Efficient Superpixel Generation for Polarimetric SAR Images with Cross-Iteration and Hexagonal Initialization

Abstract

Clustering-based methods of polarimetric synthetic aperture radar (PolSAR) image superpixel generation are popular due to their feasibility and parameter controllability. However, these methods pay more attention to improving boundary adherence and are usually time-consuming to generate satisfactory superpixels. To address this issue, a novel cross-iteration strategy is proposed to integrate various advantages of different distances with higher computational efficiency for the first time. Therefore, the revised Wishart distance (RWD), which has better boundary adherence but is time-consuming, is first integrated with the geodesic distance (GD), which has higher efficiency and more regular shape, to form a comprehensive similarity measure via the cross-iteration strategy. This similarity measure is then utilized alternately in the local clustering process according to the difference between two consecutive ratios of the current number of unstable pixels to the total number of unstable pixels, to achieve a lower computational burden and competitive accuracy for superpixel generation. Furthermore, hexagonal initialization is adopted to further reduce the complexity of searching pixels for relabelling in the local regions. Extensive experiments conducted on the AIRSAR, RADARSAT-2 and simulated data sets demonstrate that the proposed method exhibits higher computational efficiency and a more regular shape, resulting in a smooth representation of land cover in homogeneous regions and better-preserved details in heterogeneous regions.

1. Introduction

In the last few decades, synthetic aperture radar (SAR) has shown its advantages of being relatively insensitive to atmospheric and illumination conditions. To date, polarimetric SAR (PolSAR), as an advanced form of SAR, has provided abundant scattering information on observed land cover and targets. For civilian purposes, PolSAR can be used to monitor the growth of crops [1] and for the study of the Earth’s resources [2], urban planning [3], mineral resource exploration [4], and disaster monitoring [5], while from a military perspective, PolSAR can be used to identify, detect and evaluate important strategic military targets [6]. Currently, several airborne and spaceborne platforms continuously provide an enormous amount of PolSAR data. It is inappropriate to interpret these highly complicated images with pixel-based methods because a large number of pixels in large-scale images prevents many algorithms from being computationally feasible [7,8]. The term superpixel refers to a region of self-similar pixels with local characteristics and certain visual significance [9]. For PolSAR images, superpixel generation is beneficial for reducing the impact of strong speckle noise, which can, in turn, accelerate subsequent image processing to a great extent. Therefore, there is an urgent need for the development of superpixel generation methods for PolSAR image interpretation, and such methods have been widely studied. For example, Quan et al. [10] and Xiang et al. [11] adopted superpixel generation methods for PolSAR image classification and segmentation to improve the interpretation accuracy. Yin et al. [12] introduced various distance measures to improve boundary adherence, and Lin et al. [13] introduced boundary constraints when generating superpixels.

The last two decades have witnessed a growing trend toward superpixel generation, but most methods proposed to date have been developed for optical images. PolSAR images provide different data representations and exhibit different amounts of inherent speckle noise compared with optical images; consequently, when directly applied to PolSAR images, common superpixel generation methods cannot obtain satisfactory results. Therefore, some scholars have proposed corresponding improvements for PolSAR images, which can be mainly divided into the following five categories:

(1) Density-based methods [14]. Based on the mean shift (MS) algorithm, Lang et al. [15] proposed the generalized MS (GMS) algorithm, which can be applied for the filtering of PolSAR images, and then [16] proposed a new fusion criterion for the application of the GMS algorithm in generating superpixels for PolSAR images. However, density-based algorithms usually cannot offer control over the number of superpixels or their compactness.

(2) Graph-based methods [17,18]. Liu et al. [19] modified the normalized cuts algorithm by incorporating the revised Wishart distance (RWD) and an edge map for superpixel generation. Wang et al. [20] introduced an improved entropy rate algorithm for PolSAR images by incorporating two distance measures and considering the uniformity of PolSAR images. Due to the limitations of algorithm processing, graph-based methods tend to require a combination of various technical elements to improve their accuracy.

(3) Contour evolution methods [21,22]. Liu et al. [23] attempted to use a turbopixel algorithm to divide PolSAR images into superpixels and other residual pixels for final classification. Their work demonstrates the practicability of superpixel generation, but its importance is not limited to classification.

(4) Energy optimization methods. Yang et al. [24] proposed a hierarchical energy-driven method for PolSAR images based on superpixels extracted via energy-driven sampling (SEEDS) by using metrics based on histogram intersections and a novel Wishart energy.

(5) Clustering-based methods [25,26]. Most of these methods make use of the principles of clustering algorithms, such as the k-means algorithm, while using color information and spatial information as the basis for distance measures. Generally, clustering-based methods can be used to obtain a controllable number of compact regions with regular shapes, which is crucial for the interpretation of PolSAR images. Representative algorithms include simple linear iterative clustering (SLIC), linear spectral clustering (LSC), and iterative edge refinement (IER).

Feng et al. [27] introduced the Wishart distance instead of Euclidean distance for PolSAR images, and Qin et al. [28] further adopted the RWD to improve the SLIC method for PolSAR images, which is called POL-SLIC. The fast implementation of the RWD combined with spatial distance was also employed [29] with the IER framework [30], which is called WS. Li et al. [31] improved the WS through a hexagonal initialization strategy with all pixels set to be unstable. The above-mentioned algorithms all use the Wishart distance or the improved ones that perform many algebraic operations based on complex matrices; therefore, the time consumption required for these methods is unsatisfactory. Moreover, these methods show segmentation reliability in homogeneous regions while insufficient regularity in heterogeneous regions.

Among numerous superpixel generation methods, clustering-based methods can provide better-balanced trade-offs between efficiency and accuracy, and the key problem is how to determine a suitable distance for measuring the similarity between different pixels. To obtain more reliable results, an increasing number of measures with good performance are usually combined to generate superpixels for PolSAR images. Specifically, Ratha et al. [32] introduced geodesic distance (GD) based on a unit sphere into PolSAR images derived from a real symmetric Kennaugh matrix that can measure the shortest distance with high computational efficiency and has shown great potential in distinguishing different land cover in heterogeneous scenes [33]. The expression ability of RWD for the inherent statistical features and the computing performance of GD are obviously extremely necessary for PolSAR image superpixel generation.

More importantly, how to integrate these two measures to ensure full utilization of their respective strengths is the key to superpixel generation. Xie et al. [34] introduced the weighted sum of edge gradient distance, spatial distance and Wishart distance to improve the distance measure of SLIC. Zhang et al. [35] and Hou et al. [36] also improved the distance measure by extracting a variety of features and calculating the weighted sum of Euclidean distances. Therefore, the weighted sum of different distances needs to be calculated multiple times when calculating the superpixel label of each pixel. Evidently, simple weighting is a computational challenge for high-resolution PolSAR images, and the performances of measures may interfere with each other. Inspired by the specialty of clustering-based methods, we propose a cross-iteration strategy to alternatively apply the comprehensive similarity measure based on the RWD and GD. This means that only one measure rather than a weighted sum needs to be calculated for local clustering, which can reduce the time cost and mine the unique superiority of each measure with iterative characteristics. In addition, this study proposes the global threshold with respect to the difference of two consecutive ratios of the current number of unstable pixels to all of the unstable pixels.

Clustering-based methods can obtain the number of unstable pixels after each iteration, inspired by this, we propose a novel method for PolSAR image superpixel generation based on cross-iteration with hexagonal initialization (i.e., HCI). To alleviate the speckled effect and time cost to search pixels for relabeling in the local regions, the PolSAR image is initialized into multiple regular hexagonal superpixels, and then all pixels are initialized as unstable pixels based on hexagons. Considering the computational burden and accuracy, cross-iteration is proposed to integrate the GD and RWD, in which the GD is derived from the Kennaugh matrix, with higher efficiency and more satisfactory superpixels in homogeneous regions with RWD. Finally, postprocessing by the dissimilarity measure algorithm based on the GD is adopted to eliminate the isolated small regions. The main contributions of our work are summarized as follows:

• To the best of our knowledge, the cross-iteration strategy is proposed for PolSAR image superpixel generation for the first time. Without imposing redundant computational load, the cross-iteration strategy is a flexible and simple scheme that can be generalized to several other interpretation tasks, i.e., image classification and segmentation, in which various measures with different superiority can be exploited alternately in the iterative process according to corresponding criteria, which completely differs from the capabilities of other existing weighing methods.
• The similarity measure ability of RWD and the computational efficiency of GD outperform numerous other distance measures for PolSAR images, which are fully exploited in this study. It is well known that none of the existing distance measures is ideal for PolSAR image superpixel generation, such as the large time consumption for RWD and weak boundary adherence for GD. However, the RWD and GD can be innovatively integrated to reduce their respective drawbacks and positively merge the merits of various measures via cross-iteration.
• The experimental results conducted on both two simulated PolSAR data sets and two real-world PolSAR data sets effectively demonstrate that the proposed HCI is capable of obtaining reliable results in various land cover scenes. Compared with six competitive state-of-the-art methods, our proposed HCI can provide better computational performance with higher boundary adherence.

