2.1.

PolSAR Data

The basic form of PolSAR data is the Sinclair scattering matrix using horizontal and vertical matrices, expressed by Eq. (1):

Then the coherency matrix $\mathbf{T}$ can be obtained as shown in Eq. (3):

2.2.

Fuzzy Set Theory

Fuzzy set theory emerged in the nineteenth century and transformed classic two-valued logic into a continuous logic in the closed interval [0, 1], which was more in line with the human brain’s cognitive and reasoning processes. The main difference between traditional set theory and fuzzy set theory is the mechanism of element membership. In traditional set theory, there are only two kinds of collection mechanisms {Yes or No}, for which a quantitative description is $\{\mathrm{1,0}\}$; that is, an element can either belong to or not belong to a set. In fuzzy set theory, the concept of membership is introduced, and an element can be assigned a grade of membership from 0 to 1. This expansion means that an element can belong to multiple collections at the same time, with different degrees of membership in each. Compared to traditional set theory, fuzzy sets smooth the process of translation between quantitative and qualitative representations.

2.3.

$\mathsf{H}/\alpha$ Classifier Based on Target Decomposition

In 1996, Cloude and Pottier proposed a target decomposition method based on a coherency matrix.²¹ The coherency matrix $T$ is first decomposed into the sum of the products of eigenvalues and eigenvectors, as shown in Eq. (4):

H and $\alpha$ can be used to define a two-dimensional feature space. Cloude and Pottier proposed the classification boundaries of the $\mathrm{H}/\alpha$ plane on the basis of a large number of experiments.⁹ Using this approach, the PolSAR data can be divided into eight classes based on Fig. 1.

Fig. 1

$\mathrm{H}/\alpha$ plane.

2.4.

$\mathsf{H}/\alpha$-Wishart Algorithm

According to Goodman,²² the coherency matrix $T$ obeys the complex Wishart distribution, and its probability density is as shown in Eq. (6):

Lee et al. proposed the $\mathrm{H}/\alpha$-Wishart algorithm based on the maximum likelihood decision criterion.¹⁰ The Wishart distance between the coherency matrix $\langle T\rangle$ of the pixel and the cluster center of the target class $m$ can be expressed as in Eq. (7):

Fig. 2

Algorithm flow charts: (a) flow chart of $\mathrm{H}/\alpha$-Wishart and (b) flow chart of $\mathrm{H}/\alpha$-Wishart fuzzy clustering.

2.5.

Fuzzy Clustering Based on Wishart Distance

To solve the problem of inflexible clustering, an $\mathrm{H}/\alpha$-Wishart fuzzy clustering algorithm based on the Wishart distance is proposed in this paper, which allows pixels to belong to more than one class with a certain degree of membership for each. According to the concept of fuzzy set theory, the degree of membership of one point in a given class shall be 1 when the distance from this point to the class center is small enough, meaning that the point completely falls into this class and that the degree of membership of this point in other classes must be 0. When no class satisfies this condition, the degree of membership of a given point shall be set according to the distance between that point and the center of every class; the further away the point, the less will be the degree of membership. In addition, when a point completely belongs to a certain class, the sum of its memberships in every class is 1, and this condition shall still be satisfied when the point belongs to several classes.

After introducing the concept of membership, the method of computing the class center is still Eq. (6), but the meaning of $U_{ij}$ has been expanded to be the degree of the membership of pixel $i$ in class $j$, and its value range has changed from $\{\mathrm{0,1}\}$ to the interval [0,1]. The improved classification process is shown in Fig. 2(b).

3.1.

Image Preprocessing

Due to imaging system complexity and real-world imaging factors, PolSAR images usually contain much noise, but its influence can be reduced by filtering operations. In this research, a refined Lee filter with a $3\times 3$ window was applied before the classification experiments.

3.2.

Initialization

In this initial step, images are classified into eight categories to initialize the cluster centers with $\mathrm{H}/\alpha$ classifier. The initial process is as follows:

3.3.

