A Bragg-Like Point Extraction Method for Co-polarization Channel Imbalance Calibration

Calibration by distributed targets is an important part of polarimetric calibration (PolCAL) without corner reflectors (CRs). Currently, both the correlation of HH and VV (<inline-formula><tex-math notation="LaTeX">$R_{\text{hhvv}}$</tex-math></inline-formula>) and the equivalent number of looks (ENL) constitute the common method for the extraction of Bragg-like points (BLPs) in order to calibrate the co-polarization channel imbalance <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula>. However, strict assumptions about distributed targets and complex mathematical expressions regarding ENL limit the impact analysis of <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> on <inline-formula><tex-math notation="LaTeX">$R_{\text{hhvv}}$</tex-math></inline-formula> and ENL; thus, the fixed thresholds for <inline-formula><tex-math notation="LaTeX">$R_{\mathrm{hhvv}}$</tex-math></inline-formula> and ENL can be obtained only by comparing the calibration errors with the CRs or simulated values in order to calibrate <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> accurately. In some Earth observation and lunar exploration without CRs, the abovementioned conditions are not satisfied. In this article, a BLP extraction method is proposed for calibrating co-polarization channel imbalance <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> in PolCAL. <inline-formula><tex-math notation="LaTeX">${H}/{\bar{\alpha }}$</tex-math></inline-formula> decomposition is utilized to deduce the specific scattering impacts of BLPs by different <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> values first. Then, a dynamic selection method is proposed to reduce the influence of fixed thresholds and improve the extraction accuracy. Multiscene polarimetric synthetic aperture radar images from AIRSAR, ALOS, and GF-3 are utilized to verify the effectiveness of the proposed algorithm in the extraction of BLPs and the calibration results of <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> obtained by applying the extracted BLPs.


PolSAR
Polarimetric synthetic aperture radar. UZH Unitary zero helix. XT Crosstalk. Variables Covariance matrix of [S 3 ]. C ij (i, j = 1, 2, 3) ith row and jth column element of [C 3 ]. χ Cross-polarization channel imbalance. cod i (i = 1, 2) Ratio of the cross-polarization power to the co-polarization amplitude based on [O]. cod j (j = 3,4) Ratio of the cross-polarization power to the co-polarization amplitude based on [C 3 ]. error amp Calculated amplitude error of k, whose unit is 1. error amp_dB Calculated amplitude error of k, whose unit is dB. Covariance matrix corresponding to [M ].
Vector format of observation scattering matrix of a trihedral corner reflector. O ij (i, j = 1, 2, 3) ith row and jth column element of [O]. ϕ C 13 Phase of C 13 . ϕ k Phase of k. p i (i = 1, 2, 3) Pseudoprobability corresponding to H. [Q] Cross-polarization channel imbalance matrix. D UE to its backscatter sensitivity, polarimetric synthetic aperture radar (PolSAR) provides distinctive mapping ability for terrain categorization and ground-cover classification that are highly useful for the measurements and validations of snow, sea surface, forests, crops, and alternative environmental ground truth [1], [2], [3], [4]. To quantitatively analyze radar backscatter from the PolSAR system, it is important to determine the polarimetric distortion matrices (PDMs) of the PolSAR system, which is termed polarimetric calibration (PolCAL). PDMs can be separated into channel imbalance (CI) and crosstalk (XT) components when ignoring added instrument noise and Faraday rotation angle [5], [6], [7].
In practical PolCAL, corner reflectors (CRs) are considered the most accurate calibrators due to the fixed ratios among polarimetric channels [8], [9], [10], [11]. Considering different size requirements of the passive CRs in different bands and the high price of polarimetric active radar calibrators (PARCs), assumptions of distributed target scattering properties, such as reciprocity and reflection symmetry, are used in the calculations of the PDMs. However, in most PolCAL methods, appropriate natural distributed targets are used to determine the XT and cross-polarization (x-pol) CI, and the co-polarization (co-pol) CI k is determined by the use of one CR [7], [12], [13], [14], [15], [16], [17], [18], [19].
Due to the limitations of CRs, the calibration of k by distributed targets has become an interesting research field in recent years [20], [21], [22], [23], [24], [25], [26]. Significantly, Shi et al. [21], [24], [26] investigated the unitary zero helix (UZH) property of Bragg-like points (BLPs) for calibrating k that has been applied in many large-scale applications because BLPs, such as bare soil are widely distributed areas and present many stable characteristics in most PolSAR applications, such as Earth observation and lunar exploration. To the best of the our knowledge, the common and efficient method for extracting BLPs in uncalibrated k PolSAR data is to assign thresholds to the correlation of HH and VV (R hhvv ) and equivalent number of looks (ENL). However, as shown in Fig. 10 of [24], experiments indicate that the calibration error of k is greatly affected by different fixed thresholds, which may be close to 4 dB/12 • . Therefore, the BLPs extraction may make the calibration accuracy of k unstable. In some PolCAL applications without CRs or other equipment, such as lunar exploration, ENL, and R hhvv cannot directly be utilized to extract BLPs accurately. There are two main reasons for the unsatisfactory errors.
The first is that in the analysis of the effect of k on R hhvv and the ENL, less general assumptions are applied so that these two parameters cannot be used directly in some polarimetric applications. For the distortion-immune property of R hhvv , two conditions must be satisfied: 1) The second-order term of XT can be ignored if the XT of a PolSAR system is lower than −20 dB; 2) The cross-correlation coefficient is small enough, which means that the selected area should satisfy the reflection symmetry assumption as much as possible. However, this hypothesis is strong in high-resolution PolSAR, which may impact the polarization orientation angle (POA) accuracy if R hhvv is utilized to extract the BLPs [24]. For the application of the ENL to practical PolCAL, the ENL is considered to be less sensitive to system distortion [24]. In some PolSAR applications, such as lunar exploration by PolSAR systems that require accurate calibration of k without CRs, it is necessary to analyze the impact of k on the ENL as a basis for extracting BLPs. However, the expression for the ENL is quite complicated and involves second-order statistics varying with k [26]. Therefore, it is difficult to analyze the effect of k on ENL. The other reason is that only fixed thresholds on R hhvv and the ENL are set. Since the influence of k on R hhvv and the ENL is unclear, selection of the fixed thresholds can be based only on the comparison with simulated values or CR results, producing the threshold corresponding to the minimum values. However, the optimal thresholds may vary among PolSAR images, so that the application of the same thresholds in all PolCAL without CRs may cause BLPs selection errors and k calibration errors.
The first contribution of this article is that polarimetric scattering entropy H and polarimetric scatteringᾱ parameter are utilized to replace R hhvv and the ENL for BLPs extraction from uncalibrated k PolSAR images. In H/ᾱ decomposition [27], H andᾱ divide the ground targets into eight parts, where Zone 9 (Z9, H ≤ 0.5 andᾱ ≤ 42.5 • ) is the Bragg-like scattering area with low helix scattering power. In addition, the two parameters are roll invariant, making the strict reflection symmetry assumption in the extraction of BLPs unnecessary for ground targets. Furthermore, H andᾱ have clear mathematical expressions and two-dimensional representations (H/ᾱ plane), enabling convenient analysis of the impact of k.
The second contribution of this article is the dynamic extraction of BLPs from uncalibrated k PolSAR data. Since the fixed thresholds directly determine the extracted BLPs (ext. BLPs), affecting k calibration accuracy, we propose that after initially extracting most of the BLPs through thresholds, k values with different amplitudes and phases are added to the uncalibrated k PolSAR image to make the points move on the H/ᾱ plane so that some BLPs that are previously not extracted by thresholds are selected in the proposed method and that the points with wrong features selected by thresholds are eliminated. Ultimately, the selected areas are highly consistent with the BLPs. We note that we do not object to selecting the BLPs by setting thresholds for the polarimetric parameters, but an additional process is required to improve the accuracy of the ext. BLPs after the use of the thresholds for preliminary selection.
The rest of this article is organized as follows. Section II introduces the effect of k on H andᾱ; Section III proposes a method for choosing the BLPs; Section IV presents the experimental results; and Section V provides discussion. Finally, Section VI concludes this article.