The remainder of this article is organized as follows. Section 2 introduces the RWD and GD based on the multilook PolSAR data. Section 3 introduces the proposed HCI segmentation framework in detail. Section 4 demonstrates the experiments and comparisons based on two real-world PolSAR data sets and simulated data sets. The conclusion is given in Section 5.

2. PolSAR Data

PolSAR utilizes vertical and horizontal polarization for emitting and receiving polarized radar waves with four channels: HH, HV, VH and VV. Each pixel of a single-look PolSAR image can be represented by a $2\times 2$ complex scattering (or Sinclair) matrix $\mathbf{S}$ as [8]

$$\mathbf{S}=\left[\begin{array}{cc}{S}_{HH}& S_{HV}\\ S_{VH}& S_{VV}\end{array}\right]$$

(1)

according to the reciprocity medium, $S_{HV}=S_{VH}$. Therefore, the coherency matrix $\mathbf{T}$ commonly used for the multilook PolSAR image can be obtained as follows:

$$\mathbf{T}=〈{\mathbf{k}}_P{\mathbf{k}}_P^{\ast T}〉=\left[\begin{array}{ccc}{T}_{11}& T_{12}& T_{13}\\ T_{21}& T_{22}& T_{23}\\ T_{31}& T_{32}& T_{33}\end{array}\right]$$

(2)

where ${\mathbf{k}}_P={\left[S_{HH}+S_{VV},S_{HH}-S_{VV},2S_{HV}\right]}^T/\phantom{{\left[S_{HH}+S_{VV},S_{HH}-S_{VV},2S_{HV}\right]}^T\sqrt{2}}\sqrt{2}$ is the Pauli scattering vector and $\langle ·\rangle$ means ensemble averaging. The superscripts ∗ and T denote the complex conjugate and transpose operations, respectively. Similarly, the $4\times 4$ real symmetric Kennaugh matrix $\mathbf{K}$ can also describe the essential target scattering information [32]. For the coherent case, the matrix $\mathbf{K}$ can be obtained from $\mathbf{S}$ in the following manner:

$$\mathbf{K}=\frac{1}{2}{A}^{*}\left(\mathbf{S}\otimes {\mathbf{S}}^{*}\right)A^{\ast T},A=\left[\begin{array}{cccc}1& 0& 0& 1\\ 1& 0& 0& -1\\ 0& 1& 1& 0\\ 0& j& -j& 0\end{array}\right]$$

(3)

where ⊗ is the Kronecker product, and $j=\sqrt{-1}$. For the incoherent case [32], matrix $\mathbf{K}$ can be obtained from $\mathbf{T}$ as following equation.

$$\mathbf{K}=\left[\begin{array}{cccc}\frac{T_{11}+T_{22}+T_{33}}{2}& \Re \left(T_{12}\right)& \Re \left(T_{13}\right)& \Im \left(T_{23}\right)\\ \Re \left(T_{12}\right)& \frac{T_{11}+T_{22}-T_{33}}{2}& \Re \left(T_{23}\right)& \Im \left(T_{13}\right)\\ \Re \left(T_{13}\right)& \Re \left(T_{23}\right)& \frac{T_{11}-T_{22}+T_{33}}{2}& -\Im \left(T_{12}\right)\\ \Im \left(T_{23}\right)& \Im \left(T_{13}\right)& -\Im \left(T_{12}\right)& \frac{-T_{11}+T_{22}+T_{33}}{2}\end{array}\right]$$

(4)

According to the definition, the matrix $\mathbf{K}$ is a real symmetric matrix that is simple to handle in terms of computation while retaining abundant backscattering information and scale invariance [32]. In Section 2.1 and Section 2.2, we briefly introduce the RWD and GD.

2.1. RWD

Generally, matrix $\mathbf{T}$ can also be expressed by the outer product of ${\mathbf{k}}_P$ as follows:

$$\mathbf{T}=\frac{1}{L}{\displaystyle \sum _{p=1}^L}{\mathbf{k}}_P{\mathbf{k}}_P^{\ast T}$$

(5)

where L is the number of looks. Let $\mathbf{X}=L\mathbf{T}$, then $\mathbf{X}$ follows the complex Wishart distribution; that is, $\mathbf{X}\sim W_c\left(L,q,\Sigma \right)$. The $\Sigma =E\left({\mathbf{k}}_P{\mathbf{k}}_P^{\ast T}\right)$ and q is the dimension of ${\mathbf{k}}_P$, i.e., 3. Let ${\Sigma }_i$ and ${\Sigma }_j$ be the center coherency matrices of the regions $R_i$ and $R_j$, respectively. The hypothesis test [12] is

$$\left\{\begin{array}{c}{H}_0:{\Sigma }_i={\Sigma }_j\\ H_1:{\Sigma }_i\ne {\Sigma }_j\end{array}\right.$$

(6)

Then, the maximum-likelihood (ML) estimators of ${\Sigma }_i$ and ${\Sigma }_j$ are ${\widehat{\Sigma }}_i=$$\left({\displaystyle \sum _{n=1}^{N_i}}{\mathbf{T}}_n\right)/\phantom{\left({\displaystyle \sum _{n=1}^{N_i}}{\mathbf{T}}_n\right)N_i}{N}_i$ and ${\widehat{\Sigma }}_j=$$\left({\displaystyle \sum _{n=1}^{N_j}}{\mathbf{T}}_n\right)/\phantom{\left({\displaystyle \sum _{n=1}^{N_j}}{\mathbf{T}}_n\right)N_j}{N}_j$, respectively, where $N_i$ and $N_j$ are the numbers of pixels belonging to the regions $R_i$ and $R_j$. When the ${\Sigma }_j$ is given for hypotheses $H_0$ and $H_1$, the likelihood-ratio test statistic [12] shown as follows:

$$Q=\frac{{\left|{\widehat{\Sigma }}_i\right|}^{L{N}_i}}{{\left|{\widehat{\Sigma }}_j\right|}^{L{N}_j}}exp\left\{-L{N}_i\left(\mathrm{Tr}\left({{\widehat{\Sigma }}_j}^{-1}{\widehat{\Sigma }}_i\right)-q\right)\right\}$$

(7)

As a result, the distance between two regions $R_i$ and $R_j$ is the RWD defined by the following

$$d_{RW}\left(R_i,R_j\right)=\ln \left(\frac{\left|{\widehat{\Sigma }}_j\right|}{\left|{\widehat{\Sigma }}_i\right|}\right)+\mathrm{Tr}\left({{\widehat{\Sigma }}_j}^{-1}{\widehat{\Sigma }}_i\right)-q$$

(8)

Therefore, the RWD between a pixel i with the coherency matrix ${\mathbf{T}}_i$ and the jth cluster $R_j$ with the center coherency matrix ${\mathbf{C}}_j=$$\left({\displaystyle \sum _{n=1}^{N_j}}{\mathbf{T}}_n\right)/\phantom{\left({\displaystyle \sum _{n=1}^{N_j}}{\mathbf{T}}_n\right)N_j}{N}_j$ is as follows:

$$d_{RW}\left(i,j\right)=\ln \left(\frac{\left|{\mathbf{C}}_j\right|}{\left|{\mathbf{T}}_i\right|}\right)+\mathrm{Tr}\left({{\mathbf{C}}_j}^{-1}{\mathbf{T}}_i\right)-q$$

(9)

