A STABLE MODEL-BASED THREE-COMPONENT DECOMPOSITION APPROACH FOR POLARIMETRIC SAR DATA
1. A STABLE MODEL-BASED THREE-COMPONENT DECOMPOSITION APPROACH FOR POLARIMETRIC SAR DATA Zhihao Jiao, Jiong Chen, Jian Yang Tsinghua University
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5. Stable decomposition To measure the stability of decomposition: Stable: A=min {a} exists, and, A is small (e.g., A<5 ) For standard Freeman decomposition, “a” is limitless
First, I will introduce the polarimetric decomposition approaches developed in recent decades, and the Freeman decomposition, witch is a very effective incoherent decomposition. Second, the stability of a decomposition will be discussed. Then we can confirm if a decomposition approach is stable or unstable. Third, the negative powers in the freeman decomposition will be describe, which is not consistent with actual scattering mechanism. Forth, an improved three-component decomposition approach will be proposed here, and will show some experiment results. The last part is the Conclusion and expectation.
Target decomposition methods are useful tools for the interpretation of PolSAR images, by decomposing polarimetric SAR data into several parts, which represent different scattering mechanisms. We can classify these decomposition method into 2 classes, coherent decompositions , and incoherent decompositions . Coherent decomposition methods decompose the sinclair scattering matrix S, including Pauli decomposition, Krogager decomposition, Cameron decomposition and SSCM decomposition. Incoherent decomposition methods decompose the coherency matrix, covariance matrix, Mueller matrix or Stokes matrix, including ….
The Freeman decomposition decomposes a coherency matrix into surface-scattering, dihedral-scattering, and volume-scattering parts The elements T13,T23,T31 and T32 are ignored. Alpha and beta are model parameters of dihedral and surface scatterings respectively, with absolute values less than 1. And fs, fd , and fv represent the contributions of surface, dihedral and volume scatterings.
In Jian Yang‘s Ph.D. Dissertation, A stable decomposition is defined as the result of the decomposition is not sensitive to noise. After three-component decomposition, a coherency matrix is decomposed to three parts, T1, T2, and T3. We add a noise matrix delta T to the initial coherency matrix, the new coherency is also decomposed to T1’, T2’ and T3’. We can define the noise sensitivity factor like this:…, it’s the ratio of sum of Ti’s error, measured in Frobenius norm, over the Frobenius norm of delta T. If the minimum of noise sensitivity factor A exists, and A is small, e.g., A is smaller than 5, the decomposition can be called stable. Unfortunately, for standard freeman decomposition, noise sensitivity factor is limitless, so freeman decomposition is not stable.
Here is an example. We add a small noise matrix delta T to the initial coherency matrix T. We can see that, the variation of the decomposition results is much larger comparing with the variation on the observed scattering matrix. So, obviously, the standard Freeman decomposition is not a stable decomposition. The instability of the freeman decomposition may cause drastic change in the decomposition results especially in a landform area, especially where the power of surface-scattering is nearly as strong as that of dihedral-scattering.
This is a comparation between Pauli decomposition and Freeman decomposition. The AIRSAR L-band multilook polarimetric data of San Francisco is used, and the image size is 1024*900 . In the ocean-beach area, the powers of surface-scattering and dihedral-scattering are assumed to be close to each other. Because the wave become bigger as the distance to the shore decrease, the power of dihedral-scattering increase, relative to surface-scattering. So in the Pauli decomposition image, Red color component increase smoothly from the ocean to the beach. But the transition is more dramatic in Freeman decomposition image . In the ocean, the power of surface-scattering in the ocean is over estimated and the power of dihedral-scattering is under estimated. In the beach area, situation is opposite. We can see a clear bounds in the continuously descending beach area . It’s caused by the instability of freeman decomposition.
Let’s look at the Calculation of Freeman Decomposition to find the reason of the instability of freeman decomposition. Here is the equations to solve the decomposition, 4 functions with 5 variables. After simplification, we get 3 functions with 4 variables. To solve this underdetermined problem, in the Freeman decomposition, as shown in the process mapping, one model parameter of surface-scattering or dihedral-scattering is fixed to a certain value. So, the calculation of Freeman decomposition is an ill-posed problem, which leads to the instability of the decomposition result in some cases. An ill-posed problem refers to an problem without a only solution, or the only solution is not stable. Though the calculation of freeman decomposition is determined, but the process of comparing T11-2T33 and T22-T33 causes the solution is not stable.