II. EFFECT OF k ON H ANDᾱ
This section mainly introduces the effect of k on H andᾱ. First, the calibration model and the concepts of H andᾱ are briefly introduced, and then the variation range of k is limited by the preliminary calibration results of the well-known PolSAR systems. Through the properties of the Hermitian matrix and the relationship between the normalized coherence matrix [N 3 ] [28] andᾱ, the relationship of the phase of k (ϕ k ) to H andᾱ is derived. Finally, Monte Carlo simulations are utilized to obtain the change in the Z9 lateral area on the H/ᾱ plane caused by the amplitude of k.

A. PolCAL Model
During the normal operation of the PolSAR system, the receiving and transmitting channels interact with each other due to the effects of the external environment and various system factors, such as XT, CI, Faraday rotation, and system noise, resulting in an observation scattering matrix [M ]. Considering that the Faraday rotation matrix can be solved by the measurement of the total electron content and calm water can be used to estimate the noise matrix [22], [29], the rest of PDMs superimposed on the nondistorted scattering matrix [S 3 ] can be expressed by [21] [ where, u, v, w, and z are related to the XT; χ is the x-pol CI; and Y denotes the absolute radiometric gain. Notably, Y is common to all channels and is not considered to be a part of PolCAL in this article. Therefore, the covariance matrix corresponding to where, [C 3 ] is the covariance matrix corresponding to [S 3 ] and † represents the vector conjugate transpose operator.

B. H/ᾱ Decomposition
As a type category of eigenvalue-based decomposition [27], [30], [31], H/ᾱ decomposition can be described by explicit and physically meaningful mathematical expressions, providing a possibility to extract BLPs in Z9 from uncalibrated k PolSAR images.
For processing PolSAR signals, the coherency matrix [T 3 ] is obtained as a matrix producing the conjugate transpose of the Pauli basis k p with itself. In the actual situation, since the targets are coherent on PolSAR images, it is necessary to perform a multilook operation for [T 3 ]. The corresponding mathematical model can be written as where, · represents multilook processing and the superscript T is the vector transpose operator. As [27] mentioned previously, (3) can be written combining the eigenvalue λ i and the unit eigenvector k i as In the complex spherical coordinate system defined by α i , β i , σ i , γ i , and ξ i , k i (i = 1, 2, 3) can be written as where, j denotes √ −1. The pseudoprobability is defined as Furthermore, the polarimetric scatteringᾱ parameter and polarimetric scattering entropy H can be expressed as In the H/ᾱ plane [30], Z9 represents the low-entropy surface area and mainly includes geometric optic and physical optic surface scattering, namely Bragg surface scattering and specular scattering. The surfaces of natural targets, such as ice surfaces, water bodies, and very smooth land surfaces, are included. Considering that one task of PolCAL is long-term system monitoring, stable natural features in most image scenes should be utilized to calibrate PolSAR systems to avoid additional errors caused by excessive changes. Because of the variable natural climate and different penetrability of wavebands, the water body has more physical forms and complex changes than other Bragg surface scatters and thus is more suitable to remove the ice and water of Z9 for calibration.