Because the RWD will be calculated many times in the proposed HCI method, a large amount of time will be costly in the computation. Therefore, a fast implementation of the RWD is employed in this study to further increase the computational efficiency [29]. Let ${\mathbf{w}}_j=f\left({\left({{\mathbf{T}}_j}^{-1}\right)}^T\right)$ and ${\mathbf{t}}_i=f\left({\mathbf{T}}_i\right)$, where $f(·)$ is a function that arranges all the elements of the matrix into a vector. Then, it is easy to notice that $\mathrm{Tr}\left({{\mathbf{C}}_j}^{-1}{\mathbf{T}}_i\right)={\left({\mathbf{w}}_j\right)}^{\mathbf{T}}{\mathbf{t}}_i$, where both ${\mathbf{w}}_j$ and ${\mathbf{t}}_i$ are 9-dim vectors. In contrast to the high computation of ${{\mathbf{C}}_j}^{-1}{\mathbf{T}}_i$, only nine multiplication operations and eight additional operations are needed to calculate ${\left({\mathbf{w}}_j\right)}^T{\mathbf{t}}_i$, which is only one-third of what $\mathrm{Tr}\left({{\mathbf{C}}_j}^{-1}{\mathbf{T}}_i\right)$ needs in the traditional way. Therefore, the RWD can be represented by the following:

$$d_{RW}\left(i,j\right)=\ln \left(\frac{\left|{\mathbf{C}}_j\right|}{\left|{\mathbf{T}}_i\right|}\right)+{\left({\mathbf{w}}_j\right)}^{\mathbf{T}}{\mathbf{t}}_i-q$$

(10)

2.2. GD

Let M be a topological space. Then, the function $\gamma \left[0,1\right]\to M$ is said to be the simple curve joining points $\gamma \left(0\right)$ and $\gamma \left(1\right)$ in M if it is continuous and nonintersecting; i.e., $\gamma \left(t_1\right)\ne \gamma \left(t_2\right)$ for all $t\in \left(0,1\right)$. For example, a straight line segment in a plane or an arc on a sphere are simple curves [37]. If it satisfies the local Euclidean property, in such a case, M is said to be a Riemannian manifold. Therefore, curves can be compared according to the length derived from the metric. Point A and point B on a Riemannian manifold are known, and $\mathcal{F}$ is the family of smooth (differentiable) simple curves starting from point A to ending at point B [32]. Consequently, there will be a curve of the minimum length in comparison with the other curves in $\mathcal{F}$, which is called the geodesic connecting point A and point B.

For two points A and B in ${\mathbb{R}}^2$, the straight line joining the two points is the geodesic. However, suppose one has the unit sphere $S^2=\left\{\left(x_1,x_2,x_3\right)\in {\mathbb{R}}^3\left|{\left(x_1^2+x_2^2+x_3^2\right)}^{1/2}=1\right.\right\}$ in ${\mathbb{R}}^3$, the desired geodesic should be the great circle segment joining point A and point B in the case that the curve must be completely lie on the sphere. The great circle is one with a radius of 1 passing through points A and B. Similarly, the great circle is still an ideal geodesic for higher dimensional sphere $S^{N-1}=\left\{\left(x_1,x_2,\dots ,x_N\right)\in {\mathbb{R}}^N\left|{\left(x_1^2+x_2^2+\cdots +x_N^2\right)}^{1/2}=1\right.\right\}$. The length measured along the great circle is generally called the arc length distance. Then, the GD between points $A(x_1,x_2,\dots ,x_N)$ and $B(y_1,y_2,\dots ,y_N)$ belonging to $S^{N-1}$ can be defined as follows [9]:

$$GD\left(A,B\right)=arccos\left(A·B\right)=arccos\left({\displaystyle \sum _{i=1}^N}{x}_i{y}_i\right)$$

(11)

where “·” represents the scalar dot product in ${\mathbb{R}}^N$. Evidently, this notion of distance is applicable to the unit sphere of arbitrary dimension, which can also be extended to PolSAR data. Each pixel in the PolSAR image can be represented by a matrix $\mathbf{K}$; thus, the GD between two pixels can be obtained (for details of the proof process to reference [32])

$$d_{GD}\left(i,j\right)={cos}^{-1}\left(\frac{\mathrm{Tr}\left({{\mathbf{K}}_1}^{\mathbf{T}}{\mathbf{K}}_2\right)}{\sqrt{\mathrm{Tr}\left({{\mathbf{K}}_1}^{\mathbf{T}}{\mathbf{K}}_1\right)}\sqrt{\mathrm{Tr}\left({{\mathbf{K}}_2}^{\mathbf{T}}{\mathbf{K}}_2\right)}}\right)$$

(12)

3. Materials and Methods

First, the PolSAR image is initialized as a hexagonal distribution. Then, based on a hexagonal distribution, cross-iteration is adopted to integrate the RWD and GD for relabeling. Finally, postprocessing is performed to eliminate isolated small regions. Figure 1 shows a schematic of the proposed HCI.

3.1. Initialization

To alleviate the computational burden in local clustering and further adapt to the complex terrain distribution of PolSAR images, hexagonal initialization and unstable pixel initialization were proposed in our previous work [31]. In an image, unstable pixels [29] are pixels whose labels are likely to change and should be checked in the next iteration. The definition of unstable pixels is given as follows:

$$\mathrm{UP}=\left\{p|nt\left(p\right)\ne nt\left(q\right)\mathrm{and}nt\left(q\right)\ne t\left(q\right),q\in Nb\left(p\right)\right\}$$

(13)

where p and q represent pixels in the image domain. $Nb\left(p\right)$ is the neighborhood function, and a 4-connected neighborhood is utilized in the experiments. $t\left(i\right)$ represents the label of i and $nt\left(i\right)$ represents the new label after one iteration, and $i=p,q$.

In both IER and WS, the images are initialized as square distributions, as shown in Figure 2a. Rectangles in a black solid line are initialized superpixels. Centers $C_{i0}$, $C_{i1}$, $C_{i2}$, $C_{i3}$, $C_{i4}$, $C_{i5}$, $C_{i6}$, $C_{i7}$ and $C_{i8}$ are sampled with an interval of S pixels. Square superpixels display central symmetry, but the regions containing the image edges are generally asymmetric. Considering the boundary adherence of superpixels across image edges, this approach can better meet the complex terrain condition while the initialized superpixel is a polygon.

Figure 2b shows the hexagonal initialization [31] with the clustering centers $C_{j0}$, $C_{j1}$, $C_{j2}$, $C_{j3}$, $C_{j4}$, $C_{j5}$ and $C_{j6}$. Hexagons in a black solid line are initialized superpixels, which are sampled horizontally at intervals of $S_h$ pixels and $S_v$ pixels in the vertical direction. To facilitate subsequent comparisons, set the hexagon superpixel equal to the square superpixel in area, then the side length of the square and hexagon need to satisfy the following geometric relationship:

$$\frac{S^2}{H^2}=\frac{3\sqrt{3}}{2}$$

(14)

According to the geometric characteristics of the hexagonal distribution, the horizontal and vertical distances of adjacent superpixels are as follows:

$$\begin{array}{cc}{S}_h=\sqrt{3}H& S_v=\left(3/2\right)H\end{array}$$

(15)

Combining (14) and (15), the horizontal distance $S_h$ and vertical distance $S_v$ can be obtained as follows:

$$\begin{array}{cc}{S}_h=\left(\sqrt{2/\phantom{2\sqrt{3}}\sqrt{3}}\right)S& S_v=\left(\sqrt{3/\phantom{3\sqrt{2}}\sqrt{2}}\right)S\end{array}$$

(16)

Specifically, our proposed method searches the superpixel center for a certain unstable pixel within the size of $2S\times 2S$. For the square distribution, there are nine clustering centers $C_{i0}$, $C_{i1}$, $C_{i2}$, $C_{i3}$, $C_{i4}$, $C_{i5}$, $C_{i6}$, $C_{i7}$ and $C_{i8}$ in the local regions. Figure 2a depicts that the search range of the unstable pixel i is the rectangle marked by the blue dotted line. Consequently, the integrated distance needs to be calculated up to nine times for the final assignment. When the input pixel of the PolSAR image is N, the complexity of each clustering iteration is $9N$.

With regard to the hexagonal distribution shown in Figure 2b, there are six adjacent superpixels ($C_{j0}$, $C_{j1}$, $C_{j2}$, $C_{j3}$, $C_{j5}$ and $C_{j6}$) in the local regions. This means that at most just six distance calculations are needed to determine the superpixel to which the unstable pixel j belongs. Compared with the square distribution, the hexagonal distribution has a lower complexity of $6N$ for one iteration of the clustering, which can improve the computation efficiency and attach to edges for the complex terrain distribution of PolSAR images.

In addition, we initialize the unstable pixels as all the pixels based on the hexagonal distribution in PolSAR images, instead of the square edges as IER [31]. By doing so, all the potential edges in PolSAR images will be maintained, resulting in accurate boundary adherence and thus accurate superpixels.

3.2. Cross-Iteration

Specifically, the clustering-based superpixel generation methods can be used to accurately obtain the number of unstable pixels after each iteration, which can facilitate the adjustment of parameters and the evaluation of the measurement performance.