Fuzzification of Wishart Distance State Space

Based on the cluster centers calculated in Sec. 3.2, one of the key points in this paper, the design of the fuzzy membership functions (FMFs), can be done. In order to fuzzify the Wishart distance state space, let $U(x_i,c_j)$ represent the degree of the fuzzy membership with which the $i$’th point belongs to the $j$’th class. The FMFs satisfy the following conditions over the entire state space:

Because of terrain, sensor type, the wavelength of the electromagnetic waves used for measurement, and various other reasons, the values of the Wishart distance vary over a wide range. In fact, because the calculation of the Wishart distance involves a logarithmic operation, as can be seen in Eq, (7), the Wishart distance can even be negative. All these reasons prevent regular membership calculation and fuzzy partition. Therefore, to make the membership calculation feasible and to solve the problem of setting an appropriate unified threshold for the fuzzy partition of an arbitrary domain, a normalization of Wishart distance for every pixel is performed as shown in Eq. (12):

For $i$’th point, there can be two different conditions: totally belonging to one cluster or belonging to several clusters at same time. When the normalized distance between this point and the $k$’th cluster center is small enough, this point shall totally belong to the $k$’th class as shown by Eqs. (13) and (14):

3.4.

Update of Cluster Center and Iterative Refinement of Clustering

After the fuzzy membership degrees $U(x_i,c_j)\in [\mathrm{0,1}]$ have been determined in Sec. 3.3, the new cluster center can be computed based on Eq. (17):

With the cluster centers updated, the categories of every pixel can be reclassified based on Eqs. (7) and (8). And then an iteration of Secs. 3.3 and 3.4 can be done until the terminating condition is met.

3.5.

Stopping Condition

The stopping condition is different in different applications. One option is to set a fixed number of iteration times, and another option is to set a fixed threshold of the number of pixels that change categories between two consecutive iterations. When the maximum iteration number is met, or the change rate is less than the threshold, output the classification result. Otherwise, return to Sec. 3.3. Finally, the proposed algorithm outputs a classification image.

3.6.

Parameter Adjustment

Because of variations in the data distribution, the parameter $p_f$ is introduced in the fuzzy function design to adjust the range of acceptance of fuzzy operation. When the data separate well and are easy to cluster, a small $p_f$ can be used. When the data have a disorderly distribution and are difficult to classify, a larger $p_f$ can be used to expand the range of acceptance of fuzzy operation to improve algorithm performance. Figure 3 shows how the parameter $p_f$ affects the fuzzification of two categories as a simple case.

Fig. 3

Fuzzy partition sample.

4.1.

Experiment 1: L-Band PolSAR Image of San Francisco Bay

The first dataset and the classification results are shown in Fig. 4. Several basic features, such as ocean, beach, mountains, vegetation, urban areas, and noise, in the upper-right corner can be clearly seen in Fig. 4(a). The water has low-entropy surface scattering, the bare soil has medium-entropy surface scattering, the vegetation has medium-entropy vegetation scattering, and the urban areas have high-entropy volume scattering. These major ground features in San Francisco Bay are more distinguishable. Experiment 1 is done to check the performance of the proposed algorithm in classifying the ground features with different kinds of scattering types.

Fig. 4

San Francisco data and classification results. (a) PauliRGB image. (b) $\mathrm{H}/\alpha$-Wishart result. (c) $\mathrm{H}/\alpha$-Wishart fuzzy clustering result.

According to the types indicated in Fig. 4(a), in the classification results of the $\mathrm{H}/\alpha$-Wishart algorithm shown in Fig. 4(b), the first zone corresponds to noise. The corresponding feature type of the second and third zones is ocean. The fourth, fifth, and sixth zones correspond to urban areas. The seventh zone corresponds to vegetation and the eighth to bare land. The algorithm effectively distinguishes several basic features of the marine, urban, vegetation, and bare land environments, but some yellow spots representing bare land are evident in the blue zone representing the ocean, which is obviously a misclassification. Moreover, the urban areas are divided into three classes, which seem to be unreasonable and overly complex because each of the separate zones has no distinguishing characteristic.

Figure 4(c) shows the classification results for $\mathrm{H}/\alpha$-Wishart fuzzy clustering. Similar to the types indicated in Fig. 4(a), the first zone corresponds to noise, and the second and third zones correspond to ocean. The fourth and fifth zones correspond to urban areas. The sixth, seventh, and eighth zones all correspond to bare land. Comparing Fig. 4(c) with Fig. 4(b), it can be seen that in the classification results of the $\mathrm{H}/\alpha$-Wishart fuzzy clustering method, the misclassified spots, which were classified as ocean, are now correct, and urban areas and vegetation are reasonably divided into two classes. Furthermore, there is a large difference between the classification results of $\mathrm{H}/\alpha$-Wishart and of the $\mathrm{H}/\alpha$-Wishart fuzzy clustering algorithm in the oval shown in Fig. 5.