C. Phase Effect of k on the H/ᾱ Plane
Because of the existence of k, the polarimetric scattering characteristics reflected on the H/ᾱ plane are inconsistent with the nondistorted targets. Next, we focus on the examination of point changes, with Z9 as the centre, in Z7, Z8, and Z9 (vertical axis direction) and Z6 and Z9 (horizontal axis direction) [30] after adding the different values of k. Note that setting the range of k is vital because it directly determines the applicability of the proposed algorithm. In some well-known PolSAR systems, such as SIR-C [32], ALOS [33], RADARSAT-2 [34], and GF-3 [35], k or the corresponding received CI obtained during the preliminary calibration results have amplitude −2 to 2 dB and phase −180 • to 180 • . Therefore, in the following we use a −2 to 2 dB amplitude and −180 • to 180 • phase of k to illustrate the scattering characteristic variations of the targets on the Z9 of the H/ᾱ plane.
First, the influence on the H/ᾱ plane is explained by ϕ k . Since the PolCAL process usually determines u, v, w, z, and χ and then determines k, it can be assumed that u = v = w = z = 0 and χ = 1 when studying only the impact of k. Of note, calibrating u, v, w, z, and χ produces residual errors prior to calibrating k, slightly affecting the selection and use of BLPs for calibration. Analysis of this effect is described in detail in Section V. Considering that [A 4 ] does not affect the relationship between the different channels, (2) can be expressed as (10) When the amplitude of k is constant, (10) suggests that ϕ k does not affect the real eigenvalues of [O]. The detailed derivation is given in the Appendix. Since the eigenvalues of the covariance matrix and corresponding coherency matrix are the same and H consists only of the eigenvalues of the coherency matrix, H is not affected by ϕ k .
After considering the influence on H by ϕ k , in the following we use the normalized coherency matrix [N 3 ] to derive theᾱ change in targets when the amplitude of k is constant and the phase changes. N 11 , which is the first element of [N 3 ], can be obtained by , T r · represents the trace of a matrix, and C ij (i, j = 1, 2, 3) represents the element in the ith row and jth column of [C 3 ]. When the amplitude of k remains unchanged and the phase changes, the varying term in (11) can be described by The abovementioned equation indicates that N 11 is the periodic variation of the trigonometric function with the change in ϕ k and that there is at least one monotonic increasing process and one monotonic decreasing process in a period, whose value is 180 • . To the best of our knowledge, obtaining the specific relationship between N 11 andᾱ is challenging. However, the boundary can be described by three kinds of extreme conditions [36], allowing us to better understand that there is a decreasing relationship between N 11 andᾱ in the statistical sense. Therefore,ᾱ also exhibits a periodic trigonometric change, and the period is 180 • .

D. Amplitude Effect of k on the H/ᾱ Plane
Since the phase of k deduced in the previous section changes only in the longitudinal area of Z9 (Z7, Z8, and Z9), the change in the lateral area points of Z9 (Z6 and Z9) caused by the amplitude of k is analyzed in the following Monte Carlo experiments. The whole process can be divided into five steps that include first finding the boundary of Z6 (0.51 < H < 0.52 and 38 • <ᾱ < 40 • ) as the seed points, then simulating the surrounding PolSAR data of the seed points [37], [38], [39], adding an amplitude of −2 to 2 dB of k for the simulated Z6 points in the third step, observing the depth of the Z6 points to Z9 in the abscissa direction in the fourth step, and finally obtaining the influence of the Z6 points on Z9.
In Fig. 1, the experimental results after the addition of different k values are provided to simulate the impact performance on Z9 by Z6 points. Fig. 1(e) represents the nondistorted H and α characteristics of the simulation points. In addition to the generated Z6 points (rose red), some of the Z9 (blue) and Z5 (yellow) and very few Z2 points (green) are also derived from the seed points. It is observed that 1) the rules obtained by each group of points are not the same, and there is rarely a uniform rule that covers them; 2) although k affects the distance from the point on the Z6 area to theᾱ-axis, the shortest value is 0.33593. The boundary of Z6 and Z9 can be set at 0.33593 in the uncalibrated k images so that all points in the range of 0 to 0.33593 of H are not affected by the Z6 points. In the H/ᾱ plane, the area with H < 0.33593 andᾱ < 42.5 • is defined as NZ9.
Through the abovementioned analysis, on the H/ᾱ plane, with the Z9 area as the centre, the longitudinal change mainly depends on ϕ k , and the horizontal change mainly depends on the amplitude of k. In particular, as the amplitude of k varies from −2 to 2 dB, the points on Z6 have a minimum entropy of 0.33593 so that there are no points of Z9 and points of Z6 mixed together in the entropy interval of 0 to 0.33593. After clearly recognizing the law, we can use the abovementioned proof to extract the BLPs exactly.