To demonstrate this characteristic more intuitively, this section utilizes the IER method based on hexagonal initialization to perform superpixel generation for a simulated PolSAR image. Figure 3c shows the downward trend of the number of unstable pixels. In the first 4 iterations from Figure 3c, the downward trend of the broken line is obvious and the slope is very large, which means that the number of unstable pixels has dropped considerably at this time. This is because the initialized superpixel is completely segmented according to the spatial geometry, but the similarity information between pixels is not taken into account at all. Evidently, the greater the number of unstable pixels that fall, the more unstable pixels need to be relabeled. Since the basis of relabeling is the distance between pixels, the effect of the distance measure utilized in the first 4 iterations on the segmentation accuracy is particularly critical. From $iter=5$ to the end of the iteration, the number of unstable pixels sets decreases, the broken line gradually tends to be flat, and the descending number of unstable pixels decreases significantly. Compared with the results in the first 4 iterations, only a small number of unstable pixels need to be relabeled. However, the distance metric used in these iterations still has a great influence on the total running time.

It is believed that the distance metric plays a crucial role in superpixel generation. Generally, a multivariate zero-mean circular Gaussian distribution is used to model the homogeneous regions of PolSAR data, but the time cost of RWD is nonnegligible [20]. On the other hand, the GD based on the Kennaugh matrix is qualified for measuring the shortest distance, which is more beneficial for heterogeneous regions and reduces the computational burden than RWD [9]. Efficiency and accuracy are the essential elements of the PolSAR superpixel generation algorithm. Therefore, the combination of the two distance measures is a suitable choice, but it is still worth exploring how to make full use of the efficiency of GD and the accuracy of RWD.

Xie et al. [34], Zhang et al. [35] and Hou et al. [36] all combine a variety of features for superpixel generation of PolSAR images by simple weighting, which can be used to measure the similarity, but the methods adopted by these algorithms significantly increase the computational burden and cause the advantage loss of some of the measures. For a more intuitive comparison of the difference between simple weighting and the proposed cross-iteration, algebraic expressions of the time-consuming of these two kinds of methods are derived. The simple weighting (SW) of RWD, GD and spatial distance is shown as follows, which considers the spatial continuity of the superpixel.

$$D_{SW}\left(i,j\right)={\left(\frac{d_{RW}\left(i,j\right)}{m_{RW}}\right)}^2+{\left(\frac{d_{GD}\left(i,j\right)}{m_{GD}}\right)}^2+{\left(\frac{d_s\left(i,j\right)}{S}\right)}^2$$

(17)

where S is half of the width of the searching range, $m_{RW}$ and $m_{GD}$ are the compactness parameters. The spatial distance is defined as follows:

$$d_s\left(i,j\right)=\sqrt{{\left(x_j-x_i\right)}^2+{\left(y_j-y_i\right)}^2}$$

(18)

where the subscripts i and j represent the cluster center of the ith superpixel and the jth unstable pixel, respectively.

In each iteration of the local clustering process, a large number of unstable pixels need to be relabeled. When relabeling the unstable pixel j, the central pixels of adjacent superpixels are searched in a $2S\times 2S$ region, and the distance between unstable pixel j and each of multiple central pixels i is calculated, respectively. Let c be the number of searched central pixels, $t_{SW}$ be the time of running (17) once, then the total time $T_{SW}$ to generate superpixels by using simple weighting is

$$T_{SW}=t_{SW}·c·{\displaystyle \sum _{n=1}^{n_{max}}}U{P}_{SW}\left(n\right)$$

(19)

where $U{P}_{SW}\left(n\right)$ is the number of unstable pixels after the nth iteration by using simple weighting.

Considering the unique advantages of the RWD and GD measures, a cross-iteration strategy is proposed to alternatively apply the comprehensive similarity measure based on the RWD and GD. Specifically, in local clustering, the same strategy of only relabeling unstable pixels as the IER method [30] is adopted, which can reduce redundant computation. Meanwhile, considering the spatial continuity of the superpixel [25], the RWD combined with the spatial distance is utilized in the first $n_{CI}$ iterations:

$$D_{RW}\left(i,j\right)={\left(\frac{d_{RW}\left(i,j\right)}{m_{CI\_RW}}\right)}^2+{\left(\frac{d_s\left(i,j\right)}{S}\right)}^2$$

(20)

where $m_{CI\_RW}$ is the compactness parameter.

Obviously, using the RWD in the first $n_{CI}$ iterations can improve the boundary adherence performance. From the ($n_{CI}+1$)th iteration to the end of the local clustering, the GD and spatial distance are combined to measure the similarity between the cluster center i and the unstable pixel j, shown as follows:

$$D_{GD}\left(i,j\right)={\left(\frac{d_{GD}\left(i,j\right)}{m_{CI\_GD}}\right)}^2+{\left(\frac{d_s\left(i,j\right)}{S}\right)}^2$$

(21)

where $m_{CI\_GD}$ is also the compactness parameter. Moreover, the GD derived from the Kennaugh matrix can efficiently measure the shortest distance between the two pixels, which can effectively shorten the distance calculation time and further improve the segmentation accuracy. Therefore, the comprehensive similarity measure based on the RWD and GD can be obtained as follows

$$D_{CI}=\left\{\begin{array}{c}{D}_{RW}\left(i,j\right)={\left(\frac{d_{RW}\left(i,j\right)}{m_{CI\_RW}}\right)}^2+{\left(\frac{d_s\left(i,j\right)}{S}\right)}^2,1\le n\le n_{CI}\\ D_{GD}\left(i,j\right)={\left(\frac{d_{GD}\left(i,j\right)}{m_{CI\_GD}}\right)}^2+{\left(\frac{d_s\left(i,j\right)}{S}\right)}^2,n_{CI}+1\le n\le n_{max}\end{array}\right.$$

(22)

Let $t_{CI\_RW}$ and $t_{CI\_GD}$ be the time of running (20) and (21) once, respectively, then the time $T_{CI\_RW}$ in the first $n_{CI}$ iterations is

$$T_{CI\_RW}=t_{CI\_RW}·c·{\displaystyle \sum _{n=1}^{n_{CI}}}U{P}_{CI\_RW}\left(n\right)$$

(23)

And the time $T_{CI\_GD}$ from the ($n_{CI}+1$)th iteration to the end of the local clustering is

$$T_{CI\_GD}=t_{CI\_GD}·c·{\displaystyle \sum _{n=n_{CI}+1}^{n_{max}}}U{P}_{CI\_GD}\left(n\right)$$

(24)

where $U{P}_{CI\_RW}\left(n\right)$ and $U{P}_{CI\_GD}\left(n\right)$ are the number of unstable pixels after the nth iteration when RWD and GD are utilized to perform local clustering, respectively. Obviously, the following relationship can be obtained as follows:

$$\left\{\begin{array}{c}{\displaystyle \sum _{n=1}^{n_{max}}}U{P}_{SW}\left(n\right)>{\displaystyle \sum _{n=1}^{n_{CI}}}U{P}_{CI\_RW}\left(n\right)\\ {\displaystyle \sum _{n=1}^{n_{max}}}U{P}_{SW}\left(n\right)\gg {\displaystyle \sum _{n=n_{CI}+1}^{n_{max}}}U{P}_{CI\_GD}\left(n\right)\\ {\displaystyle \sum _{n=1}^{n_{max}}}U{P}_{SW}\left(n\right)>{\displaystyle \sum _{n=1}^{n_{CI}}}U{P}_{CI\_RW}\left(n\right)+{\displaystyle \sum _{n=n_{CI}+1}^{n_{max}}}U{P}_{CI\_GD}\left(n\right)\end{array}\right.$$

(25)

Moreover, $t_{SW}>t_{CI\_RW}>t_{CI\_GD}$. Combining (19) and (23)–(25), we can obtain

$$\begin{array}{c}{T}_{CI\_RW}+T_{CI\_GD}\hfill \\ =c·\left(t_{CI\_RW}·{\displaystyle \sum _{n=1}^{n_{CI}}}U{P}_{CI\_RW}\left(n\right)+\right.\left.t_{CI\_GD}·{\displaystyle \sum _{n=n_{CI}+1}^{n_{max}}}U{P}_{CI\_GD}\left(n\right)\right)\hfill \\ <c·t_{CI\_RW}·\left({\displaystyle \sum _{n=1}^{n_{CI}}}U{P}_{CI\_RW}\left(n\right)+{\displaystyle \sum _{n=n_{CI}+1}^{n_{max}}}U{P}_{CI\_GD}\left(n\right)\right)\hfill \\ <c·t_{SW}·\left({\displaystyle \sum _{n=1}^{n_{CI}}}U{P}_{CI\_RW}\left(n\right)+{\displaystyle \sum _{n=n_{CI}+1}^{n_{max}}}U{P}_{CI\_GD}\left(n\right)\right)\hfill \\ <c·t_{SW}·{\displaystyle \sum _{n=1}^{n_{max}}}U{P}_{SW}\left(n\right)=T_{SW}\hfill \end{array}$$

(26)

The superpixel generation method for PolSAR images aims to eliminate speckle noise and improve the subsequent interpretation efficiency. As a preprocessing step, if the segmentation accuracy is poor, it will directly affect the interpretation results. Obviously, the significance of superpixels will be lost if the accuracy is improved at the expense of computational efficiency. Specifically, the cross-iteration strategy can make full use of the high precision of RWD and the high efficiency of GD, as shown in Figure 4.