Fig. 5

Detailed comparison of San Francisco data classification results. (a) PauliRGB image. (b) $\mathrm{H}/\alpha$-Wishart result. (c) $\mathrm{H}/\alpha$-Wishart fuzzy clustering result.

In the area in the upper left of Fig. 4(a) marked by a red box, at the back of a mountain, there is a shadow area in the image because the higher terrain blocks the other parts of the back slope, which leads to less effective reception of information. The shaded area is similar to a calm water surface in its polarization characteristics, therefore, in the $\mathrm{H}/\alpha$-Wishart classification result, this section is almost completely classified as ocean. However, in the $\mathrm{H}/\alpha$-Wishart fuzzy clustering result, the algorithm is able to find a more reasonable cluster center, which leads to most of those pixels being classified more reasonably as bare land. Experimental results show that the classification accuracy of the improved algorithm proposed in this paper is greatly increased over that of the $\mathrm{H}/\alpha$-Wishart method.

To validate the improvements in fuzzy clustering achieved by calculating cluster centers, the movement paths of the cluster centers have been tracked.

Figures 6(a) and 6(b) show the movement paths of the cluster centers as calculated by the $\mathrm{H}/\alpha$-Wishart and the $\mathrm{H}/\alpha$-Wishart fuzzy clustering algorithm. The background is the data distribution in the $\mathrm{H}/\alpha$ plane, the transition from red to dark blue represents the density change from high to low, and the hollow points indicate the final positions of the cluster centers. The combination of the cluster center locations and the data distribution in the $\mathrm{H}/\alpha$ plane reflects the effectiveness and reasonableness of the cluster centers. Figure 6(a) shows that there were five cluster centers that were calculated by the H/a-Wishart algorithm and located in the medium-entropy multiple-scattering region, with the first and fifth cluster centers slightly distant from the data-intensive region. In Fig. 6(b), the cluster centers calculated by the proposed method eventually moved to the data-intensive region and distributed themselves evenly. The classification results indicate that the clustering density of the proposed method is more reasonable, while the clustering density of the $\mathrm{H}/\alpha$-Wishart algorithm is uneven because the urban areas were divided into three classes and the gray part contained only a few pixels, which seemed to be insignificant.

Fig. 6

Comparison of the movement paths of the cluster centers. (a) Movement path of the cluster centers of $\mathrm{H}/\alpha$-Wishart. (b) Movement path of the cluster centers of $\mathrm{H}/\alpha$-Wishart fuzzy clustering.

4.2.

Experiment 2: L-Band PolSAR Image of Flevoland

To further verify the validity of the classification algorithm, the second set of PolSAR data was used for another set of experiments and a quantitative analysis was performed using a confusion matrix. Most feature types in this experimental area are croplands, including grassland, potatoes, alfalfa, wheat, soybeans, sugar beets, peas, and other target ground features, which usually result in medium-entropy vegetation scattering. Experiment 2 is done to check the performance of the proposed algorithm in classifying the different ground features with similar kinds of scattering types.

Figure 7(a) shows an RGB composite image of the region. The red, green, and blue components of the composite image were obtained using the three parameters $|\mathrm{HH}-\mathrm{VV}|$, $\left|\mathrm{HV}\right|$, and $|\mathrm{HH}+\mathrm{VV}|$ derived from the Pauli decomposition.

Fig. 7

Flevoland data Pauli image and classification results. (a) Pauli composite image of the Flevoland experimental zone. (b) $\mathrm{H}/\alpha$-Wishart classification result. (c) $\mathrm{H}/\alpha$-Wishart fuzzy clustering result.

Comparing Figs. 7(b) and 7(c), the classification result from the fuzzy clustering method is much smoother than from the other method. In the results of the $\mathrm{H}/\alpha$-Wishart algorithm, some areas were not distinguished, such as peas, while some other types were misclassified into several classes, such as lawns and potatoes. However, the majority of surface features, such as peas and beets, can be identified correctly in the $\mathrm{H}/\alpha$-Wishart fuzzy clustering result. The stretch of road at the bottom of this area and the edges of every field in the fuzzy clustering results are both clearer than the $\mathrm{H}/\alpha$-Wishart algorithm. To evaluate the classification accuracy, Fig. 8 shows the reference image of the real surface features, and the six major class samples from the image that were selected to compute the confusion matrices.