The emergence of negative powers is another problem of the Freeman decomposition. There are three reasons which will cause the powers of surface-scattering or dihedral-scattering to be negative, as follows. However, after decomposition, the powers of sub-scatterings are supposed to be nonnegative, so the emergence of negative powers is not consistent with actual scattering mechanism.
A regularization method named Tikhonov regularization is commonly used to solve ill-posed problems for its integrity in theory and simplicity in implementation. So, we will employ it to stabilize the decomposition by adding a regularization term in the initially ill-posed problem. At the same time, the negative powers can also be eliminated. regularization method uses a series of well-posed problems with a parameter to approximate the ill-posed problem. With a appropriate parameter, the ill-posed problem could be solved well. Tikhonov regularization is a regularization method to minimize a smoothing function as follows, J(x)= 。。。 Where y-F(x) represents the ill-posed problem and omega(x) is a nonnegative function, to measure the “size” of the regularized solution. Lambda is the regularization parameter. It’s user selected, which can be regarded as a tuning value for the influence of the regulation term.
We usually use prior information to decide the omega(x). So we choose 。。。 as omega(x) , since with no prior information about the ground truth, it is inclined to select small modulus of alpha and beta as the possible solution. This regularization term also coincides with the standard Freeman decomposition in which alpha or beta is fixed to be 0 when the corresponding scattering power is secondary. So the objective function is as this, where delta T is the difference between the measured coherency matrix and the coherency matrix calculated by the candidate solution.
Then we discuss selection of the regularization parameter. The most effective value is determined by the intensity of noise, which is usually not easy to be estimated accurately. So, the regularization parameter could be fixed to a constant for a image, but it’s better to choose a more effective value for every pixel. In our proposed approach, the L-curve method is employed. It is a posterior rule to find an almost most effective by calculating the curvature of the L-curve. L-curve is a curve drawn with 。。。 as the abscissa and 。。。。 as the ordinate 。 Here X0 is the optimal solution with a candidate regularization parameter. At the position corresponding the best lambda, the L-curve has the largest curvature.
Here shows the decomposition with fixed regularization parameters. The lower two figures are the proposed decomposition results with different fixed regularization parameter, λ =0.1 and λ =6 , respectively. Upper two figures are results of pauli decomposition and freeman decomposition to compare easily. Comparing with the result of Freeman decomposition, the powers of dihedral-scattering in the urban area shown in lower figures are enhanced, which is helpful for us to classify urban area and forest area. However, the dihedral-scattering powers in the ocean area are overestimated at the same time, especially when λ =6 . So, it is necessary to employ a regularization parameter selecting method to choose an effective λ for every pixel automatically.
Here is the procedure of the proposed approach. The deorientation process is applied to the coherency matrix before all decompositions to reduce the overestimation of the volume -scattering power. Then we compare the absolute value of 。。。 And 。。。 When there is significant difference between the values of 。。。 And 。。。 , for example, one is 5 times bigger than the other, we use original Freeman decomposition for it’s small computing complexity Otherwise the proposed approach is used. To solve the optimization problem, we can use the quasi-Newton method. By introducing constraints such as fs>=0 。。。 , all negative powers are eliminated.
Here is the experiment results. Left figure is result of freeman decomposition and right figure is the proposed approach’s result. Red for dihedral-scattering , blue for surface-scattering , green for volume-scattering. It can be seen, that, by using the improved decomposition approach, the powers of dihedral-scattering are increased in the urban area and the behavior of backscatter in the ocean and forest area is described well. Therefore the detection performance of man-made targets in the forest area could be improved if using the proposed approach. Moreover, in the ocean-beach area boxed out. It is shown that the color from the beach to the ocean changes smoothly in right figure while the transition is more dramatic in left figure. This is an indication that the proposed decomposition approach is more stable . By constraining the sub-scattering powers to positive values, no pixel with negative powers exist in the result of the proposed approach, while more than one fifth pixels have negative powers if using original Freeman decomposition.
Then let us make a conclusion. To solve the instability issue of the Freeman-decomposition and eliminate the negative powers, an improved model-based three-component decomposition approach is developed. By minimizing the continuous objective function consisting of the error of coherency matrix and the regularization term, the decomposition result is more stable. The negative powers are also eliminated by introducing constraints to the solution domain of decomposition. Because we ignore neither alpha nor beta when the powers of surface-scattering and dihedral-scattering are close, surface-scattering and dihedral-scattering are considered equally. This implies the decomposition is also more reliable. However, the proposed approach has larger computing complexity for the calculation of optimization. Moreover, The decomposition based on regularization method has more margin of improvement, for example, in areas with different scattering components, we fix different regularization parameters relative to alpha and beta. Regularization term as omega = 。。。。 may be effective.
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