In addition, the parameter $n_{CI}$ undoubtedly plays a decisive role in both efficiency and segmentation accuracy. Since $n_{CI}$ is determined by the reduced number of unstable pixels, the parameter $DUR$(decreasing unstable-pixels-to-all-pixels ratio) is proposed in this study. $DUR\left(n\right)$ represents the difference of $UR$(unstable-pixels-to-all-pixels ratio) before and after the nth iteration. If n satisfies the following conditions, let $n_{CI}=n$$(n\ge 2)$.

$$\left\{\begin{array}{c}DUR\left(n\right)=R(n-1)-R\left(n\right)\ge R_{th}\\ DUR(n+1)=R\left(n\right)-R(n+1)<R_{th}\end{array}\right.$$

(27)

where $R\left(n\right)$ represents the $UR$ after the nth iteration, and the hyperparameter $R_{th}$ is 0.08 in this study.

3.3. Procedure of the Proposed Algorithm

To merge the generated small isolated regions and to preserve the strong point targets, a postprocessing procedure based on the GD combined with the dissimilarity measure [29] algorithm is adopted in this study. When the size of a superpixel is smaller than $N_{th}=S^2/4$, the dissimilarities between this superpixel and the superpixels in its 8-neighborhood will be calculated. If the smallest dissimilarity value is smaller than a predefined threshold $G_{th}$, we merge this superpixel into the neighbor with the smallest dissimilarity between them. If not, the next superpixel is selected. The dissimilarity measure between two superpixels $R_i$ and $R_j$ is defined as follows:

$$G\left(R_i,R_j\right)=\frac{1}{q}{∥\frac{{\mathbf{K}}_i^{diag}-{\mathbf{K}}_j^{diag}}{{\mathbf{K}}_i^{diag}+{\mathbf{K}}_j^{diag}}∥}_1$$

(28)

where ${\mathbf{K}}^{diag}$ means the vector consisting of the diagonal elements of the center $\mathbf{K}$ matrix of a superpixel and ${∥.∥}_1$ denotes 1-norm of a matrix. Since the dissimilarity G belongs to [0,1], $G_{th}$ is set as 0.3 in all the experiments throughout this study [29].

The details of our proposed method are summarized as follows. Moreover, the flowchart is shown in Figure 5.

(1)

Initialization. Initialize the PolSAR image as a hexagonal distribution. Then, all pixels are set as unstable pixels. Set the iteration index $n=0$.

(2)

Local relabeling. If $n\ge n_{max}$ or if the unstable pixel set is empty, then the algorithm ends and proceeds to (4). Alternatively, we adopt the comprehensive similarity measure via cross-iteration with (22) to find the clustering center in the $2S\times 2S$ searching region, and the label of this center is assigned to the current unstable pixel.

(3)

Updating. Update the superpixel models and the unstable pixel set. Set $n=n+1$ and return to (2).

(4)

Postprocessing. Search the superpixels with sizes smaller than $N_{th}$. Calculate the dissimilarity with (28) and merge this superpixel with its neighborhood based on the predefined criterion.

3.4. Evaluation Criteria

To evaluate the performance of different methods for superpixel generation, all the experiments on the PolSAR data sets in this article are quantitatively assessed with four criteria: boundary recall (BR), running time (RT), under-segmentation error (USE) and achievable segmentation accuracy (ASA) [25]. The BR, USE and ASA are defined as follows:

BR is the ratio of boundary pixels shared by the obtained superpixels and the ground truth, and it can be represented as

$$\mathrm{BR}=N_{S\cap G}/\phantom{N_{S\cap G}{N}_G}{N}_G$$

(29)

where $N_{S\cap G}$ denotes the number of superpixel boundary pixels overlapping the ground truth edges, and $N_G$ represents the number of ground truth edges. In this study, the internal boundaries of the ground truth and superpixels were employed. Thus, a larger value of BR indicates that superpixel blocks agree better with the input image edges.

If the ground truth segments are $g_1,g_2,\dots ,g_m$ and the obtained superpixels are $s_1,s_2,\dots ,s_n$, the USE can be defined by the following [29]:

$$\mathrm{USE}=\frac{1}{N}\left[{\displaystyle \sum _{i=1}^M}\left(\sum _{\left[s_j|s_j\cap g_i>B\right]}\left|s_j\right|\right)-N\right]$$

(30)

where N is the number of all the pixels in an image, and $s_j\cap g_i$ is the overlapping error of the superpixel $s_j$ relative to a ground truth segment $g_i$. The $|.|$ denotes the size of superpixels, and B represents a minimum number of pixels in $s_j$ overlapping $g_i$. Hence, the USE should be as low as possible for obtaining good superpixels.

ASA is a performance upper-bound measure and the highest achievable accuracy of object segmentation when utilizing superpixels as units. By labeling each superpixel with the label of the ground truth segment that has the largest overlap, ASA can be computed as the fraction of correctly labeled pixels, and it can be represented as

$$\mathrm{ASA}=\frac{{\sum }_j{max}_i\left|s_j\cap g_j\right|}{{\sum }_i{g}_i}$$

(31)

4. Results and Discussion

To demonstrate the effectiveness of the HCI, extensive experiments are conducted on simulated PolSAR data sets and real-world PolSAR data sets collected by AIRSAR and RADARSAT-2. In Section 4.1, we first introduce the data sets and the commonly used evaluation criteria. Section 4.2 introduces the comparison experiments of the different initialization strategies on two competitive algorithms based on the simulated PolSAR data sets. Section 4.3 introduces the parameter setting of $n_{CI}$ based on the total PolSAR data sets in this article. In Section 4.4, Section 4.5 and Section 4.6, experiments and discussions on the simulated PolSAR data sets and two real-world PolSAR data sets are contained, and the results show that the HCI method can provide better balanced trade-offs between the computational efficiency and boundary adherence compared with six state-of-the-art algorithms (i.e., POL-SLIC [27], POL-LSC [26], POL-HLT [12], WS [29], HAWS [31] and HAGS [9]). The operation of HAGS is the same as that of HAWS except that the distance measure is chosen as the GD. All the experiments were performed on a computer with a 3.30 GHz Intel Pentium CPU, 64 GB of memory and MATLAB code, with the exception that the POL-LSC method is implied by MATLAB mixed with C code [26].

4.1. Data Sets

The AIRSAR data set acquired over Flevoland, the Netherlands, on 3 July 1991 is a 4-look L-band PolSAR image with $750\times 1024$ pixels, and the Pauli color-coded image is shown in Figure 6. The second data set is from RADARSAT-2, acquired over an agriculture field in Kitchener, Canada, on 19 April 2009, which has $800\times 750$ pixels, and the Pauli color-coded image is shown in Figure 7. For visual inspection and quantitative evaluation, manual segmentation maps used as ground truth are provided for the RADARSAT-2 data set and part regions of the AIRSAR data set. Simulated PolSAR data sets with three identical regular regions but different sizes are employed, which have $400\times 400$ pixels and $500\times 500$ pixels, respectively. The generation of simulated data sets adopts the inverse transform method [38], the image of $400\times 400$ pixels is shown in Figure 8a, and the corresponding ground truth is given in Figure 8b.

4.2. Performance of Initialization

Our proposed HCI first initializes the PolSAR image into hexagons and then initializes all the pixels to unstable pixels. To validate the effectiveness of the initializations, experiments are carried out with two algorithms based on simulated data sets using two distance measures and four different initialization strategies, as shown in

Table 1. Algorithms used to verify the initialization performance.

Distance Measure	Revised Wishart Distance	Geodesic Distance
Square distribution & Edge pixels	WS	GS
Hexagonal distribution & Edge pixels	HWS	HGS
Hexagonal distribution & All pixels	HAWS	HAGS

Note: “Edge pixels” means only the edge pixels are initialized as unstable pixels. “All pixels” means all pixels based on hexagonal distribution are initialized as unstable pixels.

