free scattering matrix

$\widehat{M}$ is known. Therefore, the reciprocal bias can play the same role as the FR bias in the evaluation of different algorithms. For the experiment on real PolSAR data in

Section 3.2, we will adopt the reciprocal bias as an indicator for the performance of both algorithms.

#### 3.1.2. Result on the Second FR Image

With the FR image of

Figure 1a we have proven that TV with

$1\times 1$ looks could suppress the noise as much as AF does with

$15\times 15$ looks. Using the FR image of

Figure 1b as the true value to test both algorithms, we also obtained similar result as we did with the first FR image (i.e.,

Figure 1a). However, instead of exhibiting the statistic result of FR bias and reciprocal bias on

Figure 1b, we aim to show the FR images estimated from both methods to show the superiority of the TV method in keeping the precision and resolution of FR.

Scene 1 in

Table 1 is used as the true scattering matrix in this section. Then, the FR image of

Figure 1b and three levels of Gaussian noise are added to the scene according to Equation (1).

Figure 5 and

Figure 6 show the estimated FR images from both methods at SNR = 10 dB and SNR = 20 dB. The FR images resulting at SNR = 0 dB are not displayed here, partly because of the limitation of the paper length and partly because the noise equivalent sigma zero (NESZ) in real scene is said to be within 10 to ~20 dB [

26]. Actually, in our experiment, the FR images estimated at SNR = 0 dB also supports the conclusions given below.

In

Figure 5, the images in the left column correspond to results of the AF method and the images in the right column are obtained from the TV method. To illustrate how different looks influence the result, the FR images estimated at

$1\times 1$ looks,

$5\times 5$ looks, and

$15\times 15$ looks are shown in

Figure 5, which are distributed from the first row to the third row. In the last row of

Figure 5, the FR bias images resulting at

$15\times 15$ looks are given to show the effect of a large filtering window on the precision of the estimated FR.

From

Figure 5a, we find that the FR image derived by AF at

$1\times 1$ looks is totally contaminated by the noise. The noise is so severe that the first two slices of FR are nearly invisible. The reason is that AF at

$1\times 1$ looks would not contribute to noise removing, which means

Figure 5a is the original FR image derived directly from the noise contaminated scattering matrix. With an increase of window size, the resulting FR images suffer less and less from the noise contamination, as shown in

Figure 5c,e. However, comparing

Figure 5e to

Figure 5c, we also find that the edges in the FR images become fuzzier as the filtering window gets larger, that is to say, while a large window helps eliminate the noise from the image it also reduces the sharpness of edges, thus, reducing the image resolution. Therefore, for the AF method a critical task is to set up a suitable filtering window to obtain FR images with high resolution and high precision. However, this task is usually not easy to complete. Nevertheless, the FR image retrieved from the TV algorithm at

$1\times 1$ looks (

Figure 5b) is a good example for both noise removing and resolution keeping. Although in

Figure 5b there is still some noise left, it can be observed that by using this

$1\times 1$ window TV has eliminated as much noise as AF has in

Figure 5e with a

$15\times 15$ window. Specifically, the noise left in

Figure 5b is subjected to a Gaussian distribution [0.2711, 0.3953] and the noise in

Figure 5e is subjected to a Gaussian distribution [0.2719, 0.4794]. Apparently, TV with a

$1\times 1$ window works even a little better than AF with a

$15\times 15$ window in noise removing. Furthermore, the edges in

Figure 5b are much sharper than the edges in

Figure 5e, which proves TV has better ability to keep the spatial resolution as compared with AF.

Figure 5d and f show that if additional filtering window is adopted after TV, the noise in the resulting FR images is further suppressed, but the spatial resolution also degenerates to some extent.

Figure 5g and h is used to validate the conclusion given in

Section 3.1.1 that the large filtering window can bias the FR image if the FR values are different from pixel to pixel. If the noise is eliminated well, then all pixel values in

Figure 5g,h should be zero. However, as can be observed, the Gaussian noise in and out of the FR slices is mostly eliminated, whereas at edges of the slices the bias is prominent. The bias is mainly caused by the variation of FR at the edge, as discussed in

Section 3.1.1 with Equation (25).

Figure 6 displays the FR images resulting under SNR = 20 dB. In this high-SNR scenario, both TV and AF perform better than under SNR = 10 dB. Even with a

$1\times 1$ window, TV could remove most of the noise, as shown in

Figure 6b. The FR bias, as shown in

Figure 6b, due to the remaining noise is subjected to a Gaussian distribution [0.14, 0.4]. Using a

$15\times 15$ window, AF could achieve the same noise elimination performance as TV with

$1\times 1$ window, as

Figure 6e depicts, where the FR bias is subjected to a Gaussian distribution [0.16, 0.48]. However, the sharpness of

Figure 6e is also corrupted by the

$15\times 15$ window. From

Figure 6g and h we can see more clearly that the resulting FR images are mostly biased at the edges of each slice where the FR values vary sharply. Again, this confirms that if we want to distinguish variations in ionosphere from FR images, a large filtering window is not suggested. To suppress the noise and keep the variations, the TV algorithm is more suitable as compared with the AF.