III. METHODOLOGY
The previous section describes the theoretical basis for selecting BLPs, focusing on the impact of k on the H/ᾱ plane, particularly the horizontal and vertical regions that are the center of Z9. Based on these theories and derivations, the following focuses on a new dynamic BLPs extraction algorithm in the uncalibrated k PolSAR images to compensate for the selection and calibration error caused by only using thresholds of ENL and R hhvv . A detailed flowchart is displayed in Fig. 2.

A. Influence of Z9 Horizontal Area Point Elimination
Due to the amplitude influence of k on H, the scattering points in Z6 are moved to Z9, affecting the selection of BLPs. However, when the amplitude of k varies from -2 to 2 dB, there is a minimum H of the original points in Z6, which is 0.33593. Considering ϕ k does not affect H change of points on the H/ᾱ,  direct extraction of BLPs in the previously defined NZ9 can eliminate the errors of the selection of BLPs and calibration of k. Therefore, we extract the BLPs in NZ9 to eliminate the influence of the Z6 points. Note that there may be BLPs in other parts of Z9 except NZ9 (for example 0.33593 < H < 0.5 and α < 42.5 • ), resulting in a smaller number of BLPs selected than the one in the entire PolSAR image. However, in PolCAL, the focus of extracting BLPs is primarily on the correct rate rather than the number of selected points. The related experimental results are shown in Section IV.

B. Influence of Z9 Longitudinal Area Point Elimination
In the following process, we divide uncalibrated k PolSAR images into different bins along the range and azimuth directions after taking into account the lack of drift along the azimuth direction and the presence of drift along the range direction of the PolSAR system [24]. According to (12), when the amplitude of k is constant and the phase changes uniformly from −180 • to 180 • , the points on the H/ᾱ plane exhibit an oscillatory cosine movement with a period of 180 • . Since the cosine function does not vary monotonically from −180 • to 180 • , we cannot determine how the points change on the H/ᾱ plane specifically when ϕ k increases. Therefore, the following describes the preliminary selection of BLPs using the threshold, and more BLPs are extracted by using the change trend of the number of BLPs caused by the phase of k. The specific steps for the division into azimuth bins in a range cell are as follows.
Step 1: First, add [K pha3 ] with an amplitude of 0 dB and different phases ϕ k to [O] for the up-and-down movements of points on the H/ᾱ plane that can be expressed as Step 2: Then, in NZ9, use the polarimetric parameters to set thresholds to find the approximate BLPs, given by [21] cod 1 = 10 * log 10 and set the same threshold R xc , i.e., the meaning of "Approximate" is that the threshold cannot completely select the BLPs in NZ9 and that there may be points in some other areas, such as Z7 and Z8 of the H/ᾱ plane, that are selected by the thresholds set.
Step 3: As mentioned previously, in the process of extracting BLPs in NZ9, the proportion of BLPs to the NZ9 points and the number of NZ9 points are two key factors, with the former more important than the latter. Therefore, we set the parameter index 1 :(number of NZ9 points/number of points in a bin)*(number of BLPs in NZ9/number of NZ9 points) 2 based on several bins in a range cell to select the bin corresponding to the maximum value. At the same time, to prevent the second item of index 1 from being too small, index 2 : number of BLPs in NZ9/number of NZ9 points must be greater than 0.9.
Step 4: If index 2 is less than 0.9, which reflects excessive or insufficient angle displacement in the actual situation, there is no significant aggregation of BLPs in NZ9. Therefore, the initial values must be reset. Based on ±45 • , ±45 • /2 1 is added to each value and the operations in Steps 1 to 3 are repeated. If index 2 is still not satisfied by the current calculated angles, ±45 • /2 2 is added to each of the previous values. In addition, a termination condition is set when the number of loops is 6, which means the adding angle is less than 1 • so that the points in NZ9 no longer change. However, this condition is inconsistent with the goal of this article, which is to select the most accurate points. Therefore, this range cell does not participate in PolCAL to reduce the error of k calculated.
Step 5: Through the processing of Steps 1 to 4, most of the BLPs eventually move to NZ9. Since adjusting ϕ k implies moving the points on the H/ᾱ plane up and down as a whole, if most of the BLPs move to NZ9, the scattering feature points representing Z7 and Z8 in nondistorted PolSAR images are also moved to their corresponding area. Therefore, the next step is to repeat Steps 1 to 4 and then observe whether the number of NZ9 points is the same as the number of points calculated previously. If so, this suggests that the maximum number of BLPs have been found in this range direction; if the number is larger or smaller, we repeat Steps 1 to 4 until the maximum point number of NZ9 is found.
During the dynamic extraction process, ϕ k provides the driving force for movement, R xc provides the movement trend of BLPs, index 1 provides the suitable bin position, and index 2 provides an extra guarantee for index 1 . R xc and index 2 are the intermediate parameters of the dynamic extraction process that have no direct impact on the number of final extraction points, as will be analyzed in Section V.