.

The GS algorithm is the same as that of the WS algorithm, and only the distance measure is changed to GD. Moreover, the three algorithms for each column in Table 1 are initialized only in different ways. The compactness parameter m is set as 1.4 and 0.3 (the best) for WS and GS. The results on the four criteria are shown in Figure 9, and the abscissa is set as half of the search range, which is equal to the value of S. It can be seen from Figure 9c,d that the broken lines of various colors essentially overlap, which indicates that the hexagonal initialization does not cause insufficient superpixel generation and reduces the overall segmentation accuracy. We find that the hexagonal distribution is obviously superior to the square distribution in terms of computational efficiency for both WS and GS, and the average running time can be significantly reduced by 30%, which is the expected property for superpixels.

However, Figure 9a shows that the BR decreases slightly when the HWS (HGS) and WS (GS) are compared, and the difference is essentially approximately 0.01. This is because it faces pixels when the image is initialized to hexagons, and an ideal situation for the geometric structure cannot be achieved. In this way, it is necessary to initialize all the pixels into unstable pixels to eliminate the above drawback. When HAWS (HAGS) and WS (GS) are compared, it can be clearly seen that BR is significantly improved in the implementation of all pixels initialized to unstable pixels. Specifically, Figure 9a shows that HAGS is superior to GS with an improved BR of approximately 0.1. Moreover, Figure 9 intuitively shows that the segmentation accuracy of WS and the computational efficiency of GS are attractive, which is beneficial for the superpixel generation of PolSAR images. For different initialization strategies, the difference between ASA and USE is not significant.

4.3. Parameter Analysis of $n_{CI}$

In this subsection, the value of $n_{CI}$ for all data sets in this study and the suitability of global threshold $R_{th}$ selection are verified. By combining the unique advantages of the clustering-based methods, a cross-iteration strategy is proposed to alternatively apply the comprehensive similarity measure based on the RWD and GD. This measure is utilized alternatively according to the difference of two consecutive ratios of the current number of the unstable pixels to all of the unstable pixels. Moreover, the proposed cross-iteration strategy is capable of ensuring full utilization of the advantages of the two measures. Specifically, the parameter $n_{CI}$ controls the performance of HCI, which determines the alternating utilization of two distance measures and thus directly affects the performance of the results. Therefore, the value of the parameter $n_{CI}$ is particularly critical. To obtain a reasonable value of $n_{CI}$, the parameter $DUR$ and global threshold $R_{th}$ ($R_{th}=0.08$) are proposed in Section 3.2.

The first row of Figure 10 shows the $UR$ of HAWS and HAGS based on all simulated and real-world data sets. The compactness parameter m is set as 1.4 and 0.3 for HAWS and HAGS based on two simulated data sets, 0.4 and 0.1 for HAWS and HAGS based on the AIRSAR data set, 0.4 and 0.3 for HAWS and HAGS based on the RADARSAT-2 data set. Figure 10 clearly shows that in each iteration, the decreasing number of unstable pixels of HAGS is always more than that of HAWS, which explains the reason for the high efficiency of HAGS. Since the iteration of RWD is before the GD for our proposed HCI, the $UR$ of HAWS is adopted to calculate the parameter $DUR$, as shown in line 2 of Figure 10.

Compared with the data in Figure 10e,f, the downward trend of the number of unstable pixels is similar for the simulated data sets of different sizes. However, the comparison between Figure 10e–h shows that the difference of $DUR\left(n\right)$ for the real-world data sets is noteworthy, which illustrates that the RWD is capable of reducing the number of unstable pixels for the complex terrain distribution of real-world data sets. According to (27), taking Figure 10a as an example and then

$$\left\{\begin{array}{c}DUR\left(4\right)=R\left(3\right)-R\left(4\right)\ge R_{th}\\ DUR\left(5\right)=R\left(4\right)-R\left(5\right)<R_{th}\end{array}\right.$$

(32)

Therefore, $n_{CI}=4$ for the simulated data of $400\times 400$ pixels. By the same token, it can be concluded that $n_{CI}=4$ for the simulated data of $500\times 500$ pixels, $n_{CI}=3$ for the AIRSAR data set and $n_{CI}=4$ for the RADARSAT-2 data set.

4.4. Superpixel Generation Results of Simulated Data

To achieve fair and objective results, five comparison algorithms evaluated on the simulated data sets are chosen from the clustering-based methods, which included POL-SLIC, POL-LSC, WS, HAWS and HAGS. For these methods, the weighting factor of POL-LSC is set as 0.2, the compactness parameter m is set as 1.2 for POL-SLIC, 1.4 for WS and HAWS, and 0.3 for HAGS. Because RWD and GD are adopted by the proposed HCI, the compactness parameters $m_{CI\_RW}=1.4$ and $m_{CI\_GD}=0.3$ for HCI.

Figure 11 quantitatively illustrates the segmentation results based on the simulated data sets. As shown in Figure 11c,d, there is no large difference between the USE and ASA for these algorithms. For BR, HAWS outperforms WS and HAGS, however, the red broken line representing the HCI is always above the other broken lines, which indicates that the generated superpixels with the boundaries closest to the real edges among these algorithms. In contrast, POL-LSC and POL-SLIC display unsatisfactory BR, and our proposed HCI is approximately 12% higher than POL-LSC when S is 13 with the image of $500\times 500$ pixels.

Apparently, unsatisfactory BR gives rise to poor performance on subsequent interpretation steps, in which the significance of superpixel generation is lost. Because the POL-LSC method was applied in MATLAB in combination with Code C, the time cost requirement was lower. However, due to the low segmentation accuracy, even if it has high computational efficiency, it cannot meet the requirements of superpixel generation as preprocessing.

Figure 11 shows that the computational burden of POL-SLIC and WS is heavy compared to that of the other algorithms. Due to the shrinkage of complexity on terrain distribution in the simulated data sets, the improvement of computational efficiency for HCI is not significant compared with that of HAWS. However, the values of Figure 11 strongly indicate that the proposed HCI maintains higher computational efficiency along with greater boundary adherence.

4.5. Superpixel Generation Results of the AIRSAR Data

For the AIRSAR data set, superpixel segments generated using POL-SLIC, POL-LSC, POL-HLT, WS, HAWS, HAGS and our proposed HCI are shown in Figure 12, where the coherency matrix of each pixel is replaced by the average value of the superpixel to which this pixel belongs. Yin [12] introduces the Hotelling-Lawley trace (HLT) distance into the SLIC for PolSAR image superpixel generation, which is called POL-HLT belonging to clustering-based methods. In addition to the used measure, the operation of POL-HLT is the same as that of POL-SLIC except for the initialization part; the initialization has only a small effect on the results with a slight improvement in BR at the expense of computational burden [12]. The initial width is generally empirically set according to the complexity of the terrain.

For these methods, the weighting factor of POL-LSC is set as 0.3, the compactness parameter m is set as 0.1 for POL-SLIC, 0.7 for POL-HLT, 0.4 for WS and HAWS, and 0.1 for HAGS. Because RWD and GD are adopted by the proposed HCI, the compactness parameters $m_{CI\_RW}=0.4$ and $m_{CI\_GD}=0.1$ for HCI. Figure 12 shows that the proposed HCI performs best with the smoothest boundary of superpixels. The performance of these six competitive algorithms on the real-world PolSAR image is evaluated by visual observations and quantitative evaluation, in which region A, indicated by the red rectangle, is selected for quantitative evaluation. Figure 13 shows the corresponding quantitative evaluation, and the generated superpixels of region A are shown as Figure 14. The BR of POL-SLIC and POL-LSC is much lower than that of the other algorithms, and the BR of POL-LSC decreases obviously with increasing S. This means that the performance of POL-LSC in generating superpixels is quite unstable, which is disadvantageous for the superpixel generation of PolSAR images. Figure 13a shows that the proposed HCI, HAWS and POL-HLT obtain a higher BR than other algorithms; admittedly, the RT of the proposed HCI is 12% and 31% faster than that of HAWS and POL-HLT, as shown in Figure 13b, because of the adoption of GD by HCI.