Fig. 8

(a) Flevoland area ground truth and (b) samples of main classes.

As can be seen from Table 1, the accuracy of the $\mathrm{H}/\alpha$-Wishart fuzzy clustering algorithm is greater than that of the $\mathrm{H}/\alpha$-Wishart method, both in terms of overall accuracy and Kappa coefficient. For some categories, such as rape, bare land, and lawn, the mapping accuracy and precision of the two methods are both $>90\%$. For most classes, the accuracy of the $\mathrm{H}/\alpha$-Wishart fuzzy clustering algorithm is always a little better because the misclassified points in each zone are fewer with the fuzzy operation.

Table 1

Confusion matrices of H/α-Wishart and fuzzy clustering algorithm.

4.3.

Experiment 3: X-Band High-Resolution PolSAR Image of LingShui, Hainan

The third dataset was collected from the farmland surrounding LingShui. Fields and roads can be easily distinguished in Fig. 9(a), where the green lines mark areas where crops grow lushly and almost completely cover the land, appearing as massive objects, the blue lines mark areas where the plants are small and the soil is bare, as reflected in the strip shapes in the image, and the black blocks are pools. Experiment 3 is done to check the performance of the proposed algorithm in classifying the high-resolution ground features.

Fig. 9

LingShui, Hainan, data and classification results. (a) PauliRGB image. (b) $\mathrm{H}/\alpha$-Wishart result. (c) $\mathrm{H}/\alpha$-Wishart fuzzy clustering result.

Figure 9(b) shows that the $\mathrm{H}/\alpha$-Wishart method was less effective in generating accurate clusters. In the vegetation zone, categories are mixed, creating many small spots. As many as six categories can be seen in a small region. There was obvious misclassification in the region indicated by black circles, where parts of roads, which generate a low-intensity response, were mistakenly classified as water. The $\mathrm{H}/\alpha$-Wishart fuzzy clustering algorithm improved the classification results and yielded a clearer clustering in which few of the pixels representing roads were classified as water. Figure 10 clearly shows that, compared with the $\mathrm{H}/\alpha$-Wishart method, the improved algorithm did a better job of distinguishing shadows from water.

Fig. 10

Detailed comparison of LingShui, Hainan, data classification results. (a) PauliRGB image. (b) $\mathrm{H}/\alpha$-Wishart result. (c) $\mathrm{H}/\alpha$-Wishart fuzzy clustering result.

The region shown in Fig. 10(a) bounded by red lines [as shown in Fig. 9(a) in the corresponding red rectangle] is a grove of mangoes. The synthetic map shows that the presence of trees above the ground created shadows as shown in Fig. 10(a), where the intensity of the echo is weak in the bottom-right portion of the mango trees, which are colored in black. In the results of the $\mathrm{H}/\alpha$-Wishart method, most of the shadows were mistakenly classified as water, while in the results of the $\mathrm{H}/\alpha$-Wishart fuzzy clustering algorithm, due to the fuzzy concept, the cluster centers moved to a more reasonable position, causing the pixels in that region to be reclassified, with only a small group remaining misclassified.

To further evaluate the performance of the new algorithm, the classification results for five typical kinds of land cover according to Fig. 11 are summarized in Table 2. In the results of the $\mathrm{H}/\alpha$-Wishart algorithm, grassy plant and woody plant classes are both classified into several classes, which seem rather messy and result in low accuracy; most of the bare soil is classified as plants or paths, while only 35.88% are in the correct category. In the results of the $\mathrm{H}/\alpha$-Wishart fuzzy clustering algorithm, $>80\%$ of the grassy plant and woody plant pixels are in correct categories, which proves that fuzzy logic can lead to a softer clustering and can classify a small group of isolated pixels into a larger class. While the accuracy of the path class is lower than the $\mathrm{H}/\alpha$-Wishart result, this seems to be the price for improving the classification accuracy of water and shadow because the algorithm used more categories to differentiate water, bare soil, paths in fields, and shadows.

Fig. 11

Locations of test samples: grassy plants (green circle), woody plants (red square), bare soil (orange triangle), paths (blue diamond), water (red upside down triangle).

Table 2

Confusion matrices of H/α-Wishart and fuzzy clustering algorithm for LingShui data.

By integrating the results of these three sets of experiments, the improved algorithm proposed in this paper can effectively improve classification accuracy and achieve a reasonable classification of shadows and water.