IV. EXPERIMENTS AND RESULTS
In this section, calibrated full-PolSAR images are used to prove the practicability of the proposed algorithm, which are C-band GF-3 Quad-Polarization Strip I (QPSI) data, L-band ALOS full-PolSAR data, and L-band AIRSAR full-PolSAR data with corresponding C-band DEM data. Fig. 3 shows the location details of the experimental data. Whether CRs exist is used to divide these PolSAR images into different parts of these experiments. The PolSAR image with CRs is marked with red stars in Fig. 3(a).
For the PolSAR data without CRs, we add different amplitudes and phases of k to the images as the simulated datasets to derive BLPs in uncalibrated k PolSAR images. Then, BLPs are extracted by the proposed method and ENL and R hhvv and the position of the ext. BLPs in H/ᾱ plane is utilized to determine the correctness of the extraction algorithms. Finally, since the BLPs extraction algorithm proposed in this article aims to reduce the error of the potential calibration k caused by the inaccurate selection, these two ext. BLPs are utilized to calibrate k by using UZH [24] to confirm the validity of the extraction method.
For PolSAR images with CRs, the Ainsworth (Ans), Zero-Ans, and Quegan algorithms [12], [13] are utilized to evaluate and correct u, v, w, z, and χ first. Second, the ext. BLPs from the proposed method and ENL and R hhvv are utilized to evaluate k by using UZH [24]. Then, we can obtain [O] of trihedral CRs to compare with the CRs from the calibration models from the PolSAR images. Finally, the calculated XT and CI are added to the relative backscatter of the ideal trihedral CRs and compared with the relative backscatter of the trihedral CRs in the PolSAR data to verify the calibration k errors by the ext. BLPs. According to the description and derivation in the appendix of [26], if the noise matrix [N ] is not added to the calibration model in (1), the expression can still be written in the form of (1) for the PolSAR image after calibrating XT and CI.
In this article, R upper and B lower , which are the thresholds of R hhvv and the ENL, are set to 0.8 and 0.7 [24], respectively, and R xc is set to −20 dB. In the following section, we call the proposed method Halpha-UZH and the ENL and R hhvv Shi-UZH, respectively.