The HLT distance is adopted by the POL-HLT, which measures the similarity of two covariance matrices complying with the complex Wishart distribution. To eliminate the two-sided (nonsymmetric) effect when the distance between two pixels is measured, both the revised HLT distance and the HLT distance should be calculated, which severely increases the computational burden. The USE of POL-HLT also performs unsatisfactorily, which is on average 7% higher than that of the proposed HCI. Moreover, Figure 15 shows the inferior regularity of POL-HLT in both the homogeneous and the heterogeneous regions, especially compared with that of the proposed HCI. Figure 15c is zoomed in to show the superpixels in blue rectangles in Figure 15a,b. Generated superpixels of the proposed HCI are more regular, which can be observed as more compact and uniform superpixels based on hexagons in the homogeneous regions. However, the superpixels obtained by POL-HLT are seriously disturbed, the sizes are different, and it is even difficult to distinguish different superpixel blocks. It is an adverse effect on the performance of subsequent PolSAR image interpretation tasks.

GD is capable of measuring the shortest distance on the intrinsic manifold of data sets, and RWD is derived from the complex Wishart distribution for PolSAR images, which has the possibility of complementarity of GD and RWD. Although the BR and USE of HAGS perform moderately, the reasonable setting of the parameter $n_{CI}$ makes the proposed HCI exploit the advantage of cross-iteration to absorb the strengths of different distance metrics. Moreover, the red rectangular regions (D and E) in Figure 14 demonstrate the better preservation ability of the proposed HCI on the boundaries of ground terrain. Instead of fast implementation, POL-SLIC adopts the original calculation strategy of RWD with a large amount of matrix processing. The square initialization of WS compared with the HAWS leads to an increase in the number of unstable pixels in the relabeling. Therefore,

Table 2. RT(s) of the AIRSAR data set.

Algorithm	Clustering (s)	Postprocessing (s)	Total (s)
POL-SLIC	839	124	963
POL-LSC	371	-	371
POL-HLT	755	64	819
WS	641	65	706
HAWS	556	64	620
HAGS	530	65	595
HCI	520	63	583

directly shows that the RT of the proposed HCI is decreased by 29%, 39%, 17% and 6% compared with POL-HLT, POL-SLIC, WS and HAWS, respectively, and is almost the same as that of HAGS. HCI significantly improves the computational efficiency and maintains high segmentation accuracy, which meets the efficiency necessary for superpixel generation as a preprocessing technology.

For visual clarity, the small regions B and C in Figure 12 are enlarged and shown in Figure 16 and Figure 17, where an enlarged view of the area marked with a yellow rectangle is shown in the black dotted frame below. The results in Figure 16a,b show that POL-LSC and POL-SLIC cannot preserve the boundaries of slim regions with irregularly generated superpixels, and this shortcoming impacts the subsequent interpretation results. In addition, the results in Figure 16a,b show that POL-SLIC and POL-LSC cannot retain the detailed information of objects, nor can they preserve the strong scattering points in Figure 17a,b, which may severely reduce the accuracy of subsequent interpretation. The boundary adherence ability of POL-HLT is insufficient in heterogeneous regions; consequently, the complex distribution of the targets in region C highlights the shortcomings of the POL-HLT algorithm, as shown in Figure 17c. HAGS is sensitive to speckle noise, and its segmentation results lack smoothness at the edges of different regions, such as the roads in Figure 16f and Figure 17f. Meanwhile, although HAWS can fit the boundaries of real objects relatively well among the compared algorithms, its RT is unsatisfactory.

Therefore, it is clear that HCI can effectively reduce the impact of speckle noise with smooth homogeneous region segmentation and has the highest computational efficiency while achieving satisfactory results, which is precisely the benefit that common PolSAR image superpixel generation algorithms do not have.

4.6. Superpixel Generation Results of RADARSAT-2 Data

To further evaluate the performance of the proposed HCI, extensive experiments and discussions based on six comparison algorithms are conducted on the RADARSAT-2 PolSAR image, including the POL-SLIC, POL-LSC, POL-HLT [12], WS, HAWS and HAGS. Through the visual analysis of the Pauli-RGB in Figure 7, obviously, the Kitchener data set is seriously disturbed by speckle noise, and taking into account the experiment condition of the POL-HLT with the Lee filter [39] for the data sets, this section conducts experiments based on the unfiltered and the filtered data set by the Lee filter. Four window sizes are used: $w=3$, $w=5$, $w=7$ and $w=9$. The initial width is generally empirically set according to the complexity of the terrain. For these methods, the weighting factor of POL-LSC is set as 0.3, the compactness parameter m is set as 0.1 for POL-SLIC, 2 for POL-HLT, 0.4 for WS and HAWS, and 0.3 for HAGS. Because RWD and GD are adopted by the proposed HCI, the compactness parameters $m_{CI\_RW}=0.4$ and $m_{CI\_GD}=0.3$ for HCI.

Table 3. BR, RT(s) and USE of the RADARSAT-2 data set based on the unfiltered image.

BR of the RADARSAT-2 Data Set Based on the Unfiltered Image
S	10	11	12	13	14	15	16	17	18	19	20	Average
POL-SLIC	0.5042	0.5049	0.5137	0.5043	0.5076	0.5082	0.5084	0.5069	0.5146	0.5071	0.5071	0.5079
POL-LSC	0.5375	0.5149	0.4924	0.4815	0.4818	0.4695	0.4578	0.4453	0.4414	0.4492	0.4421	0.4739
POL-HLT	0.9682	0.9701	0.9674	0.9670	0.9666	0.9645	0.9538	0.9592	0.9556	0.9551	0.9449	0.9611
WS	0.8678	0.8705	0.8677	0.8675	0.8710	0.8698	0.8565	0.8644	0.8586	0.8554	0.8465	0.8632
HAWS	0.9264	0.9262	0.9241	0.9229	0.9190	0.9176	0.9152	0.9149	0.9067	0.9090	0.9006	0.9166
HAGS	0.8126	0.8119	0.7963	0.7850	0.7721	0.7722	0.7566	0.7556	0.7407	0.7392	0.7306	0.7703
HCI	0.9245	0.9246	0.9249	0.9233	0.9223	0.9199	0.9185	0.9192	0.9114	0.9144	0.9066	0.9191
RT(s) of the RADARSAT-2 Data Set Based on the Unfiltered Image
S	10	11	12	13	14	15	16	17	18	19	20	Average
POL-SLIC	683	690	685	683	679	673	652	670	646	645	624	666
POL-LSC	83	51	35	29	26	23	21	23	18	17	15	31
POL-HLT	6587	6849	7069	7038	7133	7237	6306	6586	6351	6511	27,495	8651
WS	464	446	436	424	418	414	394	400	390	388	376	414
HAWS	461	456	433	419	400	396	386	400	421	464	416	423
HAGS	284	276	287	302	276	239	220	219	213	209	202	248
HCI	387	373	362	359	346	351	343	343	335	337	328	351
USE of the RADARSAT-2 Data Set Based on the Unfiltered Image
S	10	11	12	13	14	15	16	17	18	19	20	Average
POL-SLIC	0.3937	0.3902	0.3910	0.3901	0.3866	0.3897	0.3927	0.3863	0.3960	0.3846	0.3881	0.3899
POL-LSC	0.3782	0.4017	0.4188	0.4368	0.4673	0.4779	0.4939	0.5199	0.5237	0.5412	0.5557	0.4741
POL-HLT	0.5286	0.5589	0.5784	0.6087	0.6369	0.6534	0.6619	0.6868	0.6966	0.6970	0.7225	0.6391
WS	0.4407	0.4660	0.4818	0.5052	0.5259	0.5402	0.5565	0.5687	0.5784	0.5827	0.5986	0.5313
HAWS	0.4529	0.4749	0.4982	0.5168	0.5395	0.5621	0.5793	0.5861	0.6098	0.6177	0.6247	0.5511
HAGS	0.4196	0.4460	0.4683	0.4884	0.5099	0.5219	0.5394	0.5544	0.5752	0.5827	0.6050	0.5192
HCI	0.4496	0.4772	0.5002	0.5235	0.5479	0.5615	0.5823	0.5894	0.6158	0.6204	0.6310	0.5544

Note: The results of the proposed HCI are in bold faces. The results of the algorithms with the lowest average of BR for POL-LSC, the highest average of RT for POL-HLT, and the highest average of USE for POL-HLT are italics.

shows three quantitative evaluation results based on the unfiltered data set, including BR, RT and USE. Figure 18 shows four quantitative evaluation results of each algorithm based on the unfiltered filter and the different filter sizes. The bold values displayed in Table 3 are the results of HCI, and the results of the algorithm that contains the most outliers are italics. Table 3 directly shows that the BR of POL-SLIC and POL-LSC is poor, and there is a large gap compared with the other five algorithms, which is approximately 45% lower than the proposed HCI. Even more importantly, the existence of improper segmentation is bound to produce unsatisfactory results in subsequent interpretation, and the significance of superpixel generation is lost. The filtering slightly changes the edge of the image; therefore, the quantitative evaluations based on the filtered data set no longer include POL-SLIC and POL-LSC, as shown in Figure 18.