A. Simulated Datasets
In this part, different values of k are added into the PolSAR images along the range direction. According to the number of range bins, we linearly add k with an amplitude of −2 to 2 dB and a phase of −180 • to 180 • . First, the AIRSAR data are analyzed using the algorithm of this article to introduce the process of the Halpha-UZH in detail; the data are marked by the red stars in Fig. 3(b).
Since there is some prominent noise in the far range and some black areas without data in the AIRSAR image, these parts have been cropped in this experiment and the final size of the data along the azimuth and range directions is 1200×800. A 7×7 multilook operation is performed on the data to obtain the Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.  covariance matrix. The H/ᾱ decomposition of the AIRSAR image is presented in Fig. 4(a), where the dark blue area represents BLPs on the H/ᾱ plane, as shown in Fig. 4(b). Fig. 4(c) shows the H/ᾱ decomposition of the PolSAR image with the addition of different k. After adding different k to the data along the range direction, the backscatter characteristics of the ground targets are quite inaccurate. The results for the H/ᾱ decomposition directly solved for the Z9 to extract BLPs is shown in Fig. 10(d). The comparison of Fig. 10(d) and (b) indicates that it is difficult to directly obtain the BLPs from the uncalibrated k PolSAR data. Fig. 5 shows the results of the BLPs selected under different azimuth bins by the algorithm mentioned in this article. Fig. 5(a)-(e) shows that the numbers of azimuth bins are 12, 15, 20, 30, and 60, respectively. Interestingly, the number of ext. BLPs, the number of real BLPs, and the percentage of real BLPs in ext. BLPs (Table I, rows 2-5, columns 2-6) are the same when dividing the AIRSAR image into different bins along the azimuth direction. This is because after the initial filtering of BLPs by the thresholds of cod 1 , cod 2 , and index 2 in NZ9, various ϕ k are added to the points on the H/ᾱ plane to move up and down in Step 5 until the number of points in NZ9 remains unchanged. Notably, it is important that the number of ext. BLPs is constant when dividing into different bins along the azimuth direction, so as not to introduce additional errors to UZH for calibrating k.  (Table I, column 7, rows 2-3) are greater than the corresponding values for Halpha-UZH. However, the percentage of real BLPs in ext. BLPs is lower and is approximately 16%. Therefore, the accuracy of Halpha-UZH is higher than that of Shi-UZH when calibrating the preliminary results of k by UZH. It should be noted that when UZH is used for calibration, first-order fitting will be performed on the preliminary results to eliminate the influence of the selected Bragg-like point error as much as possible in order to acquire the final solution. However, fitting will only work on data with small deviations. If the accuracy of the selected area is not high, resulting in a large deviation between the k results of the preliminary solution and the real values, the fitting will not reduce the error, and may even increase the error. Therefore, if the ext. BLPs by Halpha-UZH are brought into UZH conditions for calibration, the accuracy of the final solution is higher than that for Shi-UZH. The last row in Table I shows the program running times for Halpha-UZH and Shi-UZH. Clearly, Halpha-UZH is slower than Shi-UZH. However, the average selection time for BLPs by Halpha-UZH is less than 10 s, which is acceptable in practical applications, particularly when the accuracy of BLPs selection is the primary consideration.
Following the selection of BLPs by Shi-UZH and Halpha-UZH, since the numbers of range and azimuth bins influence the accuracy of k calibration, Table II shows a comparison of the results obtained by the two algorithms with setting different numbers of range and azimuth bins. The amplitude and phase errors of k are error amp_dB = mean dB Cal (amp k ) True (amp k ) where, error amp_dB is the amplitude error in dB and error pha is the phase error in degrees. An examination of the data presented in Table II that calibration based on Halpha-UZH has better accuracy than Shi-UZH in most cases for different algorithms with the same azimuth and range blocks, except that shown in bold in Table II. At the same time, since the extraction accuracy of Shi-UZH mainly depends on fixed thresholds of ENL and R hhvv , the preliminary results obtained by the calibration algorithm may deviate significantly from the real value, resulting in a large deviation after fitting, which is printed in italics in Table II. By contrast, this is not the case for Halpha-UZH. When using the same algorithm with different azimuth and range directions, there is no evident regular trend along the azimuth direction or the range direction. By calculating the mean and variance of the errors in different distance directions respectively (Table II, rows 9-10), it is demonstrated that the calibration of k based on Halpha-UZH is more accurate and more stable than that using Shi-UZH.
Since the proposed method should be suitable for polarimetric applications in different scenarios where CRs cannot be deployed but requires accurate calibration of k, next, we use 24 multiscene simulated data by ALOS and GF-3 PolSAR images without CRs to further obtain more simulated results, that are presented in Fig. 6. Before carrying out the next step, the ALOS images are sampled every three pixels in the azimuth direction to ensure nearly equal values between the range and azimuth spacing, and the GF-3 images are sampled every five pixels in the azimuth and range directions to reduce data size.
The numbers of ext. BLPs under different data obtained by Halpha-UZH and Shi-UZH are shown in Fig. 6(a). Halpha-UZH is dominant in the quantity of selected points in nearly half of the data. The plots in Fig. 6(b) and (c) validate the number and the percentage of real BLPs in the ext. BLPs for all 24 simulated data points. With regard to the number of accurate points selected shown in Fig. 6(b), Halpha-UZH is superior to Shi-UZH for 17 of the 24 data points. It is observed from Fig. 6(c) that the accuracy of the points selected by Halpha-UZH is much higher compared to the accuracy of the points selected by Shi-UZH, showing that Halpha-UZH has a certain superiority over Shi-UZH with regard to the number and percentage of accurate extracted points. Furthermore, only the No. 11 simulated PolSAR data do not reach more than 99% in the accuracy of the selected points. The No. 11 image mainly describes the urban area of Beijing, China, where trees, houses, and farmland are the main components. In Fig. 6(c), not all Z7 and Z8 points are fully distinguished from the NZ9 points, accounting for 13.02% of all selected points in No. 11 of the PolSAR images with k. This is most like because with the gradual reduction in ϕ k , it is difficult to move the points closer to the boundary of NZ9. In addition, the number of ext. BLPs and the number of real BLPs in the ext. BLPs of No. 11. of the PolSAR images are shown in Fig. 6(a) and (b), which suggests that Halpha-UZH is much better than Shi-UZH in a complex multitype scattering environment with regard to the accuracy of the selected points.
In Step 4 of Section III-B, if the threshold of index 2 is not met within a certain range, the range data are removed to reduce the fitting error after UZH is used to obtain the preliminary results. Since the No. 11 data are mainly urban areas and contains fewer Bragg-like points, the sum of delete-flags is higher than for other simulated PolSAR data. Fig. 6(e) and (f) shows the amplitude error, phase error, and the relative error of the two methods after extracting BLPs through Halpha-UZH and Shi-UZH, and using UZH to perform the calibration process. To make the amplitude error more obvious, this parameter is expressed by (17) on the basis of (16) Among the 24 selected data, the amplitude and phase results of 19 groups obtained by Halpha-UZH are comprehensively better than those obtained by Shi-UZH, showing that the calibration results of k can be improved by the quality and quantity of the selected points, with a particularly strong effect for the quality. However, not all experimental results obtained using Halpha-UZH are better than those of Shi-UZH, which are represented by orange triangles in Fig. 6(e) and (f). This is because the accuracy of the selected points is not the only factor that determines the calibration results; rather, the distribution of selected points in each range cell that affect the fitting process in [24] is also important. To summarize, based on the simulation data, the algorithm proposed in this article greatly improves the accuracy of extracting BLPs without CRs, and further strongly improves the calibration accuracy.

B. CR Dataset
Since the PolSAR products used in the simulation dataset have residual calibration errors, the calibration results obtained based on the simulation data do not provide the real calibration accuracy of the image, but rather the calibration accuracy relative to the residual calibration error of the image. In this section, we use the reconstructed trihedral CRs to compare to those obtained directly in the image, and obtain the error of the reconstructed trihedral CRs, which can be considered as the calibration error of the image. The PolSAR data with CRs were acquired by the GF-3 calibration team in Inner Mongolia on September 19, 2019 [26]. To verify the calibration performance of the GF-3 system, China set up active radar calibrators, trihedral CRs, 0 • dihedral CRs, and 45 • rotated dihedral CRs. The optical and Pauli images of the CRs data are shown in Fig. 7(a) and (b), where the position of the CR is marked by the red box. Fig. 7(d) is the H/ᾱ decomposition result corresponding to the red box in Fig. 7(b). To remove the influence of coherent speckle noise, the 7×7 multilook operator is used to estimate the coherency and covariance matrix in the subsequent image processing. Following ZeroAinsworth (Zero-Ans) calibration [13], [26] for XT and x-pol CI, Fig. 7(e) shows the BLPs of the Fig. 7(d) extracted by Halpha-UZH.
Since the resolution of the GF-3 image is 8 m and the edge length of five trihedral CRs is 1.235 m, the CR of each polarimetric channel is oversampled 16 times at the signal peak to analyse the preferments of CRs by the maximum method. As a trihedral CR, |VV/HH| is usually defined to observe the unbalanced performance of the co-pol CI, and |HV/HH| and |VH/HH| to observe the XT. Table III shows the corresponding performance analysis of the five trihedral CRs in Fig. 7. Since the calibration model used is shown in (1), if the HV channel and the VH channel of a trihedral CR are observed at the same time, the reconstructed model based on calibration model should be changed according to where, [O tri ] and [S tri ] are, respectively, the vector format of the observation scattering matrix and nondistorted scattering matrix of a trihedral CR. Tables IV, V, and VI provide the XT and CI of different [O tri ] obtained by using the Ans, Zero-Ans, and Quegan algorithms [12], [13] to determine XT and x-pol CI, and UZH to determine the co-pol CI, respectively. In addition, The BLPs for UZH are provided by Halpha-UZH and Shi-UZH. It is clearly observed that 1) under the same algorithm for extracting BLPs, the accuracy is significantly higher for the Zero-Ans algorithm and the Ans algorithm in |VV/HH| compared to that of the Quegan algorithm, and |HV/HH| and |VH/HH| is underestimated in most cases. In particular, the underestimation is more pronounced for the Ans algorithm is more prominent than for the other algorithms. 2) With the same algorithm used to calibrate XT and x-pol CI, solving

V. DISCUSSION
Because Halpha-UZH incorporates some arbitrary factors, such as threshold selection, that affect the accuracy of Halpha-UZH, a detailed discussion of these problems is presented in this section.
A. R xc : Threshold Selection of cod 1 and cod 2 In the proposed algorithm, cod 1 and cod 2 are utilized to select the preliminary BLPs. In calibrating k, different feature types and distribution characteristics directly affect how to select R xc . In Fig. 8, we use two new parameters to show the impact of R xc in experimental scenes Nos. 1-24, which can be written as The relationship between (19) and (14) is given by cod 1 = 10 * log 10 |O 22 | 2 |O 11 | = 10 * log 10 |C 22 | 2 |C 11 | cod 2 = 10 * log 10 |O 22 | 2 |O 33 | = 20 * log 10 (|k|) + 10 * log 10 It is clear that cod 1 is equal to cod 3 and that the value range of cod 2 is −2 + cod 4 ∼ 2 + cod 4 because the amplitude of k is −2 to 2 dB. Fig. 8(a) and (b) shows the number of BLPs in different cod 3 and cod 4 situations. It is observed that cod 3 and cod 4 of almost all BLPs are less than −20 dB. Similarly, cod 1 and cod 2 are the same, although the maximum of cod 2 is cod 4 +2, except for PolSAR images Nos. 11 and 17 that contain volume and double-bounce scattering targets. Considering that the dynamic extraction is proposed to select most of the BLPs in the uncalibrated k PolSAR data first and then select the accurate BLPs through the phase shift of k, it is appropriate to set R xc at −20 dB after checking all experimental data. Note that although non-BLPs may be selected by the R xc threshold, the displacement of the H/ᾱ plane by subsequently adding the ϕ k is more orderly, which means that the whole plane tends to reflect the characteristics of real features rather than being more disorderly. In other words, the threshold set by R xc provides only a reference rather than a decision for the last selected points.

B. Threshold Selection of index 2
When we select the BLPs by taking into account the influences of the selected points in each bin and the BLPs in the selected points of NZ9, we obtain the maximum value of index 1 by adding each ϕ k . The index 2 greater than 0.9 at this maximum value is considered to ensure better calibration results due to the more accurate selection of BLPs. When the stopping condition is reached, the range cell whose index 2 is less than 0.9 does not participate in the next calibration operation to reduce the calibration error of k. If the threshold of index 2 is set to a lower value, then the value of index 1 may be dominated by the number of ext. BLPs, and there may exist more non-BLPs that are on the boundary of NZ9 and Z8 in the H/ᾱ plane, resulting in low fitting accuracy for calibrating k. However, it can be clearly seen that after extracting BLPs by the threshold of index 2 , the real BLPs still move to NZ9 by adding [K pha3 ] to the uncalibrated k PolSAR images, as can be seen by the maximum number change of the ext. BLPs under different thresholds of index 2 . Fig. 9 shows different fitting error results obtained by the antithetic threshold of index 2 in Nos. 1-24 of the experimental datasets, namely 0.70, 0.80, 0.90, 0.91, 0.92, 0.93, 0.94, and 0.95. The number of range bins is 80, and the azimuth size of a bin is 80. R xc is set at −20 dB. Fig. 9(a) and (b) represents the ext. BLPs and the number of real BLPs in the ext. BLPs in the selected points according to Halpha-UZH. Fig. 9(c) shows the percentage of the real BLPs in the ext. BLPs. When the threshold of index 2 is set to the abovementioned numbers, it does not affect the number of final ext. BLPs and the number of real BLPs in most cases. In a few cases, it is possible that after adding [K pha3 ] to the uncalibrated k PolSAR images, more points in Z7 and Z8 move to or away from NZ9 since the two weights of index 1 are similar. Furthermore, even if affected, the subsequent smaller angle cycle does not have a strong impact on the selected points, and the difference in the number of points in this case is less than 200 according to the results of the experiments presented in Fig. 9(a) and (b).
In Fig. 9(a) and (b), the maximum difference between any two numbers is 135 and 152, respectively, under the same abscissa. Fig. 9(d) indicates how many range cells in the experimental data do not participate in the fitting operation because the threshold of index 2 is not reached. In Fig. 9(e) and (f), the amplitude errors and phase errors of k remain unchanged in most cases due to the invariance of the selected points and the correct points or the small number of range cells in the No. 16 experimental image that do not participate in fitting. However, in some rare cases, the amplitude errors and phase errors indeed change. According to Fig. 9, the most important reason affecting the error changes of Nos. 11 and 16 is the number of delete flags. Considering the accuracy of the selected points and the number of delete flags, we recommend setting the index x 2 threshold at 0.9 to obtain a higher accuracy of the selected points and to maximize the number of range cells used for fitting.