The BRs of our proposed HCI, HAWS and POL-HLT perform well, with values above 0.90. However, Table 3 shows that the RT of POL-HLT is severely inefficient. HCI is approximately 17 times faster than POL-HLT and 71 times faster than POL-HLT when $S=20$, which is inferior. The experiments of [12] also demonstrate the lower computational efficiency of the HLT distance. Higher accuracy inevitably leads to a heavier computational burden in the common methods. Compared with other algorithms, POL-HLT is not on the same order of magnitude as the RT based on the unfiltered data set, so Figure 18 removes the RT of POL-HLT based on the unfiltered data set. With the rapid development of imaging technology, a large number of PolSAR data sets need to be interpreted. Therefore, as a preprocessing step, the significance of superpixel generation will be lost when the accuracy is improved at the expense of computational efficiency. Excessive calculation burden may lead to the elimination of algorithms.

The BR of WS and HAGS shows a slight disadvantage, and the HAGS decreases are more sensitive to the parameter S, but the computational efficiency of the HAGS is higher than others. This is because the GD can accurately measure the shortest distance between two pixels, so it can more accurately label the current unstable pixels and quickly reduce the number of unstable pixels between two consecutive iterations, thus improving the computational efficiency. Therefore, the proposed HCI adopts the RWD and GD to implement the cross-iteration, and the RT of HAGS is also conspicuous, as shown in Table 3. To intuitively demonstrate the advantages of the proposed HCI, the data in the Table 3 were inverted and normalized to construct a radar chart, as shown in Figure 19. As a preprocessing step, both accuracy and efficiency are indispensable for PolSAR image superpixel generation. A radar chart is capable of reflecting the total evaluation criterion and comprehensive ability.

Figure 19a shows that the USE and RT of POL-HLT are both quite poor, and the BR of HAGS is also lower than that of the other algorithms. In contrast, both the segmentation accuracy and computational efficiency of the proposed HCI method are outstanding, as shown in Figure 19b. The adoption of cross-iteration and the reasonable selection of the parameter $n_{CI}$ enable the proposed HCI method to validly combine the advantages of the RWD and GD. Furthermore, the better-balanced trade-off between computational efficiency and segmentation accuracy allows the proposed method to better satisfy the needs of superpixel generation for PolSAR images. Evidently, the boundaries of real objects will be unavoidably affected by filtering to reduce noise interference. Figure 18 also shows that the BR of all algorithms decreases after filtering and decreases with increasing window size. HCI and POL-HLT always achieve higher BR than the other algorithms, as shown in Figure 18. However, regarding the RT after filtering, the color corresponding to POL-HLT in Figure 18 is the lightest. In contrast, HAGS and the proposed HCI can always maintain high computational efficiency. There are no large differences in USE and ASA among the different algorithms, although the USE decreases with increasing window size. The light yellow color in Figure 18 indicates that when S is larger, the problem of incomplete segmentation of POL-HLT is more severe. Therefore, with or without filtering, our proposed HCI is superior to the other methods at balancing computational efficiency and segmentation accuracy.

Figure 20 gives the representation maps of different algorithms, where the red lines superimposed onto the Pauli images depict the superpixel boundaries and the coherency matrix of each pixel is replaced by the average value of the superpixel to which this pixel belongs. The first row of Figure 20a–h shows the results based on the unfiltered image, the second row of Figure 20a–h shows the results based on the filtered image of $w=5$. The Kitchener data set is severely affected by speckle noise; therefore, the regularity is inferior for the results based on the original image. To make a more accurate qualitative analysis of the results, two blue rectangles in Figure 20 are enlarged and shown in Figure 21 and Figure 22, respectively. Moreover, Figure 21a,c shows the final superpixel maps of different algorithms with the red edges superimposed onto the Pauli images; Figure 21b,d show the representation maps. Figure 22 is shown in the same way as Figure 21.

Although POL-SLIC and POL-LSC can obtain regular superpixels, as shown in Figure 21a,c, the edges of superpixels do not fit the boundary of the real objects, which result in the deficient performance with BR. Generated superpixels with the loss of details and destruction of real boundaries will directly impact the performance of the subsequent interpretation tasks. In addition, the significance of superpixel generation may diminish. WS and POL-HLT adopt the approach of square distribution; however, HAWS, HAGS and the proposed HCI utilize the strategy of hexagonal distribution. It is observed that the initialization of the hexagonal distribution can be more closely adherent to the edges than that of the square distribution.

Figure 21 and Figure 22 show that the HAGS results do not adhere to some edges, but their regularity is better, even for the original image, which is severely affected by speckle noise. HAWS boasts a higher value of BR, but it cannot produce uniform and highly compact superpixels. It is well known that none of the existing distance measures is ideal for superpixel generation; therefore, various measures with different advantages can be considered and combined to achieve satisfactory results. Via cross-iteration, the proposed HCI is capable of utilizing the merits of both the RWD and GD to achieve closer boundary adherence and more regular superpixels, which can reduce the load imposed by redundant computations. The computational efficiency of our proposed HCI method is significantly better than that of POL-HLT; moreover, the blue rectangles in Figure 22c,d show that the proposed method achieves more desirable regularity and compactness of the generated superpixels than POL-HLT does.

Figure 23 shows the final statistical distance from each pixel to its corresponding class center, denoted by F-D, based on the image filtered with $w=9$. Because two distance measures are adopted in the proposed HCI method, Figure 23c shows the F-D image for HCI based on the RWD, and Figure 23e shows the F-D image for HCI based on the GD. Evidently, the statistics of Figure 23a–c are consistent with a complex Wishart distribution, and the boundaries between different regions in the HAWS and HCI results are more distinguishable than those in the POL-HLT results, especially for regions A and C.

In Figure 23e, the color of each region is uniform; that is, with the proposed HCI, pixels belonging to the same class exhibit similar statistical characteristics. Although the blurry edges of regions B and D shown in Figure 23d illustrate the poor BR of HAGS, its lighter computational burden is attractive. The GD is capable of measuring the shortest distance on the intrinsic manifold of the data sets, and the RWD is derived from the complex Wishart distribution for PolSAR images; thus, the properties of the GD and RWD should be complementary. These results strongly validate the benefits of jointly applying the RWD and GD metrics in the proposed HCI.

5. Conclusions

As a preprocessing step, PolSAR image superpixel generation should provide the ability to reduce the computational burden and further afford more promotion of the interpretation results. However, most methods focus more on boundary adherence, which neglects to account for time consumption. To address this problem, a fast superpixel generation framework based on cross-iteration for PolSAR images is proposed in this study. To meet the computational efficiency, we propose a feasible cross-iteration inspired by the characteristics of clustering-based methods. Moreover, the proposed HCI offers more possibilities for integrating different distance measures without redundant calculations; therefore, the modeling capabilities of RWD and the manifold measurement of GD can be fully incorporated. To reduce the number of distance calculations in the local clustering, the PolSAR image is initialized as a hexagonal distribution, and all pixels are set as unstable pixels. Then, according to the difference of two consecutive ratios of a current number of the unstable pixels to all of the unstable pixels, the comprehensive similarity measure based on different measures can be alternately utilized via cross-iteration to exploit their full benefits. Considering the generated small isolated regions, a postprocessing procedure based on the GD combined with the dissimilarity measure is applied to achieve the final results.

The quantitative performance evaluations on the AIRSAR data set, RADARSAT-2 data set and two simulated PolSAR data sets demonstrate the availability of the proposed HCI and the parameter $n_{CI}$ in terms of four commonly used criteria, i.e., BR, RT, USE and ASA. Among the six state-of-the-art PolSAR image superpixel generation algorithms, POL-SLIC and POL-LSC have poor BR with unsatisfactory results in slim regions, and the calculation burden of HLT distance utilized by the POL-HLT is heavy.

However, when either unfiltered or filtered with different window sizes, the proposed HCI outperforms other algorithms, with lower time consumption and more adherent edges, especially for detailed land cover scenes with better regularity.

Although being simple and easy to implement, the cross-iteration framework provides the chance to combine various distance measures with higher efficiency and competitive precision, which is attractive in real-time applications. Furthermore, this study only shows its application in superpixel generation. In the future, other more sophisticated distances could be incorporated to further improve the performance of numerous interpretation tasks.