C. ±45 • : Initial Degrees of ϕ k
According to the derivation in (12), an interesting regulation suggests that points on the H/ᾱ plane exhibit an oscillating change of a trigonometric function with the period of 180 • by changing ϕ k . In Section III-B, this derivation is implemented when extracting the BLPs. Simultaneously, when the points on the H/ᾱ plane are moved by adding ϕ k , we utilize ±45 • as the initial value. In the following, we explain why this parameter is set to 45 • . Considering that the periodic change is 180 • , it can be determined that the change interval of the independent variable becomes −90 • to 90 • . Then, we artificially divide −90 • to 90 • into two parts, that is, −90 • to 0 • and 0 • to 90 • . At each interval, the most representative angle is the intermediate angle. Therefore, it is acceptable to solve the problem by setting the initial value in the middle of the two intervals. Therefore, we set the initial degrees of ϕ k as ±45 • . This can also explain why we choose ±45/2 i as the angle of adding k for our subsequent angle moves during the ith loop.

D. Error of Calibrating k Caused by Residual XT and X-Pol CI
Since the algorithm proposed in this article extracts the BLPs for calibrating k, all formulas and solution algorithms in this article are based on the absence of XT and x-pol CI in PolSAR systems. Residual XT and x-pol CI are inevitable and should be considered when extracting BLPs. By analyzing the nominal PolCAL accuracies of PolSAR sensors [15], [23], [40], [41], the residual XT and residual CIs are better than −30 dB and ± 0.5 dB/10 • , respectively. Next, XT with −40 to 30 dB and x-pol CI with 0 dB/0 • , 0.5 dB/5 • , and 1 dB/10 • are added to the AIRSAR simulation data to analyze the impact on the ext. BLPs and calibration of k. The numbers of range bins and azimuth bins are 80 and 60, respectively.
The results are shown in Fig. 10, where the green lines indicate the results without any XT and x-pol CI. In Fig. 10(a), as the residual XT increases, an increasing number of BLPs are selected. However, the opposite is true for the residual x-pol CI. Considering that the percentage of ext. BLPs in the real BLPs shown in Fig. 10(b) is almost unchanged, it can be conducted from the comparison with the green lines that XT can increase the number of ext. BLPs in the real BLPs, while the opposite is the case for the x-pol CI. Fig. 10(c) and (d) shows the amplitude error and phase error of k, respectively. XT is the main influencing factor of the amplitude error. However, the the effects of the residual XT and x-pol CI on the phase error are unpredictable. Considering the green lines in these two figures, it may be possible to reduce the influence of residual XT and x-pol CI by increasing the quality and quantity of selected points, particularly with regard to the amplitude error. Therefore, there is a vast space to reconsider the exact impact of XT and x-pol CI on the calibration model of UZH and modify them to enhance the accuracy of k.

VI. CONCLUSION
In this article, we propose a BLP extraction method by using H andᾱ from uncalibrated k PolSAR images. We perform simulated and actual data analysis on AIRSAR, ALOS, and GF-3 data. For the experimental results, the real BLPs account for more than 99% of all ext. BLPs in the majority data. The k calibration by the ext. BLPs extracted by Halpha-UZH is better than that for Shi-UZH and is very close to the measurement of trihedral CRs. After using the Ans, Zero-Ans, and Quegan algorithms to perform calibration of XT and x-pol CI, the smallest average error amplitude is 0.0751 dB, and the smallest phase is 1.9975 • in calibrating k obtained by Halpha-UZH.
The main idea in this article of selecting BLPs is to use H and α to find some of the Z9 points with the property of zero helicity from uncalibrated k PolSAR images. In Section V, it is found that the residual XT and x-pol CI have different effects on the scatter characteristics of targets on the H/ᾱ plane, which is an interesting topic for further analysis. In the framework proposed in this article, since the points of Z9 are mainly affected by the Z6 points when we change only the amplitude of k in the H/ᾱ plane, the ext. BLPs are calculated from NZ9 rather than Z9. Therefore, some of the Z9 points are discarded after setting the limit. If the discarded points can also be fully utilized, it is quite likely that the accuracy of selecting BLPs will be improved, and we plan to examine this topic in the future.

APPENDIX
When the amplitude of k is constant and the phase is changed, we hypothesize that the eigenvalue of [O] is unaffected. This appendix presents the corresponding mathematical derivation process. Since we consider the amplitude and phase of k separately, (10) can be rewritten as where, | · | represents the absolute value operation and ϕ k is the phase of k. According to the definition of the covariance matrix, it can be obtained that [C 3 ] is a Hermitian matrix. Since we consider only the change in ϕ k and |k| is unchanged, we define [C 3 ] as